1st Edition

# Group Theoretic Cryptography

244 Pages 53 B/W Illustrations
by Chapman & Hall

244 Pages
by Chapman & Hall

Also available as eBook on:

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.

PRELIMINARIES

Mathematical background
Algebraic structures in a nutshell
Finite groups
Exercises

Basics on complexity
Complexity classes
Asymptotic notation and examples
Exercises

Cryptology: An introduction

A short historical overview
Historical encryption schemes
Public-key cryptography
Modern cryptology
Exercises

PUBLIC-KEY ENCRYPTION

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?
Exercises

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
Exercises

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
Exercises

III SECRET-KEY ENCRYPTION

Block ciphers
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
Exercises

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
Exercises

OTHER CRYPTOGRAPHIC CONSTRUCTIONS

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
Exercises

Signature and identification schemes
Definitions and terminology
RSA signatures: FDH and PSS
Identification schemes
Exercises

APPENDIX

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

References
Index

### Biography

Maria Isabel Gonzalez Vasco, Rainer Steinwandt

"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