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On the additive index of the Diffie-Hellman mapping and the discrete logarithm

Pierre-Yves Bienvenu, Arne Winterhof

TL;DR

This work introduces and analyzes the additive index as a complexity measure for cryptographic maps over finite fields, focusing on the univariate Diffie-Hellman map $d_\gamma$ and the discrete-logarithm mapping $P$. By representing near-mappings with linearised polynomials and exploiting Weil bounds, square-counting results, and subgroup-sum estimates, the authors derive substantial lower bounds on the additive index across multiple regimes, including the principal case $T=q-1$ and the presence of up to $m$ discrepancies. They show that, under broad arithmetic conditions, the additive index grows quasi-polynomially in the field size (e.g., at least $C_ε q^{1-ε}/p$ for $d_\gamma$ and at least $q/(n+2)$ for $P$), and in some cases achieves the maximum $q$, highlighting the strong additive-structural complexity of these mappings. The results extend to small $T$ regimes (e.g., $T > q^ε$) and provide a spectrum of bounds that reinforce the security-relevant unpredictability of the additive-index measure for key cryptographic mappings in finite fields.

Abstract

Several complexity measures such as degree, sparsity and multiplicative index for cryptographic functions including the Diffie-Hellman mapping and the discrete logarithm in a finite field have been studied in the literature. In 2022, Reis and Wang introduced another complexity measure, the additive index, of a self-mapping of a finite field. In this paper, under certain conditions, we determine lower bounds on the additive index of the univariate Diffie-Hellman mapping and a self-mapping of $\mathbb{F}_q$ which can be identified with the discrete logarithm in a finite field.

On the additive index of the Diffie-Hellman mapping and the discrete logarithm

TL;DR

This work introduces and analyzes the additive index as a complexity measure for cryptographic maps over finite fields, focusing on the univariate Diffie-Hellman map and the discrete-logarithm mapping . By representing near-mappings with linearised polynomials and exploiting Weil bounds, square-counting results, and subgroup-sum estimates, the authors derive substantial lower bounds on the additive index across multiple regimes, including the principal case and the presence of up to discrepancies. They show that, under broad arithmetic conditions, the additive index grows quasi-polynomially in the field size (e.g., at least for and at least for ), and in some cases achieves the maximum , highlighting the strong additive-structural complexity of these mappings. The results extend to small regimes (e.g., ) and provide a spectrum of bounds that reinforce the security-relevant unpredictability of the additive-index measure for key cryptographic mappings in finite fields.

Abstract

Several complexity measures such as degree, sparsity and multiplicative index for cryptographic functions including the Diffie-Hellman mapping and the discrete logarithm in a finite field have been studied in the literature. In 2022, Reis and Wang introduced another complexity measure, the additive index, of a self-mapping of a finite field. In this paper, under certain conditions, we determine lower bounds on the additive index of the univariate Diffie-Hellman mapping and a self-mapping of which can be identified with the discrete logarithm in a finite field.
Paper Structure (14 sections, 34 theorems, 94 equations)

This paper contains 14 sections, 34 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Diffie-Hellman mapping and discrete logarithm
  3. The additive index
  4. Statement of some results and structure of the paper
  5. Preliminaries
  6. The Weil bound
  7. Number of solutions to the equation $x^2\equiv j\bmod T$
  8. Number of squares modulo $T$
  9. Number of $\xi\in G$ with $\xi-\xi^{-1}\in G$
  10. Sums of elements of a subgroup
  11. The Diffie-Hellman mapping
  12. Discrete logarithm
  13. Some results on the mapping $x\mapsto \xi_x$
  14. Bounds for the additive index of the discrete logarithm

Key Result

Theorem 1

Suppose $p>3$ or $p=3$ and $n$ is even. Assume $\gamma$ is a generator of ${\mathbb F}_q^*$, that is $T=q-1$. Then for every $\varepsilon >0$, there exists a positive constant $C_\varepsilon$ such that the additive index of $d_\gamma$ is at least $C_\varepsilon \frac{q^{1-\varepsilon}}{p}$.

Theorems & Definitions (68)

  • Remark
  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Corollary 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • Corollary 2
  • ...and 58 more