ECDSA: Elliptic Curve Signatures

The ECDSA (Elliptic Curve Digital Signature Algorithm) is a cryptographically secure digital signature scheme, based on the elliptic-curve cryptography (ECC). ECDSA relies on the math of the cyclic groups of elliptic curves over finite fields and on the difficulty of the ECDLP problem (elliptic-curve discrete logarithm problem). The ECDSA sign / verify algorithm relies on EC point multiplication and works as described below. ECDSA keys and signatures are shorter than in RSA for the same security level. A 256-bit ECDSA signature has the same security strength like 3072-bit RSA signature.
ECDSA uses cryptographic elliptic curves (EC) over finite fields in the classical Weierstrass form. These curves are described by their EC domain parameters, specified by various cryptographic standards such as SECG: SEC 2 and Brainpool (RFC 5639). Elliptic curves, used in cryptography, define:
Generator point G, used for scalar multiplication on the curve (multiply integer by EC point)
Order n of the subgroup of EC points, generated by G, which defines the length of the private keys (e.g. 256 bits)
For example, the 256-bit elliptic curve secp256k1 has:
Order n = 115792089237316195423570985008687907852837564279074904382605163141518161494337 (prime number)
Generator point G {x = 55066263022277343669578718895168534326250603453777594175500187360389116729240, y = 32670510020758816978083085130507043184471273380659243275938904335757337482424}
Key Generation
The ECDSA key-pair consists of:
private key (integer): privKey
public key (EC point): pubKey = privKey * G
The private key is generated as a random integer in the range [0...n-1]. The public key pubKey is a point on the elliptic curve, calculated by the EC point multiplication: pubKey = privKey * G (the private key, multiplied by the generator point G).
The public key EC point {x, y} can be compressed to just one of the coordinates + 1 bit (parity). For the secp256k1 curve, the private key is 256-bit integer (32 bytes) and the compressed public key is 257-bit integer (~ 33 bytes).
ECDSA Sign
The ECDSA signing algorithm (RFC 6979) takes as input a message msg ****+ a private key privKey ****and produces as output a signature, which consists of pair of integers {r, s}. The ECDSA signing algorithm is based on the ElGamal signature scheme and works as follows (with minor simplifications):
1.Calculate the message hash, using a cryptographic hash function like SHA-256: h = hash(msg)
2.Generate securely a random number k in the range [1..n-1]
In case of deterministic-ECDSA, the value k is HMAC-derived from h + privKey (see RFC 6979)
3.Calculate the random point R = k * G and take its x-coordinate: r = R.x
4.Calculate the signature proof: s = k−1∗(h+r∗privKey)(modn)k^{-1} * (h + r * privKey) \pmod nk−1∗(h+r∗privKey)(modn)
​
The modular inverse k−1(modn)k^{-1} \pmod nk−1(modn)
 is an integer, such that k∗k−1≡1(modn)k * k^{-1} \equiv 1 \pmod nk∗k−1≡1(modn)
​
5.Return the signature {r, s}.
The calculated signature {r, s} is a pair of integers, each in the range [1...n-1]. It encodes the random point R = k * G, along with a proof s, confirming that the signer knows the message h and the private key privKey. The proof s is by idea verifiable using the corresponding pubKey.
ECDSA signatures are 2 times longer than the signer's private key for the curve used during the signing process. For example, for 256-bit elliptic curves (like secp256k1) the ECDSA signature is 512 bits (64 bytes) and for 521-bit curves (like secp521r1) the signature is 1042 bits.
