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Elliptic Curve Digital Signature Algorithm
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-size
As with elliptic-curve cryptography in general, the bit size of the public key believed to be needed for ECDSA is about twice the size of the security level, in bits. For example, at a security level of 80 bits—meaning an attacker requires a maximum of about
operations to find the private key—the size of an ECDSA private key would be 160 bits, whereas the size of a DSA private key is at least 1024 bits. On the other hand, the signature size is the same for both DSA and ECDSA: approximately
bits, where is the security level measured in bits, that is, about 320 bits for a security level of 80 bits.
Signature generation algorithm
Suppose Alice wants to send a signed message to Bob. Initially, they must agree on the curve parameters
. In addition to the field and equation of the curve, we need
, a base point of prime order on the curve; is the multiplicative order of the point
.
 Parameter CURVE the elliptic curve field and equation used G elliptic curve base point, a point on the curve that generates a subgroup of large prime order n n integer order of G, means that , where is the identity element. the private key (randomly selected) the public key (calculated by elliptic curve) m the message to send
The order of the base point
must be prime. Indeed, we assume that every nonzero element of the ring
is invertible, so that
must be a field. It implies that must be prime (cf. Bézout's identity).
Alice creates a key pair, consisting of a private key integer
, randomly selected in the interval
; and a public key curve point
. We use to denote elliptic curve point multiplication by a scalar.
For Alice to sign a message
, she follows these steps:
1. Calculate
. (Here HASH is a cryptographic hash function, such as SHA-2, with the output converted to an integer.)
2. Let be the
leftmost bits of , where
is the bit length of the group order . (Note that can be greater than but not longer.)
3. Select a cryptographically secure random integer from
.
4. Calculate the curve point
.
5. Calculate
. If
, go back to step 3.
6. Calculate
. If
, go back to step 3.
7. The signature is the pair
. (And
is also a valid signature.)

As the standard notes, it is not only required for to be secret, but it is also crucial to select different for different signatures, otherwise the equation in step 6 can be solved for
, the private key: given two signatures
and
, employing the same unknown for different known messages
and
, an attacker can calculate and
, and since
(all operations in this paragraph are done modulo ) the attacker can find
. Since
, the attacker can now calculate the private key
.
This implementation failure was used, for example, to extract the signing key used for the PlayStation 3 gaming-console.
Another way ECDSA signature may leak private keys is when 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.
To ensure that is unique for each message, one may bypass random number generation completely and generate deterministic signatures by deriving from both the message and the private key.
Signature verification algorithm
For Bob to authenticate Alice's signature, he must have a copy of her public-key curve point
. Bob can verify
is a valid curve point as follows:
1. Check that
is not equal to the identity element O, and its coordinates are otherwise valid
2. Check that
lies on the curve
3. Check that
After that, Bob follows these steps:
1. Verify that r and s are integers in
. If not, the signature is invalid.
2. Calculate
, where HASH is the same function used in the signature generation.
3. Let z be the
leftmost bits of e.
4. Calculate
and
.
5. Calculate the curve point
. If
then the signature is invalid.
6. The signature is valid if
, invalid otherwise.
Note that an efficient implementation would compute inverse
only once. Also, using Shamir's trick, a sum of two scalar multiplications
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 as C the curve point computed in step 5 of verification,
From the definition of the public key as
,
Because elliptic curve scalar multiplication distributes over addition,
Expanding the definition of
and
from verification step 4,
Collecting the common term
,
Expanding the definition of s from signature step 6,
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
From the definition of r, this is verification step 6.
This shows only that a correctly signed message will verify correctly; many other properties[which?] are required for a secure signature algorithm.
Public key recovery
Given a message m and Alice's signature
on that message, Bob can (potentially) recover Alice's public key:
1. Verify that r and s are integers in
. If not, the signature is invalid.
2. Calculate a curve point
where
is one of ,
,
, etc. (provided
is not too large for a field element) and
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
, where HASH is the same function used in the signature generation.
4. Let z be the
leftmost bits of e.
5. Calculate
and
.
6. Calculate the curve point
.
7. The signature is valid if
, 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
from recovery step 6,
From the definition
from signing step 4,
Because elliptic curve scalar multiplication distributes over addition,
Expanding the definition of
and
from recovery step 5,
Expanding the definition of s from signature step 6,
Since the product of an element's inverse and the element is the identity, we are left with
The first and second terms cancel each other out,
From the definition of
, 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
from signature value r.
Security
In December 2010, a group calling itself fail0verflow announced 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 was static instead of random. As pointed out in the Signature generation algorithm section above, this makes
solvable, rendering the entire algorithm useless.
On March 29, 2011, two researchers published an IACR paper 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. The vulnerability was fixed in OpenSSL 1.0.0e.
In August 2013, it was revealed that bugs in some implementations of the Java class SecureRandom sometimes generated collisions in the 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.
This issue can be prevented by an unpredictable generation of , e.g., a deterministic procedure as described by RFC 6979.
