Low-Latency Boolean Functions and Bijective S-boxes

Abstract

In this paper, we study the gate depth complexity of (vectorial) Boolean functions in the basis of {NAND, NOR, INV} as a new metric, called latency complexity, to mathematically measure the latency of Boolean functions. We present efficient algorithms to find all Boolean functions with low-latency complexity, or to determine the latency complexity of the (vectorial) Boolean functions, and to find all the circuits with the minimum latency complexity for a given Boolean function. Then, we present another algorithm to build bijective S-boxes with low-latency complexity which with respect to the computation cost, this algorithm overcomes the previous methods of building S-boxes.
As a result, for latency complexity 3, we present n-bit S-boxes of 3 ≤ n ≤ 8 with linearity 2ⁿ⁻¹ and uniformity 2ⁿ⁻² (except for 5-bit S-boxes for whose the minimum achievable uniformity is 6). Besides, for latency complexity 4, we present several n-bit S-boxes of 5 ≤ n < 8 with linearity 2ⁿ⁻² and uniformity 2ⁿ⁻⁴.