EdDSA and Ed25519
EdDSA (Edwards-curve Digital Signature Algorithm) is a modern and secure digital signature algorithm based on performance-optimized elliptic curves, such as the 255-bit curve Curve25519 and the 448-bit curve Curve448-Goldilocks. The EdDSA signatures use the Edwards form of the elliptic curves (for performance reasons), respectively edwards25519 and edwards448. The EdDSA algorithm is based on the Schnorr signature algorithm and relies on the difficulty of the ECDLP problem.
The EdDSA signature algorithm and its variants Ed25519 and Ed448 are technically described in the RFC 8032.
EdDSA Key Generation
Ed25519 and Ed448 use small private keys (32 or 57 bytes respectively), small public keys (32 or 57 bytes) and small signatures (64 or 114 bytes) with high security level at the same time (128-bit or 224-bit respectively).
Assume the elliptic curve for the EdDSA algorithm comes with a generator point G and a subgroup order q for the EC points, generated from G.
The EdDSA key-pair consists of:
private key (integer): privKey
public key (EC point): pubKey = privKey * G
The private key is generated from a random integer, known as seed (which should have similar bit length, like the curve order). The seed is first hashed, then the last few bits, corresponding to the curve cofactor (8 for Ed25519 and 4 for X448) are cleared, then the highest bit is cleared and the second highest bit is set. These transformations guarantee that the private key will always belong to the same subgroup of EC points on the curve and that the private keys will always have similar bit length (to protect from timing-based side-channel attacks). For Ed25519 the private key is 32 bytes. For Ed448 the private key is 57 bytes.
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 for the curve). The public key is encoded as compressed EC point: the y-coordinate, combined with the lowest bit (the parity) of the x-coordinate. For Ed25519 the public key is 32 bytes. For Ed448 the public key is 57 bytes.
EdDSA Sign
The EdDSA signing algorithm (RFC 8032) takes as input a text message msg + the signer's EdDSA private key privKey and produces as output a pair of integers {R, s}. EdDSA signing works as follows (with minor simplifications):
EdDSA_sign(msg, privKey) --> { R, s }
Calculate pubKey = privKey * G
Deterministically generate a secret integer r = hash(hash(privKey) + msg) mod q (this is a bit simplified)
Calculate the public key point behind r by multiplying it by the curve generator: R = r * G
Calculate h = hash(R + pubKey + msg) mod q
Calculate s = (r + h * privKey) mod q
Return the signature { R, s }
The produced digital signature is 64 bytes (32 + 32 bytes) for Ed25519 and 114 bytes (57 + 57 bytes) for Ed448. It holds a compressed point R + the integer s (confirming that the signer knows the msg and the privKey).
EdDSA Verify Signature
The EdDSA signature verification algorithm (RFC 8032) takes as input a text message msg + the signer's EdDSA public key pubKey + the EdDSA signature {R, s} and produces as output a boolean value (valid or invalid signature). EdDSA verification works as follows (with minor simplifications):
EdDSA_signature_verify(msg, pubKey, signature { R, s } ) --> valid / invalid
Calculate h = hash(R + pubKey + msg) mod q
Calculate P1 = s * G
Calculate P2 = R + h * pubKey
Return P1 == P2
How Does it Work?
During the verification the point P1 is calculated as: P1 = s * G.
During the signing s = (r + h * privKey) mod q. Now replace s in the above equation:
P1 = s * G = (r + h * privKey) mod q * G = r * G + h * privKey * G = R + h * pubKey
The above is exactly the other point P2. If these points P1 and P2 are the same EC point, this proves that the point P1, calculated by the private key matches the point P2, created by its corresponding public key.
ECDSA vs EdDSA
If we compare the signing and verification for EdDSA, we shall find that EdDSA is simpler than ECDSA, easier to understand and to implement. Both signature algorithms have similar security strength for curves with similar key lengths. For the most popular curves (liked edwards25519 and edwards448) the EdDSA algorithm is slightly faster than ECDSA, but this highly depends on the curves used and on the certain implementation. Unlike ECDSA the EdDSA signatures do not provide a way to recover the signer's public key from the signature and the message. Generally, it is considered that EdDSA is recommended for most modern apps.
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