Independent Submission
Request for Comments: 6979
Category: Informational
ISSN: 2070-1721
T. Pornin
August 2013

Deterministic Usage of the Digital Signature Algorithm (DSA) and

Elliptic Curve Digital Signature Algorithm (ECDSA)


This document defines a deterministic digital signature generation procedure. Such signatures are compatible with standard Digital Signature Algorithm (DSA) and Elliptic Curve Digital Signature Algorithm (ECDSA) digital signatures and can be processed with unmodified verifiers, which need not be aware of the procedure described therein. Deterministic signatures retain the cryptographic security features associated with digital signatures but can be more easily implemented in various environments, since they do not need access to a source of high-quality randomness.

Status of This Memo

This document is not an Internet Standards Track specification; it is published for informational purposes.

This is a contribution to the RFC Series, independently of any other RFC stream. The RFC Editor has chosen to publish this document at its discretion and makes no statement about its value for implementation or deployment. Documents approved for publication by the RFC Editor are not a candidate for any level of Internet Standard; see Section 2 of RFC 5741.

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Table of Contents

   1. Introduction ....................................................3
      1.1. Requirements Language ......................................4
   2. DSA and ECDSA Notations .........................................4
      2.1. Key Parameters .............................................5
      2.2. Key Pairs ..................................................5
      2.3. Integer Conversions ........................................6
           2.3.1. Bits and Octets .....................................6
           2.3.2. Bit String to Integer ...............................6
           2.3.3. Integer to Octet String .............................7
           2.3.4. Bit String to Octet String ..........................7
           2.3.5. Usage ...............................................8
      2.4. Signature Generation .......................................8
   3. Deterministic DSA and ECDSA ....................................10
      3.1. Building Blocks ...........................................10
           3.1.1. HMAC ...............................................10
      3.2. Generation of k ...........................................10
      3.3. Alternate Description of the Generation of k ..............12
      3.4. Usage Notes ...............................................13
      3.5. Rationale .................................................13
      3.6. Variants ..................................................14
   4. Security Considerations ........................................15
   5. Intellectual Property Status ...................................17
   6. References .....................................................17
      6.1. Normative References ......................................17
      6.2. Informative References ....................................18
   Appendix A.  Examples .............................................20
     A.1.  Detailed Example ..........................................20
       A.1.1.  Key Pair ..............................................20
       A.1.2.  Generation of k .......................................20
       A.1.3.  Signature .............................................23
     A.2.  Test Vectors ..............................................24
       A.2.1.  DSA, 1024 Bits ........................................25
       A.2.2.  DSA, 2048 Bits ........................................27
       A.2.3.  ECDSA, 192 Bits (Prime Field) .........................29
       A.2.4.  ECDSA, 224 Bits (Prime Field) .........................31
       A.2.5.  ECDSA, 256 Bits (Prime Field) .........................33
       A.2.6.  ECDSA, 384 Bits (Prime Field) .........................35
       A.2.7.  ECDSA, 521 Bits (Prime Field) .........................38
       A.2.8.  ECDSA, 163 Bits (Binary Field, Koblitz Curve) .........42
       A.2.9.  ECDSA, 233 Bits (Binary Field, Koblitz Curve) .........44
       A.2.10. ECDSA, 283 Bits (Binary Field, Koblitz Curve) .........46
       A.2.11. ECDSA, 409 Bits (Binary Field, Koblitz Curve) .........49
       A.2.12. ECDSA, 571 Bits (Binary Field, Koblitz Curve) .........52
       A.2.13. ECDSA, 163 Bits (Binary Field, Pseudorandom Curve) ....56
       A.2.14. ECDSA, 233 Bits (Binary Field, Pseudorandom Curve) ....58
       A.2.15. ECDSA, 283 Bits (Binary Field, Pseudorandom Curve) ....60
       A.2.16. ECDSA, 409 Bits (Binary Field, Pseudorandom Curve) ....63
       A.2.17. ECDSA, 571 Bits (Binary Field, Pseudorandom Curve) ....66
     A.3.  Sample Code ...............................................70

1. Introduction

DSA [FIPS-186-4] and ECDSA [X9.62] are two standard digital signature schemes. They provide data integrity and verifiable authenticity in various protocols.

One characteristic of DSA and ECDSA is that they need to produce, for each signature generation, a fresh random value (hereafter designated as k). For effective security, k must be chosen randomly and uniformly from a set of modular integers, using a cryptographically secure process. Even slight biases in that process may be turned into attacks on the signature schemes.

The need for a cryptographically secure source of randomness proves to be a hindrance to deployment of DSA and ECDSA signature schemes in some architectures in which secure random number generation is challenging, in particular, embedded systems such as smartcards. In those systems, the RSA signature algorithm, used as specified in Public-Key Cryptography Standards (PKCS) #1 [RFC3447] (with "type 1" padding, not the Probabilistic Signature Scheme (PSS)) and ISO 9796-2 [ISO-9796-2], is often preferred, even though it is computationally more expensive, because RSA (with such padding schemes) is deterministic and thus does not require a source of randomness.

The randomized nature of DSA and ECDSA also makes implementations harder to test. Automatic tests cannot reliably detect whether the implementation uses a source of randomness of high enough quality. This makes the implementation process more vulnerable to catastrophic failures, often discovered after the system has been deployed and successfully attacked.

It is possible to turn DSA and ECDSA into deterministic schemes by using a deterministic process for generating the "random" value k. That process must fulfill some cryptographic characteristics in order to maintain the properties of verifiability and unforgeability expected from signature schemes; namely, for whoever does not know the signature private key, the mapping from input messages to the corresponding k values must be computationally indistinguishable from what a randomly and uniformly chosen function (from the set of messages to the set of possible k values) would return.

This document describes such a procedure. It has the following features:

  • Produced signatures remain fully compatible with plain DSA and ECDSA. Entities that verify the signatures need not be changed or even be aware of the process used to generate k.
  • Key pair generation is not altered. Existing private keys can be used with deterministic DSA and ECDSA.
  • Using deterministic DSA and ECDSA implies no extra storage requirement of any secret or public value.
  • Deterministic DSA and ECDSA can be applied over the same inputs as plain DSA and ECDSA, namely a hash value computed over the message that is to be signed, with a cryptographically secure hash function.

Some relatively arbitrary choices were taken in the definition of deterministic (EC)DSA as specified in this document; this was done in order to make it as universally applicable as possible, so as to maximize usefulness of included test vectors. See Section 3.6 for a discussion of some possible variants.

It shall be noted that key pair generation still requires a source of randomness. In embedded systems where quality of randomness is an issue, it can often be arranged that key pair generation occurs within more controlled conditions (e.g., during a special smartcard initialization procedure or under physical control of sworn agents) or the key might even be generated elsewhere and imported in the device. Deterministic DSA and ECDSA only deal with the need for randomness at the time of signature generation.

1.1. Requirements Language

The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in RFC 2119 [RFC2119].

2. DSA and ECDSA Notations

In this section, we succinctly describe DSA and ECDSA and define our notations. The complete specifications for DSA and ECDSA can be found in [FIPS-186-4] and [X9.62], respectively.

2.1. Key Parameters

DSA and ECDSA work over a large group of prime size, in which the group operation is easy to compute, but the discrete logarithm is computationally infeasible with existing and foreseeable technology. The definition of the group is called the "key parameters". Key parameters may be shared between different key pairs with no ill effect on security; this is the usual case with ECDSA in particular.

DSA uses the following key parameters:

   p    a large prime number (at least 1024 bits)
   q    a sufficiently large prime number (at least 160 bits) that is
        also a divisor of p-1
   g    a generator for the multiplicative subgroup of order q of
        integers modulo p

The group on which DSA will be computed consists of the values 'g^j mod p', where '^' denotes exponentiation and j ranges from 0 to q-1 (inclusive). The size of the group is q.

ECDSA uses the following key parameters:

   E    an elliptic curve, defined over a given finite field
   q    a sufficiently large prime number (at least 160 bits) that is a
        divisor of the curve order
   G    a point of E, of order q

The group on which ECDSA will be computed consists of the curve points jG (multiplication of point G by integer j) where j ranges from 0 to q-1. G is such that qG = 0 (the "point at infinity" on the curve E). The size of the group is q. Note that these notations slightly differ from those described in [X9.62]; we use them in order to match those used for DSA.

2.2. Key Pairs

A DSA or ECDSA private key is an integer x taken modulo q. The relevant standards prescribe that x shall not be 0; hence, x is an integer in the range [1, q-1].

A DSA or ECDSA public key is computed from the private key x and the key parameters:

  • For DSA, the public key is the integer: y = g^x mod p
  • For ECDSA, the public key is the curve point: U = xG

2.3. Integer Conversions

Let qlen be the binary length of q. qlen is the smallest integer such that q is less than 2^qlen. This is the size of the binary representation of q without a sign bit (note that q, being a big prime, is odd, thus avoiding any ambiguity about the length of any integer equal to a power of 2). We define five conversion functions, which work on strings of bits, octets, and integers modulo q. qlen is the main parameter for these conversions.

In the following subsections, we use two other lengths, called blen and rlen. rlen is equal to qlen, rounded up to the next multiple of 8 (if qlen is already a multiple of 8, then rlen equals qlen; otherwise, rlen is slightly larger, up to qlen+7). Note that rlen is unrelated to the value r, the first half of a generated signature. blen is the length (in bits) of an input sequence of bits and may vary between calls. blen may be smaller than, equal to, or larger than qlen.

2.3.1. Bits and Octets

Formally, all operations are defined on sequences of bits. A sequence is ordered; the first bit is said to be leftmost, while the last bit is rightmost.

On most software systems, bits are grouped into octets (sequences of eight bits). Binary data, e.g., the output of a hash function, is available as a sequence of octets. Whenever applicable, we consider that bits within an octet are ordered from most significant to least significant: the first (leftmost) bit within an octet has numerical value 128, while the last (rightmost) has numerical value 1.

2.3.2. Bit String to Integer

The bits2int transform takes as input a sequence of blen bits and outputs a non-negative integer that is less than 2^qlen. It consists of the following steps:

  1. The sequence is first truncated or expanded to length qlen:
  • if qlen < blen, then the qlen leftmost bits are kept, and subsequent bits are discarded;
  • otherwise, qlen-blen bits (of value zero) are added to the left of the sequence (i.e., before the input bits in the sequence order).
  1. The resulting sequence is then converted to an integer value using the big-endian convention: if input bits are called b_0 (leftmost) to b_(qlen-1) (rightmost), then the resulting value is:

b_0*2^(qlen-1) + b_1*2^(qlen-2) + ... + b_(qlen-1)*2^0

The bits2int transform can also be described in the following way: the input bit sequence (of length blen) is transformed into an integer using the big-endian convention. Then, if blen is greater than qlen, the resulting integer is divided by two to the power blen-qlen (Euclidian division: the remainder is discarded); in many software implementations of arithmetics on big integers, that division is equivalent to a "right shift" by blen-qlen bits.

2.3.3. Integer to Octet String

An integer value x less than q (and, in particular, a value that has been taken modulo q) can be converted into a sequence of rlen bits, where rlen = 8*ceil(qlen/8). This is the sequence of bits obtained by big-endian encoding. In other words, the sequence bits x_i (for i ranging from 0 to rlen-1) are such that:

      x = x_0*2^(rlen-1) + x_1*2^(rlen-2) + ... + x_(rlen-1)

We call this transform int2octets. Since rlen is a multiple of 8 (the smallest multiple of 8 that is not smaller than qlen), then the resulting sequence of bits is also a sequence of octets, hence the name.

2.3.4. Bit String to Octet String

The bits2octets transform takes as input a sequence of blen bits and outputs a sequence of rlen bits. It consists of the following steps:

  1. The input sequence b is converted into an integer value z1 through the bits2int transform:
          z1 = bits2int(b)
  1. z1 is reduced modulo q, yielding z2 (an integer between 0 and q-1, inclusive):
          z2 = z1 mod q

Note that since z1 is less than 2^qlen, that modular reduction can be implemented with a simple conditional subtraction: z2 = z1-q if that value is non-negative; otherwise, z2 = z1.

  1. z2 is transformed into a sequence of octets (a sequence of rlen bits) by applying int2octets.

2.3.5. Usage

It is worth noting that int2octets is not the reverse of bits2int, even for input sequences of length qlen: int2octets will add some bits on the left, while bits2int will discard some bits on the right. int2octets is the reverse of bits2int only when qlen is a multiple of 8 and bit sequences already have length qlen.

bits2int is used during signature generation and verification in standard DSA and ECDSA to transform a hash value (computed over the input message) into an integer modulo q. That is, the integer obtained through bits2int is further reduced modulo q; since that integer is less than 2^qlen, that reduction can be performed with at most one subtraction.

int2octets is defined under the name "Integer-to-OctetString" in Section 2.3.7 of SEC 1 [SEC1]. It is used in the specification of the encoding of an ECDSA private key (x) within an ASN.1-based structure.

bits2octets is not used in standard DSA or ECDSA. We will use it in the specification of deterministic (EC)DSA.

2.4. Signature Generation

Signature generation uses a cryptographic hash function H and an input message m. The message is first processed by H, yielding the value H(m), which is a sequence of bits of length hlen. Normally, H is chosen such that its output length hlen is roughly equal to qlen, since the overall security of the signature scheme will depend on the smallest of hlen and qlen; however, the relevant standards support all combinations of hlen and qlen.

The following steps are then applied:

  1. H(m) is transformed into an integer modulo q using the bits2int transform and an extra modular reduction:
          h = bits2int(H(m)) mod q

As was noted in the description of bits2octets, the extra modular reduction is no more than a conditional subtraction.

  1. A random value modulo q, dubbed k, is generated. That value shall not be 0; hence, it lies in the [1, q-1] range. Most of the remainder of this document will revolve around the process used to generate k. In plain DSA or ECDSA, k should be selected through a random selection that chooses a value among the q-1 possible values with uniform probability.
  1. A value r (modulo q) is computed from k and the key parameters:
  • For DSA:
             r = g^k mod p mod q

(The exponentiation is performed modulo p, yielding a number between 0 and p-1, which is then further reduced modulo q.)

  • For ECDSA: the point kG is computed; its X coordinate (a member of the field over which E is defined) is converted to an integer, which is reduced modulo q, yielding r.

If r turns out to be zero, a new k should be selected and r computed again (this is an utterly improbable occurrence).

  1. The value s (modulo q) is computed:
          s = (h+x*r)/k mod q

The pair (r, s) is the signature. How a signature is to be encoded is not covered by the DSA and ECDSA standards themselves; a common way is to use a DER-encoded ASN.1 structure (a SEQUENCE of two INTEGERs, for r and s, in that order).

3. Deterministic DSA and ECDSA

Deterministic (EC)DSA is the process of generating an (EC)DSA signature over an input message m by using the standard (EC)DSA signature generation process (discussed in the previous section), except that the value k, instead of being randomly generated, is obtained through the process described in this section.

We use the notations described in Section 2.

3.1. Building Blocks

3.1.1. HMAC

HMAC [RFC2104] is a construction of a Message Authentication Code using a hash function and a secret key. Here, we use HMAC with the same hash function H as the one used to process the input message prior to signature generation or verification.

We denote the process of applying HMAC with key K over data V by:


which returns a sequence of bits of length hlen (the output length of the underlying hash function H).

