244 Pages 53 B/W Illustrations
    by Chapman & Hall

    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.


    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



    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