Bilinear cryptography using Lie algebras from p-groups

Document Type : Original Article


Department of Mathematics, Islamic Azad university, Shahr-e-Qods Branch, Tehran, Iran.


Pairings are particular bilinear maps, and they have been defined based on elliptic curves which
are abelian groups. In cryptography and security problems use these pairings. Mrabet et al. proposed
pairings from a tensor product of groups in 2013. Also Mahalanobis et al. proposed bilinear cryptography
using groups of nilpotency class two in 2017. In this paper, I develop a novel idea of a bilinear cryptosystem
using Lie algebras from p-groups. First the researcher proposes pairing on Lie algebras from elliptic curves,
and then pairings that can be constructed on Lie algebras from some of the non-abelian p-groups.


[1] R. Balasubramanian, N.Koblitz, The improbability than an elliptic curve has subexponential discrete
log problem under the Menezes-Okamoto-Vanstone algorithm, Journal of Cryptology 11(2) 1998, 141-
[2] R. Barua, R. Dutta, P. Sarkar, Extending Jouxs protocol to multi party key agreement, International
Conference on Cryptology in India, Springer, Berlin, Heidelberg, 2003.
[3] R. Dutta, R. Barua, P. Sarkar, Pairing based cryptographic protocols: A survey, IACR Cryptol. ePrint
Arch. 2004, 64
[4] I.F. Blake, G. Seroussi, N.P. Smart, Advances in elliptic curve cryptography. London Mathematical
Society, Lecture Note Series, Cambridge University Press 2005.
[5] D. Boneh, Twenty years of attacks on the RSA cryptosystem, Notices Amer. Math. Soc. 46 1999,
[6] D. Boneh, H. Shacham, B. Lynn, Short signatures from theWeil pairing. Journal of Cryptology, 17(4)
2004, 297-319.
[7] D. Boneh, M. K. Franklin, Identity-based encryption from the Weil pairing. SIAM Journal of Com-
puting, 32(3) 2003, 586-617.
[8] J. Boxall, A. Enge, Some security aspects of pairing-based cryptography. Technical report of the ANR
Project PACE, 2009, 243-258.
[9] W. Burnside, Theory of Groups of Finite Order, second ed, Cambridge Univ. Press, 1911.
[10] S. Chatterjee, P. Sarkar, Identity-Based Encryption, Springer, 2011.
[11] C. Costello, Pairing for beginners, A Note, 2013.
[12] W.A. Graff, Lie algebras: theory and algorithms, Elsevier, 2000.
[13] R. James, The groups of order p6 (p an odd prime), Math. Comput. 34 1980, 613-637.
[14] A. Joux, A one round protocol for Diffie-Hellman, Proceedings of the 4th International Symposium
on Algorithmic Number Theory, 2000, 385394.
[15] M. Joye, G. Neven, Identity-based cryptography, 2 of Cryptology and Information Security Series,
IOS Press, 2009.
[16] MD. Huang, W. Raskind , A multilinear Generalization of the Tate Pairing. Contemporary Mathe-
matics, 2010, 225-263.
[17] B. Huppert, N. Blackbum, Finite Groups II, Springer-Verlag Berlin Heidelberg New York, 1982.
[18] N. Koblitz, Algebraic aspects of cryptography, Algorithms and Computation in Mathematics, Algo-
rithms and Computation in Mathematics, 1998.
[19] S. Lee, A class of descendant p-groups of order p9 and Higmans PORC conjecture, Journal of Algebra,
468 2016 440-447.
[20] D. Lubicz, D. Robert, Efficient pairing computations with theta functions, Proceedings of the 9th
International Symposium in Algorithmic Number Theory, Nancy, France, July 19-23. Lecture Notes
in Computer Science 6197 2010, 251-269.
[21] A. Mahalanobis, P. Shinde, Bilinear cryptography using groups of nilpotency class 2, IMA Interna-
tional Conference on Cryptography and Coding, 2017, 127-134.
[22] N.E. Mrabet, L. Poinsot,, Pairings from a tensor product point of view, arXiv preprint
arXiv:1304.5779, 2013.
[23] A. Menezes, T. Okamoto, S.A. Vanstone, Reducing elliptic curve logarithms to logarithms in a  nite
 eld, IEEE Transactions on Information Theory 39(5)1993, 163-1646.
[24] N.E. Mrabet, L. Poinsot, Elementary group-theoretic approach to pairings, Liebniz International
Proceeding Informatics, 2012, 1-13.
[25] N.E. Mrabet, A. Guillevi, Sorina Ionica, Efficient Multiplication in Finite Field Extensions of Degree
5, International Conference on Cryptology in Africa. Springer, Berlin, Heidelberg, 2011.
[26] M.F. Newman, Determination of groups of prime-power order, in Group Theory, Lecture Notes in
Mathematics 573, Canberra, 1975, Springer-Verlag, Berlin, Heidelberg, New York, 1977, 7{84.
[27] E.A. O'Brien, The p-group generation algorithm, Journal of symbolic computation, 9(5-6) 1990, 677-
[28] E.A. O'Brien, M.R. Vaughan-Lee, The groups with order p7 for odd prime p, Journal of Algebra
292(1) 2005, 243-258.
[29] T. Okamoto, K. Takashima, Homomorphic encryption and signatures from vector decomposition,
International conference on pairing-based cryptography. Springer, Berlin, Heidelberg, 2008.
[30] T. Okamoto, K. Takashima, Hierarchical predicate encryption for inner-products, International Con-
ference on the Theory and Application of Cryptology and Information Security. Springer, Berlin,
Heidelberg, 2009.
[31] V.A. Roman kov, Discrete logarithm for nilpotent group and cryptanalysis of polylinear cryptographic
system, Prikle. Mat. Suppl, 2019(12) 2019, 154-160.
[32] V.A. Roman kov, Algebraic cryptanalysis and new security enhancements, Moscow Journal of com-
binatorics and Number Theory, 9(2) 2020, 123-146.
[33] J.H. Silverman, The arithmetic of elliptic curves, Volume 106 of Graduate Texts in Mathematics,
Springer, 1986.
[34] P.C. Van Oorschot, M.J. Wiener, Parallel collision search with cryptanalytic applications, Journal of
cryptology, 12(1) 1999, 1-28.
[35] M.R. Vaughan-Lee, Groups of order p8 and exponent p, International Journal of Group Theory, 4(4)
2015, 25-42.
Volume 2, Issue 1
January 2021
Pages 71-77
  • Receive Date: 01 January 2021
  • Revise Date: 09 February 2021
  • Accept Date: 20 February 2021
  • First Publish Date: 20 February 2021