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Introduction ECDSA Algorithm A Naive Algorithm for ECDSA Batch Verification A New Batch-verification Algorithm for ECDSA (S1) A More Efficient Batch-verification Algorithm (S2) Efficient Variants of S1 and S2 Experimental Results Conclusion References Solving the Multivariate Equations A Strategy for Faster Equation Generation Retrieving the Unknown y-coordinates Analysis of Algorithm S1 A Strategy for Faster Equation Generation ◮ Let ρ 1 z 1 + ρ 2 z 2 + · · · + ρ µ z µ = γ (12) ◮ Let f (z 1 , z 2 , . . . , z µ ) be any F q -linear combination of the monomials z 1 , z 2 , . . . , z µ . ◮ (ρ 1 z 1 + ρ 2 z 2 + · · · + ρ µ z µ )f (z 1 , z 2 , . . . , z µ ) = γf (z 1 , z 2 , . . . , z µ ) and use Eqn (3). ◮ This again yields a linear equation in z 1 , z 2 , . . . , z µ . ◮ Choice f (z 1 , z 2 , . . . , z µ ) = z i with a small degree of z i typically leads to a faster generation full-rank system. ◮ Only R x suffices to generate a uniquely solvable linearized system. Sabyasachi Karati Batch Verification of ECDSA Signatures AfricaCrypt 2012 Ifrane, Morocco
Introduction ECDSA Algorithm A Naive Algorithm for ECDSA Batch Verification A New Batch-verification Algorithm for ECDSA (S1) A More Efficient Batch-verification Algorithm (S2) Efficient Variants of S1 and S2 Experimental Results Conclusion References Retrieving the Unknown y-coordinates Solving the Multivariate Equations A Strategy for Faster Equation Generation Retrieving the Unknown y-coordinates Analysis of Algorithm S1 ◮ The final step is the determination of the y-coordinates y i of the points R i . ◮ Solving the multivariate linear system for monomials. ◮ Use R y [Eqn (10)] to retrieve all the unknown y-coordinates. ◮ We accept all the signatures if and only x(R (1) ) = x(R (2) ) if yi 2 = ri 3 + ar i + b (mod q) for all i = 1, 2, . . . , t. Sabyasachi Karati Batch Verification of ECDSA Signatures AfricaCrypt 2012 Ifrane, Morocco
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