Elliptic Curve Digital Signature Algorithm

Not to be confused withEdDSA.

Incryptography,theElliptic Curve Digital Signature Algorithm(ECDSA) offers a variant of theDigital Signature Algorithm(DSA) which useselliptic-curve cryptography.

Key and signature sizes

As with elliptic-curve cryptography in general, the bitsizeof theprivate keybelieved to be needed for ECDSA is about twice the size of thesecurity level,in bits.For example, at a security level of 80 bits—meaning an attacker requires a maximum of about$2^{80}$operations to find the private key—the size of an ECDSA private key would be 160 bits. On the other hand, the signature size is the same for both DSA and ECDSA: approximately$4t$bits, where$t$is the exponent in the formula$2^{t}$,that is, about 320 bits for a security level of 80 bits, which is equivalent to$2^{80}$operations.

Signature generation algorithm

SupposeAlicewants to send a signed message toBob.Initially, they must agree on the curve parameters$({\textrm {CURVE}},G,n)$.In addition to thefieldand equation of the curve, we need$G$,a base point of prime order on the curve;$n$is the multiplicative order of the point$G$.

Parameter
CURVE	the elliptic curve field and equation used
G	elliptic curve base point, a point on the curve that generates asubgroup of large prime order n
n	integer order ofG,means that$n\times G=O$,where$O$is the identity element.
$d_{A}$	the private key (randomly selected)
$Q_{A}$	the public key$d_{A}\times G$(calculated by elliptic curve)
m	the message to send

The order$n$of the base point$G$must be prime.Indeed, we assume that every nonzero element of thering$\mathbb {Z} /n\mathbb {Z}$is invertible, so that$\mathbb {Z} /n\mathbb {Z}$must be afield.It implies that$n$must be prime (cf.Bézout's identity).

Alice creates a key pair, consisting of a private key integer$d_{A}$,randomly selected in the interval$[1,n-1]$;and a public key curve point$Q_{A}=d_{A}\times G$.We use$\times$to denoteelliptic curve point multiplication by a scalar.

For Alice to sign a message$m$,she follows these steps:

1. Calculate$e={\textrm {HASH}}(m)$.(Here HASH is acryptographic hash function,such asSHA-2,with the output converted to an integer.)
2. Let$z$be the$L_{n}$leftmost bits of$e$,where$L_{n}$is the bit length of the group order$n$.(Note that$z$can begreaterthan$n$but notlonger.)
3. Select acryptographically secure randominteger$k$from$[1,n-1]$.
4. Calculate the curve point$(x_{1},y_{1})=k\times G$.
5. Calculate$r=x_{1}\,{\bmod {\,}}n$.If$r=0$,go back to step 3.
6. Calculate$s=k^{-1}(z+rd_{A})\,{\bmod {\,}}n$.If$s=0$,go back to step 3.
7. The signature is the pair$(r,s)$.(And$(r,-s\,{\bmod {\,}}n)$is also a valid signature.)

As the standard notes, it is not only required for$k$to be secret, but it is also crucial to select different$k$for different signatures. Otherwise, the equation in step 6 can be solved for$d_{A}$,the private key: given two signatures$(r,s)$and$(r,s')$,employing the same unknown$k$for different known messages$m$and$m'$,an attacker can calculate$z$and$z'$,and since$s-s'=k^{-1}(z-z')$(all operations in this paragraph are done modulo$n$) the attacker can find$k={\frac {z-z'}{s-s'}}$.Since$s=k^{-1}(z+rd_{A})$,the attacker can now calculate the private key$d_{A}={\frac {sk-z}{r}}$.

This implementation failure was used, for example, to extract the signing key used for thePlayStation 3gaming-console.

Another way ECDSA signature may leak private keys is when$k$is generated by a faultyrandom number generator.Such a failure in random number generation caused users of Android Bitcoin Wallet to lose their funds in August 2013.

To ensure that$k$is unique for each message, one may bypass random number generation completely and generate deterministic signatures by deriving$k$from both the message and the private key.

Signature verification algorithm

For Bob to authenticate Alice's signature$r,s$on a message$m$,he must have a copy of her public-key curve point$Q_{A}$.Bob can verify$Q_{A}$is a valid curve point as follows:

1. Check that$Q_{A}$is not equal to the identity elementO,and its coordinates are otherwise valid.
2. Check that$Q_{A}$lies on the curve.
3. Check that$n\times Q_{A}=O$.