3.2. Generation of k

Given the input message m, the following process is applied:

a. Process m through the hash function H, yielding:

          h1 = H(m)

(h1 is a sequence of hlen bits).

b. Set:

          V = 0x01 0x01 0x01 ... 0x01

such that the length of V, in bits, is equal to 8*ceil(hlen/8). For instance, on an octet-based system, if H is SHA-256, then V is set to a sequence of 32 octets of value 1. Note that in this step and all subsequent steps, we use the same H function as the one used in step 'a' to process the input message; this choice will be discussed in more detail in Section 3.6.

c. Set:

          K = 0x00 0x00 0x00 ... 0x00

such that the length of K, in bits, is equal to 8*ceil(hlen/8).

d. Set:

          K = HMAC_K(V || 0x00 || int2octets(x) || bits2octets(h1))

where '||' denotes concatenation. In other words, we compute HMAC with key K, over the concatenation of the following, in order: the current value of V, a sequence of eight bits of value 0, the encoding of the (EC)DSA private key x, and the hashed message (possibly truncated and extended as specified by the bits2octets transform). The HMAC result is the new value of K. Note that the private key x is in the [1, q-1] range, hence a proper input for int2octets, yielding rlen bits of output, i.e., an integral number of octets (rlen is a multiple of 8).

e. Set:

          V = HMAC_K(V)

f. Set:

          K = HMAC_K(V || 0x01 || int2octets(x) || bits2octets(h1))

Note that the "internal octet" is 0x01 this time.

g. Set:

          V = HMAC_K(V)

h. Apply the following algorithm until a proper value is found for


  1. Set T to the empty sequence. The length of T (in bits) is denoted tlen; thus, at that point, tlen = 0.
  1. While tlen < qlen, do the following:
              V = HMAC_K(V)
              T = T || V
  1. Compute:
              k = bits2int(T)

If that value of k is within the [1,q-1] range, and is suitable for DSA or ECDSA (i.e., it results in an r value that is not 0; see Section 3.4), then the generation of k is finished. The obtained value of k is used in DSA or ECDSA. Otherwise, compute:

              K = HMAC_K(V || 0x00)
              V = HMAC_K(V)

and loop (try to generate a new T, and so on).

Please note that when k is generated from T, the result of bits2int is compared to q, not reduced modulo q. If the value is not between 1 and q-1, the process loops. Performing a simple modular reduction would induce biases that would be detrimental to signature security.

3.3. Alternate Description of the Generation of k

The process described in the previous section is actually derived from the "HMAC_DRBG" pseudorandom number generator, described in [SP800-90A] and Annex D of [X9.62]. Using the terminology from [SP800-90A], the generation of k can be described as such:

a. Instantiate HMAC_DRBG using HMAC parameterized with the same hash

function H as the one used for processing the message that is to be signed. Instantiation parameters are:


Set this parameter to any value that the HMAC_DRBG

implementation will accept, when using H as base hash



Set this parameter to "false".


Set this parameter to "Null" (the empty bit sequence).


Use int2octets(x) as entropy string.


Use bits2octets(H(m)) as nonce.

Note that the last two parameters are not parameters to the HMAC_DRBG instantiation function per se; instead, those values are requested from the internal Get_entropy_input function during instantiation. For deterministic (EC)DSA, we want HMAC_DRBG to run with the entropy string and nonce that we specify, without accessing an actual entropy source.

b. Generate a candidate value for k by requesting qlen bits from

HMAC_DRBG and converting the resulting bits into an integer with the bits2int transform. Repeat this step until a value is obtained, which is non-zero, less than q, and suitable for (EC)DSA (see Section 3.4).

Note that we instantiate a new HMAC_DRBG instance for each signature generation process. There is no "personalization string" and no "additional input" when generating bits. The reseed function of HMAC_DRBG is never invoked, neither externally nor as a consequence of the internal HMAC_DRBG processing.

As shown above, we use the encoding of the private key as "entropy string" and the hashed message (truncated and expanded by bits2octets) as "nonce". In HMAC_DRBG, the entropy string and nonce are simply concatenated into the initial seed; hence, the split between "entropy" and "nonce" is quite arbitrary. Using qlen bits for each ought to be compatible with most HMAC_DRBG implementation input requirements.

3.4. Usage Notes

With DSA or ECDSA, the value k is used to compute the first half of the signature, dubbed r (see Section 2.4). The DSA and ECDSA standards mandate that, if r is zero, then a new k should be selected. In that situation, this document specifies that the value k is "unsuitable", and the generation process shall keep on looping.

This occurrence is utterly improbable. Actually, it would require considerable computational effort (similar to breaking preimage resistance of the hash function) to find a private key and a message that lead to a zero value for r; hitting such a case by pure chance is thus deemed implausible, and an attacker cannot force it with carefully crafted messages. In practice, such a code path will not be triggered and thus can be implemented with little optimization.

3.5. Rationale

The process described in the previous sections mimics the "Approved" generation process of k described in Annex D of [X9.62], with the "HMAC_DRBG" pseudorandom number generator. The main difference is that we use the concatenation of the private key x and the hashed message H(m) as the pseudorandom number generator (PRNG) seed. If using a "security level" of n bits, then HMAC_DRBG should be used with seed entropy at least n+64 bits; however, the key x should also have been generated with that much entropy, and the length of x is qlen, which is at least equal to 2*n and thus larger than n+64 (DSA and ECDSA, as specified by the standards, require qlen >= 160). It can then be argued that deterministic ECDSA fulfills the entropy requirements of Annex D of [X9.62].

We use bits2octets(H(m)) instead of H(m) in order to ease integration. Indeed, many existing signature systems offload the message hashing; the signature engine (which has access to the private key) receives only H(m). In some applications, where data bandwidth is constrained, only the first qlen bits of H(m) are transferred to the signature engine, on the basis that the bits2int transform will ignore subsequent bits anyway. Possibly, in some systems, the truncated H(m) could be externally reduced modulo q, since that is the first thing that (EC)DSA performs on the hashed message. With the definition of bits2octets, deterministic (EC)DSA can be applied with the same input.

3.6. Variants

Many parts of the specification of deterministic (EC)DSA are quite arbitrary. It is possible to define variants that are NOT "deterministic (EC)DSA" but that may nonetheless be useful in some contexts:

  • It is possible to use H(m) directly, instead of bits2octets(H(m)), as part of the HMAC input. As explained in Section 3.5, we use bits2octets(H(m)) in order to ease integration into systems that already use an (EC)DSA signature engine by sending it an already- truncated hash value. Using the whole H(m) does not introduce any vulnerability.
  • Additional data may be added to the input of HMAC, concatenated after bits2octets(H(m)):
         K = HMAC_K(V || 0x00 || int2octets(x) || bits2octets(h1) || k')

A use case may be a protocol that requires a non-deterministic signature algorithm on a system that does not have access to a high-quality random source. It suffices that the additional data k' is non-repeating (e.g., a signature counter or a monotonic clock) to ensure "random-looking" signatures are indistinguishable, in a cryptographic way, from plain (EC)DSA signatures. In [SP800-90A] terminology, k' is the "additional input" that can be set as a parameter when generating pseudorandom bits. This variant can be thought of as a "strengthening" of the randomness of the source of the additional data k'.

  • Instead of using x (the private key) as input to HMAC, it is possible to use additional secret data, stored along with the private key with the same security measures. The entropy of that additional data SHALL be at least n bits, preferably n+64 bits or more, where n is the target security level. Having additional secret data may help in formally proving the security of derandomization, but it implies an extra storage cost and incompatibility with already-generated (EC)DSA private keys.
  • Similarly, the private key could be a value z, from which both x (the "private key" in the plain (EC)DSA sense) and another value x', to be used as input to HMAC in the generation of k, would be derived through a suitable Pseudorandom Function (PRF) (such as HMAC_DRBG). This would keep private key storage requirements to a minimum while providing a more easily proven security, but it would impact private key generation and would not be compatible with already-generated key pairs.
  • In this document, we use the same hash function H for processing the input message and as a parameter to HMAC. Two distinct hash functions could be used, provided that both are adequately secure. The overall security will be limited by the weaker of the two hash functions, i.e., the one with the smaller output. Using a specific, constant hash function for HMAC may be useful for constrained implementations that accept externally hashed messages, regardless of what hash function was used for that, but have resources for implementing only one hash function for HMAC.

The main disadvantage of any variant is that it ceases to be verifiable against the test vectors published in this document.

4. Security Considerations

Proper implementation and usage of a cryptographic signature algorithm require taking into account many parameters. In particular, private key generation, storage, access control, and disposal are sensitive operations, which this document does not address in any way. Deterministic (EC)DSA shows how to achieve the security characteristics of a standard DSA or ECDSA signature scheme while removing the need for a source of strong randomness, or even any source of randomness, during signature generation.

Private key generation, however, absolutely requires such a strongly random source. In situations where deterministic (EC)DSA is to be used due to the lack of an appropriate source of randomness, one must assume that the private key has been generated externally and imported into the signature generation system or was generated in a context where randomness was available. For instance, one can imagine a smartcard that generates its private key while still in the factory under controlled environmental conditions, but for which random data generation cannot be guaranteed once deployed in the field, when physically in the hands of potential attackers.

Both removal of the random source requirement and the ability to test an implementation against test vectors enhance security of DSA and ECDSA signer implementations, in that they help avoid hard-to-test failure conditions. Deterministic signature schemes may also help in other situations, e.g., to avoid spurious duplicates, when the same data element is signed several times with the same key: with a deterministic signature scheme, the same signature is generated every time, making duplicate detection much easier.

Conversely, lack of randomization may have adverse effects in some advanced protocols, e.g., related to anonymity in some voting schemes. As a rule of thumb, deterministic DSA or ECDSA can be used in lieu of the genuine DSA or ECDSA, with no additional security issues, if the overall protocol would tolerate another deterministic signature scheme, in particular RSA as specified in PKCS #1 [RFC3447] (with "type 1" padding, not PSS) or ISO 9796-2 [ISO-9796-2]. The list of protocols in which deterministic DSA or ECDSA is appropriate includes Transport Layer Security (TLS) [RFC5246], the Secure SHell (SSH) Protocol [RFC4251], Cryptographic Message Syntax (CMS) [RFC5652] and derivatives, X.509 public key infrastructures [RFC5280], and many others.

The construction described in this document is known as a "derandomization". This has been proposed for various signature schemes. Security relies on whether the generation of k is indistinguishable from the output of a random oracle. Roughly speaking, HMAC_DRBG is secure in that role as long as HMAC behaves as a PRF (Pseudorandom Function). For details on the security of HMAC and HMAC_DRBG, please refer to [H2008] and [B2006]. For a more formal treatment of derandomization, see [LN2009].

One remaining issue with deterministic (EC)DSA, as presented in this document, is the "double use" of the private key x, both as the private key in the signature generation algorithm itself and as input to the HMAC_DRBG-based pseudorandom oracle for producing the k value. This requires HMAC_DRBG to keep on being a random oracle, even when the public key (which is computed from x) is also known. Given the lack of common structure between HMAC and discrete logarithms, this seems a reasonable assumption.

Side-channel attacks are an important consideration whenever an attacker can accurately measure aspects of an implementation such as the length of time that it takes to perform a signing operation or the power consumed at each point of a signing operation. The determinism of the algorithms described in this note may be useful to an attacker in some forms of side-channel attacks, so implementations SHOULD use defensive measures to avoid leaking the private key through a side channel.

5. Intellectual Property Status

To the best of our knowledge, deterministic (EC)DSA is not covered by any active patent. The paper [BDLSY2011] points to two independent publications of the idea of derandomization by Barwood and Wigley, both in early 1997, and also to a patent application by Naccache, M'Raihi, and Levy-dit-Vehel a few months later [NML1997], but the application was withdrawn in 2003. We are not aware of any other patent on the subject.

6. References

6.1. Normative References

   [FIPS-186-4]  National Institute of Standards and Technology,
                 "Digital Signature Standard (DSS)", Federal Information
                 Processing Standards Publication (FIPS PUB) 186-4,
                 July 2013.
   [RFC2104]     Krawczyk, H., Bellare, M., and R. Canetti, "HMAC:
                 Keyed-Hashing for Message Authentication", RFC 2104,
                 February 1997.
   [RFC2119]     Bradner, S., "Key words for use in RFCs to Indicate
                 Requirement Levels", BCP 14, RFC 2119, March 1997.
   [SEC1]        Certicom Research, "SEC 1: Elliptic Curve Cryptography
                 (Version 2.0)", May 2009.
   [SP800-90A]   National Institute of Standards and Technology,
                 "Recommendation for Random Number Generation Using
                 Deterministic Random Bit Generators (Revised)", NIST
                 Special Publication 800-90A, January 2012.
   [X9.62]       American National Standards Institute, "Public Key
                 Cryptography for the Financial Services Industry: The
                 Elliptic Curve Digital Signature Algorithm (ECDSA)",
                 ANSI X9.62-2005, November 2005.

6.2. Informative References

   [B2006]       Bellare, M., "New Proofs for NMAC and HMAC: Security
                 without Collision-Resistance", Crypto 2006, LNCS 4117,
                 August 2006.
   [BDLSY2011]   Bernstein, D., Duif, N., Lange, T., Schwabe, P., and B.
                 Yang, "High-speed high-security signatures", Cryptology
                 ePrint Archive Report 2011/368, September 2011.
   [FIPS-180-4]  National Institute of Standards and Technology, "Secure
                 Hash Standard (SHS)", Federal Information Processing
                 Standards Publication (FIPS PUB) 180-4, March 2012.
   [H2008]       Hirose, S., "Security Analysis of DRBG Using HMAC in
                 NIST SP 800-90", Information Security Applications
                 (WISA 2008), LNCS 5379, September 2008.
   [ISO-9796-2]  International Organization for Standardization,
                 "Information technology -- Security techniques --
                 Digital signature schemes giving message recovery --
                 Part 2: Integer factorization based mechanisms", ISO/
                 IEC 9796-2:2010, December 2010.
   [LN2009]      Leurent, G. and P. Nguyen, "How Risky is the Random-
                 Oracle Model?", Cryptology ePrint Archive Report 2008/
                 441, July 2009, <>.
   [NML1997]     Naccache, D., M'Raihi, D., and F. Levy-dit-Vehel,
                 DRAWING", WIPO patent publication WO/1998/051038,
                 May 1998.
   [RFC3447]     Jonsson, J. and B. Kaliski, "Public-Key Cryptography
                 Standards (PKCS) #1: RSA Cryptography Specifications
                 Version 2.1", RFC 3447, February 2003.
   [RFC4251]     Ylonen, T. and C. Lonvick, "The Secure Shell (SSH)
                 Protocol Architecture", RFC 4251, January 2006.
   [RFC5246]     Dierks, T. and E. Rescorla, "The Transport Layer
                 Security (TLS) Protocol Version 1.2", RFC 5246,
                 August 2008.
   [RFC5280]     Cooper, D., Santesson, S., Farrell, S., Boeyen, S.,
                 Housley, R., and W. Polk, "Internet X.509 Public Key
                 Infrastructure Certificate and Certificate Revocation
                 List (CRL) Profile", RFC 5280, May 2008.
   [RFC5652]     Housley, R., "Cryptographic Message Syntax (CMS)",
                 STD 70, RFC 5652, September 2009.

Appendix A. Examples

A.1. Detailed Example

We detail here the intermediate values obtained during the generation of k on an example message and key. We use a binary curve because that specific curve is standard and has a group order length (qlen) that is not a multiple of 8; this illustrates the fine details of how conversions are performed between integers and bit sequences.

