Not to be confused with EdDSA.

In cryptography, the Elliptic Curve Digital Signature Algorithm (ECDSA) offers a variant of the Digital Signature Algorithm (DSA) which uses elliptic-curve cryptography.

Key and signature sizes

As with elliptic-curve cryptography in general, the bit size of the private key believed to be needed for ECDSA is about twice the size of the security level, in bits.[1] 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 4 t 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

Suppose Alice wants to send a signed message to Bob. Initially, they must agree on the curve parameters (\textrm{CURVE}, G, n). In addition to the field and equation of the curve, we need G, a base point of prime order on the curve; n is the additive order of the point G.

Parameter
CURVEthe elliptic curve field and equation used
Gelliptic curve base point, a point on the curve that generates a subgroup of large prime order n
ninteger order of G, means that n \times G = O, where O is the identity element.
d_Athe private key (randomly selected)
Q_Athe public key d_A \times G (calculated by elliptic curve)
mthe message to send

The order n of the base point G must be prime. Indeed, we assume that every nonzero element of the ring \mathbb{Z}/n\mathbb{Z} is invertible, so that \mathbb{Z}/n\mathbb{Z} must be a field. 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 denote elliptic 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 a cryptographic hash function, such as SHA-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 be greater than n but not longer.[2])
  3. Select a cryptographically secure random integer 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 + r d_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 + r d_A), the attacker can now calculate the private key d_A = \frac{s k - z}{r}.

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

Another way ECDSA signature may leak private keys is when k is generated by a faulty random number generator. Such a failure in random number generation caused users of Android Bitcoin Wallet to lose their funds in August 2013.[4]

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.[5]

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 element O, 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 that r and s are 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 of e.
  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.[6]

Correctness of the algorithm

It is not immediately obvious why verification even functions correctly. To see why, denote as C the 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 = (z s^{-1} + r d_A s^{-1}) \times G

Collecting the common term s^{-1},

C = (z + r d_A) s^{-1} \times G

Expanding the definition of s from signature step 6,

C = (z + r d_A) (z + r d_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 of r, 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 to cryptanalytic attacks are required for a secure signature algorithm.

Public key recovery

Given a message m and Alice's signature r, s on that message, Bob can (potentially) recover Alice's public key:[7]

  1. Verify that r and s are 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 the field of 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 different R value results in a distinct recovered key.
  3. Calculate e = \textrm{HASH}(m), where HASH is the same function used in the signature generation.
  4. Let z be the L_n leftmost bits of e.
  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 possible R points 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 = (-z r^{-1} + s k r^{-1}) \times G

Expanding the definition of s from signature step 6,

Q_A = (-z r^{-1} + k^{-1} (z + r d_A) k r^{-1}) \times G

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

Q_A = (-z r^{-1} + (z r^{-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 value r.

Security

In December 2010, a group calling itself fail0verflow announced the recovery of the ECDSA private key used by Sony to sign software for the PlayStation 3 game 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 the Signature generation algorithm section above, this makes d_A solvable, rendering the entire algorithm useless.[8]

On March 29, 2011, two researchers published an IACR paper[9] demonstrating that it is possible to retrieve a TLS private key of a server using OpenSSL that authenticates with Elliptic Curves DSA over a binary field via a timing attack.[10] The vulnerability was fixed in OpenSSL 1.0.0e.[11]

In August 2013, it was revealed that bugs in some implementations of the Java class SecureRandom sometimes 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 some Android app implementations, which use Java and rely on ECDSA to authenticate transactions.[12]

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 of NIST-produced curves being questioned after revelations were made that the NSA willingly inserts backdoors into software, hardware components and published standards; well-known cryptographers[13] have expressed[14][15] doubts about how the NIST curves were designed, and voluntary tainting has already been proved in the past.[16][17] (See also the libssh curve25519 introduction.[18]) Nevertheless, a proof that the named NIST curves exploit a rare weakness is still missing.
  2. Technical concerns: the difficulty of properly implementing the standard, its slowness, and design flaws which reduce security in insufficiently defensive implementations.[19]

