Elliptic Curve Digital Signature Algorithm

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.{{cite journal

| title=The Elliptic Curve Digital Signature Algorithm (ECDSA)

| year=1999

| first1=Don

| last1=Johnson

| first2=Alfred

| last2=Menezes

| citeseerx=10.1.1.38.8014

| journal=Certicom Research. Canada

}} 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.

Suppose Alice wants to send a signed message to Bob. Initially, they must agree on the curve parameters $(\backslash textrm\{CURVE\},\; G,\; n)$. 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 $G$.

The order $n$ of the base point $G$ must be prime. Indeed, we assume that every nonzero element of the ring $\backslash mathbb\{Z\}/n\backslash mathbb\{Z\}$ is invertible, so that $\backslash mathbb\{Z\}/n\backslash mathbb\{Z\}$ 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 $Q\_A\; =\; d\_A\; \backslash times\; G$. We use $\backslash times$ to denote elliptic curve point multiplication by a scalar.

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

1. Calculate $e\; =\; \backslash textrm\{HASH\}(m)$. (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.

[http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf NIST FIPS 186-4, July 2013, pp. 19 and 26])

1. Select a cryptographically secure random integer $k$ from $[1,\; n-1]$.
2. Calculate the curve point $(x\_1,\; y\_1)\; =\; k\; \backslash times\; G$.
3. Calculate $r\; =\; x\_1\backslash ,\backslash bmod\backslash ,n$. If $r\; =\; 0$, go back to step 3.
4. Calculate $s\; =\; k^\{-1\}(z\; +\; r\; d\_A)\backslash ,\backslash bmod\backslash ,n$. If $s\; =\; 0$, go back to step 3.
5. The signature is the pair $(r,\; s)$. (And $(r,-s\backslash ,\backslash bmod\backslash ,n)$ 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 $(r,\; s\text{'})$, employing the same unknown $k$ for different known messages $m$ and $m\text{'}$, an attacker can calculate $z$ and $z\text{'}$, and since $s\; -\; s\text{'}\; =\; k^\{-1\}(z\; -\; z\text{'})$ (all operations in this paragraph are done modulo $n$) the attacker can find $k\; =\; \backslash frac\{z\; -\; z\text{'}\}\{s\; -\; s\text{'}\}$. Since $s\; =\; k^\{-1\}(z\; +\; r\; d\_A)$, the attacker can now calculate the private key $d\_A\; =\; \backslash frac\{s\; k\; -\; z\}\{r\}$.

This implementation failure was used, for example, to extract the signing key used for the PlayStation 3 gaming-console.[https://events.ccc.de/congress/2010/Fahrplan/attachments/1780_27c3_console_hacking_2010.pdf Console Hacking 2010 - PS3 Epic Fail] {{Webarchive|url=https://web.archive.org/web/20141215140847/http://events.ccc.de/congress/2010/Fahrplan/attachments/1780%5F27c3%5Fconsole%5Fhacking%5F2010.pdf |date=December 15, 2014 }}, page 123–128

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.{{cite web|url=https://bitcoin.org/en/alert/2013-08-11-android|title=Android Security Vulnerability|access-date=February 24, 2015}}

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.{{cite tech report|url=https://www.rfc-editor.org/rfc/rfc6979.html|title=RFC 6979 - Deterministic Usage of the Digital Signature Algorithm (DSA) and Elliptic Curve Digital Signature Algorithm (ECDSA)|year=2013 |doi=10.17487/RFC6979 |access-date=February 24, 2015|last1=Pornin |first1=T. |doi-access=free }}

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 {{mvar|O}}, and its coordinates are otherwise valid.
2. Check that $Q\_A$ lies on the curve.
3. Check that $n\; \backslash times\; Q\_A\; =\; O$.

After that, Bob follows these steps:

1. Verify that {{mvar|r}} and {{mvar|s}} are integers in $[1,\; n-1]$. If not, the signature is invalid.
2. Calculate $e\; =\; \backslash textrm\{HASH\}(m)$, where HASH is the same function used in the signature generation.
3. Let $z$ be the $L\_n$ leftmost bits of {{mvar|e}}.
4. Calculate $u\_1\; =\; zs^\{-1\}\backslash ,\backslash bmod\backslash ,n$ and $u\_2\; =\; rs^\{-1\}\backslash ,\backslash bmod\backslash ,n$.
5. Calculate the curve point $(x\_1,\; y\_1)\; =\; u\_1\; \backslash times\; G\; +\; u\_2\; \backslash times\; Q\_A$. If $(x\_1,\; y\_1)\; =\; O$ then the signature is invalid.
6. The signature is valid if $r\; \backslash equiv\; x\_1\; \backslash pmod\{n\}$, invalid otherwise.