A.1.1. Key Pair

We consider ECDSA on the curve K-163 described in [FIPS-186-4] (also known as "ansix9t163k1" in [X9.62]). The curve is defined over a field GF(2^163): field elements are encoded into 163-bit strings. The order of the conventional base point is the prime value:

      q = 0x4000000000000000000020108A2E0CC0D99F8A5EF

which has length qlen = 163 bits.

Our private key is:

      x = 0x09A4D6792295A7F730FC3F2B49CBC0F62E862272F

The corresponding public key is the curve point U = xG. This point has two coordinates, which are elements of the field GF(2^163). These elements can be converted to integers using the procedure described in Section A.5.6 of [X9.62], yielding the two public point coordinates:

      Ux = 0x79AEE090DB05EC252D5CB4452F356BE198A4FF96F
      Uy = 0x782E29634DDC9A31EF40386E896BAA18B53AFA5A3

A.1.2. Generation of k

In this example, we use the hash function SHA-256 [FIPS-180-4]. The input message is the UTF-8 encoding of the string "sample" (6 octets, i.e., 48 bits).

The hashed input message h1 = SHA-256(m) is:


      AF 2B DB E1 AA 9B 6E C1 E2 AD E1 D6 94 F4 1F C7
      1A 83 1D 02 68 E9 89 15 62 11 3D 8A 62 AD D1 BF

(32 octets; each octet value is listed in hexadecimal notation).

We convert the private key x to a sequence of octets using the int2octets transform:


      00 9A 4D 67 92 29 5A 7F 73 0F C3 F2 B4 9C BC 0F
      62 E8 62 27 2F

Note: Although the specific value of x would numerically fit in 160 bits, i.e., 20 octets, we still encode x into 21 octets, because the encoding length is driven by the length of q, which is 163 bits.

We also truncate and/or expand the hashed message using bits2octets:


      01 79 5E DF 0D 54 DB 76 0F 15 6D 0D AC 04 C0 32
      2B 3A 20 42 24

The steps b to g (see Section 3.2) then compute the values for the K and V variables. These variables are sequences of 256 bits (the hash function output length, rounded up to a multiple of 8). We reproduce here the successive values:

V after step b:

      01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01
      01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

K after step c:

      00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
      00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

K after step d:

      09 99 9A 9B FE F9 72 D3 34 69 11 88 3F AD 79 51
      D2 3F 2C 8B 47 F4 20 22 2D 11 71 EE EE AC 5A B8

V after step e:

      D5 F4 03 0F 75 5E E8 6A A1 0B BA 8C 09 DF 11 4F
      F6 B6 11 1C 23 85 00 D1 3C 73 43 A8 C0 1B EC F7

K after step f:

      0C F2 FE 96 D5 61 9C 9E F5 3C B7 41 7D 49 D3 7E
      A6 8A 4F FE D0 D7 E6 23 E3 86 89 28 99 11 BD 57

V after step g:

      78 34 57 C1 CF 31 48 A8 F2 A9 AE 73 ED 47 2F A9
      8E D9 CD 92 5D 8E 96 4C E0 76 4D EF 3F 84 2B 9A

In step h, we perform the final loop. Since we use HMAC with SHA- 256, which produces 256 bits worth of output, and we need only 163 bits for T, a single HMAC invocation yields the following T:

T (first try)

      93 05 A4 6D E7 FF 8E B1 07 19 4D EB D3 FD 48 AA
      20 D5 E7 65 6C BE 0E A6 9D 2A 8D 4E 7C 67 31 4A

which, when converted to an integer with bits2int, yields a first candidate for k:

      k1 = 0x4982D236F3FFC758838CA6F5E9FEA455106AF3B2B

Since that value is greater than q-1, we have to loop. This first entails computing new values for K and V:

new K

      75 CB 5C 05 B2 A7 8C 3D 81 DF 12 D7 4D 7B E0 A0
      E9 4A B1 98 15 78 1D 4D 8E 29 02 A7 9D 0A 66 99

new V

      DC B9 CA 12 61 07 A9 C2 7C E7 7B A5 8E A8 71 C8
      C9 12 D8 35 EA DD C3 05 F2 44 5D 88 F6 6C 4C 43

then a new T:

T (second try)

      C7 0C 78 60 8A 3B 5B E9 28 9B E9 0E F6 E8 1A 9E
      2C 15 16 D5 75 1D 2F 75 F5 00 33 E4 5F 73 BD EB

and a new candidate for k:

      k2 = 0x63863C30451DADF4944DF4877B740D4F160A8B6AB

Since k2 is also greater than q-1, we loop again:

new K (2)

      0A 5A 64 B9 9C 05 95 20 10 36 86 CB 6F 36 BC FC
      A7 88 EB 3B CF 69 BA 66 A5 BB 08 0B 05 93 BA 53

new V (2)

      0B 3B 19 68 11 B1 9F 6C 6F 72 9C 43 F3 5B CF 0D
      FD 72 5F 17 CA 34 30 E8 72 14 53 E5 55 50 A1 8F

T (third try)

      47 5E 80 E9 92 14 05 67 FC C3 A5 0D AB 90 FE 84
      BC D7 BB 03 63 8E 9C 46 56 A0 6F 37 F6 50 8A 7C

and we finally get an acceptable value for k:

      k = 0x23AF4074C90A02B3FE61D286D5C87F425E6BDD81B

A.1.3. Signature

With our private key and the value of k that we just generated, we can now compute the signature using the standard ECDSA mechanisms. First, the point kG is computed, and the X coordinate of that point is converted to an integer and then reduced modulo q, yielding the first signature half:

      r = 0x113A63990598A3828C407C0F4D2438D990DF99A7F

which we use, together with x (the private key), k (which we computed above), and h = bits2int(h1), to compute the second signature half:

      s = 0x1313A2E03F5412DDB296A22E2C455335545672D9F

An ECDSA signature is a pair of integers. In many protocols that require a signature to be a sequence of bits (or octets), it is customary to encode the signature as an ASN.1 SEQUENCE of two INTEGER values, with DER rules. This results in the following 48-octet signature:

      30 2E 02 15 01 13 A6 39 90 59 8A 38 28 C4 07 C0
      F4 D2 43 8D 99 0D F9 9A 7F 02 15 01 31 3A 2E 03
      F5 41 2D DB 29 6A 22 E2 C4 55 33 55 45 67 2D 9F

A.2. Test Vectors

In the following sections, we give test vectors for various key sizes and hash functions, both for DSA and ECDSA.

All numbers are given in hexadecimal notation. Each signature consists of two integers, named r and s; many implementations will encode those integers into a single ASN.1 structure or with some other encoding convention, which is outside of the scope of this document. We also show the k value used internally.

For every key, we list ten signatures, corresponding to two distinct input messages, and five of the SHA [FIPS-180-4] functions: SHA-1, SHA-224, SHA-256, SHA-384, and SHA-512. The two input messages are the UTF-8 encoding of the strings "sample" and "test" (without the quotes), of length 48 and 32 bits, respectively.

The ECDSA examples use the standard curves described in [FIPS-186-4].

A.2.1. DSA, 1024 Bits

Key pair:

key parameters:

p = 86F5CA03DCFEB225063FF830A0C769B9DD9D6153AD91D7CE27F787C43278B447

E6533B86B18BED6E8A48B784A14C252C5BE0DBF60B86D6385BD2F12FB763ED88 73ABFD3F5BA2E0A8C0A59082EAC056935E529DAF7C610467899C77ADEDFC846C 881870B7B19B2B58F9BE0521A17002E3BDD6B86685EE90B3D9A1B02B782B1779

   q = 996F967F6C8E388D9E28D01E205FBA957A5698B1

g = 07B0F92546150B62514BB771E2A0C0CE387F03BDA6C56B505209FF25FD3C133D

89BBCD97E904E09114D9A7DEFDEADFC9078EA544D2E401AEECC40BB9FBBF78FD 87995A10A1C27CB7789B594BA7EFB5C4326A9FE59A070E136DB77175464ADCA4 17BE5DCE2F40D10A46A3A3943F26AB7FD9C0398FF8C76EE0A56826A8A88F1DBD

private key:

   x = 411602CB19A6CCC34494D79D98EF1E7ED5AF25F7

public key:

y = 5DF5E01DED31D0297E274E1691C192FE5868FEF9E19A84776454B100CF16F653

92195A38B90523E2542EE61871C0440CB87C322FC4B4D2EC5E1E7EC766E1BE8D 4CE935437DC11C3C8FD426338933EBFE739CB3465F4D3668C5E473508253B1E6 82F65CBDC4FAE93C2EA212390E54905A86E2223170B44EAA7DA5DD9FFCFB7F3B


With SHA-1, message = "sample":
k = 7BDB6B0FF756E1BB5D53583EF979082F9AD5BD5B
r = 2E1A0C2562B2912CAAF89186FB0F42001585DA55
s = 29EFB6B0AFF2D7A68EB70CA313022253B9A88DF5

With SHA-224, message = "sample":
k = 562097C06782D60C3037BA7BE104774344687649
r = 4BC3B686AEA70145856814A6F1BB53346F02101E
s = 410697B92295D994D21EDD2F4ADA85566F6F94C1

With SHA-256, message = "sample":
k = 519BA0546D0C39202A7D34D7DFA5E760B318BCFB
r = 81F2F5850BE5BC123C43F71A3033E9384611C545
s = 4CDD914B65EB6C66A8AAAD27299BEE6B035F5E89

With SHA-384, message = "sample":
k = 95897CD7BBB944AA932DBC579C1C09EB6FCFC595
r = 07F2108557EE0E3921BC1774F1CA9B410B4CE65A
s = 54DF70456C86FAC10FAB47C1949AB83F2C6F7595

With SHA-512, message = "sample":
k = 09ECE7CA27D0F5A4DD4E556C9DF1D21D28104F8B
r = 16C3491F9B8C3FBBDD5E7A7B667057F0D8EE8E1B
s = 02C36A127A7B89EDBB72E4FFBC71DABC7D4FC69C

With SHA-1, message = "test":
k = 5C842DF4F9E344EE09F056838B42C7A17F4A6433
r = 42AB2052FD43E123F0607F115052A67DCD9C5C77
s = 183916B0230D45B9931491D4C6B0BD2FB4AAF088

With SHA-224, message = "test":
k = 4598B8EFC1A53BC8AECD58D1ABBB0C0C71E67297
r = 6868E9964E36C1689F6037F91F28D5F2C30610F2
s = 49CEC3ACDC83018C5BD2674ECAAD35B8CD22940F

With SHA-256, message = "test":
k = 5A67592E8128E03A417B0484410FB72C0B630E1A
r = 22518C127299B0F6FDC9872B282B9E70D0790812
s = 6837EC18F150D55DE95B5E29BE7AF5D01E4FE160

With SHA-384, message = "test":
k = 220156B761F6CA5E6C9F1B9CF9C24BE25F98CD89
r = 854CF929B58D73C3CBFDC421E8D5430CD6DB5E66
s = 91D0E0F53E22F898D158380676A871A157CDA622

With SHA-512, message = "test":
k = 65D2C2EEB175E370F28C75BFCDC028D22C7DBE9C
r = 8EA47E475BA8AC6F2D821DA3BD212D11A3DEB9A0
s = 7C670C7AD72B6C050C109E1790008097125433E8

A.2.2. DSA, 2048 Bits

Key pair:

key parameters:

p = 9DB6FB5951B66BB6FE1E140F1D2CE5502374161FD6538DF1648218642F0B5C48

C8F7A41AADFA187324B87674FA1822B00F1ECF8136943D7C55757264E5A1A44F FE012E9936E00C1D3E9310B01C7D179805D3058B2A9F4BB6F9716BFE6117C6B5 B3CC4D9BE341104AD4A80AD6C94E005F4B993E14F091EB51743BF33050C38DE2 35567E1B34C3D6A5C0CEAA1A0F368213C3D19843D0B4B09DCB9FC72D39C8DE41 F1BF14D4BB4563CA28371621CAD3324B6A2D392145BEBFAC748805236F5CA2FE 92B871CD8F9C36D3292B5509CA8CAA77A2ADFC7BFD77DDA6F71125A7456FEA15 3E433256A2261C6A06ED3693797E7995FAD5AABBCFBE3EDA2741E375404AE25B

   q = F2C3119374CE76C9356990B465374A17F23F9ED35089BD969F61C6DDE9998C1F

g = 5C7FF6B06F8F143FE8288433493E4769C4D988ACE5BE25A0E24809670716C613

D7B0CEE6932F8FAA7C44D2CB24523DA53FBE4F6EC3595892D1AA58C4328A06C4 6A15662E7EAA703A1DECF8BBB2D05DBE2EB956C142A338661D10461C0D135472 085057F3494309FFA73C611F78B32ADBB5740C361C9F35BE90997DB2014E2EF5 AA61782F52ABEB8BD6432C4DD097BC5423B285DAFB60DC364E8161F4A2A35ACA 3A10B1C4D203CC76A470A33AFDCBDD92959859ABD8B56E1725252D78EAC66E71 BA9AE3F1DD2487199874393CD4D832186800654760E1E34C09E4D155179F9EC0 DC4473F996BDCE6EED1CABED8B6F116F7AD9CF505DF0F998E34AB27514B0FFE7

private key:

   x = 69C7548C21D0DFEA6B9A51C9EAD4E27C33D3B3F180316E5BCAB92C933F0E4DBC

public key:

y = 667098C654426C78D7F8201EAC6C203EF030D43605032C2F1FA937E5237DBD94

9F34A0A2564FE126DC8B715C5141802CE0979C8246463C40E6B6BDAA2513FA61 1728716C2E4FD53BC95B89E69949D96512E873B9C8F8DFD499CC312882561ADE CB31F658E934C0C197F2C4D96B05CBAD67381E7B768891E4DA3843D24D94CDFB 5126E9B8BF21E8358EE0E0A30EF13FD6A664C0DCE3731F7FB49A4845A4FD8254 687972A2D382599C9BAC4E0ED7998193078913032558134976410B89D2C171D1 23AC35FD977219597AA7D15C1A9A428E59194F75C721EBCBCFAE44696A499AFA 74E04299F132026601638CB87AB79190D4A0986315DA8EEC6561C938996BEADF


With SHA-1, message = "sample":
k = 888FA6F7738A41BDC9846466ABDB8174C0338250AE50CE955CA16230F9CBD53E r = 3A1B2DBD7489D6ED7E608FD036C83AF396E290DBD602408E8677DAABD6E7445A s = D26FCBA19FA3E3058FFC02CA1596CDBB6E0D20CB37B06054F7E36DED0CDBBCCF

With SHA-224, message = "sample":
k = BC372967702082E1AA4FCE892209F71AE4AD25A6DFD869334E6F153BD0C4D806 r = DC9F4DEADA8D8FF588E98FED0AB690FFCE858DC8C79376450EB6B76C24537E2C s = A65A9C3BC7BABE286B195D5DA68616DA8D47FA0097F36DD19F517327DC848CEC

With SHA-256, message = "sample":
k = 8926A27C40484216F052F4427CFD5647338B7B3939BC6573AF4333569D597C52 r = EACE8BDBBE353C432A795D9EC556C6D021F7A03F42C36E9BC87E4AC7932CC809 s = 7081E175455F9247B812B74583E9E94F9EA79BD640DC962533B0680793A38D53