ECDSA Verify Signature
The algorithm to verify a ECDSA signature takes as input the signed message msg + the signature {r, s} produced from the signing algorithm + the public key pubKey, corresponding to the signer's private key. The output is boolean value: valid or invalid signature. The ECDSA signature verify algorithm works as follows (with minor simplifications):
1.Calculate the message hash, with the same cryptographic hash function used during the signing: h = hash(msg)
2.Calculate the modular inverse of the signature proof: s1 = s−1(modn)s^{-1} \pmod ns−1(modn)
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3.Recover the random point used during the signing: R' = (h * s1) * G + (r * s1) * pubKey
4.Take from R' its x-coordinate: r' = R'.x
5.Calculate the signature validation result by comparing whether r' == r
The general idea of the signature verification is to recover the point R' using the public key and check whether it is same point R, generated randomly during the signing process.
How Does it Work?
The ECDSA signature {r, s} has the following simple explanation:
The signing signing encodes a random point R (represented by its x-coordinate only) through elliptic-curve transformations using the private key privKey and the message hash h into a number s, which is the proof that the message signer knows the private key privKey. The signature {r, s} cannot reveal the private key due to the difficulty of the ECDLP problem.
The signature verification decodes the proof number s from the signature back to its original point R, using the public key pubKey and the message hash h and compares the x-coordinate of the recovered R with the r value from the signature.
The Math behind the ECDSA Sign / Verify
Read this section only if you like math. Most developer may skip it.
How does the above sign / verify scheme work? It is not obvious, but let's play a bit with the equations.
The equation behind the recovering of the point R', calculated during the signature verification, can be transformed by replacing the pubKey with privKey * G as follows:
R' = (h * s1) * G + (r * s1) * pubKey =
**** = (h * s1) * G + (r * s1) * privKey * G ****=
**** = (h + r * privKey) * s1 * G
If we take the number s = k−1∗(h+r∗privKey)(modn)k^{-1} * (h + r * privKey) \pmod nk−1∗(h+r∗privKey)(modn)
, calculated during the signing process, we can calculate s1 = s−1(modn)s^{-1} \pmod ns−1(modn)
 like this:
s1 = s−1(modn)s^{-1} \pmod ns−1(modn)
 =
= (k−1∗(h+r∗privKey))−1(modn)(k^{-1} * (h + r * privKey))^{-1} \pmod n(k−1∗(h+r∗privKey))−1(modn)
 =
= k∗(h+r∗privKey)−1(modn)k * (h + r * privKey)^{-1} \pmod nk∗(h+r∗privKey)−1(modn)
​
Now, replace s1 in the point R'.
R' = (h + r * privKey) * s1 * G =
= (h+r∗privKey)∗k∗(h+r∗privKey)−1(modn)(h + r * privKey) * k * (h + r * privKey)^{-1} \pmod n(h+r∗privKey)∗k∗(h+r∗privKey)−1(modn)
 * G =
**** = k * G
The final step is to compare the point R' (decoded by the pubKey) with the point R (encoded by the privKey). The algorithm in fact compares only the x-coordinates of R' and R: the integers r' and r.
It is expected that r' == r if the signature is valid and r' ≠ r if the signature or the message or the public key is incorrect.
ECDSA: Public Key Recovery from Signature
It is important to know that the ECDSA signature scheme allows the public key to be recovered from the signed message together with the signature. The recovery process is based on some mathematical computations (described in the SECG: SEC 1 standard) and returns 0, 1 or 2 possible EC points that are valid public keys, corresponding to the signature. To avoid this ambiguity, some ECDSA implementations add one additional bit v to the signature during the signing process and it takes the form {r, s, v}. From this extended ECDSA signature {r, s, v} + the signed message, the signer's public key can be restored with confidence.
The public key recovery from the ECDSA signature is very useful in bandwidth constrained or storage constrained environments (such as blockchain systems), when transmission or storage of the public keys cannot be afforded. For example, the Ethereum blockchain uses extended signatures {r, s, v} for the signed transactions on the chain to save storage and bandwidth.
Public key recovery is possible for signatures, based on the ElGamal signature scheme (such as DSA and ECDSA).
Exercises: RSA Sign and Verify
ECDSA: Sign / Verify - Examples
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