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Zk
@risotto
ZK Scholars Assembly Revision 2 - Elliptic Curve & Schnorr Signature Elliptic curve can be simply described as the equation y^2 = x^3 + ax + b (mod p). This normal form is also called Weierstrass Form. Different elliptic curves are formed by changing the value of a, b and p. We can define elliptic curve points as a group but more interestingly, we can also do geometric operations with it. For example, the inverse -P of a point P is the one symmetric across the other side of x-axis. We can also do geometric addition to compute the sum of two points P and Q by drawing a line passing through both of them, the 3rd intersection point of this line will be R, and the inverse -R is the sum of P + Q because P + Q + R = 0 thus P + Q = -R in abelian group. To compute P + P, we draw a tangent line on point P and the inverse of next intersection point is the sum of those.
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Zk
@risotto
By picking a generator point G from elliptic curve, we can now hide a scalar value a in G by multiplying them together aG or as an exponent G^a. Thanks to discrete logarithm problem, it is hard to extract a from aG / G^a. This enables encryption of secret value into a public sharable point and creates cryptography applications such as ELGAMAL, Schnorr Signature and ECDSA. In ELGAMAL, to encrypt an encoded message M, a random k is chosen and compute ciphertext pairs (C1, C2) as follows: C1 = k * G C2 = M + k * B B is the public key of Bob with secret key b s.t. B = bG To decrypt the message M, Bob just has to do C2 - b * C1. By substituting (C1, C2) and B we get: M + k * bG - b * k * G Notice that now the equation cancel off everything and leaving only M!
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