With SHA-384, message = "sample":
k = C345D5AB3DA0A5BCB7EC8F8FB7A7E96069E03B206371EF7D83E39068EC564920 r = B2DA945E91858834FD9BF616EBAC151EDBC4B45D27D0DD4A7F6A22739F45C00B s = 19048B63D9FD6BCA1D9BAE3664E1BCB97F7276C306130969F63F38FA8319021B

With SHA-512, message = "sample":
k = 5A12994431785485B3F5F067221517791B85A597B7A9436995C89ED0374668FC r = 2016ED092DC5FB669B8EFB3D1F31A91EECB199879BE0CF78F02BA062CB4C942E s = D0C76F84B5F091E141572A639A4FB8C230807EEA7D55C8A154A224400AFF2351

With SHA-1, message = "test":
k = 6EEA486F9D41A037B2C640BC5645694FF8FF4B98D066A25F76BE641CCB24BA4F r = C18270A93CFC6063F57A4DFA86024F700D980E4CF4E2CB65A504397273D98EA0 s = 414F22E5F31A8B6D33295C7539C1C1BA3A6160D7D68D50AC0D3A5BEAC2884FAA

With SHA-224, message = "test":
k = 06BD4C05ED74719106223BE33F2D95DA6B3B541DAD7BFBD7AC508213B6DA6670 r = 272ABA31572F6CC55E30BF616B7A265312018DD325BE031BE0CC82AA17870EA3 s = E9CC286A52CCE201586722D36D1E917EB96A4EBDB47932F9576AC645B3A60806

With SHA-256, message = "test":
k = 1D6CE6DDA1C5D37307839CD03AB0A5CBB18E60D800937D67DFB4479AAC8DEAD7 r = 8190012A1969F9957D56FCCAAD223186F423398D58EF5B3CEFD5A4146A4476F0 s = 7452A53F7075D417B4B013B278D1BB8BBD21863F5E7B1CEE679CF2188E1AB19E

With SHA-384, message = "test":
k = 206E61F73DBE1B2DC8BE736B22B079E9DACD974DB00EEBBC5B64CAD39CF9F91C r = 239E66DDBE8F8C230A3D071D601B6FFBDFB5901F94D444C6AF56F732BEB954BE s = 6BD737513D5E72FE85D1C750E0F73921FE299B945AAD1C802F15C26A43D34961

With SHA-512, message = "test":
k = AFF1651E4CD6036D57AA8B2A05CCF1A9D5A40166340ECBBDC55BE10B568AA0AA r = 89EC4BB1400ECCFF8E7D9AA515CD1DE7803F2DAFF09693EE7FD1353E90A68307 s = C9F0BDABCC0D880BB137A994CC7F3980CE91CC10FAF529FC46565B15CEA854E1

A.2.3. ECDSA, 192 Bits (Prime Field)

Key pair:


          NIST P-192

(qlen = 192 bits)

private key:

   x = 6FAB034934E4C0FC9AE67F5B5659A9D7D1FEFD187EE09FD4
   public key: U = xG
   Ux = AC2C77F529F91689FEA0EA5EFEC7F210D8EEA0B9E047ED56
   Uy = 3BC723E57670BD4887EBC732C523063D0A7C957BC97C1C43


With SHA-1, message = "sample":
k = 37D7CA00D2C7B0E5E412AC03BD44BA837FDD5B28CD3B0021
r = 98C6BD12B23EAF5E2A2045132086BE3EB8EBD62ABF6698FF
s = 57A22B07DEA9530F8DE9471B1DC6624472E8E2844BC25B64

With SHA-224, message = "sample":
k = 4381526B3FC1E7128F202E194505592F01D5FF4C5AF015D8
r = A1F00DAD97AEEC91C95585F36200C65F3C01812AA60378F5
s = E07EC1304C7C6C9DEBBE980B9692668F81D4DE7922A0F97A

With SHA-256, message = "sample":
k = 32B1B6D7D42A05CB449065727A84804FB1A3E34D8F261496
r = 4B0B8CE98A92866A2820E20AA6B75B56382E0F9BFD5ECB55
s = CCDB006926EA9565CBADC840829D8C384E06DE1F1E381B85

With SHA-384, message = "sample":
k = 4730005C4FCB01834C063A7B6760096DBE284B8252EF4311
r = DA63BF0B9ABCF948FBB1E9167F136145F7A20426DCC287D5
s = C3AA2C960972BD7A2003A57E1C4C77F0578F8AE95E31EC5E

With SHA-512, message = "sample":
k = A2AC7AB055E4F20692D49209544C203A7D1F2C0BFBC75DB1
r = 4D60C5AB1996BD848343B31C00850205E2EA6922DAC2E4B8
s = 3F6E837448F027A1BF4B34E796E32A811CBB4050908D8F67

With SHA-1, message = "test":
k = D9CF9C3D3297D3260773A1DA7418DB5537AB8DD93DE7FA25
r = 0F2141A0EBBC44D2E1AF90A50EBCFCE5E197B3B7D4DE036D
s = EB18BC9E1F3D7387500CB99CF5F7C157070A8961E38700B7

With SHA-224, message = "test":
k = F5DC805F76EF851800700CCE82E7B98D8911B7D510059FBE
r = 6945A1C1D1B2206B8145548F633BB61CEF04891BAF26ED34
s = B7FB7FDFC339C0B9BD61A9F5A8EAF9BE58FC5CBA2CB15293

With SHA-256, message = "test":
k = 5C4CE89CF56D9E7C77C8585339B006B97B5F0680B4306C6C
r = 3A718BD8B4926C3B52EE6BBE67EF79B18CB6EB62B1AD97AE
s = 5662E6848A4A19B1F1AE2F72ACD4B8BBE50F1EAC65D9124F

With SHA-384, message = "test":
k = 5AFEFB5D3393261B828DB6C91FBC68C230727B030C975693
r = B234B60B4DB75A733E19280A7A6034BD6B1EE88AF5332367
s = 7994090B2D59BB782BE57E74A44C9A1C700413F8ABEFE77A

With SHA-512, message = "test":
k = 0758753A5254759C7CFBAD2E2D9B0792EEE44136C9480527
r = FE4F4AE86A58B6507946715934FE2D8FF9D95B6B098FE739
s = 74CF5605C98FBA0E1EF34D4B5A1577A7DCF59457CAE52290

A.2.4. ECDSA, 224 Bits (Prime Field)

Key pair:


          NIST P-224

q = FFFFFFFFFFFFFFFFFFFFFFFFFFFF16A2E0B8F03E13DD29455C5C2A3D (qlen = 224 bits)

private key:

   x = F220266E1105BFE3083E03EC7A3A654651F45E37167E88600BF257C1
   public key: U = xG
   Ux = 00CF08DA5AD719E42707FA431292DEA11244D64FC51610D94B130D6C
   Uy = EEAB6F3DEBE455E3DBF85416F7030CBD94F34F2D6F232C69F3C1385A


With SHA-1, message = "sample":
k = 7EEFADD91110D8DE6C2C470831387C50D3357F7F4D477054B8B426BC
r = 22226F9D40A96E19C4A301CE5B74B115303C0F3A4FD30FC257FB57AC
s = 66D1CDD83E3AF75605DD6E2FEFF196D30AA7ED7A2EDF7AF475403D69

With SHA-224, message = "sample":
k = C1D1F2F10881088301880506805FEB4825FE09ACB6816C36991AA06D
s = A6694FD7718A21053F225D3F46197CA699D45006C06F871808F43EBC

With SHA-256, message = "sample":
k = AD3029E0278F80643DE33917CE6908C70A8FF50A411F06E41DEDFCDC
r = 61AA3DA010E8E8406C656BC477A7A7189895E7E840CDFE8FF42307BA
s = BC814050DAB5D23770879494F9E0A680DC1AF7161991BDE692B10101

With SHA-384, message = "sample":
k = 52B40F5A9D3D13040F494E83D3906C6079F29981035C7BD51E5CAC40
r = 0B115E5E36F0F9EC81F1325A5952878D745E19D7BB3EABFABA77E953
s = 830F34CCDFE826CCFDC81EB4129772E20E122348A2BBD889A1B1AF1D

With SHA-512, message = "sample":
k = 9DB103FFEDEDF9CFDBA05184F925400C1653B8501BAB89CEA0FBEC14
r = 074BD1D979D5F32BF958DDC61E4FB4872ADCAFEB2256497CDAC30397
s = A4CECA196C3D5A1FF31027B33185DC8EE43F288B21AB342E5D8EB084

With SHA-1, message = "test":
k = 2519178F82C3F0E4F87ED5883A4E114E5B7A6E374043D8EFD329C253
r = DEAA646EC2AF2EA8AD53ED66B2E2DDAA49A12EFD8356561451F3E21C
s = 95987796F6CF2062AB8135271DE56AE55366C045F6D9593F53787BD2

With SHA-224, message = "test":
k = DF8B38D40DCA3E077D0AC520BF56B6D565134D9B5F2EAE0D34900524
r = C441CE8E261DED634E4CF84910E4C5D1D22C5CF3B732BB204DBEF019
s = 902F42847A63BDC5F6046ADA114953120F99442D76510150F372A3F4

With SHA-256, message = "test":
k = FF86F57924DA248D6E44E8154EB69F0AE2AEBAEE9931D0B5A969F904
r = AD04DDE87B84747A243A631EA47A1BA6D1FAA059149AD2440DE6FBA6
s = 178D49B1AE90E3D8B629BE3DB5683915F4E8C99FDF6E666CF37ADCFD

With SHA-384, message = "test":
k = 7046742B839478C1B5BD31DB2E862AD868E1A45C863585B5F22BDC2D
r = 389B92682E399B26518A95506B52C03BC9379A9DADF3391A21FB0EA4
s = 414A718ED3249FF6DBC5B50C27F71F01F070944DA22AB1F78F559AAB

With SHA-512, message = "test":
k = E39C2AA4EA6BE2306C72126D40ED77BF9739BB4D6EF2BBB1DCB6169D
r = 049F050477C5ADD858CAC56208394B5A55BAEBBE887FDF765047C17C
s = 077EB13E7005929CEFA3CD0403C7CDCC077ADF4E44F3C41B2F60ECFF

A.2.5. ECDSA, 256 Bits (Prime Field)

Key pair:


          NIST P-256

q = FFFFFFFF00000000FFFFFFFFFFFFFFFFBCE6FAADA7179E84F3B9CAC2FC632551 (qlen = 256 bits)

private key:

   x = C9AFA9D845BA75166B5C215767B1D6934E50C3DB36E89B127B8A622B120F6721
   public key: U = xG
   Ux = 60FED4BA255A9D31C961EB74C6356D68C049B8923B61FA6CE669622E60F29FB6
   Uy = 7903FE1008B8BC99A41AE9E95628BC64F2F1B20C2D7E9F5177A3C294D4462299


With SHA-1, message = "sample":
k = 882905F1227FD620FBF2ABF21244F0BA83D0DC3A9103DBBEE43A1FB858109DB4 r = 61340C88C3AAEBEB4F6D667F672CA9759A6CCAA9FA8811313039EE4A35471D32 s = 6D7F147DAC089441BB2E2FE8F7A3FA264B9C475098FDCF6E00D7C996E1B8B7EB

With SHA-224, message = "sample":
k = 103F90EE9DC52E5E7FB5132B7033C63066D194321491862059967C715985D473 r = 53B2FFF5D1752B2C689DF257C04C40A587FABABB3F6FC2702F1343AF7CA9AA3F s = B9AFB64FDC03DC1A131C7D2386D11E349F070AA432A4ACC918BEA988BF75C74C

With SHA-256, message = "sample":
k = A6E3C57DD01ABE90086538398355DD4C3B17AA873382B0F24D6129493D8AAD60 r = EFD48B2AACB6A8FD1140DD9CD45E81D69D2C877B56AAF991C34D0EA84EAF3716 s = F7CB1C942D657C41D436C7A1B6E29F65F3E900DBB9AFF4064DC4AB2F843ACDA8

With SHA-384, message = "sample":
k = 09F634B188CEFD98E7EC88B1AA9852D734D0BC272F7D2A47DECC6EBEB375AAD4 r = 0EAFEA039B20E9B42309FB1D89E213057CBF973DC0CFC8F129EDDDC800EF7719 s = 4861F0491E6998B9455193E34E7B0D284DDD7149A74B95B9261F13ABDE940954

With SHA-512, message = "sample":
k = 5FA81C63109BADB88C1F367B47DA606DA28CAD69AA22C4FE6AD7DF73A7173AA5 r = 8496A60B5E9B47C825488827E0495B0E3FA109EC4568FD3F8D1097678EB97F00 s = 2362AB1ADBE2B8ADF9CB9EDAB740EA6049C028114F2460F96554F61FAE3302FE

With SHA-1, message = "test":
k = 8C9520267C55D6B980DF741E56B4ADEE114D84FBFA2E62137954164028632A2E r = 0CBCC86FD6ABD1D99E703E1EC50069EE5C0B4BA4B9AC60E409E8EC5910D81A89 s = 01B9D7B73DFAA60D5651EC4591A0136F87653E0FD780C3B1BC872FFDEAE479B1

With SHA-224, message = "test":
k = 669F4426F2688B8BE0DB3A6BD1989BDAEFFF84B649EEB84F3DD26080F667FAA7 r = C37EDB6F0AE79D47C3C27E962FA269BB4F441770357E114EE511F662EC34A692 s = C820053A05791E521FCAAD6042D40AEA1D6B1A540138558F47D0719800E18F2D

With SHA-256, message = "test":
k = D16B6AE827F17175E040871A1C7EC3500192C4C92677336EC2537ACAEE0008E0 r = F1ABB023518351CD71D881567B1EA663ED3EFCF6C5132B354F28D3B0B7D38367 s = 019F4113742A2B14BD25926B49C649155F267E60D3814B4C0CC84250E46F0083

With SHA-384, message = "test":
k = 16AEFFA357260B04B1DD199693960740066C1A8F3E8EDD79070AA914D361B3B8 r = 83910E8B48BB0C74244EBDF7F07A1C5413D61472BD941EF3920E623FBCCEBEB6 s = 8DDBEC54CF8CD5874883841D712142A56A8D0F218F5003CB0296B6B509619F2C

With SHA-512, message = "test":
k = 6915D11632ACA3C40D5D51C08DAF9C555933819548784480E93499000D9F0B7F r = 461D93F31B6540894788FD206C07CFA0CC35F46FA3C91816FFF1040AD1581A04 s = 39AF9F15DE0DB8D97E72719C74820D304CE5226E32DEDAE67519E840D1194E55

A.2.6. ECDSA, 384 Bits (Prime Field)

Key pair:


