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Zk

@risotto

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ZK Scholars Assembly Revision 3 - More Elliptic Curves, Parings and KGZ Other than Weierstrass form of elliptic curve that is used in Bitcoin & Ethereum (secp256k1) and TEE & Secure Enclave (secp256r1), there are also other forms of elliptic curves such as Montgomery Curves, Edwards Curves and Twisted Edwards Curves. Montgomery curves have no point at infinity, they can also do arithmetic operations that are more computational efficient such as differential addition and Montgomery ladder. For example scalar multiplication kP where k is a large integer. In the Weierstrass form, this involves numerous point additions and doublings, each requiring multiple field inversions. In the Montgomery form, the Montgomery ladder performs this operation using only field multiplications and squaring, significantly reducing the computational overhead.
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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 Scholars Assembly Revision 1 - Group Theory Group is a set of elements that satisfied interesting properties that are useful in cryptography such as - Closure - Associativity - Identity Element - Inverse Element Closure means for any elements a, b ∈ G (group), the result of the operation between them will be result in an element that is of the same group, example a β€’ b = c ∈ G. Associativity means sequence of running operation doesn’t matter for any elements a, b, c ∈ G, for example the result of (a β€’ b) β€’ c will be equal to a β€’ (b β€’ c). Identity Element means there exists an element e ∈ G such that for any element a ∈ G, the operation with e will always result in itself, example e β€’ a = a β€’ e = a. One can imagine e as 0 in simple additive group or 1 in simple multiplicative group as 3 + 0 = 3 and 3 * 1 = 3.
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