Abstract
Our purpose is to describe a promising linear algebraic attack on the AAFG1 braid group cryptosystem proposed in [2] employing parameters suggested by the authors. Our method employs the well known Burau matrix representation of the braid group and techniques from computational linear algebra and provide evidence which shows that at least a certain class of keys are weak. We argue that if AAFG1 is to be viable the parameters must be fashioned to defend against this attack.
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References
I. Anshel, M. Anshel, and D. Goldfeld. “An algebraic method for public-key cryptography”. Mathematical Research Letters 6 (1999), 1–5
I. Anshel, M. Anshel, B. Fisher, and D. Goldfeld. “New Key Agreement Protocol in Braid Group Cryptography”. Topics in Cryptology-CT-RSA2001. Lecture Notes in Computer Science, Vol. 2020. (Springer-Verlag, 2001), 13–27.
I. Anshel, M. Anshel, and D. Goldfeld. “A Linear Time Matrix Key Agreement Protocol”. Contemporary Methods in Cryptography. Institute for Pure and Applied Mathematics (IPAM), Winter 2002. From URL http://www.ipam.ucla.edu/programs/cry2002/abstracts/cry2002_dgoldfeld_abstract.html
E. Artin. “Theorie der Zopfe”. Hamburg Abh 4 (1925), 47–72
A. Cleary, and J. Dongarra. “Implementation in ScaLAPACK of Divide-and-Conquer Algorithms for Banded and Tridiagonal Systems”. Technical Report CS-97-358, University of Tennessee, Knoxville, TN, April 1997. Available as LAPACK Working Note #125 from URL http://www.netlib.org/lapack/lawns/
BOO BARKEE, DEH CAC CAN, JULIA ECKS, THEO MORIARITY, R. F. REE. “Why You Cannot Even Hope to use Grobner Bases in Public Key Cryptography: An Open Letter to Those Who Have Not Yet Failed”. J. Symbolic Computation 18 (1994), 497–501
S. Bigelow. “Homological representation of Braid groups”. Ph.D. Thesis, Dept. of Mathematics, Berkeley Univ., 2000
S. Bigelow. “Braid Groups Are Linear”. From URL http://citeseer.nj.nec.com/465605.html
S. Bigelow. “The Burau representation is not faithful for n = 5”. Geometry and Topology. 3 (1999), 397–404
E. Brieskorn, and K. Saito. “Artin Gruppen und Coxeter Gruppen”. Invent. Math. 17 (1972), 245–271
J. Birman. “Braids, Links, and Mapping Class Groups”. Annals of Mathematics Studies. Princeton University Press, Princeton, New Jersey, 1975
J. Birman, K. Ko, and S. Lee. “A new approach to the word and conjugacy problems in the braid groups”. Advances in Math. 139 (1998), 322–353
A.V. Borovik, A.G. Myasnikov, and V. Shpilrain. “Measuring sets in infinite groups”, From URL http://www.ma.umist.ac.uk/avb/pdf/measurePrep.pdf
W. Burau. “Ũber Zopfgruppen und gleichsinning verdrillte Verkettungen”. Abh. Math. Sem. Ham. II (1936), 171–178
P. Dehornoy. “A fast method for comparing braids”. Advances in Math. 127 (1997), 200–235
E. A. Elrifai and H. R. Morton. “Algorithms for positive braids”. Quart. J. Math. Oxford. 45 (1994), 479–497
H. Garside. “The braid group and other groups”. Quart. J. Math. Oxford. 20 (1969), 235–254
D. Goldfeld, Private Correspondence, November 17, 2001, Message-ID: < 3BF6E636.40195953@veriomail.com >
S.G. Hahn, E.K. Lee, J.H. Park. “The Generalized Conjugacy Search Problem and the Burau Representation”. Preprint, February, 2001, From URL http://crypt.kaist.ac.kr/pre_papers/hlp_revised1.ps
J. Hughes, and A. Tannenbaum. “Length-based attacks for certain group based encryption rewriting systems”. Institute for Mathematics and Its Applications, April, 2000, Minneapolis, MN, Preprint number 1696
J. Hughes. “The LeftSSS attack on Ko-Lee-Cheon-Han-Kang-Park Key Agreement Protocol in B45”, Rump Session Crypto 2000, Santa Barbara, CA, May, 2000. From URL http://www.network.com/hughes/Crypt2000.pdf
K. Ko, S. Lee, J. Cheon, J. Han, J. Kang, and C. Park. “New public-key cryptosystem using braid groups”. Technical Report, Korea Advance Institute of Science and Technology, Taejon, Korea, February 2000
R.J. Lawrence. “Homological representations of the Hecke algebra”. Comm. Math. Phys. 135 (1990), pp. 141–191.
D. Long and M. Paton. “The Burau representation is not faithful for n=6”. Topology 32 (1993), 439–447.
A. Odlyzko. “Cryptanalytic attacks on the multiplicative knapsack cryptosystem and on Shamir’s fast signature scheme”. IEEE Trans. Inform. Theory. 30 (1984), 594–601.
R.L. Rivest. “Cryptography”, Chapter 13 of Handbook of Theoretical Computer Science, (ed. J. Van Leeuwen). 1 (Elsevier, 1990), 717–755. http://theory.lcs.mit.edu/rivest/Rivest-Cryptography.pdf
M. Abadi, and P. Rogaway. “Reconciling Two Views of Cryptography (The Computational Soundness of Formal Encryption)”. Journal of Cryptology. 15 (2002), 103–127
V Shpilrain. “Average Case Complexity of the Word and Conjugacy Problems in the Braid Groups”. From URL http://zebra.sci.ccny.cuny.edu/web/shpil/complexity.ps
N. Franco, and J. Gonzalez-Meneses. “Computation of Normalizers in Braid groups and Garside Groups”. From URL http://xxx.lanl.gov/abs/math.GT/0201243
A. Joux and J. Stern. “Cryptanalysis of another knapsack cryptosystem”. Advances in Cryptology: Proceedings of AsiaCrypt’91, Volume 739 Lecture Notes in Computer Science, (Springer Verlag, 1991), 470–476
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Hughes, J. (2002). A Linear Algebraic Attack on the AAFG1 Braid Group Cryptosystem. In: Batten, L., Seberry, J. (eds) Information Security and Privacy. ACISP 2002. Lecture Notes in Computer Science, vol 2384. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45450-0_15
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DOI: https://doi.org/10.1007/3-540-45450-0_15
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