          NIST P-384
   (qlen = 384 bits)

private key:

x = 6B9D3DAD2E1B8C1C05B19875B6659F4DE23C3B667BF297BA9AA47740787137D8

   public key: U = xG

Ux = EC3A4E415B4E19A4568618029F427FA5DA9A8BC4AE92E02E06AAE5286B300C64


Uy = 8015D9B72D7D57244EA8EF9AC0C621896708A59367F9DFB9F54CA84B3F1C9DB1



   With SHA-1, message = "sample":
   k = 4471EF7518BB2C7C20F62EAE1C387AD0C5E8E470995DB4ACF694466E6AB09663
   r = EC748D839243D6FBEF4FC5C4859A7DFFD7F3ABDDF72014540C16D73309834FA3
   s = A3BCFA947BEEF4732BF247AC17F71676CB31A847B9FF0CBC9C9ED4C1A5B3FACF
   With SHA-224, message = "sample":
   k = A4E4D2F0E729EB786B31FC20AD5D849E304450E0AE8E3E341134A5C1AFA03CAB
   r = 42356E76B55A6D9B4631C865445DBE54E056D3B3431766D0509244793C3F9366
   s = 9DA0C81787064021E78DF658F2FBB0B042BF304665DB721F077A4298B095E483
   With SHA-256, message = "sample":
   k = 180AE9F9AEC5438A44BC159A1FCB277C7BE54FA20E7CF404B490650A8ACC414E
   r = 21B13D1E013C7FA1392D03C5F99AF8B30C570C6F98D4EA8E354B63A21D3DAA33
   s = F3AA443FB107745BF4BD77CB3891674632068A10CA67E3D45DB2266FA7D1FEEB
   With SHA-384, message = "sample":
   k = 94ED910D1A099DAD3254E9242AE85ABDE4BA15168EAF0CA87A555FD56D10FBCA
   r = 94EDBB92A5ECB8AAD4736E56C691916B3F88140666CE9FA73D64C4EA95AD133C
   s = 99EF4AEB15F178CEA1FE40DB2603138F130E740A19624526203B6351D0A3A94F
   With SHA-512, message = "sample":
   k = 92FC3C7183A883E24216D1141F1A8976C5B0DD797DFA597E3D7B32198BD35331
   r = ED0959D5880AB2D869AE7F6C2915C6D60F96507F9CB3E047C0046861DA4A799C
   s = 512C8CCEEE3890A84058CE1E22DBC2198F42323CE8ACA9135329F03C068E5112
   With SHA-1, message = "test":
   k = 66CC2C8F4D303FC962E5FF6A27BD79F84EC812DDAE58CF5243B64A4AD8094D47
   r = 4BC35D3A50EF4E30576F58CD96CE6BF638025EE624004A1F7789A8B8E43D0678
   s = D5A6326C494ED3FF614703878961C0FDE7B2C278F9A65FD8C4B7186201A29916
   With SHA-224, message = "test":
   k = 18FA39DB95AA5F561F30FA3591DC59C0FA3653A80DAFFA0B48D1A4C6DFCBFF6E
   r = E8C9D0B6EA72A0E7837FEA1D14A1A9557F29FAA45D3E7EE888FC5BF954B5E624
   s = 07041D4A7A0379AC7232FF72E6F77B6DDB8F09B16CCE0EC3286B2BD43FA8C614
   With SHA-256, message = "test":
   k = 0CFAC37587532347DC3389FDC98286BBA8C73807285B184C83E62E26C401C0FA
   r = 6D6DEFAC9AB64DABAFE36C6BF510352A4CC27001263638E5B16D9BB51D451559
   s = 2D46F3BECBCC523D5F1A1256BF0C9B024D879BA9E838144C8BA6BAEB4B53B47D
   With SHA-384, message = "test":
   k = 015EE46A5BF88773ED9123A5AB0807962D193719503C527B031B4C2D225092AD
   r = 8203B63D3C853E8D77227FB377BCF7B7B772E97892A80F36AB775D509D7A5FEB
   s = DDD0760448D42D8A43AF45AF836FCE4DE8BE06B485E9B61B827C2F13173923E0
   With SHA-512, message = "test":
   k = 3780C4F67CB15518B6ACAE34C9F83568D2E12E47DEAB6C50A4E4EE5319D1E8CE
   r = A0D5D090C9980FAF3C2CE57B7AE951D31977DD11C775D314AF55F76C676447D0
   s = 976984E59B4C77B0E8E4460DCA3D9F20E07B9BB1F63BEEFAF576F6B2E8B22463

A.2.7. ECDSA, 521 Bits (Prime Field)

Key pair:


          NIST P-521
   (qlen = 521 bits)

private key:

x = 0FAD06DAA62BA3B25D2FB40133DA757205DE67F5BB0018FEE8C86E1B68C7E75C

AA896EB32F1F47C70855836A6D16FCC1466F6D8FBEC67DB89EC0C08B0E996B83 538

   public key: U = xG

Ux = 1894550D0785932E00EAA23B694F213F8C3121F86DC97A04E5A7167DB4E5BCD3

71123D46E45DB6B5D5370A7F20FB633155D38FFA16D2BD761DCAC474B9A2F502 3A4

Uy = 0493101C962CD4D2FDDF782285E64584139C2F91B47F87FF82354D6630F746A2

8A0DB25741B5B34A828008B22ACC23F924FAAFBD4D33F81EA66956DFEAA2BFDF CF5


   With SHA-1, message = "sample":
   k = 089C071B419E1C2820962321787258469511958E80582E95D8378E0C2CCDB3CB
   r = 0343B6EC45728975EA5CBA6659BBB6062A5FF89EEA58BE3C80B619F322C87910
   s = 0E7B0E675A9B24413D448B8CC119D2BF7B2D2DF032741C096634D6D65D0DBE3D
   With SHA-224, message = "sample":
   k = 121415EC2CD7726330A61F7F3FA5DE14BE9436019C4DB8CB4041F3B54CF31BE0
   r = 1776331CFCDF927D666E032E00CF776187BC9FDD8E69D0DABB4109FFE1B5E2A3
   s = 050CB5265417FE2320BBB5A122B8E1A32BD699089851128E360E620A30C7E17B
   With SHA-256, message = "sample":
   k = 0EDF38AFCAAECAB4383358B34D67C9F2216C8382AAEA44A3DAD5FDC9C3257576
   r = 1511BB4D675114FE266FC4372B87682BAECC01D3CC62CF2303C92B3526012659
   s = 04A171143A83163D6DF460AAF61522695F207A58B95C0644D87E52AA1A347916
   With SHA-384, message = "sample":
   k = 1546A108BC23A15D6F21872F7DED661FA8431DDBD922D0DCDB77CC878C8553FF
   r = 1EA842A0E17D2DE4F92C15315C63DDF72685C18195C2BB95E572B9C5136CA4B4
   s = 1F21A3CEE066E1961025FB048BD5FE2B7924D0CD797BABE0A83B66F1E35EEAF5
   With SHA-512, message = "sample":
   k = 1DAE2EA071F8110DC26882D4D5EAE0621A3256FC8847FB9022E2B7D28E6F1019
   r = 0C328FAFCBD79DD77850370C46325D987CB525569FB63C5D3BC53950E6D4C5F1
   s = 0617CCE7CF5064806C467F678D3B4080D6F1CC50AF26CA209417308281B68AF2
   With SHA-1, message = "test":
   k = 0BB9F2BF4FE1038CCF4DABD7139A56F6FD8BB1386561BD3C6A4FC818B20DF5DD
   r = 13BAD9F29ABE20DE37EBEB823C252CA0F63361284015A3BF430A46AAA80B87B0
   s = 1E9BB81FF7944CA409AD138DBBEE228E1AFCC0C890FC78EC8604639CB0DBDC90
   With SHA-224, message = "test":
   k = 040D09FCF3C8A5F62CF4FB223CBBB2B9937F6B0577C27020A99602C25A011369
   r = 1C7ED902E123E6815546065A2C4AF977B22AA8EADDB68B2C1110E7EA44D42086
   s = 177336676304FCB343CE028B38E7B4FBA76C1C1B277DA18CAD2A8478B2A9A9F5
   With SHA-256, message = "test":
   k = 01DE74955EFAABC4C4F17F8E84D881D1310B5392D7700275F82F145C61E84384
   r = 00E871C4A14F993C6C7369501900C4BC1E9C7B0B4BA44E04868B30B41D807104
   s = 0CD52DBAA33B063C3A6CD8058A1FB0A46A4754B034FCC644766CA14DA8CA5CA9
   With SHA-384, message = "test":
   k = 1F1FC4A349A7DA9A9E116BFDD055DC08E78252FF8E23AC276AC88B1770AE0B5D
   r = 14BEE21A18B6D8B3C93FAB08D43E739707953244FDBE924FA926D76669E7AC8C
   s = 133330865C067A0EAF72362A65E2D7BC4E461E8C8995C3B6226A21BD1AA78F0E
   With SHA-512, message = "test":
   k = 16200813020EC986863BEDFC1B121F605C1215645018AEA1A7B215A564DE9EB1
   r = 13E99020ABF5CEE7525D16B69B229652AB6BDF2AFFCAEF38773B4B7D08725F10
   s = 1FBD0013C674AA79CB39849527916CE301C66EA7CE8B80682786AD60F98F7E78

A.2.8. ECDSA, 163 Bits (Binary Field, Koblitz Curve)

Key pair:


          NIST K-163

q = 4000000000000000000020108A2E0CC0D99F8A5EF
(qlen = 163 bits)

private key:

   x = 09A4D6792295A7F730FC3F2B49CBC0F62E862272F
   public key: U = xG
   Ux = 79AEE090DB05EC252D5CB4452F356BE198A4FF96F
   Uy = 782E29634DDC9A31EF40386E896BAA18B53AFA5A3


With SHA-1, message = "sample":
k = 09744429FA741D12DE2BE8316E35E84DB9E5DF1CD
r = 30C45B80BA0E1406C4EFBBB7000D6DE4FA465D505
s = 38D87DF89493522FC4CD7DE1553BD9DBBA2123011

With SHA-224, message = "sample":
k = 323E7B28BFD64E6082F5B12110AA87BC0D6A6E159
r = 38A2749F7EA13BD5DA0C76C842F512D5A65FFAF32
s = 064F841F70112B793FD773F5606BFA5AC2A04C1E8

With SHA-256, message = "sample":
k = 23AF4074C90A02B3FE61D286D5C87F425E6BDD81B
r = 113A63990598A3828C407C0F4D2438D990DF99A7F
s = 1313A2E03F5412DDB296A22E2C455335545672D9F

With SHA-384, message = "sample":
k = 2132ABE0ED518487D3E4FA7FD24F8BED1F29CCFCE
r = 34D4DE955871BB84FEA4E7D068BA5E9A11BD8B6C4
s = 2BAAF4D4FD57F175C405A2F39F9755D9045C820BD

With SHA-512, message = "sample":
k = 00BBCC2F39939388FDFE841892537EC7B1FF33AA3
r = 38E487F218D696A7323B891F0CCF055D895B77ADC
s = 0972D7721093F9B3835A5EB7F0442FA8DCAA873C4

With SHA-1, message = "test":
k = 14CAB9192F39C8A0EA8E81B4B87574228C99CD681
r = 1375BEF93F21582F601497036A7DC8014A99C2B79
s = 254B7F1472FFFEE9002D081BB8CE819CCE6E687F9

With SHA-224, message = "test":
k = 091DD986F38EB936BE053DD6ACE3419D2642ADE8D
r = 110F17EF209957214E35E8C2E83CBE73B3BFDEE2C
s = 057D5022392D359851B95DEC2444012502A5349CB

With SHA-256, message = "test":
k = 193649CE51F0CFF0784CFC47628F4FA854A93F7A2
r = 0354D5CD24F9C41F85D02E856FA2B0001C83AF53E
s = 020B200677731CD4FE48612A92F72A19853A82B65

With SHA-384, message = "test":
k = 37C73C6F8B404EC83DA17A6EBCA724B3FF1F7EEBA
r = 11B6A84206515495AD8DBB2E5785D6D018D75817E
s = 1A7D4C1E17D4030A5D748ADEA785C77A54581F6D0

With SHA-512, message = "test":
k = 331AD98D3186F73967B1E0B120C80B1E22EFC2988
r = 148934745B351F6367FF5BB56B1848A2F508902A9
s = 36214B19444FAB504DBA61D4D6FF2D2F9640F4837

A.2.9. ECDSA, 233 Bits (Binary Field, Koblitz Curve)

Key pair:


          NIST K-233

q = 8000000000000000000000000000069D5BB915BCD46EFB1AD5F173ABDF (qlen = 232 bits)

private key:

   x = 103B2142BDC2A3C3B55080D09DF1808F79336DA2399F5CA7171D1BE9B0
   public key: U = xG
   Ux = 0682886F36C68473C1A221720C2B12B9BE13458BA907E1C4736595779F2
   Uy = 1B20639B41BE0927090999B7817A3B3928D20503A39546044EC13A10309


With SHA-1, message = "sample":
k = 273179E3E12C69591AD3DD9C7CCE3985820E3913AB6696EB14486DDBCF r = 5474541C988A9A1F73899F55EF28963DFFBBF0C2B1A1EE787C6A76C6A4 s = 46301F9EC6624257BFC70D72186F17898EDBD0A3522560A88DD1B7D45A

With SHA-224, message = "sample":
k = 71626A309D9CD80AD0B975D757FE6BF4B84E49F8F34C780070D7746F19 r = 667F2FCE3E1C497EBD8E4B7C6372A8234003FE4ED6D4515814E7E11430 s = 6A1C41340DAA730320DB9475F10E29A127D7AE3432F155E1F7954E1B57

With SHA-256, message = "sample":
k = 73552F9CAC5774F74F485FA253871F2109A0C86040552EAA67DBA92DC9 r = 38AD9C1D2CB29906E7D63C24601AC55736B438FB14F4093D6C32F63A10 s = 647AAD2599C21B6EE89BE7FF957D98F684B7921DE1FD3CC82C079624F4

With SHA-384, message = "sample":
k = 17D726A67539C609BD99E29AA3737EF247724B71455C3B6310034038C8 r = 0C6510F57559C36FBCFF8C7BA4B81853DC618AD0BAAB03CFFDF3FD09FD s = 0AD331EE1C9B91A88BA77997235769C60AD07EE69E11F7137E17C5CF67

With SHA-512, message = "sample":
k = 0E535C328774CDE546BE3AF5D7FCD263872F107E807435105BA2FDC166 r = 47C4AC1B344028CC740BA7BB9F8AA59D6390E3158153D4F2ADE4B74950 s = 26CE0CDE18A1B884B3EE1A879C13B42F11BB7C85F7A3745C8BECEC8E6E

With SHA-1, message = "test":
k = 1D8BBF5CB6EFFA270A1CDC22C81E269F0CC16E27151E0A460BA9B51AFF r = 4780B2DE4BAA5613872179AD90664249842E8B96FCD5653B55DD63EED4 s = 6AF46BA322E21D4A88DAEC1650EF38774231276266D6A45ED6A64ECB44

With SHA-224, message = "test":
k = 67634D0ABA2C9BF7AE54846F26DCD166E7100654BCE6FDC96667631AA2 r = 61D9CC8C842DF19B3D9F4BDA0D0E14A957357ADABC239444610FB39AEA s = 66432278891CB594BA8D08A0C556053D15917E53449E03C2EF88474CF6

With SHA-256, message = "test":
k = 2CE5AEDC155ACC0DDC5E679EBACFD21308362E5EFC05C5E99B2557A8D7 r = 05E4E6B4DB0E13034E7F1F2E5DBAB766D37C15AE4056C7EE607C8AC7F4 s = 5FC46AA489BF828B34FBAD25EC432190F161BEA8F60D3FCADB0EE3B725

With SHA-384, message = "test":
k = 1B4BD3903E74FD0B31E23F956C70062014DFEFEE21832032EA5352A055 r = 50F1EFEDFFEC1088024620280EE0D7641542E4D4B5D61DB32358FC571B s = 4614EAE449927A9EB2FCC42EA3E955B43D194087719511A007EC9217A5

