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Identification Codes via Prime Numbers

Emad Zinoghli, Mohammad Javad Salariseddigh

TL;DR

This work addresses identification codes over noiseless and BSC channels by revisiting Ahlswede’s 3-step RI scheme and mitigating its computational bottleneck through probabilistic prime generation. It develops a polynomial-time prime-generation approach based on Miller–Rabin tests, links RI codes to almost-universal hash families, and introduces a generalized, hash-based 3-step construction that preserves zero Type I error while achieving doubly-exponential message sizes with controlled Type II error. The paper provides non-asymptotic bounds and asymptotic rate analyses, illuminating trade-offs between block length, error probability, and computation. Practically, the results enable scalable RI coding with improved encoder efficiency and open avenues for alternative number-theoretic key choices and deterministic primality tests, potentially broadening applicability in post-Shannon identification tasks.

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.

Identification Codes via Prime Numbers

TL;DR

This work addresses identification codes over noiseless and BSC channels by revisiting Ahlswede’s 3-step RI scheme and mitigating its computational bottleneck through probabilistic prime generation. It develops a polynomial-time prime-generation approach based on Miller–Rabin tests, links RI codes to almost-universal hash families, and introduces a generalized, hash-based 3-step construction that preserves zero Type I error while achieving doubly-exponential message sizes with controlled Type II error. The paper provides non-asymptotic bounds and asymptotic rate analyses, illuminating trade-offs between block length, error probability, and computation. Practically, the results enable scalable RI coding with improved encoder efficiency and open avenues for alternative number-theoretic key choices and deterministic primality tests, potentially broadening applicability in post-Shannon identification tasks.

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 first prime numbers for identification codes with block length 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.
Paper Structure (23 sections, 15 theorems, 49 equations, 4 figures, 3 algorithms)

This paper contains 23 sections, 15 theorems, 49 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notations
  3. Organization
  4. System Model and Preliminaries
  5. System Model
  6. Identification Coding for the Noiseless Binary Symmetric Channel
  7. Preliminaries on Number Theory and Hash Functions
  8. Non-Asymptotic Bounds for the Prime-Counting Function and the $n$-th Prime Number
  9. Prime Number Generation
  10. Uniform Prime Generation
  11. Miller-Rabin Analysis
  12. Previous Results - 3-Step Identification Scheme for the Noiseless BSC
  13. Main Results -Part I - Modified 3-Step Identification Scheme
  14. Rate Analysis
  15. Error Analysis
  16. ...and 8 more sections

Key Result

Theorem 1

Let $\pi(x)$ denote the prime-counting function for real $x > 0 \,.$ Then,

Figures (4)

  • Figure 1: Theoretical upper bound and empirical type II error probability corresponding to parameters $\alpha = 1.1$ and $R = 100$
  • Figure 2: Theoretical upper bound and empirical type II error probability corresponding to parameters $\alpha = 1.1$ and $R = 500$
  • Figure 3: Theoretical upper bound and empirical type II error probability corresponding to parameters $\alpha = 1.01$ and $R = 500$
  • Figure 4: Theoretical upper bound and empirical type II error probability corresponding to parameters $\alpha = 1.5$ and $R = 500$

Theorems & Definitions (33)

  • Definition 1: Identification Code for Noiseless Channels
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1: Apostol76
  • proof
  • Theorem 2: Apostol76
  • proof
  • Theorem 3: Apostol76
  • proof
  • ...and 23 more