Low-Latency ASIC Algorithms of Modular Squaring of Large Integers for VDF Evaluation

This article is an attempt in quest of the fastest hardware algorithms for the computation of the evaluation component of verifiable delay functions (VDFs),
$a^{2^T} \bmod N$
, proposed for use in various distributed protocols, in which no party is assumed to compute it significantly faster than other participants. To this end, we propose a class of modular squaring algorithms suitable for low-latency ASIC implementations. The proposed algorithms aim to achieve highest levels of parallelization that have not been explored in previous works in the literature, which usually pursue more balanced optimization of speed and area. For this, we utilize redundant representations of integers and introduce three modular squaring algorithms that work with integers in redundant forms: i) Montgomery algorithm, ii) memory-based algorithm and iii) direct reduction algorithm for fixed moduli. All algorithms enable
$O(\log k)$
depth circuit implementations, where
$k$
is the bit-size of the modulus
$N$
in the VDF function. We analyze and compare gate level-circuits of the proposed algorithms and provide estimates for their critical path delay and gate count.