With SHA-512, message = "test":
k = 1775ED919CA491B5B014C5D5E86AF53578B5A7976378F192AF665CB705 r = 6FE6D0D3A953BB66BB01BC6B9EDFAD9F35E88277E5768D1B214395320F s = 7C01A236E4BFF0A771050AD01EC1D24025D3130BBD9E4E81978EB3EC09

A.2.10. ECDSA, 283 Bits (Binary Field, Koblitz Curve)

Key pair:


          NIST K-283
   (qlen = 281 bits)

private key:

x = 06A0777356E87B89BA1ED3A3D845357BE332173C8F7A65BDC7DB4FAB3C4CC79A

   public key: U = xG

Ux = 25330D0A651D5A20DC6389BC02345117725640AEC3C126612CE444EDD19649BD


Uy = 505BD60A4B67182474EC4D1C668A73140F70504A68F39EFCD972487E9530E050



   With SHA-1, message = "sample":
   k = 0A96F788DECAF6C9DBE24DC75ABA6EAAE85E7AB003C8D4F83CB1540625B2993B
   r = 1B66D1E33FBDB6E107A69B610995C93C744CEBAEAF623CB42737C27D60188BD1
   s = 02E45B62C9C258643532FD536594B46C63B063946494F95DAFF8759FD5525023
   With SHA-224, message = "sample":
   k = 1B4C4E3B2F6B08B5991BD2BDDE277A7016DA527AD0AAE5BC61B64C5A0EE63E8B
   r = 018CF2F371BE86BB62E02B27CDE56DDAC83CCFBB3141FC59AEE022B66AC1A60D
   s = 1854E02A381295EA7F184CEE71AB7222D6974522D3B99B309B1A8025EB84118A
   With SHA-256, message = "sample":
   k = 1CEB9E8E0DFF53CE687DEB81339ACA3C98E7A657D5A9499EF779F887A934408E
   r = 19E90AA3DE5FB20AED22879F92C6FED278D9C9B9293CC5E94922CD952C9DBF20
   s = 135AA7443B6A25D11BB64AC482E04D47902D017752882BD72527114F46CF8BB5
   With SHA-384, message = "sample":
   k = 1460A5C41745A5763A9D548AE62F2C3630BBED71B6AA549D7F829C22442A728C
   r = 0F8C1CA9C221AD9907A136F787D33BA56B0495A40E86E671C940FD767EDD75EB
   s = 1071A56915DEE89E22E511975AA09D00CDC4AA7F5054CBE83F5977EE6F8E1CC3
   With SHA-512, message = "sample":
   k = 00F3B59FCB5C1A01A1A2A0019E98C244DFF61502D6E6B9C4E957EDDCEB258EF4
   r = 1D0008CF4BA4A701BEF70771934C2A4A87386155A2354140E2ED52E18553C35B
   s = 0D15F4FA1B7A4D41D9843578E22EF98773179103DC4FF0DD1F74A6B5642841B9
   With SHA-1, message = "test":
   k = 168B5F8C0881D4026C08AC5894A2239D219FA9F4DA0600ADAA56D5A1781AF81F
   r = 140932FA7307666A8CCB1E1A09656CC40F5932965841ABD5E8E43559D93CF231
   s = 16A2FD46DA497E5E739DED67F426308C45C2E16528BF2A17EB5D65964FD88B77
   With SHA-224, message = "test":
   k = 045E13EA645CE01D9B25EA38C8A8A170E04C83BB7F231EE3152209FE10EC8B2E
   r = 0E72AF7E39CD72EF21E61964D87C838F977485FA6A7E999000AFA97A381B2445
   s = 1644FF7D848DA1A040F77515082C27C763B1B4BF332BCF5D08251C6B57D80631
   With SHA-256, message = "test":
   k = 0B585A7A68F51089691D6EDE2B43FC4451F66C10E65F134B963D4CBD4EB844B0
   r = 158FAEB2470B306C57764AFC8528174589008449E11DB8B36994B607A65956A5
   s = 0521BC667CA1CA42B5649E78A3D76823C678B7BB3CD58D2E93CD791D53043A6F
   With SHA-384, message = "test":
   k = 1E88738E14482A09EE16A73D490A7FE8739DF500039538D5C4B6C8D6D7F208D6
   r = 1CC4DC5479E0F34C4339631A45AA690580060BF0EB518184C983E0E618C3B93A
   s = 0284D72FF8AFA83DE364502CBA0494BB06D40AE08F9D9746E747EA87240E589B
   With SHA-512, message = "test":
   k = 00E5F24A223BD459653F682763C3BB322D4EE75DD89C63D4DC61518D543E7658
   r = 1E7912517C6899732E09756B1660F6B96635D638283DF9A8A11D30E008895D7F
   s = 0887E75CBD0B7DD9DE30ED79BDB3D78E4F1121C5EAFF5946918F594F88D36364

A.2.11. ECDSA, 409 Bits (Binary Field, Koblitz Curve)

Key pair:


          NIST K-409
   (qlen = 407 bits)

private key:

x = 29C16768F01D1B8A89FDA85E2EFD73A09558B92A178A2931F359E4D70AD853E5

   public key: U = xG

Ux = 0CF923F523FE34A6E863D8BA45FB1FE6D784C8F219C414EEF4DB8362DBBD3CA7


Uy = 13B1C374D5132978A1B1123EBBE9A5C54D1A9D56B09AFDB4ADE93CCD7C4D332E



   With SHA-1, message = "sample":
   k = 7866E5247F9A3556F983C86E81EDA696AC8489DB40A2862F278603982D304F08
   r = 7192EE99EC7AFE23E02CB1F9850D1ECE620475EDA6B65D04984029408EC1E5A6
   s = 1DE75DE97CBE740FC79A6B5B22BC2B7832C687E6960F0B8173D5D8BE2A75AC6C
   With SHA-224, message = "sample":
   k = 512340DB682C7B8EBE407BF1AA54194DFE85D49025FE0F632C9B8A06A996F2FC
   r = 41C8EDF39D5E4E76A04D24E6BFD4B2EC35F99CD2483478FD8B0A03E99379576E
   s = 659652EEAC9747BCAD58034B25362B6AA61836E1BA50E2F37630813050D43457
   With SHA-256, message = "sample":
   k = 782385F18BAF5A36A588637A76DFAB05739A14163BF723A4417B74BD1469D37A
   r = 49EC220D6D24980693E6D33B191532EAB4C5D924E97E305E2C1CCFE6F1EAEF96
   s = 1A4AB1DD9BAAA21F77C503E1B39E770FFD44718349D54BA4CF08F688CE89D7D7
   With SHA-384, message = "sample":
   k = 4DA637CB2E5C90E486744E45A73935DD698D4597E736DA332A06EDA8B26D5ABC
   r = 562BB99EE027644EC04E493C5E81B41F261F6BD18FB2FAE3AFEAD91FAB8DD44A
   s = 25BA5F28047DDDBDA7ED7E49DA31B62B20FD9C7E5B8988817BBF738B3F4DFDD2
   With SHA-512, message = "sample":
   k = 57055B293ECFDFE983CEF716166091E573275C53906A39EADC25C89C5EC8D7A7
   r = 16C7E7FB33B5577F7CF6F77762F0F2D531C6E7A3528BD2CF582498C1A48F2007
   s = 2729617EFBF80DA5D2F201AC7910D3404A992C39921C2F65F8CF4601392DFE93
   With SHA-1, message = "test":
   k = 545453D8DC05D220F9A12EF322D0B855E664C72835FABE8A41211453EB8A7CFF
   r = 565648A5BAD24E747A7D7531FA9DBDFCB184ECFEFDB00A319459242B68D0989E
   s = 7420BA6FF72ECC5C92B7CA0309258B5879F26393DB22753B9EC5DF905500A042
   With SHA-224, message = "test":
   k = 3C5352929D4EBE3CCE87A2DCE380F0D2B33C901E61ABC530DAF3506544AB0930
   r = 251DFE54EAEC8A781ADF8A623F7F36B4ABFC7EE0AE78C8406E93B5C3932A8120
   s = 77854C2E72EAA6924CC0B5F6751379D132569843B1C7885978DBBAA6678967F6
   With SHA-256, message = "test":
   k = 251E32DEE10ED5EA4AD7370DF3EFF091E467D5531CA59DE3AA791763715E1169
   r = 58075FF7E8D36844EED0FC3F78B7CFFDEEF6ADE5982D5636552A081923E24841
   s = 0A737469D013A31B91E781CE201100FDE1FA488ABF2252C025C678462D715AD3
   With SHA-384, message = "test":
   k = 11C540EA46C5038FE28BB66E2E9E9A04C9FE9567ADF33D56745953D44C1DC8B5
   r = 1C5C88642EA216682244E46E24B7CE9AAEF9B3F97E585577D158C3CBC3C59825
   s = 1D3FD721C35872C74514359F88AD983E170E5DE5B31AFC0BE12E9F4AB2B2538C
   With SHA-512, message = "test":
   k = 59527CE953BC09DF5E85155CAE7BB1D7F342265F41635545B06044F844ECB4FA
   r = 1A32CD7764149DF79349DBF79451F4585BB490BD63A200700D7111B45DDA4140
   s = 582AB1076CAFAE23A76244B82341AEFC4C6D8D8060A62A352C33187720C8A37F

A.2.12. ECDSA, 571 Bits (Binary Field, Koblitz Curve)

Key pair:


          NIST K-571
   q = 2000000000000000000000000000000000000000000000000000000000000000
   (qlen = 570 bits)

private key:

x = 0C16F58550D824ED7B95569D4445375D3A490BC7E0194C41A39DEB732C29396C

DF1D66DE02DD1460A816606F3BEC0F32202C7BD18A32D87506466AA92032F131 4ED7B19762B0D22

   public key: U = xG

Ux = 6CFB0DF7541CDD4C41EF319EA88E849EFC8605D97779148082EC991C463ED323

19596F9FDF4779C17CAF20EFD9BEB57E9F4ED55BFC52A2FA15CA23BC62B7BF01 9DB59793DD77318

Uy = 1CFC91102F7759A561BD8D5B51AAAEEC7F40E659D67870361990D6DE29F6B4F7

E18AE13BDE5EA5C1F77B23D676F44050C9DBFCCDD7B3756328DDA059779AAE84 46FC5158A75C227


   With SHA-1, message = "sample":
   k = 17F7E360B21BEAE4A757A19ACA77FB404D273F05719A86EAD9D7B3F4D5ED7B46
   r = 0767913F96C82E38B7146A505938B79EC07E9AA3214377651BE968B52C039D3E
   s = 109F89F55FA39FF465E40EBCF869A9B1DB425AEA53AB4ECBCE3C310572F79315
   With SHA-224, message = "sample":
   k = 0B599D068A1A00498EE0B9AD6F388521F594BD3F234E47F7A1DB6490D7B57D60
   r = 010774B9F14DE6C9525131AD61531FA30987170D43782E9FB84FF0D70F093946
   s = 06DFE9AA5FEA6CF2CEDC06EE1F9FD9853D411F0B958F1C9C519C90A85F6D24C1
   With SHA-256, message = "sample":
   k = 0F79D53E63D89FB87F4D9E6DC5949F5D9388BCFE9EBCB4C2F7CE497814CF40E8
   r = 1604BE98D1A27CEC2D3FA4BD07B42799E07743071E4905D7DCE7F6992B21A27F
   s = 18249377C654B8588475510F7B797081F68C2F8CCCE49F730353B2DA3364B1CD
   With SHA-384, message = "sample":
   k = 0308253C022D25F8A9EBCD24459DD6596590BDEC7895618EEE8A2623A98D2A2B
   r = 1E6D7FB237040EA1904CCBF0984B81B866DE10D8AA93B06364C4A46F6C9573FA
   s = 04F94550072ADA7E8C82B7E83577DD39959577799CDABCEA60E267F36F1BEB98
   With SHA-512, message = "sample":
   k = 0C5EE7070AF55F84EBC43A0D481458CEDE1DCEBB57720A3C92F59B4941A044FE
   r = 086C9E048EADD7D3D2908501086F3AF449A01AF6BEB2026DC381B39530BCDDBE
   s = 09FEE0A68F322B380217FCF6ABFF15D78C432BD8DD82E18B6BA877C01C860E24
   With SHA-1, message = "test":
   k = 1D056563469E933E4BE064585D84602D430983BFBFD6885A94BA484DF9A7AB03
   r = 1D055F499A3F7E3FC73D6E7D517B470879BDCB14ABC938369F23643C7B96D024
   s = 1621376C53CFE3390A0520D2C657B1FF0EBB10E4B9C2510EDC39D04FEBAF12B8
   With SHA-224, message = "test":
   k = 1DA875065B9D94DBE75C61848D69578BCC267935792624F9887B53C9AF9E43CA
   r = 18709BDE4E9B73D046CE0D48842C97063DA54DCCA28DCB087168FA37DA2BF5FD
   s = 12D8B9E98FBF1D264D78669E236319D8FFD8426C56AFB10C76471EE88D7F0AB1
   With SHA-256, message = "test":
   k = 04DDD0707E81BB56EA2D1D45D7FAFDBDD56912CAE224086802FEA1018DB306C4
   r = 1F5BF6B044048E0E310309FFDAC825290A69634A0D3592DBEE7BE71F69E45412
   s = 1B44CBFB233BFA2A98D5E8B2F0B2C27F9494BEAA77FEB59CDE3E7AE9CB2E385B
   With SHA-384, message = "test":
   k = 0141B53DC6E569D8C0C0718A58A5714204502FDA146E7E2133E56D19E905B794
   r = 11F61A6EFAB6D83053D9C52665B3542FF3F63BD5913E527BDBA07FBAF34BC766
   s = 16BF6341876F051DF224770CC8BA0E4D48B3332568A2B014BC80827BAA89DE18
   With SHA-512, message = "test":
   k = 14842F97F263587A164B215DD0F912C588A88DC4AB6AF4C530ADC1226F16E086
   r = 0F1E50353A39EA64CDF23081D6BB4B2A91DD73E99D3DD5A1AA1C49B4F6E34A66
   s = 0B385004D7596625028E3FDE72282DE4EDC5B4CE33C1127F21CC37527C90B730

A.2.13. ECDSA, 163 Bits (Binary Field, Pseudorandom Curve)

Key pair:


          NIST B-163

q = 40000000000000000000292FE77E70C12A4234C33
(qlen = 163 bits)

private key:

   x = 35318FC447D48D7E6BC93B48617DDDEDF26AA658F
   public key: U = xG
   Ux = 126CF562D95A1D77D387BA75A3EA3A1407F23425A
   Uy = 7D7CB5273C94DA8CA93049AFDA18721C24672BD71


With SHA-1, message = "sample":
k = 0707A94C3D352E0A9FE49FB12F264992152A20004
r = 153FEBD179A69B6122DEBF5BC61EB947B24C93526
s = 37AC9C670F8CF18045049BAE7DD35553545C19E49

With SHA-224, message = "sample":
k = 3B24C5E2C2D935314EABF57A6484289B291ADFE3F
r = 0A379E69C44F9C16EA3215EA39EB1A9B5D58CC955
s = 04BAFF5308DA2A7FE2C1742769265AD3ED1D24E74

With SHA-256, message = "sample":
k = 3D7086A59E6981064A9CDB684653F3A81B6EC0F0B
r = 134E00F78FC1CB9501675D91C401DE20DDF228CDC
s = 373273AEC6C36CB7BAFBB1903A5F5EA6A1D50B624

