A Fast Large-Integer Extended GCD Algorithm and Hardware Design for Verifiable Delay Functions and Modular Inversion

Abstract

The extended GCD (XGCD) calculation, which computes Bézout coefficients b_a, b_b such that b_a ∗ a₀ + b_b ∗ b₀ = GCD(a₀, b₀), is a critical operation in many cryptographic applications. In particular, large-integer XGCD is computationally dominant for two applications of increasing interest: verifiable delay functions that square binary quadratic forms within a class group and constant-time modular inversion for elliptic curve cryptography. Most prior work has focused on fast software implementations. The few works investigating hardware acceleration build on variants of Euclid’s division-based algorithm, following the approach used in optimized software. We show that adopting variants of Stein’s subtraction-based algorithm instead leads to significantly faster hardware. We quantify this advantage by performing a large-integer XGCD accelerator design space exploration comparing Euclid- and Stein-based algorithms for various application requirements. This exploration leads us to an XGCD hardware accelerator that is flexible and efficient, supports fast average and constant-time evaluation, and is easily extensible for polynomial GCD. Our 16nm ASIC design calculates 1024-bit XGCD in 294ns (8x faster than the state-of-the-art ASIC) and constant-time 255-bit XGCD for inverses in the field of integers modulo the prime 2²⁵⁵−19 in 85ns (31× faster than state-of-the-art software). We believe our design is the first high-performance ASIC for the XGCD computation that is also capable of constant-time evaluation. Our work is publicly available at https://github.com/kavyasreedhar/sreedhar-xgcd-hardware-ches2022.