Unified formulas for some deterministic almost-injective encodings into hyperelliptic curves

Abstract

Recently, efficient deterministic and invertible encodings on some hyperelliptic curves in genus 1 and 2 using the technique in Elligator 2 (ACM CCS 2013) have been proposed. We have successfully generalized their encodings for hyperelliptic curves of genus 3, 4 and 5. We have found unified formulas (using Mersenne numbers) for the encodings into the hyperelliptic curves of genus g ≤ 5: ℍg: y2 = fg(x) = x(2g+1) + a(2g1)x(2g1) + a(2g3)x(2g3) +… + a1x + a0We have conjectured that our method works on arbitrary genus.