Identification Codes via Prime Numbers

TL;DR

This paper improves prime number-based identification codes by introducing probabilistic prime generation for efficiency and proposing a hash function-based generalization of Ahlswede's scheme.

Contribution

It presents a polynomial-time prime generation method and a generalized identification coding scheme using hash functions, enhancing previous prime-based methods.

Findings

Prime generation can be achieved in polynomial time using probabilistic algorithms.

Enhanced upper bounds for code performance are derived.

A generalized identification coding scheme based on hash functions is proposed.

Abstract

We introduce a method for construction of identification codes based on prime number generation over the noiseless channels. The earliest method for such construction based on prime numbers was proposed by Ahlswede which relies on algorithms for generation of prime numbers. This method requires knowledge of $2^n$ first prime numbers for identification codes with block length $n,$ which is not computationally efficient. In this work, we revisit Ahlswede's scheme and propose a number of modifications. In particular, employing probabilistic prime generation algorithm, we guarantee that the prime keys generation is possible in polynomial time. Furthermore, additional improvements in terms of type II upper bound are derived and presented. Finally, we propose a method for identification coding based on hash functions which generalizes the Ahlswede's scheme.