Gaussian elimination in split unitary groups with an application to public-key cryptography
Keywords:
Unitary groups, Gaussian elimination, Row-column operationsAbstract
Gaussian elimination is used in special linear groups to solve the word problem. In this paper, we extend Gaussian elimination to split unitary groups. These algorithms have an application in building a public-key cryptosystem, we demonstrate that.
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S. Ambrose, S. Murray, C. E. Praeger, and C. Schneider, Constructive membership testing in black–box classical groups, in: K. Fukuda, J. van der Hoeven, M. Joswig, and N. Takayama (Eds.), Mathematical Software – ICMS 2010, Lecture Notes in Computer Science, vol. 6327, Springer, Berlin, Heidelberg, 2010, pp. 54–57.
H. Bäärnhielm, D. Holt, C. R. Leedham-Green, and E. A. O’Brien, A practical model for computation with matrix groups, J. Symbolic Comput. 68 (2015) 27–60.
W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I: The user language, J. Symbolic Comput. 24(3–4) (1997) 235–265.
P. A. Brooksbank, Constructive recognition of classical groups in their natural representation, J. Symbolic Comput. 35(2) (2003) 195–239.
P. A. Brooksbank, Fast constructive recognition of black–box unitary groups, LMS J. Comput. Math. 6 (2003) 162–197.
R. W. Carter, Simple Groups of Lie Type, Pure and Applied Mathematics, vol. 28, John Wiley & Sons, London, 1972.
A. M. Cohen, S. H. Murray, and D. E. Taylor, Computing in groups of Lie type, Math. Comp. 73(247) (2004) 1477–1498.
E. Costi, Constructive Membership Testing in Classical Groups, Ph.D. thesis, Queen Mary, University of London, 2009.
J. Dieudonné, On the Automorphisms of the Classical Groups, Mem. Amer. Math. Soc., no. 2, American Mathematical Society, Providence, RI, 1951.
J. F. Grcar, Mathematicians of Gaussian elimination, Notices Amer. Math. Soc. 58(6) (2011) 782–792.
L. C. Grove, Classical Groups and Geometric Algebra, Graduate Studies in Mathematics, vol. 39, American Mathematical Society, Providence, RI, 2002.
N. Jacobson, Basic Algebra I, 2nd ed., W. H. Freeman and Company, New York, 1985.
W. M. Kantor and Á. Seress, Black Box Classical Groups, Mem. Amer. Math. Soc., vol. 149, no. 708, American Mathematical Society, Providence, RI, 2001.
C. R. Leedham-Green and E. A. O’Brien, Constructive recognition of classical groups in odd characteristic, J. Algebra 322(3) (2009) 833–881.
A. Mahalanobis, A simple generalization of the ElGamal cryptosystem to non–abelian groups II, Comm. Algebra 40(9) (2012) 3583–3596.
C. Monico, Cryptanalysis of a matrix–based MOR system, Comm. Algebra 44(1) (2016) 218–227.
A. C. Niemeyer and C. E. Praeger, A recognition algorithm for classical groups over finite fields, Proc. London Math. Soc. 77(1) (1998) 117–169.
E. A. O’Brien, Algorithms for matrix groups, in: C. M. Campbell et al. (Eds.), Groups St Andrews 2009 in Bath, London Mathematical Society Lecture Note Series, vol. 388, Cambridge University Press, Cambridge, 2011, pp. 297–323.
S.-H. Paeng, K.-C. Ha, J. H. Kim, S. Chee, and C. Park, New public key cryptosystem using finite non–abelian groups, in: J. Kilian (Ed.), Advances in Cryptology – CRYPTO 2001, Lecture Notes in Computer Science, vol. 2139, Springer, Berlin, Heidelberg, 2001, pp. 470–485.
Á. Seress, An introduction to computational group theory, Notices Amer. Math. Soc. 44(6) (1997) 671–679.
R. Steinberg, Variations on a theme of Chevalley, Pacific J. Math. 9(3) (1959) 875–891.