Group theory in cryptography
2009
Abstract
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.
References (92)
- Iris Anshel, Michael Anshel and Dorian Goldfeld, An algebraic method for public-key cryptography, Math. Res. Lett. 6 (1999), 287-291.
- Iris Anshel, Michael Anshel, Dorian Goldfeld and Stephane Lemieux, Key agreement, the Algebraic Eraser TM , and lightweight cryptography, Contemp. Math. 418 (2006) 1-34.
- Emil Artin, The theory of braids, Annals of Math. 48 (1947) 101-126.
- Daniel J. Bernstein, Johannes Buchmann, Erik Dahmen (Eds.) Post- Quantum Cryptography, Springer-Verlag, Berlin Heidelberg, 2009.
- Norman Biggs, The critical group from a cryptographic perspective, Bull. London Math. Soc. 39 (2007) 829-836.
- Joan S. Birman, Volker Gebhardt and Juan González-Meneses, Conju- gacy in Garside groups I: cycling, powers and rigidity, Groups Geom. Dynamics 1 (2007), 221-279.
- Joan S. Birman, Volker Gebhardt and Juan González-Meneses, Con- jugacy in Garside groups II: structure of the ultra-summit set, Groups Geom. Dynamics 2 (2008), 13-61.
- Joan S. Birman, Volker Gebhardt and Juan González-Meneses, Con- jugacy in Garside groups III: periodic braids, J. Algebra 316 (2007), 746-776.
- Joan Birman, Ki Hyoung Ko and Sang Jin Lee, A new approach to the word and conjugacy problems in the braid groups, Adv. Math 139 (1998) 322-353.
- Simon R. Blackburn, Cryptanalysing the critical group: efficiently solv- ing Biggs's discrete logarithm problem, J. Math. Cryptology, to appear.
- Simon R. Blackburn, Carlos Cid, Ciaran Mullan, Cryptanalysis of the M ST 3 cryptosystem, J. Math. Cryptology, to appear.
- Simon Blackburn, Sean Murphy and Jacques Stern, The cryptanalysis of a public key implementation of Finite Group Mappings, J. Cryptology 8 (1995), 157-166.
- Jens-Matthias Bohli, Rainer Steinwandt, María Isabel González Vasco and Consuelo Martinez, Weak keys in M ST 1 , Designs, Codes and Cryp- tography 37 (2005) 509-524.
- Ryan D. Budney, On the image of the Lawrence-Krammer representa- tion, J. Knot Theory and its Ramifications 14 (2005), 1-17.
- Keith W. Campbell and Michael J. Wiener, DES is not a group, in Advances in Cryptology -CRYPTO '92 (E.F. Brickell, ed), Lecture Notes in Computer Science 740 (Springer-Verlag, Berlin, 1993) 512- 520.
- V. Canda, T. van Trung, S. S. Magliveras and T. Horvath, Symmetric block ciphers based on group bases, in Selected Areas in Cryptography, SAC 2000 (D.R. Stinson and S.E. Tavares, eds.), Lecture Notes in Computer Science 2012 (Springer-Verlag, Berlin, 2001) 89-105.
- A Caranti, F. Dalla Volta, The round functions of cryptosystem PGM generate the symmetric group, Designs, Codes and Cryptography, 38 (2006), 147-155.
- A. Caranti, Francesca Dalla Volta and M. Sala, An application of the O'Nan-Scott theorem to the group generated by the round functions of an AES-like cipher, Designs, Codes and Cryptography 52 (2009), 293-301.
- A. Caranti, Francesca Dalla Volta and M. Sala, On some block ciphers and imprimitive groups, http://arxiv.org/abs/0806.4135.
- CGC Bulletin -Combinatorial Group Theory and Cryptography. http://u.cs.biu.ac.il/ ~tsaban/CGC/cgc.html.
- Jae Choon Cha, Ki Hyong Ko, Sang Jin Lee, Jae Woo Han and Jung Hee Cheon, An efficient implementation of braid groups, in Advances in Cryptology -Asiacrypt 2001 (C. Boyd, ed.), Lecture Notes in Computer Science 2248 (Springer, Berlin, 2001) 144-156.
- Jung Hee Cheon and Byungheup Jun, A polynomial time algorithm for the braid Diffie-Hellman conjugacy problem, in Advances in Cryptology -CRYPTO 2003 (D. Boneh, ed), Lecture Notes in Computer Science 2729 (Springer, Berlin, 2003) 212-225
- D. Coppersmith, The Data Encryption Standard (DES) and its strength against attacks, IBM Research Report RC 18613 (IBM, 1992).
- Patrick Dehornoy, Braid-based cryptography, Contemporary Mathe- matics 360 (2004), 5-33.
- Whitfield Diffie and Martin E. Hellman, New directions in cryptogra- phy, IEEE Trans. Information Theory 22 (1976) 644-654.
- Steven Galbraith and Alfred Menezes, Algebraic curves and cryptogra- phy, Finite Fields and Applications 11 (2005), 544-577.
- David Garber, Braid group cryptography, in Braids: Introductory Lec- tures on Braids, Configurations and Their Applications (J. Berrick, F.R. Cohen, E. Hanbury, eds) (World Scientific, Singapore, 2009) http://arxiv.org/abs/0711.3941.
- David Garber, Shmuel Kaplan, Mina Teicher, Boaz Tsaban and Uzi Vishne, Probabilistic solutions of equations in the braid group, Adv. Appl. Math. 35 (2005), 323-334.
- F.A. Garside, The braid group and other groups, Quart. J. Math. Ox- ford 20 (1969), 235-254.
- Robert Gilman, Alex D. Miasnikov, Alexei G. Myasnikov and Alexan- der Ushakov, New developments in commutator key exchange, in Proc. First Int. Conf. on Symbolic Computation and Cryptography (SCC- 2008), Bejing, 2008. http://www.math.stevens.edu/ ~rgilman/.
- Oded Goldreich, Shafi Goldwasser, and Shai Halevi, Public-key cryp- tosystems from lattice reduction problems, in Advances in Cryptology - CRYPTO 97 (B.S. Kaliski Jr, ed.), Lecture Notes in Computer Science 1294 (Springer, Berlin, 1997) 112-131.
- Oded Goldreich et al., Letters to the editor, Notices of the A.M.S. 54 (2007) 1454-1456.
- María Isabel González Vasco, Spyros Magliveras and Rainer Stein- wandt, Group-theoretic cryptography, Chapman & Hall / CRC Press, to appear.
- María Isabel Gonzalez Vasco, Martin Rötteler and Rainer Steinwandt, On minimal length factorizations of finite groups, J. Exp. Math. 12 (2003), 1-12.
- María Isabel González Vasco and Rainer Steinwandt, A reaction attack on a public key cryptosystem based on the word problem, Applica- ble Algebra in Engineering, Communication and Computing 14 (2004), 335-340.
- María Isabel González Vasco and Rainer Steinwandt, Obstacles in two public-key cryptosystems based on group factorizations, Tatra Mt. Math. Pub. 25 (2002) 23-37.
- Markus Grassl, Ivana Ilić, Spyros Magliveras and Rainer Steinwandt, Cryptanalysis of the Tillich-Zémor hash function, http://eprint.iacr.org/2009/229
- D. Hofheinz and R. Steinwandt, A practical attack on some braid group based cryptographic primitives, in Public Key Cryptography -PKC 2003 (Y.G. Desmedt, ed.), Lecture Notes in Computer Science 2384 (Springer, Berlin, 2002), 176-189.
- P. E. Holmes, On minimal factorisations of sporadic groups, J. Exp. Math. 13 (2004) 435-440.
- G. Hornauer, W. Stephan, and R. Wernsdorf, Markov ciphers and alter- nating groups, in Advances in Cryptology -EUROCRYPT '93 (T. Helle- seth, ed) Lecture Notes in Computer Science 765 (Springer-Verlag, Berlin, 1994), 453-460.
- James Hughes, A linear algebraic attack on the AAFG1 braid group cryptosystem, in Information Security and Privacy (G. Goos, J. Hart- manis and J. van Leeuwen, eds), Lecture Notes in Computer Science 2384 (Springer-Verlag, Berlin, 2002), 176-189.
- J. Hughes and A. Tannenbaum, Length-based attacks for certain group based encryption rewriting systems, http://arxiv.org/PS_cache/cs/pdf/0306/0306032v1.pdf.
- IACR Calendar of Events in Cryptology, http://www.iacr.org/events/.
- IACR Cryptology ePrint Archive, http://eprint.iacr.org/.
- David Kahn, The Codebreakers: The Comprehensive History of Secret Communication from Ancient Times to the Internet (2nd Edition, Si- mon & Schuster, London, 1997).
- Burton S. Kaliski Jr, Ronald L. Rivest, and Alan T. Sherman, Is the Data Encryption Standard a group? (Results of cycling experiments on DES), J. Cryptology 1 (1988), 3-36.
- Arkadius Kalka, Mina Teicher and Boaz Tsaban, 'Cryptanalysis of the Algebraic Eraser and short expressions of permutations as products', preprint. See http://arxiv.org/abs/0804.0629.
