Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Public-key encryption from a trapdoor one-way embedding of $SL_2(\mathbb{N}$)

Robert Hines

TL;DR

The paper proposes a public-key encryption scheme built from a trapdoor one-way embedding of words into $SL_2(\mathbb{N})$, mapping $\lambda$-bit strings to words and obfuscating them via conjugation and higher-dimensional block embeddings. It formalizes a concrete instantiation using generators $L=(1011)$ and $R=(1101)$, randomized generators $G_0,G_1$, and a secret $S$ in $GL_{2n}(\mathbb{Z}/m\mathbb{Z})$, with public keys $P_0,S^{-1}\widetilde{G_0}S$ and $P_1,S^{-1}\widetilde{G_1}S$, and plaintexts encrypted by multiplying public-key matrices. Decryption inverts the embedding by conjugating back with $S$, then reduces to a factorization task in $SL_2^{(\lambda)}(\mathbb{N})$ to recover the message; the paper provides a reference Python implementation and discusses parameter regimes, practical timings, and significant open questions on security. Critical observations include explicit security gaps (notably potential trace leakage and lack of randomness), a discussion of parameter trade-offs (dimension $n$, word-length $l$, and modulus $m$), and proposed avenues for improvement and further cryptanalysis, especially for reliable IND-CPA or KEM deployment.

Abstract

We obfuscate words of a given length in a free monoid on two generators with a simple factorization algorithm (namely $SL_2(\mathbb{N})$) to create a public-key encryption scheme. We provide a reference implementation in Python and suggested parameters. The security analysis is between weak and non-existent, left to future work.

Public-key encryption from a trapdoor one-way embedding of $SL_2(\mathbb{N}$)

TL;DR

The paper proposes a public-key encryption scheme built from a trapdoor one-way embedding of words into , mapping -bit strings to words and obfuscating them via conjugation and higher-dimensional block embeddings. It formalizes a concrete instantiation using generators and , randomized generators , and a secret in , with public keys and , and plaintexts encrypted by multiplying public-key matrices. Decryption inverts the embedding by conjugating back with , then reduces to a factorization task in to recover the message; the paper provides a reference Python implementation and discusses parameter regimes, practical timings, and significant open questions on security. Critical observations include explicit security gaps (notably potential trace leakage and lack of randomness), a discussion of parameter trade-offs (dimension , word-length , and modulus ), and proposed avenues for improvement and further cryptanalysis, especially for reliable IND-CPA or KEM deployment.

Abstract

We obfuscate words of a given length in a free monoid on two generators with a simple factorization algorithm (namely ) to create a public-key encryption scheme. We provide a reference implementation in Python and suggested parameters. The security analysis is between weak and non-existent, left to future work.
Paper Structure (22 sections, 17 equations, 3 figures)

This paper contains 22 sections, 17 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Prior art
  3. The medium: SL_2(\\ mathbb{N})
  4. The trapdoor: modular reduction, conjugation, increasing dimension, and random generators.
  5. First attempt (conjugate)
  6. Strike two (increase the dimension)
  7. Third time's a charm (use random generators)
  8. Four (all of the above)?
  9. Public key encryption scheme
  10. Key generation
  11. Encryption
  12. Decryption
  13. Security and parameter selection
  14. Attacking conjugation
  15. Attacking the secret key S
  16. ...and 7 more sections

Figures (3)

  • Figure 1: Trace distributions for words of length 22, 23, and 24. Histograms with 1000 bins of equal size.
  • Figure 2: $L^{\infty}$ distributions for words of length 22, 23, and 24. Histograms with 1000 bins of equal size.
  • Figure 3: Joint distributions for words of length 22, 23, and 24. Histgrams with 1000 bins in each dimension.