Mathematical foundations of modern cryptography: computational complexity perspective

TL;DR

This paper surveys the mathematical foundations of modern cryptography through the lens of computational complexity, highlighting how complexity theory underpins key cryptographic protocols and concepts.

Contribution

It provides a comprehensive overview of how computational complexity theory informs the development of cryptographic methods and protocols, emphasizing unifying paradigms and proof techniques.

Findings

Complexity-theoretic foundations underpin cryptographic tasks

Unified paradigms and proof techniques are central to modern cryptography

Complexity theory has integrated into mainstream cryptographic research

Abstract

Theoretical computer science has found fertile ground in many areas of mathematics. The approach has been to consider classical problems through the prism of computational complexity, where the number of basic computational steps taken to solve a problem is the crucial qualitative parameter. This new approach has led to a sequence of advances, in setting and solving new mathematical challenges as well as in harnessing discrete mathematics to the task of solving real-world problems. In this talk, I will survey the development of modern cryptography -- the mathematics behind secret communications and protocols -- in this light. I will describe the complexity theoretic foundations underlying the cryptographic tasks of encryption, pseudo-randomness number generators and functions, zero knowledge interactive proofs, and multi-party secure protocols. I will attempt to highlight the paradigms and proof techniques which unify these foundations, and which have made their way into the mainstream of complexity theory.