Improving Algorithmic Efficiency using Cryptography

Cryptographic primitives have been used for various non-cryptographic objectives, such as eliminating or reducing randomness and interaction. We show how to use cryptography to improve the time complexity of solving computational problems. Specifically, we show that under standard cryptographic assumptions, we can design algorithms that are asymptotically faster than existing ones while maintaining correctness. As a concrete demonstration, we construct a distribution of trapdoored matrices with the following properties: (a) computationally bounded adversaries cannot distinguish a random matrix from one drawn from this distribution (under computational hardness assumptions), and (b) given a trapdoor, we can multiply such an $n \times n$ matrix with any vector in near-linear (in $n$) time. We provide constructions both over finite fields and over the reals. This enables a broad speedup technique: any algorithm relying on a random matrix -- such as those that use various notions of dimensionality reduction -- can replace it with a matrix from our distribution, achieving computational speedups while preserving correctness. Using these trapdoored matrices, we present the first uniform reduction from worst-case to approximate and average-case matrix multiplication with optimal parameters (improving on Hirahara--Shimizu STOC 2025, albeit under computational assumptions), the first worst-case to average-case reductions for matrix inversion, solving a linear system, and computing a determinant, as well as a speedup of inference time in classification models.