Paper 2022/1052

Double-Odd Jacobi Quartic

Abstract

Double-odd curves are curves with order equal to 2 modulo 4. A prime order group with complete formulas and a canonical encoding/decoding process could previously be built over a double-odd curve. In this paper, we reformulate such curves as a specific case of the Jacobi quartic. This allows using slightly faster formulas for point operations, as well as defining a more efficient encoding format, so that decoding and encoding have the same cost as classic point compression (decoding is one square root, encoding is one inversion). We define the prime-order groups jq255e and jq255s as the application of that modified encoding to the do255e and do255s groups. We furthermore define an optimized signature mechanism on these groups, that offers shorter signatures (48 bytes instead of the usual 64 bytes, for 128-bit security) and makes signature verification faster (down to less than 83000 cycles on an Intel x86 Coffee Lake core).