Concerns
There exist two sorts of concerns with ECDSA:
1. Political concerns: the trustworthiness of NIST-produced curves being questioned after revelations that the NSA willingly inserts backdoors into software, hardware components and published standards were made; well-known cryptographers have expressed doubts about how the NIST curves were designed, and voluntary tainting has already been proved in the past. 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 of the Dual_EC_DRBG random number generator.
Both of those concerns are summarized in libssh curve25519 introduction.
Implementations
Below is a list of cryptographic libraries that provide support for ECDSA:
Example usage
Wikipedia.org uses ECDSA in a TLS ciphersuite to authenticate itself to web browsers, which the following abbreviated transcript shows.
\$ date Wed Mar 4 10:24:52 EST 2020\$ openssl s_client -connect wikipedia.org:443 # output below has DELETIONS for brevity​CONNECTED(00000003)​depth=2 O = Digital Signature Trust Co., CN = DST Root CA X3verify return:1depth=1 C = US, O = Let's Encrypt, CN = Let's Encrypt Authority X3verify return:1depth=0 CN = *.wikipedia.orgverify return:1---Certificate chain 0 s:/CN=*.wikipedia.org i:/C=US/O=Let's Encrypt/CN=Let's Encrypt Authority X3 1 s:/C=US/O=Let's Encrypt/CN=Let's Encrypt Authority X3 i:/O=Digital Signature Trust Co./CN=DST Root CA X3---Server certificate-----BEGIN CERTIFICATE-----​MIIHOTCCBiGgAwIBAgISA4srJU6bpT7xpINN6bbGO2/mMA0GCSqGSIb3DQEBCwUA ... many lines DELETED ....​kTOXMoKzBkJCU8sCdeziusJtNvWXW6p8Z3UpuTw=​-----END CERTIFICATE-----​subject=/CN=*.wikipedia.org​issuer=/C=US/O=Let's Encrypt/CN=Let's Encrypt Authority X3---No client certificate CA names sentPeer signing digest: SHA256Server Temp Key: ECDH, P-256, 256 bits---SSL handshake has read 3353 bytes and written 431 bytes---New, TLSv1/SSLv3, Cipher is ECDHE-ECDSA-AES256-GCM-SHA384​Server public key is 256 bitSecure Renegotiation IS supported​Compression: NONEExpansion: NONENo ALPN negotiated​SSL-Session: Protocol  : TLSv1.2 Cipher  : ECDHE-ECDSA-AES256-GCM-SHA384 Session-ID: ... DELETED ... Session-ID-ctx: Master-Key: ... DELETED ... Key-Arg  : None PSK identity: None PSK identity hint: None SRP username: None Start Time: 1583335210 Timeout  : 300 (sec) Verify return code: 0 (ok)---DONE
See also
References
1. ^ Johnson, Don; Menezes, Alfred (1999). "The Elliptic Curve Digital Signature Algorithm (ECDSA)". CiteSeerX. Retrieved May 9, 2021.
2. ^ NIST FIPS 186-4, July 2013, pp. 19 and 26
3. ^ Console Hacking 2010 - PS3 Epic FailArchived December 15, 2014, at the Wayback Machine, page 123–128
4. ^ "Android Security Vulnerability". Retrieved February 24, 2015.
5. ^ "RFC 6979 - Deterministic Usage of the Digital Signature Algorithm (DSA) and Elliptic Curve Digital Signature Algorithm (ECDSA)". Retrieved February 24, 2015.
6. ^ "The Double-Base Number System in Elliptic Curve Cryptography" (PDF). Retrieved April 22, 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 (December 29, 2010). "Hackers Describe PS3 Security As Epic Fail, Gain Unrestricted Access". Exophase.com. Retrieved January 5, 2011.
9. ^ "Cryptology ePrint Archive: Report 2011/232". Retrieved February 24, 2015.
10. ^ "Vulnerability Note VU#536044 - OpenSSL leaks ECDSA private key through a remote timing attack". www.kb.cert.org.
11. ^ "ChangeLog". OpenSSL Project. Retrieved April 22, 2014.
12. ^ "Android bug batters Bitcoin wallets". The Register. August 12, 2013.
13. ^ Schneier, Bruce (September 5, 2013). "The NSA Is Breaking Most Encryption on the Internet". Schneier on Security.
14. ^ "SafeCurves: choosing safe curves for elliptic-curve cryptography". October 25, 2013.
15. ^ Bernstein, Daniel J.; Lange, Tanja (May 31, 2013). "Security dangers of the NIST curves"(PDF).
16. ^ Schneier, Bruce (November 15, 2007). "The Strange Story of Dual_EC_DRBG". Schneier on Security.
17. ^ Greenemeier, Larry (September 18, 2013). "NSA Efforts to Evade Encryption Technology Damaged U.S. Cryptography Standard". Scientific American.
18. ^ Bernstein, Daniel J. (March 23, 2014). "How to design an elliptic-curve signature system". The cr.yp.to blog.
19. ^ "New key type (ed25519) and private key format".
20. ^ "curve25519-sha256@libssh.org.txt\doc - projects/libssh.git". libssh shared repository.
Further reading

External links
Last edited on 26 May 2021, at 20:36
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