[Resource Topic] 2021/1292: A Fast Large-Integer Extended GCD Algorithm and Hardware Design for Verifiable Delay Functions and Modular Inversion

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

Authors: Kavya Sreedhar, Mark Horowitz, and Christopher Torng


The extended GCD (XGCD) calculation, which computes B'ezout coefficients b_a, b_b such that b_a * a_0 + b_b * b_0 = GCD(a_0, b_0), 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 2^{255} - 19 in 85ns (32X 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 GitHub - kavyasreedhar/sreedhar-xgcd-hardware-ches2022: Artifact associated with CHES 2022 paper https://eprint.iacr.org/2021/1292.

ePrint: https://eprint.iacr.org/2021/1292

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