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Hard Instances of Discrete Logarithm Problem and Cryptographic Applications

Christopher Battarbee, Arman Darbinyan, Delaram Kahrobaei

TL;DR

The paper addresses the classical Discrete Logarithm Problem (DLP) by moving to infinite yet finitely generated groups and showing that DLP hardness can be made as hard as evaluating any computable function $f$. It constructs a countable abelian group $A_f$ in which the word problem runs in time related to $T_{f^{-1}}$ and the DLP is polynomially equivalent to computing $f$, then embeds $A_f$ into a two-generator group $G_f$ so that DLP in $G_f$ mirrors $f$ while preserving polynomial-time word problem. A corollary yields two-generated groups with polynomial-time word problem and NP-hard DLP, and the paper also proposes a generic cryptographic protocol within this group-theoretic framework. While quantum resistance is not resolved, the work opens a new avenue for controllable hardness in cryptographic constructions and invites further exploration of post-quantum properties and concrete instantiations in finitely generated infinite groups.

Abstract

Let f be an arbitrary positive integer valued function. The goal of this note is to show that one can construct a finitely generated group in which the discrete log problem is polynomially equivalent to computing the function f. In particular, we provide infinite, but finitely generated groups, in which the discrete logarithm problem is arbitrarily hard. As another application, we construct a family of two-generated groups that have polynomial time word problem and NP-complete discrete log problem. Additionally, using our framework, we propose a generic scheme of cryptographic protocols, which might be of independent interest.

Hard Instances of Discrete Logarithm Problem and Cryptographic Applications

TL;DR

The paper addresses the classical Discrete Logarithm Problem (DLP) by moving to infinite yet finitely generated groups and showing that DLP hardness can be made as hard as evaluating any computable function . It constructs a countable abelian group in which the word problem runs in time related to and the DLP is polynomially equivalent to computing , then embeds into a two-generator group so that DLP in mirrors while preserving polynomial-time word problem. A corollary yields two-generated groups with polynomial-time word problem and NP-hard DLP, and the paper also proposes a generic cryptographic protocol within this group-theoretic framework. While quantum resistance is not resolved, the work opens a new avenue for controllable hardness in cryptographic constructions and invites further exploration of post-quantum properties and concrete instantiations in finitely generated infinite groups.

Abstract

Let f be an arbitrary positive integer valued function. The goal of this note is to show that one can construct a finitely generated group in which the discrete log problem is polynomially equivalent to computing the function f. In particular, we provide infinite, but finitely generated groups, in which the discrete logarithm problem is arbitrarily hard. As another application, we construct a family of two-generated groups that have polynomial time word problem and NP-complete discrete log problem. Additionally, using our framework, we propose a generic scheme of cryptographic protocols, which might be of independent interest.

Paper Structure

This paper contains 15 sections, 11 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Related work.
  3. The main premise behind the group theoretical constructions in our paper
  4. Quantum resistance.
  5. Enumerated abelian groups and algorithmic problems on them
  6. The Discrete Log problem in countable abelian groups
  7. Finitely generated groups with controllable Discrete Log Problem via group embeddings
  8. The general idea
  9. An effective embedding of $A_f$ into $G_f$.
  10. The Discrete Log Problem in $G_f$
  11. A new cryptographic protocol
  12. A cryptographic key sharing scheme
  13. Adaptation in the setting of our group theoretical construction:
  14. Possible candidates for $f, g: \mathop{\mathrm{\mathbb{N}}}\nolimits \times \mathop{\mathrm{\mathbb{N}}}\nolimits \rightarrow \mathop{\mathrm{\mathbb{N}}}\nolimits$
  15. Conclusions

Key Result

Theorem 1.1

For any computable function $f: \mathop{\mathrm{\mathbb{N}}}\nolimits \rightarrow \mathop{\mathrm{\mathbb{N}}}\nolimits$, there exists a two-generator group $G_f$ such that the discrete log problem in $G_f$can be solved in time $T_f$ and the word problem can be solved in time $T_{f^{-1}},$ both up t

Theorems & Definitions (23)

  • Theorem 1.1: See Section \ref{['section-the-embedding']}
  • Theorem 1.2: Corollary \ref{['cor-np-complete-discrete-log']}
  • Definition 2.1: Reduced word
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 3.1
  • ...and 13 more