Abstract
The paper considers digital signature algorithms with a hidden group, security of which is based on the computational difficulty of solving a system of many quadratic equations with many unknowns. The attention is paid to an implementation of the said type of algorithms on finite non-commutative associative algebras set over the finite fields of characteristics two. The use of the latter type of algebraic support is aimed to improving the performance and reducing the hardware implementation cost. A new algebraic algorithm with a hidden group is introduced, in which a four-dimensional non-commutative algebra is used as algebraic support. In the used algebra the vector multiplication operation is defined by a sparse basis vector multiplication table. Decomposition of the non-commutative algebra into set of commutative subalgebras is studied. The formulas describing the number of the subalgebras of every type are also presented. It is shown that the factorization of the order of the hidden group is non-critical for the security of the signature algorithm, so one can apply the GF(2z) fields with a sufficiently large number of different values of the degree z, including those that are equal to a Mersenne exponent.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on quantum computer. SIAM J. Comput. 26, 1484–1509 (2017)
Smolin, J.A., Smith, G., Vargo, A.: Oversimplifying quantum factoring. Nature 499(7457), 163–165 (2013)
Moldovyan, A.A., Moldovyan, N.A.: Post-quantum signature algorithms based on the hidden discrete logarithm problem. Comput. Sci. J. Moldova 26(3(78)), 301–313 (2018)
Moldovyan, D., Moldovyan, A., Moldovyan, N.: A new concept for designing post-quantum digital signature algorithms on non-commutative algebras. Voprosy kiberbezopasnosti 1(47), 18–25 (2022). https://doi.org/10.21681/2311-3456-2022-1-18-25
Shuaiting, Q., Wenbao, H., Yifa, L., Luyao, J.: Construction of extended multivariate public key cryptosystems. Int. J. Netw. Secur. 18(1), 60–67 (2016)
Jintai, D., Dieter, S.: Multivariable Public Key Cryptosystems. https://eprint.iacr.org/2004/350.pdf (2004). Accessed 30 June 2022
Moldovyan, D.N.: A practical digital signature scheme based on the hidden logarithm problem. Comput. Sci. J. Moldova 29(2(86)), 206–226 (2021)
Moldovyan, N.A., Moldovyan, A.A.: Digital signature scheme on the 2×2 matrix algebra. Vestnik of Saint Petersburg Univ., Appl. Math. Comput. Sci. Control Processes 17(3), 254–261 (2021)
Moldovyan, N.A., Moldovyan, A.A.: Finite non-commutative associative algebras as carriers of hidden discrete logarithm problem. In: Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming & Computer Software (Bulletin SUSU MMCS), vol. 12, no. 1, pp. 66–81 (2019)
Moldovyan, N.A., Moldovyan, A.A.: Candidate for practical post-quantum signature scheme. Vestnik of Saint Petersburg Univ., Appl. Math. Comput. Sci. Control Processes 16(4), 455–461 (2020)
Moldovyan, N.A., Abrosimov, I.K.: Post-quantum electronic digital signature scheme based on the enhanced form of the hidden discrete logarithm problem. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes 15(2), 212–220 (2019). https://doi.org/10.21638/11702/spbu10.2019.205
Moldovyan, N.A.: Signature schemes on algebras, satisfying enhanced criterion of post-quantum security. Bull. Acad. Sci. Moldova, Math. 2(93), 62–67 (2020)
Moldovyan, N.A.: Unified method for defining finite associative algebras of arbitrary even dimensions. Quasigroups Relat. Syst. 26(2), 263–270 (2018)
Moldovyan, N.A., Moldovyanu, P.A.: New primitives for digital signature algorithms. Quasigroups Relat. Syst. 17(2), 271–282 (2009)
Moldovyan, N.A.: Fast signatures based on non-cyclic finite groups. Quasigroups Relat. Syst. 18(1), 83–94 (2010)
Crandall, R., Pomerance, C.: Prime Numbers - A Computational Perspective. Springer, New York (2002)
Ding, J., Schmidt, D.: Rainbow, a new multivariable polynomial signature scheme. In: Ioannidis, J., Keromytis, A., Yung, M. (eds.) ACNS 2005. LNCS, vol. 3531, pp. 164–175. Springer, Heidelberg (2005). https://doi.org/10.1007/11496137_12
Rainbow Signature: One of three NIST post-quantum signature finalists. https://www.pqcrainbow.org/,(2021). Accessed 30 June 2022
Funding
This research is partially supported by RFBR (project # 21–57-54001-Vietnam) and by Vietnam Academy of Science and Technology (project # QTRU01.13/21–22).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
May, D.T., Bac, D.T., Minh, N.H., Kurysheva, A.A., Kostina, A.A., Moldovyan, D.N. (2022). Signature Algorithms on Non-commutative Algebras Over Finite Fields of Characteristic Two. In: Dang, T.K., Küng, J., Chung, T.M. (eds) Future Data and Security Engineering. Big Data, Security and Privacy, Smart City and Industry 4.0 Applications. FDSE 2022. Communications in Computer and Information Science, vol 1688. Springer, Singapore. https://doi.org/10.1007/978-981-19-8069-5_18
Download citation
DOI: https://doi.org/10.1007/978-981-19-8069-5_18
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-19-8068-8
Online ISBN: 978-981-19-8069-5
eBook Packages: Computer ScienceComputer Science (R0)