Group-based Cryptography

  • Alexei Myasnikov
  • Alexander Ushakov
  • Vladimir Shpilrain

Part of the Advanced Courses in Mathematics - CRM Barcelona book series (ACMBIRK)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Background on Groups, Complexity, and Cryptography

  3. Non-commutative Cryptography

  4. Generic Complexity and Cryptanalysis

  5. Asymptotically Dominant Properties and Cryptanalysis

    1. Front Matter
      Pages 135-138
  6. Back Matter
    Pages 169-183

About this book


This book is about relations between three different areas of mathematics and theoretical computer science: combinatorial group theory, cryptography, and complexity theory. It is explored how non-commutative (infinite) groups, which are typically studied in combinatorial group theory, can be used in public key cryptography. It is also shown that there is a remarkable feedback from cryptography to combinatorial group theory because some of the problems motivated by cryptography appear to be new to group theory, and they open many interesting research avenues within group theory.

Then, complexity theory, notably generic-case complexity of algorithms, is employed for cryptanalysis of various cryptographic protocols based on infinite groups, and the ideas and machinery from the theory of generic-case complexity are used to study asymptotically dominant properties of some infinite groups that have been applied in public key cryptography so far.

Its elementary exposition makes the book accessible to graduate as well as undergraduate students in mathematics or computer science.


Finite Group theory average complexity cryptography mathematics

Authors and affiliations

  • Alexei Myasnikov
    • 1
  • Alexander Ushakov
    • 2
  • Vladimir Shpilrain
    • 3
  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  2. 2.Department of MathematicsStevens Institute of TechnologyHobokenUSA
  3. 3.Department of MathematicsThe City College of New YorkNew YorkUSA

Bibliographic information