Abstract
We present a computational approach based on algorithmic techniques to obtain maximal cyclic subgroups of the groups of units in Galois rings. The objective of this work was to provide an automated methodology to obtain such maximal cyclic subgroups for minimizing the human effort in such calculations. Necessity of getting a stock of maximal cyclic subgroups is due to their novel role in the formation of cyclic codes over finite commutative rings and prophesied shifting of S-box construction through binary field extensions \(\mathrm{GF}( {2^h}),1\le h\le 8\) to these maximal cyclic subgroups of the groups of units of finite Galois rings \(\mathrm{GR}(2^{k},{h})\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adams C, Tavares S (1989) Good S-boxes are easy to find. In: Advances in cryptology: proceedings of CRYPTO\(\_\)89. Lecture notes in computer science, pp 612–615
Andrade AA, Shah T (2011a) Goppa codes through polynomials of \(B[{X;(1/(2^{2}))Z_0 } ]\) and its decoding principle. J Adv Res Appl Math 3(4):12–20
Andrade AA, Shah T (2011b) Goppa codes via \(B[{X;(1/(3^{2}))Z_0 } ]\) and its decoding principle. Int J Appl Math 24(4):563–572
Andrade AA, Shah T, Khan A (2010a) Goppa codes through generalized polynomials and its decoding principle. Int J Appl Math 23(3):515–526
Andrade AA, Shah T, Khan A (2010b) Goppa codes through generalized polynomials and its decoding principle. Int J Appl Math 23(3):517–526
Andrade AA, Shah T, Khan A (2011) A note on linear codes over semigroup rings. TEMA Tend Mat Apl Comput 12(2):79–89
Andrade AA, Palazzo R Jr (1999) Construction and decoding of BCH codes over finite rings. Linear Algebra Appl 286:69–85
Andrade AA, Palazzo R Jr (2005) Linear codes over finite rings. TEMA Tend Mat Apl Comput 6(2):207–217
Cohen S (2009) Finite fields and applications. Cambridge University Press, England
Cui L, Cao Y (2007) A new S-box structure named Affine-Power-Affine. Int J Innov Comput I 3(3):45–53
Hussain I, Shah T (2013) Literature survey on nonlinear components and chaotic nonlinear components of block cipher. Nonlinear Dyn. 74:869–904
Hussain I, Shah T, Mahmood H, Gondal MA, Bhatti UY (2011) Some analysis of S-box based on residue of prime number. Proc Pak Acad Sci 48(2):111–115
Hussain I, Shah T, Gondal MA (2012a) Image encryption algorithm based on PGL (2, GF (2\(^{8}))\) S-boxes and TD-ERCS chaotic sequence. Nonlinear Dyn 70(1):181–187. doi:10.1007/s11071-012-0440-0
Hussain I, Shah T, Gondal MA, Mahmood H (2012b) Generalized majority logic criterion to analyze the statistical strength of S-boxes. Z Naturforsch. 67a:282–288. doi:10.5560/ZNA.2012-0022
Hussain I, Shah T, Mahmood H (2013) A group theoretic approach to construct cryptographically strong substitution boxes. Neural Comput Appl 23:97–104. doi:10.1007/s00521-012-0914-5
Kim J, Phan RCW (2009) Advanced differential-style crypt-analysis of the NSA’s skipjack block cipher. Cryptologia 33(3):246–270
McDonald BR (1974) Finite rings with identity. Marcel Dekker. Inc., New York
Shah T, Andrade AA (2012a) Cyclic codes through \(B[X], B[X;(1/(kp))Z_0 ]\): a comparison. J Algebra Appl 11(4):19. doi:10.1142/S0219498812500788
Shah T, Andrade AA (2012b) Cyclic codes through \(B[{X;( {\frac{a}{b}})Z_0 } ],(a/b\in Q^+, b=a+1)\) and encoding. Discret Math Algorithms Appl (DMAA) 04(04). doi:10.1142/S1793830912500590
Shah T, Andrade AA (2012c) Linear codes over finite local rings in a chain. J Adv Res Appl Math 4(4):66–77
Shah T, Khan A, Andrade AA (2011a) Encoding through generalized polynomial codes. Comput Appl Math 30(2):1–18
Shah T, Khan A, Andrade AA (2011b) Constructions of codes through semigroup ring \(B[X;(1/(2^{2}))Z_{0}]\)and encoding. Comput Math Appl 62(4):1645–1654
Shah T, Qamar A, Andrade AA (2012a) Construction and decoding of BCH codes over chain of commutative rings. Math Sci 6:51. doi:10.1186/2251-7456-6-51
Shah T, Qamar A, Andrade AA (2012b) Constructions and decoding of a sequence of BCH codes. Math Sci Res J 16(9):234–250
Shah T, Qamar A, Hussain I (2013) Substitution box on maximal cyclic subgroup of units of a Galois ring. Naturforsch A 68a:567–572. doi:10.5560/ZNA.2013-0021
Shanbhag AG, Kumar PV Helleseth T (1996) Upper bound for a hybrid sum over Galois rings with applications to the aperiodic correlation of some q-ary sequences. IEEE Trans Inf Theory 42(1):250–254
Shankar P (1979) On BCH codes over arbitrary integer rings. IEEE Trans Inf 25(4):480–483
Tran MT, Bui DK, