Groups St Andrews 2009 in Bath: Group theory in cryptography

This paper is a guide for the pure mathematician who would like to know more about cryptography based on group theory. The paper gives a brief overview of the subject, and provides pointers to good textbooks, key research papers and recent survey papers in the area.

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[18]  Norman Biggs.  The Critical Group from a Cryptographic Perspective , 2007 .

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[21]  Rüdiger Sparr,et al.  Group theoretic properties of Rijndael-like ciphers , 2008, Discret. Appl. Math..

[22]  Ralph Wernsdorf,et al.  Markov Ciphers and Alternating Groups , 1994, EUROCRYPT.

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[24]  Ludovic Perret,et al.  Security analysis of word problem-based cryptosystems , 2010, Des. Codes Cryptogr..

[25]  Ralph Wernsdorf,et al.  The Round Functions of RIJNDAEL Generate the Alternating Group , 2002, FSE.

[26]  Alfred Menezes,et al.  Algebraic curves and cryptography , 2005, Finite Fields Their Appl..

[27]  Markus Grassl,et al.  Cryptanalysis of the Tillich–Zémor Hash Function , 2010, Journal of Cryptology.

[28]  A. Myasnikov,et al.  New Developments in Commutator Key Exchange , 2008 .

[29]  Gilles Zémor,et al.  Hashing with SL_2 , 1994, CRYPTO.

[30]  Ludovic Perret,et al.  On the Wagner-Magyarik Cryptosystem , 2005, WCC.

[31]  Thomas Beth,et al.  Weaknesses in the SL 2 (IF s 2 ) Hashing Scheme. , 2000, CRYPTO 2000.

[32]  Vladimir Shpilrain.  Cryptanalysis of Stickel's Key Exchange Scheme , 2008, CSR.

[33]  L. Washington.  Elliptic Curves: Number Theory and Cryptography , 2003 .

[34]  Jung Hee Cheon,et al.  New Public-Key Cryptosystem Using Braid Groups , 2000, CRYPTO.

[35]  Gilles Zémor,et al.  Collisions for the LPS Expander Graph Hash Function , 2008, EUROCRYPT.

[36]  Jung Hee Cheon,et al.  A Polynomial Time Algorithm for the Braid Diffie-Hellman Conjugacy Problem , 2003, CRYPTO.

[37]  Ralph Wernsdorf,et al.  The One-Round Functions of the DES Generate the Alternating Group , 1992, EUROCRYPT.

[38]  Douglas R. Stinson.  Cryptography: Theory and Practice, Third Edition , 2005 .

[39]  Spyros S. Magliveras,et al.  Symmetric Block Ciphers Based on Group Bases , 2000, Selected Areas in Cryptography.

[40]  Adi Shamir,et al.  Length-based cryptanalysis: the case of Thompson's group , 2007, J. Math. Cryptol..

[41]  David Garber,et al.  Probabilistic Solutions of Equations in the Braid Group , 2005, Adv. Appl. Math..

[42]  Rainer Steinwandt,et al.  A Reaction Attack on a Public Key Cryptosystem Based on the Word Problem , 2003, Applicable Algebra in Engineering, Communication and Computing.

[43]  Nasir D. Memon,et al.  Algebraic properties of cryptosystem PGM , 1992, Journal of Cryptology.

[44]  Claude E. Shannon,et al.  Communication theory of secrecy systems , 1949, Bell Syst. Tech. J..

[45]  Sangjin Lee,et al.  Potential Weaknesses of the Commutator Key Agreement Protocol Based on Braid Groups , 2002, EUROCRYPT.

[46]  N. Koblitz.  The Uneasy Relationship Between Mathematics and Cryptography , 2007 .

[47]  Dennis Hofheinz,et al.  A Practical Attack on Some Braid Group Based Cryptographic Primitives , 2003, Public Key Cryptography.

[48]  Simon R. Blackburn.  Cryptanalysing the critical group: efficiently solving Biggs's discrete logarithm problem , 2009, J. Math. Cryptol..

[49]  Jung Hee Cheon,et al.  An Efficient Implementation of Braid Groups , 2001, ASIACRYPT.

[50]  Alexander Ushakov,et al.  Length Based Attack and Braid Groups: Cryptanalysis of Anshel-Anshel-Goldfeld Key Exchange Protocol , 2007, Public Key Cryptography.

[51]  Eberhard Stickel,et al.  A New Method for Exchanging Secret Keys , 2005, Third International Conference on Information Technology and Applications (ICITA'05).

[52]  Volker Gebhardt,et al.  Conjugacy in Garside groups III: Periodic braids , 2006 .

[53]  Jacques Stern,et al.  The cryptanalysis of a public-key implementation of finite group mappings , 2004, Journal of Cryptology.

[54]  Don Coppersmith,et al.  The Data Encryption Standard (DES) and its strength against attacks , 1994, IBM J. Res. Dev..

[55]  Massimiliano Sala,et al.  An application of the O’Nan-Scott theorem to the group generated by the round functions of an AES-like cipher , 2009, Des. Codes Cryptogr..

[56]  Volker Gebhardt,et al.  Conjugacy in Garside groups II: structure of the ultra summit set , 2006 .

[57]  Andrea Caranti,et al.  The Round Functions of Cryptosystem PGM Generate the Symmetric Group , 2006, Des. Codes Cryptogr..

[58]  Douglas R. Stinson,et al.  New Approaches to Designing Public Key Cryptosystems Using One-Way Functions and Trapdoors in Finite Groups , 2001, Journal of Cryptology.

[59]  Whitfield Diffie,et al.  New Directions in Cryptography , 1976, IEEE Trans. Inf. Theory.

[60]  Allen R. Tannenbaum,et al.  Length-Based Attacks for Certain Group Based Encryption Rewriting Systems , 2003, IACR Cryptol. ePrint Arch..

[61]  Andrei V. Kelarev,et al.  Braid-based cryptography , 2005 .

[62]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[63]  Douglas R. Stinson,et al.  Cryptography: Theory and Practice , 1995 .

[64]  Alexei G. Myasnikov,et al.  Random subgroups and analysis of the length-based and quotient attacks , 2007, J. Math. Cryptol..

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[67]  Oded Goldreich,et al.  Public-Key Cryptosystems from Lattice Reduction Problems , 1996, CRYPTO.

[68]  D. Goldfeld,et al.  An algebraic method for public-key cryptography , 1999 .

[69]  Dale K. Pace,et al.  The Codebreakers: The Comprehensive History of Secret Communication from Ancient Times to the Internet , 1998 .

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[71]  Spyros S. Magliveras,et al.  A Public Key Cryptosystem Based on Non-abelian Finite Groups , 2008, Journal of Cryptology.

[72]  Joan S. Birman,et al.  A new approach to the word and conjugacy problems in the braid groups , 1997 .

[73]  ON THE IMAGE OF THE LAWRENCE–KRAMMER REPRESENTATION , 2002, math/0202246.

[74]  Neal R. Wagner,et al.  A Public Key Cryptosystem Based on the Word Problem , 1985, CRYPTO.

[75]  D. Kahn.  The Codebreakers: The Comprehensive History of Secret Communication from Ancient Times to the Internet , 1967 .

[76]  Alfred Menezes,et al.  A note on cyclic groups, finite fields, and the discrete logarithm problem , 2005, Applicable Algebra in Engineering, Communication and Computing.