Paper 2025/1121

On randomness complexity of 1-private protocols

Abstract

In the field of information-theoretic cryptography, randomness complexity is a key metric for protocols for private computation, that is, the number of random bits needed to realize the protocol. Although some general bounds are known, even for the relatively simple example of $1$-private computation of $n$-party AND, the exact complexity is unknown. We study two settings. First, we consider the model of Goyal, Ishai, and Song (Crypto '22) where helper parties without any inputs are allowed to assist in the computation. In this setting, we show that two random bits always suffice to compute an arbitrary Boolean circuit $C$ $1$-privately: a single designated inputless helper flips the two bits and privately distributes the derived one-time bits to the other helper parties and the input parties as they are needed. We give an explicit construction using seven helper parties per AND gate and three helper parties per XOR gate (plus the single global randomness dealer). Moreover, two random bits are necessary already for the AND functionality (by a reduction to the standard no-helper model together with the lower bound of Kushilevitz, Ostrovsky, Prouff, Ros\'en, Thillard and Vergnaud (TCC '19), and therefore the worst-case helper-party randomness complexity is exactly $2$ bits. Second, in the setting without helper parties, we improve the upper bound from Couteau and Ros\'en (Asiacrypt '22) on the (asymptotic) randomness complexity of $n$-party AND from $6$ to $5$ bits. That is, we give a $1$-private protocol for computing the AND of $n$ parties' inputs requiring $5$ bits of randomness, for all $n\ge 6$. Our construction, like that of Couteau and Ros\'en, uses a single party to flip the $5$ bits and distribute the required derived values during the execution. Our approach to both problems is built around a more systematic exploration of techniques for recycling randomness across sub-computations. As part of resolving the second problem, we isolate an exact local-independence combinatorial object called a Sliding-Window Independence Generator, or a SWIG. A $(k,m)$-SWIG is a linear generator from a $k$-bit seed to $m\geq k$ output bits, where every cyclic length-$k$ sliding window chosen from $m$ output bits is perfectly uniform. We give an explicit $(k,m)$-SWIG for every $k\ge1$ and every $m\geq k$ and use a $(5,n-1)$-SWIG in our no-helper AND protocol.