Note that an efficient implementation would compute inverse $s^\{-1\}\backslash ,\backslash bmod\backslash ,n$ only once. Also, using Shamir's trick, a sum of two scalar multiplications $u\_1\; \backslash times\; G\; +\; u\_2\; \backslash times\; Q\_A$ can be calculated faster than two scalar multiplications done independently.{{cite web | url=http://www.lirmm.fr/~imbert/talks/laurent_Asilomar_08.pdf | title=The Double-Base Number System in Elliptic Curve Cryptography | access-date=22 April 2014}}

It is not immediately obvious why verification even functions correctly. To see why, denote as {{mvar|C}} the curve point computed in step 5 of verification,

: $C\; =\; u\_1\; \backslash times\; G\; +\; u\_2\; \backslash times\; Q\_A$

From the definition of the public key as $Q\_A\; =\; d\_A\; \backslash times\; G$,

: $C\; =\; u\_1\; \backslash times\; G\; +\; u\_2\; d\_A\; \backslash times\; G$

Because elliptic curve scalar multiplication distributes over addition,

: $C\; =\; (u\_1\; +\; u\_2\; d\_A)\; \backslash times\; G$

Expanding the definition of $u\_1$ and $u\_2$ from verification step 4,

: $C\; =\; (z\; s^\{-1\}\; +\; r\; d\_A\; s^\{-1\})\; \backslash times\; G$

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

: $C\; =\; (z\; +\; r\; d\_A)\; s^\{-1\}\; \backslash times\; G$

Expanding the definition of {{mvar|s}} from signature step 6,

: $C\; =\; (z\; +\; r\; d\_A)\; (z\; +\; r\; d\_A)^\{-1\}\; (k^\{-1\})^\{-1\}\; \backslash times\; G$

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

: $C\; =\; k\; \backslash times\; G$

From the definition of {{mvar|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.

Given a message {{mvar|m}} and Alice's signature $r,\; s$ on that message, Bob can (potentially) recover Alice's public key:Daniel R. L. Brown SECG SEC 1: Elliptic Curve Cryptography (Version 2.0) https://www.secg.org/sec1-v2.pdf

1. Verify that {{mvar|r}} and {{mvar|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 {{mvar|R}} value results in a distinct recovered key.
3. Calculate $e\; =\; \backslash textrm\{HASH\}(m)$, where HASH is the same function used in the signature generation.
4. Let {{mvar|z}} be the $L\_n$ leftmost bits of {{mvar|e}}.
5. Calculate $u\_1\; =\; -zr^\{-1\}\backslash ,\backslash bmod\backslash ,n$ and $u\_2\; =\; sr^\{-1\}\backslash ,\backslash bmod\backslash ,n$.
6. Calculate the curve point $Q\_A\; =\; (x\_A,\; y\_A)\; =\; u\_1\; \backslash times\; G\; +\; u\_2\; \backslash times\; R$.
7. The signature is valid if $Q\_A$, matches Alice's public key.
8. The signature is invalid if all the possible {{mvar|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.

Start with the definition of $Q\_A$ from recovery step 6,

: $Q\_A\; =\; (x\_A,\; y\_A)\; =\; u\_1\; \backslash times\; G\; +\; u\_2\; \backslash times\; R$

From the definition $R\; =\; (x\_1,\; y\_1)\; =\; k\; \backslash times\; G$ from signing step 4,

: $Q\_A\; =\; u\_1\; \backslash times\; G\; +\; u\_2\; k\; \backslash times\; G$

Because elliptic curve scalar multiplication distributes over addition,

: $Q\_A\; =\; (u\_1\; +\; u\_2\; k)\; \backslash times\; G$

Expanding the definition of $u\_1$ and $u\_2$ from recovery step 5,

: $Q\_A\; =\; (-z\; r^\{-1\}\; +\; s\; k\; r^\{-1\})\; \backslash times\; G$

Expanding the definition of {{mvar|s}} from signature step 6,

: $Q\_A\; =\; (-z\; r^\{-1\}\; +\; k^\{-1\}\; (z\; +\; r\; d\_A)\; k\; r^\{-1\})\; \backslash times\; G$

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

: $Q\_A\; =\; (-z\; r^\{-1\}\; +\; (z\; r^\{-1\}\; +\; d\_A))\; \backslash times\; G$

The first and second terms cancel each other out,

: $Q\_A\; =\; d\_A\; \backslash times\; G$

From the definition of $Q\_A\; =\; d\_A\; \backslash times\; G$, 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 {{mvar|r}}.

