Secure Computation Lecture 6 - Secrets Sharing and Circuit Evaluation
This content discusses secure computation techniques such as secret sharing and circuit evaluation. It covers linear and non-linear aspects of Shamir Secret Sharing, multiplication of shared secrets, and secure circuit evaluation with examples.
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Secure Computation (Lecture 6) Arpita Patra
(n,t) Secret Sharing s : (n,t) Secret Sharing of secret s Linear For MPC: Linear (n,t) Secret Sharing Linearity: The parties can do the following s1 s2 s1 s2 from c: public constant c s s from
Linearity of (n, t) Shamir Secret Sharing each party does locally a a1 + a2 + a3 + b3 b2 b1 b1 b b2 b3 b a3 a2 a1 a c1 c2 c3 1 2 3
Linearity of (n, t) Shamir Secret Sharing c3 c2 a a1 + a2 + a3 + c1 b3 a+b b2 b1 b1 b b2 b3 b a3 a2 a1 a c1 c2 c3 1 2 3 a+b Addition is Absolutely free
Linearity of (n, t) Shamir Secret Sharing a1 a2 a3 d3 a d2 d1 c a a3 a2 c c c a1 a c a 1 2 3 d1 d2 d3 c is a publicly known constant Multiplication by public constants is Absolutely free
Non-linearity of (n, t) Shamir Secret Sharing d2 d3 a b a1 a2 a3 a d1 b3 b2 b1 b1 b2 b3 b b a3 a2 a1 a d1 d2 d3 a b 1 2 3 Come up with example. Multiplication of shared secrets is not free
Secure Circuit Evaluation x1 x2 x3 x4 c y
Secure Circuit Evaluation 2 1 5 9 3 y
Secure Circuit Evaluation 1. (n, t)- secret share each input 1 5 9 2 3
Secure Circuit Evaluation 1. (n, t)- secret share each input 1 5 9 2 2. Find (n, t)-sharing of each intermediate value 3
Secure Circuit Evaluation 1. (n, t)- secret share each input 1 5 9 2 2. Find (n, t)-sharing of each intermediate value 45 3 3 48 144
Secure Circuit Evaluation 1. (n, t)- secret share each input 1 5 9 2 2. Find (n, t)-sharing of each intermediate value Linear gates: Linearity of Shamir Sharing - Non-Interactive 45 3 3 48 144
Secure Circuit Evaluation 1. (n, t)- secret share each input 1 5 9 2 2. Find (n, t)-sharing of each intermediate value Linear gates: Linearity of Shamir Sharing - Non-Interactive 45 3 3 Non-linear gate: Require degree- reduction Technique. Interactive 48 144
Secure Multiplication Gate Evaluation x y x y P1 x1 y1 =z1 y1 x1 Shamir-share z1 P2 y2 x2 Shamir-share x2 y2 =z2 z2 z3 y3 P3 x3 y3 =z3 x3 Shamir-share zn Shamir-share xn yn = zn f1 (x) f2 (x) of degree 2t with x y as its constant term Pn yn xn f1 (x) f2 (x)
Secure Circuit Evaluation Privacy follows (intuitively) because: 1 5 9 2 1. No inputs of the honest parties are leaked. 45 3 3 2. No intermediate value is leaked. 48 144