- Jonathan Katz and Yehuda Lindell, Introduction to Modern Cryptog- raphy (Chapman & Hall / CRC Press, Boca Raton, 2007).
- Aviad Kipnis and Adi Shamir, Cryptanalysis of the HFE public key cryptosystem, in Advances in Cryptology -CRYPTO '99 (M. Weiner, ed.), Lecture Notes in Computer Science 1666 (Springer, Berlin, 1999) 19-30.
- Ki Hyoung Ko, Sang Jin Lee, Jung Hee Cheon, Jae Woo Han, Ju-sung Kang, and Choonsik Park, New public-key cryptosystem using braid group, in Advances in Cryptology -CRYPTO 2000 (M. Bellare, ed.), Lecture Notes in Computer Science 1880 (Springer, Berlin, 2000) 166- 183.
- Neal Koblitz, The uneasy relationship between mathematics and cryp- tography, Notices Amer. Math. Soc. 54 (2007), 972-979.
- Sang Jin Lee and Eonkyung Lee, Potential weaknesses of the commu- tator key agreement protocol based on braid groups, in Advances in Cryptology -EUROCRYPT 2002, (L. Knudsen, ed.), Lecture Notes in Comp. Science 2332 (Springer, Berlin, 2002) 14-28.
- Wolfgang Lempken and Tran van Trung, On minimal logarithmic sig- natures of finite groups, J. Exp. Math. 14 (2005) 257-269.
- Wolfgang Lempken, Tran van Trung, Spyros S. Magliveras and Wandi Wei, A public key cryptosystem based on non-abelian finite groups, J. Cryptology 22 (2009) 62-74.
- Françoise Levy-dit-Vehel and Ludovic Perret, On the Wagner-Magyarik cryptosystem, in Coding and Cryptography (Ø. Ytrehus, ed) (Springer, Berlin, 2006), 316-329.
- Françoise Levy-dit-Vehel and Ludovic Perret, Security analysis of word problem-based cryptosystems, Designs, Codes and Cryptography 54 (2010), 29-41.
- Helger Lipmaa, Multiparty computa- tions, http://research.cyber.ee/~lipmaa/ crypto/link/mpc/.
- S. S. Magliveras, A cryptosystem from logarithmic signatures of finite groups, Proceedings of the 29'th Midwest Symposium on Circuits and Systems, Elsevier Publishing Company (1986), 972-975.
- S. S. Magliveras, Secret and public-key cryptosystems from group fac- torizations, Tatra Mt. Math. Publ. 25 (2002). 1-12.
- Spyros S. Magliveras and Nasir D. Memon, The algebraic properties of cryptosystem PGM, J. Cryptology 5 (1992), 167-183.
- S. S. Magliveras, D. R. Stinson and Tran van Trung, New approaches to designing public key cryptosystems using one-way functions and trap- doors in finite groups, J. Cryptology 15 (2002) 167-183.
- R.J. McEliece, A public key cryptosystem based on algebraic coding theory, DSN Progress Report 42 -44 (Jet Propulsion Lab, Pasadena, 1978) 114-116.
- Alfred J. Menezes and Scott A. Vanstone, A note on cyclic groups, finite fields and the discrete logarithm problem, Applicable Algebra in Engineering, Communication and Computing, 3 (1992), 67-74.
- Sean Murphy, Kenneth Paterson, and Peter Wild, A weak cipher that generates the symmetric group, J. Cryptology 7 (1994) 61-65.
- Alexei Myasnikov, Vladimir Shpilrain and Alexander Ushakov, Group- based Cryptography, Advanced Courses in Mathematics CRM Barcelona (Birkhäuser, Basel, 2008).
- Alex D. Myasnikov and Alexander Ushakov, Length based attack and braid groups: cryptanalysis of Anshel-Anshel-Goldfeld key exchange protocol, in Public Key Cryptography -PKC 2007 (T. Okamoto, X. Wang, eds.), Lecture Notes in Computer Science 4450 (Springer, Berlin, 2007) 76-88.
- A.G. Myasnikov and A. Ushakov, Random subgroups and analysis of the length-based and quotient attacks, J. Math. Cryptology 2 (2008) 29-61.
- Phong Q. Nguyen, Cryptanalysis of the Goldreich-Goldwasser-Halevi cryptosystem from CRYPTO 97', in Advances in Cryptology - CRYPTO '99 (M. Weiner, ed.), Lecture Notes in Computer Science 1666 (Springer, Berlin, 1999) 288-304,
- National Bureau of Standards, The Data Encryption Standard, Federal Information Processing Standards Publication (FIPS) 46, 1977.