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.{{Cite news|last=Bendel|first=Mike|title=Hackers Describe PS3 Security As Epic Fail, Gain Unrestricted Access|publisher=Exophase.com|date=2010-12-29|url=http://exophase.com/20540/hackers-describe-ps3-security-as-epic-fail-gain-unrestricted-access/|access-date=2011-01-05}}

On March 29, 2011, two researchers published an IACR paper{{cite web|url=http://eprint.iacr.org/2011/232|title=Cryptology ePrint Archive: Report 2011/232|access-date=February 24, 2015}} 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.{{cite web|url=https://www.kb.cert.org/vuls/id/536044|title=Vulnerability Note VU#536044 - OpenSSL leaks ECDSA private key through a remote timing attack|website=www.kb.cert.org}} The vulnerability was fixed in OpenSSL 1.0.0e.{{cite web | url=http://www.openssl.org/news/changelog.html | title=ChangeLog | publisher=OpenSSL Project | access-date=22 April 2014}}

In August 2013, it was revealed that bugs in some implementations of the Java class [https://docs.oracle.com/javase/10/docs/api/java/security/SecureRandom.html 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.{{cite web |title=Android bug batters Bitcoin wallets

| url= https://www.theregister.co.uk/2013/08/12/android_bug_batters_bitcoin_wallets/ |publisher=The Register |date=12 August 2013}}

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

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{{cite web|url=https://www.schneier.com/blog/archives/2013/09/the_nsa_is_brea.html#c1675929|title=The NSA Is Breaking Most Encryption on the Internet|website=Schneier on Security|first=Bruce|last=Schneier|date=September 5, 2013}} have expressed{{cite web|url=http://safecurves.cr.yp.to/rigid.html|title=SafeCurves: choosing safe curves for elliptic-curve cryptography|date=Oct 25, 2013}}{{cite web|url=https://www.hyperelliptic.org/tanja/vortraege/20130531.pdf|title=Security dangers of the NIST curves|first1=Daniel J.|last1=Bernstein|first2=Tanja|last2=Lange|author2-link=Tanja Lange|date=May 31, 2013}} doubts about how the NIST curves were designed, and voluntary tainting has already been proved in the past.{{cite web|url=https://www.schneier.com/blog/archives/2007/11/the_strange_sto.html|title=The Strange Story of Dual_EC_DRBG|website=Schneier on Security|first=Bruce|last=Schneier|date=November 15, 2007}}{{cite web|url=http://www.scientificamerican.com/article/nsa-nist-encryption-scandal/|title=NSA Efforts to Evade Encryption Technology Damaged U.S. Cryptography Standard|first=Larry|last=Greenemeier|publisher=Scientific American|date=September 18, 2013}} (See also the libssh curve25519 introduction.{{cite web|url=https://git.libssh.org/projects/libssh.git/tree/doc/curve25519-sha256@libssh.org.txt#n4|title=curve25519-sha256@libssh.org.txt\doc - projects/libssh.git|website=libssh shared repository}}) 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.{{cite web|url=http://blog.cr.yp.to/20140323-ecdsa.html|title=How to design an elliptic-curve signature system|first=Daniel J.|last=Bernstein|date=March 23, 2014|website=The cr.yp.to blog}}

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

• EdDSA
• RSA (cryptosystem)

{{Reflist|30em}}

• Accredited Standards Committee [http://www.x9.org X9], ASC X9 Issues New Standard for Public Key Cryptography/ECDSA, Oct. 6, 2020. [https://x9.org/asc-x9-issues-new-standard-for-public-key-cryptography-ecdsa/ Source]
• Accredited Standards Committee [http://www.x9.org X9], 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, [https://www.secg.org/sec1-v2.pdf Standards for efficient cryptography, SEC 1: Elliptic Curve Cryptography], Version 2.0, May 21, 2009.
• López, J. and Dahab, R. [http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.37.2771 An Overview of Elliptic Curve Cryptography], Technical Report IC-00-10, State University of Campinas, 2000.
• Daniel J. Bernstein, [http://cr.yp.to/papers/pippenger.pdf Pippenger's exponentiation algorithm], 2002.
• Daniel R. L. Brown, Generic Groups, Collision Resistance, and ECDSA, Designs, Codes and Cryptography, 35, 119–152, 2005. [http://eprint.iacr.org/2002/026 ePrint version]
• Ian F. Blake, Gadiel Seroussi, and Nigel Smart, editors, Advances in Elliptic Curve Cryptography, London Mathematical Society Lecture Note Series 317, Cambridge University Press, 2005.
• {{Cite book | last1 = Hankerson | first1 = D. | last2 = Vanstone | first2 = S. | author-link2 = Scott Vanstone | last3 = Menezes | first3 = A. | author-link3 = Alfred Menezes | doi = 10.1007/b97644 | title = Guide to Elliptic Curve Cryptography | series = Springer Professional Computing | publisher = Springer| location = New York| year = 2004 | isbn = 0-387-95273-X | s2cid = 720546 }}

• [https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf Digital Signature Standard; includes info on ECDSA]
• [https://web.archive.org/web/20160304101319/http://cs.ucsb.edu/~koc/ccs130h/notes/ecdsa-cert.pdf The Elliptic Curve Digital Signature Algorithm (ECDSA); provides an in-depth guide on ECDSA]. [https://web.archive.org/web/20100627011540/http://cs.ucsb.edu/~koc/ccs130h/notes/ecdsa-cert.pdf Wayback link]

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