With SHA-384, message = "sample":
k = 3B1E4443443486C7251A68EF184A936F05F8B17C7
r = 29430B935AF8E77519B0CA4F6903B0B82E6A21A66
s = 1EA1415306E9353FA5AA54BC7C2581DFBB888440D

With SHA-512, message = "sample":
k = 2EDF5CFCAC7553C17421FDF54AD1D2EF928A879D2
r = 0B2F177A99F9DF2D51CCAF55F015F326E4B65E7A0
s = 0DF1FB4487E9B120C5E970EFE48F55E406306C3A1

With SHA-1, message = "test":
k = 10024F5B324CBC8954BA6ADB320CD3AB9296983B4
r = 256D4079C6C7169B8BC92529D701776A269D56308

With SHA-224, message = "test":
k = 34F46DE59606D56C75406BFB459537A7CC280AA62
r = 28ECC6F1272CE80EA59DCF32F7AC2D861BA803393
s = 0AD4AE2C06E60183C1567D2B82F19421FE3053CE2

With SHA-256, message = "test":
k = 38145E3FFCA94E4DDACC20AD6E0997BD0E3B669D2
r = 227DF377B3FA50F90C1CB3CDCBBDBA552C1D35104
s = 1F7BEAD92583FE920D353F368C1960D0E88B46A56

With SHA-384, message = "test":
k = 375813210ECE9C4D7AB42DDC3C55F89189CF6DFFD
r = 11811DAFEEA441845B6118A0DFEE8A0061231337D
s = 36258301865EE48C5C6F91D63F62695002AB55B57

With SHA-512, message = "test":
k = 25AD8B393BC1E9363600FDA1A2AB6DF40079179A3
r = 3B6BB95CA823BE2ED8E3972FF516EB8972D765571
s = 13DC6F420628969DF900C3FCC48220B38BE24A541

A.2.14. ECDSA, 233 Bits (Binary Field, Pseudorandom Curve)

Key pair:


          NIST B-233

q = 1000000000000000000000000000013E974E72F8A6922031D2603CFE0D7 (qlen = 233 bits)

private key:

   x = 07ADC13DD5BF34D1DDEEB50B2CE23B5F5E6D18067306D60C5F6FF11E5D3
   public key: U = xG
   Ux = 0FB348B3246B473AA7FBB2A01B78D61B62C4221D0F9AB55FC72DB3DF478
   Uy = 1162FA1F6C6ACF7FD8D19FC7D74BDD9104076E833898BC4C042A6E6BEBF


With SHA-1, message = "sample":
k = 0A4E0B67A3A081C1B35D7BECEB5FE72A918B422B907145DB5416ED751CE r = 015CC6FD78BB06E0878E71465515EA5A21A2C18E6FC77B4B158DBEB3944 s = 0822A4A6C2EB2DF213A5E90BF40377956365EE8C4B4A5A4E2EB9270CB6A

With SHA-224, message = "sample":
k = 0F2B1C1E80BEB58283AAA79857F7B83BDF724120D0913606FD07F7FFB2C r = 05D9920B53471148E10502AB49AB7A3F11084820A074FD89883CF51BC1A s = 04D3938900C0A9AAA7080D1DFEB56CFB0FADABE4214536C7ED5117ED13A

With SHA-256, message = "sample":
k = 034A53897B0BBDB484302E19BF3F9B34A2ABFED639D109A388DC52006B5 r = 0A797F3B8AEFCE7456202DF1E46CCC291EA5A49DA3D4BDDA9A4B62D5E0D s = 01F6F81DA55C22DA4152134C661588F4BD6F82FDBAF0C5877096B070DC2

With SHA-384, message = "sample":
k = 04D4670B28990BC92EEB49840B482A1FA03FE028D09F3D21F89C67ECA85 r = 015E85A8D46225DD7E314A1C4289731FC14DECE949349FE535D11043B85 s = 03F189D37F50493EFD5111A129443A662AB3C6B289129AD8C0CAC85119C

With SHA-512, message = "sample":
k = 0DE108AAADA760A14F42C057EF81C0A31AF6B82E8FBCA8DC86E443AB549 r = 03B62A4BF783919098B1E42F496E65F7621F01D1D466C46940F0F132A95 s = 0F4BE031C6E5239E7DAA014CBBF1ED19425E49DAEB426EC9DF4C28A2E30

With SHA-1, message = "test":
k = 0250C5C90A4E2A3F8849FEBA87F0D0AE630AB18CBABB84F4FFFB36CEAC0 r = 02F1FEDC57BE203E4C8C6B8C1CEB35E13C1FCD956AB41E3BD4C8A6EFB1F s = 05738EC8A8EDEA8E435EE7266AD3EDE1EEFC2CEBE2BE1D614008D5D2951

With SHA-224, message = "test":
k = 07BDB6A7FD080D9EC2FC84BFF9E3E15750789DC04290C84FED00E109BBD r = 0CCE175124D3586BA7486F7146894C65C2A4A5A1904658E5C7F9DF5FA5D s = 08804B456D847ACE5CA86D97BF79FD6335E5B17F6C0D964B5D0036C867E

With SHA-256, message = "test":
k = 00376886E89013F7FF4B5214D56A30D49C99F53F211A3AFE01AA2BDE12D r = 035C3D6DFEEA1CFB29B93BE3FDB91A7B130951770C2690C16833A159677 s = 0600F7301D12AB376B56D4459774159ADB51F97E282FF384406AFD53A02

With SHA-384, message = "test":
k = 03726870DE75613C5E529E453F4D92631C03D08A7F63813E497D4CB3877 r = 061602FC8068BFD5FB86027B97455D200EC603057446CCE4D76DB8EF42C s = 03396DD0D59C067BB999B422D9883736CF9311DFD6951F91033BD03CA8D

With SHA-512, message = "test":
k = 09CE5810F1AC68810B0DFFBB6BEEF2E0053BB937969AE7886F9D064A8C4 r = 07E12CB60FDD614958E8E34B3C12DDFF35D85A9C5800E31EA2CC2EF63B1 s = 0E8970FD99D836F3CC1C807A2C58760DE6EDAA23705A82B9CB1CE93FECC

A.2.15. ECDSA, 283 Bits (Binary Field, Pseudorandom Curve)

Key pair:


          NIST B-283
   (qlen = 282 bits)

private key:

x = 14510D4BC44F2D26F4553942C98073C1BD35545CEABB5CC138853C5158D2729E

   public key: U = xG

Ux = 17E3409A13C399F0CA8A192F028D46E3446BCFFCDF51FF8A905ED2DED786E74F


Uy = 47EFCBCC31C01D86D1992F7BFAC0277DBD02A6D289274099A2C0F039C8F59F31



   With SHA-1, message = "sample":
   k = 277F389559667E8AE4B65DC056F8CE2872E1917E7CC59D17D485B0B98343206F
   r = 201E18D48C6DB3D5D097C4DCE1E25587E1501FC3CF47BDB5B4289D79E273D6A9
   s = 151AE05712B024CE617358260774C8CA8B0E7A7E72EF8229BF2ACE7609560CB3
   With SHA-224, message = "sample":
   k = 14CC8FCFEECD6B999B4DC6084EBB06FDED0B44D5C507802CC7A5E9ECF36E69DA
   r = 143E878DDFD4DF40D97B8CD638B3C4706501C2201CF7108F2FB91478C11D6947
   s = 0CBF1B9717FEEA3AABB09D9654110144267098E0E1E8D0289A6211BE0EEDFDD8
   With SHA-256, message = "sample":
   k = 38C9D662188982943E080B794A4CFB0732DBA37C6F40D5B8CFADED6FF31C5452
   r = 29FD82497FB3E5CEF65579272138DE59E2B666B8689466572B3B69A172CEE83B
   s = 05A89D9166B40795AF0FE5958201B9C0523E500013CA12B4840EA2BC53F25F9B
   With SHA-384, message = "sample":
   k = 21B7265DEBF90E6F988CFFDB62B121A02105226C652807CC324ED6FB119A287A
   r = 2F00689C1BFCD2A8C7A41E0DE55AE182E6463A152828EF89FE3525139B660329
   s = 1744514FE0A37447250C8A329EAAADA81572226CABA16F39270EE5DD03F27B1F
   With SHA-512, message = "sample":
   k = 20583259DC179D9DA8E5387E89BFF2A3090788CF1496BCABFE7D45BB120B0C81
   r = 0DA43A9ADFAA6AD767998A054C6A8F1CF77A562924628D73C62761847AD8286E
   s = 1D118733AE2C88357827CAFC6F68ABC25C80C640532925E95CFE66D40F8792F3
   With SHA-1, message = "test":
   k = 0185C57A743D5BA06193CE2AA47B07EF3D6067E5AE1A6469BCD3FC510128BA56
   r = 05A408133919F2CDCDBE5E4C14FBC706C1F71BADAFEF41F5DE4EC27272FC1CA9
   s = 012966272872C097FEA7BCE64FAB1A81982A773E26F6E4EF7C99969846E67CA9
   With SHA-224, message = "test":
   k = 2E5C1F00677A0E015EC3F799FA9E9A004309DBD784640EAAF5E1CE64D3045B9F
   r = 08F3824E40C16FF1DDA8DC992776D26F4A5981AB5092956C4FDBB4F1AE0A711E
   s = 0A64B91EFADB213E11483FB61C73E3EF63D3B44EEFC56EA401B99DCC60CC28E9
   With SHA-256, message = "test":
   k = 018A7D44F2B4341FEFE68F6BD8894960F97E08124AAB92C1FFBBE90450FCC935
   r = 3597B406F5329D11A79E887847E5EC60861CCBB19EC61F252DB7BD549C699951
   s = 0A6A100B997BC622D91701D9F5C6F6D3815517E577622DA69D3A0E8917C1CBE6
   With SHA-384, message = "test":
   k = 3C75397BA4CF1B931877076AF29F2E2F4231B117AB4B8E039F7F9704DE1BD352
   r = 1BB490926E5A1FDC7C5AA86D0835F9B994EDA315CA408002AF54A298728D422E
   s = 36C682CFC9E2C89A782BFD3A191609D1F0C1910D5FD6981442070393159D65FB
   With SHA-512, message = "test":
   k = 14E66B18441FA54C21E3492D0611D2B48E19DE3108D915FD5CA08E786327A267
   r = 19944AA68F9778C2E3D6E240947613E6DA60EFCE9B9B2C063FF5466D72745B5A
   s = 03F1567B3C5B02DF15C874F0EE22850824693D5ADC4663BAA19E384E550B1DD4

A.2.16. ECDSA, 409 Bits (Binary Field, Pseudorandom Curve)

Key pair:


          NIST B-409
   q = 10000000000000000000000000000000000000000000000000001E2AAD6A612F
   (qlen = 409 bits)

private key:

x = 0494994CC325B08E7B4CE038BD9436F90B5E59A2C13C3140CD3AE07C04A01FC4

   public key: U = xG

Ux = 1A7055961CF1DA4B9A015B18B1524EF01FDD9B93FAEFC26FB1F2F828A7227B70


Uy = 18105C042F290736088F30AEC7AE7732A45DE47BCE0940113AB8132516D1E059



   With SHA-1, message = "sample":
   k = 042D8A2B34402757EB2CCFDDC3E6E96A7ADD3FDA547FC10A0CB77CFC720B4F9E
   r = 0D8783188E1A540E2022D389E1D35B32F56F8C2BB5636B8ABF7718806B27A713
   s = 03A6B4A80E204DB0DE12E7415C13C9EC091C52935658316B4A0C591216A38791
   With SHA-224, message = "sample":
   k = 0C933F1DC4C70838C2AD16564715ACAF545BCDD8DC203D25AF3EC63949C65CB2
   r = 0EE4F39ACC2E03CE96C3D9FCBAFA5C22C89053662F8D4117752A9B10F09ADFDA
   s = 00A2B83265B456A430A8BF27DCC8A9488B3F126C10F0D6D64BF7B8A218FAAF20
   With SHA-256, message = "sample":
   k = 08EC42D13A3909A20C41BEBD2DFED8CACCE56C7A7D1251DF43F3E9E289DAE00E
   r = 02D8B1B31E33E74D7EB46C30FDE5AD2CA04EC8FE08FBA0E73BA5E568953AC5EA
   s = 079F7D471E6CB73234AF7F7C381D2CE15DE35BAF8BB68393B73235B3A26EC2DF
   With SHA-384, message = "sample":
   k = 0DA881BCE3BA851485879EF8AC585A63F1540B9198ECB8A1096D70CB25A104E2
   r = 07BC638B7E7CE6FEE5E9C64A0F966D722D01BB4BC3F3A35F30D4CDDA92DFC5F7
   s = 06D904429850521B28A32CBF55C7C0FDF35DC4E0BDA2552C7BF68A171E970E67
   With SHA-512, message = "sample":
   k = 0750926FFAD7FF5DE85DF7960B3A4F9E3D38CF5A049BFC89739C48D42B34FBEE
   r = 05D178DECAFD2D02A3DA0D8BA1C4C1D95EE083C760DF782193A9F7B4A8BE6FC5
   s = 013B7581E98F6A63FBBCB3E49BCDA60F816DB230B888506D105DC229600497C3
   With SHA-1, message = "test":
   k = 017E167EAB1850A3B38EE66BFE2270F2F6BFDAC5E2D227D47B20E75F0719161E
   r = 049F54E7C10D2732B4638473053782C6919218BBEFCEC8B51640FC193E832291
   s = 0499E267DEC84E02F6F108B10E82172C414F15B1B7364BE8BFD66ADC0C5DE23F
   With SHA-224, message = "test":
   k = 01ADEB94C19951B460A146B8275D81638C07735B38A525D76023AAF26AA8A058
   r = 0B1527FFAA7DD7C7E46B628587A5BEC0539A2D04D3CF27C54841C2544E1BBDB4
   s = 0442C68C044868DF4832C807F1EDDEBF7F5052A64B826FD03451440794063F52
   With SHA-256, message = "test":
   k = 06EBA3D58D0E0DFC406D67FC72EF0C943624CF40019D1E48C3B54CCAB0594AFD
   r = 0BB27755B991D6D31757BCBF68CB01225A38E1CFA20F775E861055DD108ED7EA
   s = 0C5BE90980E7F444B5F7A12C9E9AC7A04CA81412822DD5AD1BE7C45D5032555E
   With SHA-384, message = "test":
   k = 0A45B787DB44C06DEAB846511EEDBF7BFCFD3BD2C11D965C92FC195F67328F36
   r = 04EFEB7098772187907C87B33E0FBBA4584226C50C11E98CA7AAC6986F8D3BE0
   s = 09574102FEB3EF87E6D66B94119F5A6062950FF4F902EA1E6BD9E2037F33FF99
   With SHA-512, message = "test":
   k = 0B90F8A0E757E81D4EA6891766729C96A6D01F9AEDC0D334932D1F81CC4E1973
   r = 07E0249C68536AE2AEC2EC30090340DA49E6DC9E9EEC8F85E5AABFB234B6DA7D
   s = 08125B5A03FB44AE81EA46D446130C2A415ECCA265910CA69D55F2453E16CD7B

A.2.17. ECDSA, 571 Bits (Binary Field, Pseudorandom Curve)

Key pair:


