The 2-Adic CM Method for Genus 2 Curves with Application to Cryptography

Abstract

Abstract. The complex multiplication (CM) method for genus 2 is currently the most efficient way of generating genus 2 hyperelliptic curves defined over large prime fields and suitable for cryptography. Since low class number might be seen as a potential threat, it is of interest to push the method as far as possible. We have thus designed a new algorithm for the construction of CM invariants of genus 2 curves, using 2-adic lifting of an input curve over a small finite field. This provides a numerically stable alternative to the complex analytic method in the first phase of the CM method for genus 2. As an example we compute an irreducible factor of the Igusa class polynomial system for the quartic CM field Q(i Ô 75 + 12 √ 17), whose class number is 50. We also introduce a new representation to describe the CM curves: a set of polynomials in (j1, j2, j3) which vanish on the precise set of triples which are the Igusa invariants of curves whose Jacobians have CM by a prescribed field. The new representation provides a speedup in the second phase, which uses Mestre's algorithm to construct a genus 2 Jacobian of prime order over a large prime field for use in cryptography.

Isogeny graphs of ordinary abelian varieties

“…(1) Complex analytic techniques [26,[35][36][37] (2) p-adic lifting techniques [38,39] (3) A technique based on the Chinese Remainder Theorem [40][41][42] (the CRT method).…”

Modular Polynomials for Genus 2

“…For M ranging over the cosets S = Sp(4, Z)/Γ (2) 0 (p), the functions j i,p (M τ ) are distinct by Lemma 4.2. Inspired by the formulas in [16,Sec. 7.1] we note that, by Lagrange interpolation, the polynomials…”

Generating pairing-friendly parameters for the CM construction of genus 2 curves over prime fields

“…2. Compute the Igusa class polynomials H i (x), i = 1, 2, 3 of K. This step can be done using the methods as described in one of [22], [24], [5], [13]. 3.…”