Remarks on Gödel’s code as a hash function

Petr Savick In this paper we analyze a simple hash function introduced in a popular book PopCo by Scarlett Thomas that is based on well known G¨odel’s numbering function. The numbering function is very efficient for practical use, however it is widely used in foundations of logic and computability theory. We show that the properties of the suggested hash function (computing the hash as a “shorter digest” of the long G¨odel’s number code) are not sufficient for cryptography. We introduce two ways how to construct meaningful collisions and in special cases also second-preimages. Further we propose a simple improvement of this hash function which prevents the simplest of the attacks, however it does not eliminate the other attacks.