Implementations

Below is a list of cryptographic libraries that provide support for ECDSA:[citation needed]

See also

References

  1. ^ Johnson, Don & Menezes, Alfred (1999). "The Elliptic Curve Digital Signature Algorithm (ECDSA)". Certicom Research. Canada. CiteSeerX 10.1.1.38.8014
  2. ^ "NIST FIPS 186-4, July 2013, pp. 19 and 26". Archived December 27, 2016 at the Wayback Machine. Retrieved March 17, 2014.
  3. ^ Console Hacking 2010 - PS3 Epic Fail Archived December 15, 2014 at the Wayback Machine, page 123–128
  4. ^ "Android Security Vulnerability". Archived April 7, 2019 at the Wayback Machine. Retrieved February 24, 2015.
  5. ^ Pornin, T. (2013). RFC 6979 - Deterministic Usage of the Digital Signature Algorithm (DSA) and Elliptic Curve Digital Signature Algorithm (ECDSA). doi:10.17487/RFC6979. Retrieved February 24, 2015.
  6. ^ "The Double-Base Number System in Elliptic Curve Cryptography". Archived July 26, 2011 at the Wayback Machine. Retrieved 22 April 2014.
  7. ^ Daniel R. L. Brown SECG SEC 1: Elliptic Curve Cryptography (Version 2.0) https://www.secg.org/sec1-v2.pdf
  8. ^ Bendel, Mike (2010-12-29). "Hackers Describe PS3 Security As Epic Fail, Gain Unrestricted Access". Exophase.com. Archived April 7, 2019 at the Wayback Machine. Retrieved 2011-01-05.
  9. ^ "Cryptology ePrint Archive: Report 2011/232". Archived December 8, 2018 at the Wayback Machine. Retrieved February 24, 2015.
  10. ^ "Vulnerability Note VU#536044 - OpenSSL leaks ECDSA private key through a remote timing attack". www.kb.cert.org. Archived April 7, 2019 at the Wayback Machine. Retrieved May 24, 2011.
  11. ^ "ChangeLog". OpenSSL Project. Archived August 9, 2020 at the Wayback Machine. Retrieved 22 April 2014.
  12. ^ "Android bug batters Bitcoin wallets". The Register. 12 August 2013. Archived August 15, 2013 at the Wayback Machine. Retrieved August 27, 2017.
  13. ^ Schneier, Bruce (September 5, 2013). "The NSA Is Breaking Most Encryption on the Internet". Schneier on Security. Archived December 15, 2017 at the Wayback Machine. Retrieved January 11, 2018.
  14. ^ "SafeCurves: choosing safe curves for elliptic-curve cryptography". Oct 25, 2013. Archived April 7, 2019 at the Wayback Machine. Retrieved January 11, 2018.
  15. ^ Bernstein, Daniel J. & Lange, Tanja (May 31, 2013). "Security dangers of the NIST curves". Archived May 28, 2019 at the Wayback Machine. Retrieved January 11, 2018.
  16. ^ Schneier, Bruce (November 15, 2007). "The Strange Story of Dual_EC_DRBG". Schneier on Security. Archived April 23, 2019 at the Wayback Machine. Retrieved January 11, 2018.
  17. ^ Greenemeier, Larry (September 18, 2013). "NSA Efforts to Evade Encryption Technology Damaged U.S. Cryptography Standard". Scientific American. Archived December 24, 2017 at the Wayback Machine. Retrieved January 11, 2018.
  18. ^ "[email protected]\doc - projects/libssh.git". libssh shared repository. Archived March 23, 2019 at the Wayback Machine. Retrieved January 11, 2018.
  19. ^ Bernstein, Daniel J. (March 23, 2014). "How to design an elliptic-curve signature system". The cr.yp.to blog. Archived March 23, 2014 at the Wayback Machine. Retrieved January 11, 2018.

Further reading