After that, Bob follows these steps:

1. Verify thatrandsare integers in$[1,n-1]$.If not, the signature is invalid.
2. Calculate$e={\textrm {HASH}}(m)$,where HASH is the same function used in the signature generation.
3. Let$z$be the$L_{n}$leftmost bits ofe.
4. Calculate$u_{1}=zs^{-1}\,{\bmod {\,}}n$and$u_{2}=rs^{-1}\,{\bmod {\,}}n$.
5. Calculate the curve point$(x_{1},y_{1})=u_{1}\times G+u_{2}\times Q_{A}$.If$(x_{1},y_{1})=O$then the signature is invalid.
6. The signature is valid if$r\equiv x_{1}{\pmod {n}}$,invalid otherwise.

Note that an efficient implementation would compute inverse$s^{-1}\,{\bmod {\,}}n$only once. Also, using Shamir's trick, a sum of two scalar multiplications$u_{1}\times G+u_{2}\times Q_{A}$can be calculated faster than two scalar multiplications done independently.

Correctness of the algorithm

It is not immediately obvious why verification even functions correctly. To see why, denote asCthe curve point computed in step 5 of verification,

$C=u_{1}\times G+u_{2}\times Q_{A}$

From the definition of the public key as$Q_{A}=d_{A}\times G$,

$C=u_{1}\times G+u_{2}d_{A}\times G$

Because elliptic curve scalar multiplication distributes over addition,

$C=(u_{1}+u_{2}d_{A})\times G$

Expanding the definition of$u_{1}$and$u_{2}$from verification step 4,

$C=(zs^{-1}+rd_{A}s^{-1})\times G$

Collecting the common term$s^{-1}$,

$C=(z+rd_{A})s^{-1}\times G$

Expanding the definition ofsfrom signature step 6,

$C=(z+rd_{A})(z+rd_{A})^{-1}(k^{-1})^{-1}\times G$

Since the inverse of an inverse is the original element, and the product of an element's inverse and the element is the identity, we are left with

$C=k\times G$

From the definition ofr,this is verification step 6.

This shows only that a correctly signed message will verify correctly; other properties such as incorrectly signed messages failing to verify correctly and resistance tocryptanalyticattacks are required for a secure signature algorithm.

Public key recovery

Given a messagemand Alice's signature$r,s$on that message, Bob can (potentially) recover Alice's public key:

1. Verify thatrandsare integers in$[1,n-1]$.If not, the signature is invalid.
2. Calculate a curve point$R=(x_{1},y_{1})$where$x_{1}$is one of$r$,$r+n$,$r+2n$,etc. (provided$x_{1}$is not too large for thefieldof the curve) and$y_{1}$is a value such that the curve equation is satisfied. Note that there may be several curve points satisfying these conditions, and each differentRvalue results in a distinct recovered key.
3. Calculate$e={\textrm {HASH}}(m)$,where HASH is the same function used in the signature generation.
4. Letzbe the$L_{n}$leftmost bits ofe.
5. Calculate$u_{1}=-zr^{-1}\,{\bmod {\,}}n$and$u_{2}=sr^{-1}\,{\bmod {\,}}n$.
6. Calculate the curve point$Q_{A}=(x_{A},y_{A})=u_{1}\times G+u_{2}\times R$.
7. The signature is valid if$Q_{A}$,matches Alice's public key.
8. The signature is invalid if all the possibleRpoints have been tried and none match Alice's public key.

Note that an invalid signature, or a signature from a different message, will result in the recovery of an incorrect public key. The recovery algorithm can only be used to check validity of a signature if the signer's public key (or its hash) is known beforehand.

Correctness of the recovery algorithm

Start with the definition of$Q_{A}$from recovery step 6,

$Q_{A}=(x_{A},y_{A})=u_{1}\times G+u_{2}\times R$

From the definition$R=(x_{1},y_{1})=k\times G$from signing step 4,

$Q_{A}=u_{1}\times G+u_{2}k\times G$

Because elliptic curve scalar multiplication distributes over addition,

$Q_{A}=(u_{1}+u_{2}k)\times G$

Expanding the definition of$u_{1}$and$u_{2}$from recovery step 5,

$Q_{A}=(-zr^{-1}+skr^{-1})\times G$

Expanding the definition ofsfrom signature step 6,

$Q_{A}=(-zr^{-1}+k^{-1}(z+rd_{A})kr^{-1})\times G$

Since the product of an element's inverse and the element is the identity, we are left with

$Q_{A}=(-zr^{-1}+(zr^{-1}+d_{A}))\times G$

The first and second terms cancel each other out,

$Q_{A}=d_{A}\times G$

From the definition of$Q_{A}=d_{A}\times G$,this is Alice's public key.