          NIST B-571
   (qlen = 570 bits)

private key:

x = 028A04857F24C1C082DF0D909C0E72F453F2E2340CCB071F0E389BCA2575DA19

124198C57174929AD26E348CF63F78D28021EF5A9BF2D5CBEAF6B7CCB6C4DA82 4DD5C82CFB24E11

   public key: U = xG

Ux = 4B4B3CE9377550140B62C1061763AA524814DDCEF37B00CD5CDE94F7792BB0E9

6758E55DA2E9FEA8FF2A8B6830AE1D57A9CA7A77FCB0836BF43EA5454CDD9FEA D5CCFE7375C6A83

Uy = 4453B18F261E7A0E7570CD72F235EA750438E43946FBEBD2518B696954767AA7

849C1719E18E1C51652C28CA853426F15C09AA4B579487338ABC7F33768FADD6 1B5A3A6443A8189


   With SHA-1, message = "sample":
   k = 2669FAFEF848AF67D437D4A151C3C5D3F9AA8BB66EDC35F090C9118F95BA0041
   r = 147D3EB0EDA9F2152DFD014363D6A9CE816D7A1467D326A625FC4AB0C786E1B7
   s = 17319571CAF533D90D2E78A64060B9C53169AB7FC908947B3EDADC54C79CCF0A
   With SHA-224, message = "sample":
   k = 2EAFAD4AC8644DEB29095BBAA88D19F31316434F1766AD4423E0B54DD2FE0C05
   r = 10F4B63E79B2E54E4F4F6A2DBC786D8F4A143ECA7B2AD97810F6472AC6AE2085
   s = 3BBEA07C6B269C2B7FE9AE4DDB118338D0C2F0022920A7F9DCFCB7489594C03B
   With SHA-256, message = "sample":
   k = 15C2C6B7D1A070274484774E558B69FDFA193BDB7A23F27C2CD24298CE1B22A6
   r = 213EF9F3B0CFC4BF996B8AF3A7E1F6CACD2B87C8C63820000800AC787F17EC99
   s = 3D32322559B094E20D8935E250B6EC139AC4AAB77920812C119AF419FB62B332
   With SHA-384, message = "sample":
   k = 0FEF0B68CB49453A4C6ECBF1708DBEEFC885C57FDAFB88417AAEFA5B1C35017B
   r = 375D8F49C656A0BBD21D3F54CDA287D853C4BB1849983CD891EF6CD6BB56A62B
   s = 1CDEC6F46DFEEE44BCE71D41C60550DC67CF98D6C91363625AC2553E4368D2DF
   With SHA-512, message = "sample":
   k = 3FF373833A06C791D7AD586AFA3990F6EF76999C35246C4AD0D519BFF180CA18
   r = 1C26F40D940A7EAA0EB1E62991028057D91FEDA0366B606F6C434C361F04E545
   s = 3691DE4369D921FE94EDDA67CB71FBBEC9A436787478063EB1CC778B3DCDC1C4
   With SHA-1, message = "test":
   k = 019B506FD472675A7140E429AA5510DCDDC21004206EEC1B39B28A688A8FD324
   r = 133F5414F2A9BC41466D339B79376038A64D045E5B0F792A98E5A7AA87E0AD01
   s = 3D16743AE9F00F0B1A500F738719C5582550FEB64689DA241665C4CE4F328BA0
   With SHA-224, message = "test":
   k = 333C711F8C62F205F926593220233B06228285261D34026232F6F729620C6DE1
   r = 3048E76506C5C43D92B2E33F62B33E3111CEEB87F6C7DF7C7C01E3CDA28FA5E8
   s = 2C99078CCFE5C82102B8D006E3703E020C46C87C75163A2CD839C885550BA5CB
   With SHA-256, message = "test":
   k = 328E02CF07C7B5B6D3749D8302F1AE5BFAA8F239398459AF4A2C859C7727A812
   r = 184BC808506E11A65D628B457FDA60952803C604CC7181B59BD25AEE1411A66D
   s = 27280D45F81B19334DBDB07B7E63FE8F39AC7E9AE14DE1D2A6884D2101850289
   With SHA-384, message = "test":
   k = 2A77E29EAD9E811A9FDA0284C14CDFA1D9F8FA712DA59D530A06CDE54187E250
   r = 319EE57912E7B0FAA1FBB145B0505849A89C6DB1EC06EA20A6A7EDE072A6268A
   s = 2CF3EA27EADD0612DD2F96F46E89AB894B01A10DF985C5FC099CFFE0EA083EB4
   With SHA-512, message = "test":
   k = 21CE6EE4A2C72C9F93BDB3B552F4A633B8C20C200F894F008643240184BE57BB
   r = 2AA1888EAB05F7B00B6A784C4F7081D2C833D50794D9FEAF6E22B8BE728A2A90
   s = 0AA5371FE5CA671D6ED9665849C37F394FED85D51FEF72DA2B5F28EDFB2C6479

A.3. Sample Code

We include here a sample implementation of deterministic DSA. It is meant for illustration purposes; for instance, this code makes no attempt at avoiding side-channel leakage of the private key. It is written in the Java programming language. The actual generation of the "random" value k is done in the computek() method. The Java virtual machine (JVM) is assumed to provide the implementation of the hash function and of HMAC.

  // ==================================================================
  import java.math.BigInteger;
  import javax.crypto.Mac;
  import javax.crypto.spec.SecretKeySpec;
   * Deterministic DSA signature generation.  This is a sample
   * implementation designed to illustrate how deterministic DSA
   * chooses the pseudorandom value k when signing a given message.
   * This implementation was NOT optimized or hardened against
   * side-channel leaks.
   * An instance is created with a hash function name, which must be
   * supported by the underlying Java virtual machine ("SHA-1" and
   * "SHA-256" should work everywhere).  The data to sign is input
   * through the {@code update()} methods.  The private key is set with
   * {@link #setPrivateKey}.  The signature is obtained by calling
   * {@link #sign}; alternatively, {@link #signHash} can be used to
   * sign some data that has been externally hashed.  The private key
   * MUST be set before generating the signature itself, but message
   * data can be input before setting the key.
   * Instances are NOT thread-safe.  However, once a signature has
   * been generated, the same instance can be used again for another
   * signature; {@link #setPrivateKey} need not be called again if the
   * private key has not changed.  {@link #reset} can also be called to
   * cancel previously input data.  Generating a signature with {@link
   * #sign} (not {@link #signHash}) also implicitly causes a
   * reset.
   * ------------------------------------------------------------------
   * Copyright © 2013 IETF Trust and the persons identified as
   * authors of the code.  All rights reserved.
   * Redistribution and use in source and binary forms, with or without
   * modification, is permitted pursuant to, and subject to the license
   * terms contained in, the Simplified BSD License set forth in Section
   * 4.c of the IETF Trust's Legal Provisions Relating to IETF Documents
   * (
   * Technical remarks and questions can be addressed to:
   * ------------------------------------------------------------------

public class DeterministicDSA {

          private String macName;
          private MessageDigest dig;
          private Mac hmac;
          private BigInteger p, q, g, x;
          private int qlen, rlen, rolen, holen;
          private byte[] bx;
           * Create an instance, using the specified hash function.
           * The name is used to obtain from the JVM an implementation
           * of the hash function and an implementation of HMAC.
           * @param hashName   the hash function name
           * @throws IllegalArgumentException  on unsupported name
          public DeterministicDSA(String hashName)
                  try {
                          dig = MessageDigest.getInstance(hashName);
                  } catch (NoSuchAlgorithmException nsae) {
                          throw new IllegalArgumentException(nsae);
                  if (hashName.indexOf('-') < 0) {
                          macName = "Hmac" + hashName;
                  } else {
                          StringBuilder sb = new StringBuilder();
                          int n = hashName.length();
                          for (int i = 0; i < n; i ++) {
                                  char c = hashName.charAt(i);
                                  if (c != '-') {
                          macName = sb.toString();
                  try {
                          hmac = Mac.getInstance(macName);
                  } catch (NoSuchAlgorithmException nsae) {
                          throw new IllegalArgumentException(nsae);
                  holen = hmac.getMacLength();
           * Set the private key.
           * @param p   key parameter: field modulus
           * @param q   key parameter: subgroup order
           * @param g   key parameter: generator
           * @param x   private key
          public void setPrivateKey(BigInteger p, BigInteger q,
                  BigInteger g, BigInteger x)
                   * Perform some basic sanity checks.  We do not
                   * check primality of p or q because that would
                   * be too expensive.
                   * We reject keys where q is longer than 999 bits,
                   * because it would complicate signature encoding.
                   * Normal DSA keys do not have a q longer than 256
                   * bits anyway.
                  if (p == null || q == null || g == null || x == null
                          || p.signum() <= 0 || q.signum() <= 0
                          || g.signum() <= 0 || x.signum() <= 0
                          || x.compareTo(q) >= 0 || q.compareTo(p) >= 0
                          || q.bitLength() > 999
                          || g.compareTo(p) >= 0 || g.bitLength() == 1
                          || g.modPow(q, p).bitLength() != 1) {
                          throw new IllegalArgumentException(
                                  "invalid DSA private key");
                  this.p = p;
                  this.q = q;
                  this.g = g;
                  this.x = x;
                  qlen = q.bitLength();
                  if (q.signum() <= 0 || qlen < 8) {
                          throw new IllegalArgumentException(
                                  "bad group order: " + q);
                  rolen = (qlen + 7) >>> 3;
                  rlen = rolen * 8;
                   * Convert the private exponent (x) into a sequence
                   * of octets.
                  bx = int2octets(x);
          private BigInteger bits2int(byte[] in)
                  BigInteger v = new BigInteger(1, in);
                  int vlen = in.length * 8;
                  if (vlen > qlen) {
                          v = v.shiftRight(vlen - qlen);
                  return v;
          private byte[] int2octets(BigInteger v)
                  byte[] out = v.toByteArray();
                  if (out.length < rolen) {
                          byte[] out2 = new byte[rolen];
                          System.arraycopy(out, 0,
                                  out2, rolen - out.length,
                          return out2;
                  } else if (out.length > rolen) {
                          byte[] out2 = new byte[rolen];
                          System.arraycopy(out, out.length - rolen,
                                  out2, 0, rolen);
                          return out2;
                  } else {
                          return out;
          private byte[] bits2octets(byte[] in)
                  BigInteger z1 = bits2int(in);
                  BigInteger z2 = z1.subtract(q);
                  return int2octets(z2.signum() < 0 ? z1 : z2);
           * Set (or reset) the secret key used for HMAC.
           * @param K   the new secret key
          private void setHmacKey(byte[] K)
                  try {
                          hmac.init(new SecretKeySpec(K, macName));
                  } catch (InvalidKeyException ike) {
                          throw new IllegalArgumentException(ike);
           * Compute the pseudorandom k for signature generation,
           * using the process specified for deterministic DSA.
           * @param h1   the hashed message
           * @return  the pseudorandom k to use
          private BigInteger computek(byte[] h1)
                   * Convert hash value into an appropriately truncated
                   * and/or expanded sequence of octets.  The private
                   * key was already processed (into field bx[]).
                  byte[] bh = bits2octets(h1);
                   * HMAC is always used with K as key.
                   * Whenever K is updated, we reset the
                   * current HMAC key.
                  /* step b. */
                  byte[] V = new byte[holen];
                  for (int i = 0; i < holen; i ++) {
                          V[i] = 0x01;
                  /* step c. */
                  byte[] K = new byte[holen];
                  /* step d. */
                  K = hmac.doFinal();
                  /* step e. */
                  V = hmac.doFinal();
                  /* step f. */
                  K = hmac.doFinal();
                  /* step g. */
                  V = hmac.doFinal();
                  /* step h. */
                  byte[] T = new byte[rolen];
                  for (;;) {
                           * We want qlen bits, but we support only
                           * hash functions with an output length
                           * multiple of 8;acd hence, we will gather
                           * rlen bits, i.e., rolen octets.
                          int toff = 0;
                          while (toff < rolen) {
                                  V = hmac.doFinal();
                                  int cc = Math.min(V.length,
                                          T.length - toff);
                                  System.arraycopy(V, 0, T, toff, cc);
                                  toff += cc;
                          BigInteger k = bits2int(T);
                          if (k.signum() > 0 && k.compareTo(q) < 0) {
                                  return k;
                           * k is not in the proper range; update
                           * K and V, and loop.
                          K = hmac.doFinal();
                          V = hmac.doFinal();
           * Process one more byte of input data (message to sign).
           * @param in   the extra input byte
          public void update(byte in)
           * Process some extra bytes of input data (message to sign).
           * @param in   the extra input bytes
          public void update(byte[] in)
                  dig.update(in, 0, in.length);
           * Process some extra bytes of input data (message to sign).
           * @param in    the extra input buffer
           * @param off   the extra input offset
           * @param len   the extra input length (in bytes)
          public void update(byte[] in, int off, int len)
                  dig.update(in, off, len);
           * Produce the signature.  {@link #setPrivateKey} MUST have
           * been called.  The signature is computed over the data
           * that was input through the {@code update*()} methods.
           * This engine is then reset (made ready for a new
           * signature generation).
           * @return  the signature
          public byte[] sign()
                  return signHash(dig.digest());
           * Produce the signature.  {@link #setPrivateKey} MUST
           * have been called.  The signature is computed over the
           * provided hash value (data is assumed to have been hashed
           * externally).  The data that was input through the
           * {@code update*()} methods is ignored, but kept.
           * If the hash output is longer than the subgroup order
           * (the length of q, in bits, denoted 'qlen'), then the
           * provided value {@code h1} can be truncated, provided that
           * at least qlen leading bits are preserved.  In other words,
           * bit values in {@code h1} beyond the first qlen bits are
           * ignored.
           * @param h1   the hash value
           * @return  the signature
          public byte[] signHash(byte[] h1)
                  if (p == null) {
                          throw new IllegalStateException(
                                  "no private key set");
                  try {
                          BigInteger k = computek(h1);
                          BigInteger r = g.modPow(k, p).mod(q);
                          BigInteger s = k.modInverse(q).multiply(
                           * Signature encoding: ASN.1 SEQUENCE of
                           * two INTEGERs.  The conditions on q
                           * imply that the encoded version of r and
                           * s is no longer than 127 bytes for each,
                           * including DER tag and length.
                          byte[] br = r.toByteArray();
                          byte[] bs = s.toByteArray();
                          int ulen = br.length + bs.length + 4;
                          int slen = ulen + (ulen >= 128 ? 3 : 2);
                          byte[] sig = new byte[slen];
                          int i = 0;
                          sig[i ++] = 0x30;
                          if (ulen >= 128) {
                                  sig[i ++] = (byte)0x81;
                                  sig[i ++] = (byte)ulen;
                          } else {
                                  sig[i ++] = (byte)ulen;
                          sig[i ++] = 0x02;
                          sig[i ++] = (byte)br.length;
                          System.arraycopy(br, 0, sig, i, br.length);
                          i += br.length;
                          sig[i ++] = 0x02;
                          sig[i ++] = (byte)bs.length;
                          System.arraycopy(bs, 0, sig, i, bs.length);
                          return sig;
                  } catch (ArithmeticException ae) {
                          throw new IllegalArgumentException(
                                  "DSA error (bad key ?)", ae);
           * Reset this engine.  Data input through the {@code
           * update*()} methods is discarded.  The current private key,
           * if one was set, is kept unchanged.
          public void reset()
  // ==================================================================

Author's Address

Thomas Pornin
Quebec, QC