- National Institute of Standards and Technology, Secure Hash Standard, Federal Information Processing Standards Publication (FIPS) 180-1, 1995.
- National Institute of Standards and Technology, Secure Hash Standard, Federal Information Processing Standards Publication (FIPS) 180-2 with Change Notice, 2002.
- National Institute of Standards and Technology, The Advanced Encryp- tion Standard, Federal Information Processing Standards Publication (FIPS) 197, 2001.
- Jacques Patarin, Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP): two new families of Asymmetric Algorithms' in: Ad- vances in Cryptology -Eurocrypt'96 (U. Maurer, ed.), Lecture Notes in Computer Science 1440 (Springer, Berlin, 1999) 33-48.
- Kenneth G. Paterson, Imprimitive permutation groups and trapdoors in iterated block ciphers, in Fast Software Encryption (L.R. Knudsen, ed), Lecture Notes in Computer Science 1636 (Springer-Verlag, Berlin, 1999), 201-214.
- Mingua Qu and Scott Vanstone, New public-key cryptosystems based on factorizations of finite groups, AUSCRYPT '92 Preproceedings.
- Dima Ruinskiy, Adi Shamir and Boaz Tsaban, Length-based crypt- analysis: The case of Thompson's Group, J. Mathematical Cryptology, 1 (2007), 359-372.
- C.E. Shannon, Communication theory of secrecy systems, Bell System Technical Journal 28 (1949), 656-715.
- Peter W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Computing 26 (1997) 1484-1509.
- Vladimir Shpilrain, Cryptanalysis of Stickel's key exchange scheme, in Computer Science -Theory and Applications (E.A. Hirsch, A.A. Razborov, A. Semenov, A, Slissenko, eds.), Lecture Notes in Computer Science 5010 (Springer, Berlin, 2008) 283-288.
- Nigel Smart, Cryptography: An Introduction, Third Edition. Available online: http://www.cs.bris.ac.uk/ ~nigel/Crypto_Book/
- Rüdiger Sparr and Ralph Wernsdorf, Group theoretic properties of RIJNDAEL-like ciphers, Discrete Appl. Math. 156 (2008), 3139-3149.
- Rainer Steinwandt, Markus Grassl, Willi Geiselmann and Thomas Beth, Weaknesses in the SL 2 (F 2 n ) hashing scheme, in Advances in Cryp- tology -CRYPTO 2000 (M. Bellare, ed), Lecture Notes in Computer Science 1880 (Springer, Berlin, 2000), 287-299.
- Eberhard Stickel, A new method for exchanging secret keys, in Proc. Third International Conference on Information Technology and Appli- cations (ICITA '05) (IEEE Computer Society, Piscataway, 2005), 426- 430.
- Douglas R. Stinson, Cryptography: Theory and Practice, Third Edition (Chapman & Hall, Boca Raton, 2005).
- The Hash Function Zoo, http://ehash.iaik.tugraz.at/wiki/The_Hash_Function_Zoo.
- Jean-Pierre Tillich and Gilles Zémor, Hashing with SL 2 , in Advances in Cryptology -CRYPTO '94 (Y. Desmedt, ed.), Lecture Notes in Computer Science, 839 (Springer, 1994), 40-49.
- Jean-Pierre Tillich and Gilles Zémor, Collisions for the LPS expander graph hash function, in Advances in Cryptology -EUROCRYPT 2008 (N. Smart, ed.), Lecture Notes in Computer Science, 4965 (Springer, 2008), 254-269.
- Neal R. Wagner and Marianne R. Magyarik, A public key cryptosystem based on the word problem, in Advances in Cryptology -CRYPTO '84 (G.R. Blakley and David Chaum, eds.) Lecture Notes in Computer Science 196 (Springer, Berlin, 1985), 19-36.
- Laurence C. Washington, Elliptic Curves: Number Theory and Cryp- tography, Second Edition (CRC Press, Boca Raton, 2008).
- Ralph Wernsdorf, The one-round functions of the DES generate the alternating group, in Advances in Cryptology -EUROCRYPT 1992 (R.A. Rueppel, ed.), Lecture Notes in Computer Science 658 (Springer- Verlag, Berlin, 1993), 99-112.
- Ralph Wernsdorf, The round functions of RIJNDAEL generate the al- ternating group, in Fast Software Encryption (J. Daemen and V. Rij- men, eds.), Lecture Notes in Computer Science 2365, (Springer-Verlag, Berlin, 2002), 143-148.
- Gilles Zémor, Hash functions and Cayley graphs, Designs, Codes and Cryptography 4 (1994), 381-394.