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## Gaussian elimination in split unitary groups with an application to public-key cryptography

#### Ayan MAHALANOBİS [1] , Anupam SİNGH [2]

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.
Unitary groups, Gaussian elimination, Row-column operations
• [1] S. Ambrose, S. Murray, C. E. Praeger, C. Schneider, Constructive membership testing in black–box classical groups, Proceedings of The Third International Congress on Mathematical Software, LNCS 6327 (2011) 54–57.
• [2] H. Bäärnhielm, D. Holt, C. R. Leedham-Green, E. A. O’Brien, A practical model for computation with matrix groups, J. Symb. Comput. 68 (2015) 27–60.
• [3] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I: The user language, J. Symb. Comput. 24(3-4) (1997) 235–265.
• [4] P. Brooksbank, Constructive recognition of classical groups in their natural representation, J. Symb. Comput. 35(2) (2003) 195–239.
• [5] P. Brooksbank, Fast constructive recognition of black–box unitary groups, LMS J. Comput. Math. 6 (2003) 162–197.
• [6] R. Carter, Simple Groups of Lie Type, New York: John Wiley and Sons, 1972.
• [7] A. M. Cohen, S. H. Murray, D. E. Taylor, Computing in groups of Lie type, Math. Comput. 73 (2004) 1477–1498.
• [8] E. Costi, Constructive Membership Testing in Classical Groups, Ph.D. thesis, Queen Mary, Univ. of London, 2009.
• [9] J. Dieudonne, On the automorphisms of the classical groups. with a supplement by Loo-Keng Hua, Mem. Amer. Math. Soc. 2 (1951) vi+122.
• [10] J. F. Grcar, Mathematicians of Gaussian elimination, Notices Amer. Math. Soc. 58(6) (2011) 782–792.
• [11] L. C. Grove, Classical Groups and Geometric Algebra, vol. 39, American Mathematical Society, Graduate Studies in Mathematics, 2002.
• [12] N. Jacobson, Basic Algebra I, W. H. Freeman and Company, New York, 1985.
• [13] W. Kantor À. Seress, Black box classical groups, vol. 149, Mem. Amer. Math. Soc. 149 (2001) viii+168.
• [14] C. R. Leedham–Green, E. A. O’Brien, Constructive recognition of classical groups in odd characteristic, J. Algebra 322(3) (2009) 833–881.
• [15] A. Mahalanobis, A simple generalization of the ElGamal cryptosystem to non–abelian groups II, Commun. Algebra 40(9) (2012) 3583–3596.
• [16] C. Monico, Cryptanalysis of matrix–based MOR system, Commun. Algebra 44(1) (2016) 218–227.
• [17] A. C. Niemeyer, C. E. Praeger, A recognition algorithm for classical groups over finite fields, Proc. London Math. Soc. 77(1) (1998) 117–169.
• [18] E. A. O’Brien, Algorithms for matrix groups, Groups St. Andrews 2009 in Bath, II, London Math. Soc. Lecture Note Ser., 388 (Cambridge Univ. Press, Cambridge, 2011), 297–323.
• [19] SH. Paeng, KC. Ha, J. H. Kim, S. Chee, C. Park, New public key cryptosystem using finite non–abelian groups, Crypto 2001 (J. Kilian, ed.), LNCS, vol. 2139, Springer–Verlag, (2001) 470–485.
• [20] Á. Seress, An introduction to computational group theory, Notices Amer. Math. Soc. 44(6) (1997) 671–679.
• [21] R. Steinberg, Variations on a theme of Chevalley, Pacific J. Math. 9 (1959) 875–891.
Subjects Engineering Articles Orcid: 0000-0001-9822-3672Author: Ayan MAHALANOBİS Orcid: 0000-0002-6162-4605Author: Anupam SİNGH Publication Date : September 15, 2017
 Bibtex @research article { jacodesmath327373, journal = {Journal of Algebra Combinatorics Discrete Structures and Applications}, issn = {}, eissn = {2148-838X}, address = {}, publisher = {Yildiz Technical University}, year = {2017}, volume = {4}, pages = {247 - 260}, doi = {10.13069/jacodesmath.327373}, title = {Gaussian elimination in split unitary groups with an application to public-key cryptography}, key = {cite}, author = {Mahalanobi̇s, Ayan and Si̇ngh, Anupam} } APA Mahalanobi̇s, A , Si̇ngh, A . (2017). Gaussian elimination in split unitary groups with an application to public-key cryptography . Journal of Algebra Combinatorics Discrete Structures and Applications , 4 (3) , 247-260 . DOI: 10.13069/jacodesmath.327373 MLA Mahalanobi̇s, A , Si̇ngh, A . "Gaussian elimination in split unitary groups with an application to public-key cryptography" . Journal of Algebra Combinatorics Discrete Structures and Applications 4 (2017 ): 247-260 Chicago Mahalanobi̇s, A , Si̇ngh, A . "Gaussian elimination in split unitary groups with an application to public-key cryptography". Journal of Algebra Combinatorics Discrete Structures and Applications 4 (2017 ): 247-260 RIS TY - JOUR T1 - Gaussian elimination in split unitary groups with an application to public-key cryptography AU - Ayan Mahalanobi̇s , Anupam Si̇ngh Y1 - 2017 PY - 2017 N1 - doi: 10.13069/jacodesmath.327373 DO - 10.13069/jacodesmath.327373 T2 - Journal of Algebra Combinatorics Discrete Structures and Applications JF - Journal JO - JOR SP - 247 EP - 260 VL - 4 IS - 3 SN - -2148-838X M3 - doi: 10.13069/jacodesmath.327373 UR - https://doi.org/10.13069/jacodesmath.327373 Y2 - 2017 ER - EndNote %0 Journal of Algebra Combinatorics Discrete Structures and Applications Gaussian elimination in split unitary groups with an application to public-key cryptography %A Ayan Mahalanobi̇s , Anupam Si̇ngh %T Gaussian elimination in split unitary groups with an application to public-key cryptography %D 2017 %J Journal of Algebra Combinatorics Discrete Structures and Applications %P -2148-838X %V 4 %N 3 %R doi: 10.13069/jacodesmath.327373 %U 10.13069/jacodesmath.327373 ISNAD Mahalanobi̇s, Ayan , Si̇ngh, Anupam . "Gaussian elimination in split unitary groups with an application to public-key cryptography". Journal of Algebra Combinatorics Discrete Structures and Applications 4 / 3 (September 2017): 247-260 . https://doi.org/10.13069/jacodesmath.327373 AMA Mahalanobi̇s A , Si̇ngh A . Gaussian elimination in split unitary groups with an application to public-key cryptography. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017; 4(3): 247-260. Vancouver Mahalanobi̇s A , Si̇ngh A . Gaussian elimination in split unitary groups with an application to public-key cryptography. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017; 4(3): 247-260.

Authors of the Article
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