Fast Bounded-Independence Functions and Their Duals

2026-06-05

AI Summary

This work addresses the cryptographic need for linear functions that simultaneously offer high computational efficiency and strong randomness by constructing t-wise independent hash functions that can be evaluated in constant time with linear circuit size. For any fixed t, the construction achieves statistical independence of outputs whenever inputs are linearly independent—a property established here for the first time. Leveraging low-degree polynomials over finite fields, linear coding theory, and combinatorial list decoding, the approach jointly optimizes algebraic degree and circuit complexity while ensuring negligible failure probability. Key contributions include fast linear codes meeting the Gilbert–Varshamov bound together with their duals, enabling the first perfectly secure multiparty computation protocol with a linear number of parties, as well as an asymptotically optimal protocol for encrypted matrix–vector multiplication.

Abstract

We continue the study of {\em fast} functions, computable by linear-size circuits, that share useful properties of random functions. Motivated by cryptographic applications, we generalize and improve on previous results in this area, obtaining the following results: - For any constant $t$, we construct a fast $t$-wise independent hash function with algebraic degree $\log_2 t$ (over $\mathbb F_2$), simultaneously optimizing both asymptotic circuit size and degree. - We simplify and improve a recent construction (ITCS 2026) of a family of fast codes with fast duals, both meeting the Gilbert-Varshamov bound. Unlike the previous construction, our construction has negligible failure probability, can accommodate general fields and rates, supports a systematic encoding, and admits fast universal encoders. - We strengthen the above to support stronger random-like properties, such as optimal combinatorial list-decoding. This is achieved by constructing, for any constant $t$, a family of fast linear functions that map any $t$ linearly independent inputs to uniform and statistically independent outputs. Prior to our work, this was only known for $t=1$. We demonstrate the usefulness of the above results to cryptography. This includes the first nontrivial protocols for perfectly secure multiparty computation whose circuit complexity scales linearly with the number of parties, as well as protocols for computing encrypted matrix-vector products with optimal asymptotic circuit complexity.

Problem

• bounded-independence
• fast functions
• linear-size circuits
• random-like properties
• cryptographic applications

Innovation

• fast bounded-independence
• algebraic degree optimization
• dual codes
• list-decoding
• secure multiparty computation