Paper 2010/001

Transfinite Cryptography

Jacques Patarin

Abstract

\begin{abstract} Let assume that Alice, Bob, and Charlie, the three classical people of cryptography are not limited anymore to perform a finite number of computations on real computers, but are limited to $\alpha$ computations and to $\alpha$ bits of memory, where $\alpha$ is a fixed infinite cardinal. For example $\alpha = \aleph _0$ (the countable cardinal, i.e. the cardinal of $\mathbb {N}$ the set of integers), or $\alpha = \mathfrak {C}$ (the cardinal of the set $\mathbb {R}$ of real numbers). Is it possible to do secret key cryptography? Public key cryptography? Encryption? Authentication? Signatures? Is it possible to generalize the notion of one way function? The aim of this paper is to give some elements of answers to these questions. We will see for example that for secret key cryptography there are some simple solutions. However for public key cryptography the results are much less clear. \end{abstract}