Index Calculus Method Small char Finite Fields 9234 bits Impact on Pairings Discrete logarithms Definition

Given a cyclic group (G, ·) of order m and a generator α ∈ G, the Discrete Logarithm Problem (DLP) asks, given β ∈ G, to find x ∈ Zm such that β = αx. Notation: logα β: = x. Commonly used groups: • The multiplicative group of a finite field Fq. • The group over an elliptic curve over Fq. • The Jacobian over a hyperelliptic curve over Fq. L-Notation for running time: Lm(α, c): = exp (c + o(1)) (lnm)α (ln lnm)1−α for some α ∈ [0, 1] and c> 0. Index Calculus Method Small char Finite Fields 9234 bits Impact on Pairings Finite field DLP milestones (larger field and/or improved complexity) bitlength char who/when running tim