Generators for Certain Alternating Groups With Applications to Cryptography

Step-by-Step Coding Implementation of Coppersmith and Grossman's 1975 paper

Frontmatter for the 1975 paper, Generators for Certain Alternating Groups with Applications to Cryptography by Don Coppersmith and Edna Grossman

Quick intro

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This paper¹ features in both our Reversible Computing Series and Cryptography Series :

• Reversible Computing Series

  • Logical Reversibilty of Computation : This paper laid the foundation for Quantum, Thermodynamic and Catalytic computing.

  • Generators for Certain Alternating Groups With Applications to Cryptography(we are here) - Introduces the idea of symmetric groups for Catalytic computers and proves that all boolean functions can be transparently computed.

• Cryptography Series

  • Lehmer’s Continued Factorization Algorithm : This paper introduces a factorization algorithm using Continued Fractions.

  • Asymptotically Fast Factorization of Integers : This paper introduces Dixon’s algorithm for integer factorization.

  • Generators for Certain Alternating Groups With Applications to Cryptography(we are here) - Proves that simple binary permutations make great, cryptographically secure ciphers.

If you’re coding along then here’s the gist.

1.0 Introduction

Page 1 introduces symmetric groups

In group theory, the symmetric group, denoted by S_n, is the set of all permutations. The size* of the symmetric group is n!

*The word, ‘order’ means the size of the group while ‘!’ is the factorial.

For example, the symmetric group S₃, has an order of 3! = 6 and is the set of all permutations of {1,2,3}

All 6 permutations in S₃

In C, we can use simple recursion to generate the symmetric group of n.

*Note that 12! is the largest value an int takes.

Generating the Symmetric Group of n

The authors also introduce the concept of basic functions : subsets of the symmetric group of n whose members are part of the set of simple reversible transformations.