Group Theoretic Cryptography: 1st Edition (Hardback) book cover

Group Theoretic Cryptography

1st Edition

By Maria Isabel González Vasco, Rainer Steinwandt

Chapman and Hall/CRC

244 pages | 53 B/W Illus.

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Hardback: 9781584888369
pub: 2015-04-01
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Group theoretic problems have propelled scientific achievements across a wide range of fields, including mathematics, physics, chemistry, and the life sciences. Many cryptographic constructions exploit the computational hardness of group theoretical problems, and the area is viewed as a potential source of quantum-resilient cryptographic primitives for the future.

Group Theoretic Cryptography supplies an ideal introduction to cryptography for those who are interested in group theory and want to learn about the possible interplays between the two fields. Assuming an undergraduate-level understanding of linear algebra and discrete mathematics, it details the specifics of using non-Abelian groups in the field of cryptography.

Moreover, the book evidences how group theoretic techniques help us gain new insight into well known, seemingly unrelated, cryptographic constructions, such as DES.

The book starts with brief overviews of the fundamentals of group theory, complexity theory, and cryptography. Part two is devoted to public-key encryption, including provable security guarantees, public-key encryption in the standard model, and public-key encryption using infinite groups.

The third part of the book covers secret-key encryption. It examines block ciphers, like the Advanced Encryption Standard, and cryptographic hash functions and message authentication codes. The last part delves into a number of cryptographic applications which are nowadays as relevant as encryption—identification protocols, key establishment, and signature schemes are covered.

The book supplies formal security analyses and highlights potential vulnerabilities for cryptographic constructions involving group theory. Summaries and references for further reading, as well as exercises, are included at the end of each chapter. Selected solutions for exercises are provided in the back of the book.


"Group Theoretic Cryptography is highly welcome. It provides an excellent introduction in group-based cryptography where algebraic properties of the platform groups, mainly from combinatorial group theory, are used prominently in both devising cryptosystems and in cryptanalysis. In particular the difficulty, in a complexity sense, of certain algorithmic problems in finitely presented groups has been crucial in encryption and decryption. … I highly recommend the book under review. It is of great value for researchers in the area as well as for advanced students which start to work in cryptology. The excellent figures and algorithmic descriptions are clear and good to understand. They help the readers to see the important points.

—Gerhard Rosenberger (Hamburg), writing in Zentralblatt MATH 1321 – 1

Table of Contents


Mathematical background

Algebraic structures in a nutshell

Finite groups

Summary and further reading


Basics on complexity

Complexity classes

Asymptotic notation and examples

Summary and further reading


Cryptology: An introduction

A short historical overview

Historical encryption schemes

Public-key cryptography

Modern cryptology

Summary and further reading



Provable security guarantees

Public-key encryption revisited

Characterizing secure public-key encryption

One-way functions and random oracles

The general Bellare-Rogaway construction

IND-CCA security with an Abelian group: RSA-OAEP

One-way functions from non-Abelian groups?

Summary and further reading


Public-key encryption in the standard model

The Crame-Shoup encryption scheme from 1998

Going beyond: Tools

Projective hash families

Subset membership problems

Hash proof systems

General Cramer-Shoup encryption scheme

A concrete instantiation

Projective hash families from (non-Abelian) groups

Group action systems

Group action projective hash families

Summary and further reading


Public-key encryption using infinite groups

The word problem in finitely presented groups

The encryption scheme of Wagner and Magyarik

Polly Cracker

A successor of the Wagner-Magyarik scheme

Using a group that is not finitely presentable?

Braid groups in cryptography

Basics on braid groups

Some computational problems in the braid group Bn

Summary and further reading



Block ciphers

Advanced Encryption Standard

Specifying the round function

Key schedule

Encryption and decryption with AES

Data Encryption Standard

General structure of DES: A Feistel cipher

Round function of DES

Key schedule

Permutation Group Mappings

Modes of operation

Electronic codebook (ECB) mode

Cipher block chaining (CBC) mode

Cipher feedback (CFB) mode

Output feedback (OFB) mode

Counter (CTR) mode

Summary and further reading


Cryptographic hash functions and message authentication codes

Cryptographic hash functions

Deriving a hash function from a block cipher

Cayley hash functions

Message authentication codes

Keyed-Hash Message Authentication Code

Cipher-based Message Authentication Code

Summary and further reading



Key establishment protocols

Setting the stage

Provable security for key exchange protocols

A secure construction

Anshel-Anshel-Goldfeld key exchange

Braid-based key exchange

Constructions over matrix groups

Summary and further reading


Signature and identification schemes

Definitions and terminology

RSA signatures: FDH and PSS

Identification schemes

Summary and further reading



Solutions to selected exercises

Solutions to selected exercises of Part I

Solutions to selected exercises of Part II

Solutions to selected exercises of Part III

Solutions to selected exercises of Part IV



About the Series

Chapman & Hall/CRC Cryptography and Network Security Series

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Subject Categories

BISAC Subject Codes/Headings:
COMPUTERS / Operating Systems / General
COMPUTERS / Security / General
MATHEMATICS / Combinatorics