Improving Algorithmic Efficiency using Cryptography

2025-02-18

AI Summary

This paper addresses the fundamental question of whether cryptography can reduce algorithmic time complexity. We propose the first framework that systematically leverages cryptographic trapdoor mechanisms to accelerate deterministic computation. Our core method constructs a provably secure trapdoor matrix distribution: under standard assumptions (LWE or DDH), it is computationally indistinguishable from a random matrix distribution, yet a holder of the trapdoor key can compute an $n imes n$ matrix–vector product in nearly linear time $ ilde{O}(n)$, breaking the classical $Omega(n^2)$ lower bound. The construction supports both finite fields and the real domain, integrating randomness elimination and efficient key derivation techniques. In canonical applications such as dimensionality reduction, it achieves asymptotic speedup while preserving exact correctness. Both the formal security guarantees and the asymptotic acceleration are rigorously proven.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Distributed Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

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, and (b) given a secret key, we can multiply such a n-by-n matrix with any vector in near-linear (in n) time. We provide constructions both over finite fields and the reals. This enables a broad speedup technique: any algorithm relying on a random matrix - such as those using various notions of dimensionality reduction - can replace it with a matrix from our distribution, achieving computational speedups while preserving correctness.

Problem

• Enhancing time complexity via cryptography
• Constructing trapdoored matrices for efficiency
• Achieving computational speedups with correctness

Innovation

• Cryptography enhances algorithmic efficiency
• Trapdoored matrices ensure computational speedups
• Speedup technique maintains algorithm correctness