Group Theoretic Cryptography

Group Theoretic Cryptography

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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 more

Product details

  • Electronic book text | 244 pages
  • Taylor & Francis Ltd
  • Chapman & Hall/CRC
  • London, United Kingdom
  • 53 Illustrations, black and white
  • 1584888377
  • 9781584888376

Review quote

"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 - 1show more

Table of contents

PRELIMINARIES Mathematical background Algebraic structures in a nutshellFinite groupsSummary and further readingExercisesBasics on complexity Complexity classesAsymptotic notation and examplesSummary and further readingExercisesCryptology: An introductionA short historical overview Historical encryption schemes Public-key cryptographyModern cryptologySummary and further readingExercisesPUBLIC-KEY ENCRYPTIONProvable security guarantees Public-key encryption revisitedCharacterizing secure public-key encryptionOne-way functions and random oraclesThe general Bellare-Rogaway constructionIND-CCA security with an Abelian group: RSA-OAEPOne-way functions from non-Abelian groups?Summary and further readingExercisesPublic-key encryption in the standard modelThe Crame-Shoup encryption scheme from 1998Going beyond: Tools Projective hash families Subset membership problems Hash proof systemsGeneral Cramer-Shoup encryption schemeA concrete instantiationProjective hash families from (non-Abelian) groups Group action systems Group action projective hash familiesSummary and further readingExercisesPublic-key encryption using infinite groupsThe word problem in finitely presented groups The encryption scheme of Wagner and Magyarik Polly Cracker A successor of the Wagner-Magyarik schemeUsing a group that is not finitely presentable?Braid groups in cryptography Basics on braid groups Some computational problems in the braid group BnSummary and further readingExercisesIII SECRET-KEY ENCRYPTION Block ciphersAdvanced Encryption Standard Specifying the round function Key schedule Encryption and decryption with AESData Encryption Standard General structure of DES: A Feistel cipher Round function of DES Key schedulePermutation Group MappingsModes 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 readingExercisesCryptographic hash functions and message authentication codes Cryptographic hash functionsDeriving a hash function from a block cipherCayley hash functionsMessage authentication codes Keyed-Hash Message Authentication Code Cipher-based Message Authentication CodeSummary and further readingExercisesOTHER CRYPTOGRAPHIC CONSTRUCTIONSKey establishment protocols Setting the stage Provable security for key exchange protocols A secure constructionAnshel-Anshel-Goldfeld key exchangeBraid-based key exchangeConstructions over matrix groupsSummary and further readingExercisesSignature and identification schemes Definitions and terminologyRSA signatures: FDH and PSSIdentification schemesSummary and further readingExercisesAPPENDIX Solutions to selected exercises Solutions to selected exercises of Part ISolutions to selected exercises of Part IISolutions to selected exercises of Part IIISolutions to selected exercises of Part IVReferences Indexshow more