This shows that a correctly signed message will recover the correct public key, provided additional information was shared to uniquely calculate curve point$R=(x_{1},y_{1})$from signature valuer.

Security

In December 2010, a group calling itselffail0verflowannounced the recovery of the ECDSA private key used bySonyto sign software for thePlayStation 3game console. However, this attack only worked because Sony did not properly implement the algorithm, because$k$was static instead of random. As pointed out in theSignature generation algorithmsection above, this makes$d_{A}$solvable, rendering the entire algorithm useless.

On March 29, 2011, two researchers published anIACRpaperdemonstrating that it is possible to retrieve a TLS private key of a server usingOpenSSLthat authenticates with Elliptic Curves DSA over a binaryfieldvia atiming attack.The vulnerability was fixed in OpenSSL 1.0.0e.

In August 2013, it was revealed that bugs in some implementations of theJavaclassSecureRandomsometimes generated collisions in the$k$value. This allowed hackers to recover private keys giving them the same control over bitcoin transactions as legitimate keys' owners had, using the same exploit that was used to reveal the PS3 signing key on someAndroidapp implementations, which use Java and rely on ECDSA to authenticate transactions.

This issue can be prevented by deterministic generation of k, as described by RFC 6979.

Concerns

Some concerns expressed about ECDSA:

1. Political concerns:the trustworthiness ofNIST-produced curves being questioned after revelations were made that theNSAwillingly insertsbackdoorsinto software, hardware components and published standards; well-known cryptographershave expresseddoubts about how the NIST curves were designed, and voluntary tainting has already been proved in the past.(See also thelibsshcurve25519introduction.) Nevertheless, a proof that the named NIST curves exploit a rare weakness is missing yet.
2. Technical concerns:the difficulty of properly implementing the standard, its slowness, and design flaws which reduce security in insufficiently defensive implementations.

Implementations

Below is a list of cryptographic libraries that provide support for ECDSA:

• Botan
• Bouncy Castle
• cryptlib
• Crypto++
• Crypto API (Linux)
• GnuTLS
• libgcrypt
• LibreSSL
• mbed TLS
• Microsoft CryptoAPI
• OpenSSL
• wolfCrypt

See also

• EdDSA
• RSA (cryptosystem)

Further reading

• Accredited Standards CommitteeX9,ASC X9 Issues New Standard for Public Key Cryptography/ECDSA,Oct. 6, 2020.Source
• Accredited Standards CommitteeX9,American National Standard X9.62-2005, Public Key Cryptography for the Financial Services Industry, The Elliptic Curve Digital Signature Algorithm (ECDSA),November 16, 2005.
• Certicom Research,Standards for efficient cryptography, SEC 1: Elliptic Curve Cryptography,Version 2.0, May 21, 2009.
• López, J. and Dahab, R.An Overview of Elliptic Curve Cryptography,Technical Report IC-00-10, State University of Campinas, 2000.
• Daniel J. Bernstein,Pippenger's exponentiation algorithm,2002.
• Daniel R. L. Brown,Generic Groups, Collision Resistance, and ECDSA,Designs, Codes and Cryptography,35,119–152, 2005.ePrint version
• Ian F. Blake, Gadiel Seroussi, andNigel Smart,editors,Advances in Elliptic Curve Cryptography,London Mathematical Society Lecture Note Series 317, Cambridge University Press, 2005.
• Hankerson, D.;Vanstone, S.;Menezes, A.(2004).Guide to Elliptic Curve Cryptography.Springer Professional Computing. New York:Springer.doi:10.1007/b97644.ISBN0-387-95273-X.S2CID720546.

External links

• Digital Signature Standard; includes info on ECDSA
• The Elliptic Curve Digital Signature Algorithm (ECDSA); provides an in-depth guide on ECDSA.Wayback link