Exploring the Potential of Threshold Logic for Cryptography-Related Operations

Abstract

Motivated by the emerging interest in new VLSI processes and technologies, such as Resonant Tunneling Diodes (RTDs), Single-Electron Tunneling (SET), Quantum Cellular Automata (QCA), and Tunneling Phase Logic (TPL), this paper explores the application of the non-Boolean computational paradigms enabled by such new technologies. In particular, we consider Threshold Logic functions, directly implementable as primitive gates in the above-mentioned technologies, and study their application to the domain of cryptographic computing. From a theoretical perspective, we present a study on the computational power of linear threshold functions related to modular reduction and multiplication, the central operations in many cryptosystems such as RSA and Elliptic Curve Cryptography. We establish an optimal bound to the delay of a threshold logic circuit implementing Montgomery modular reduction and multiplication. In particular, we show that fixed-modulus Montgomery reduction can be implemented as a polynomial-size depth-2 threshold circuit, while Montgomery multiplication can be implemented as a depth-3 circuit. We also propose an architecture for Montgomery modular reduction and multiplication, which ensures feasible O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) area requirements, preserving the properties of constant latency and a low architectural critical path independent of the input size n. We compare this result with existing polynomial-size solutions based on the Boolean computational model, showing that the presented approach has intrinsically better architectural delay and latency, both O(1).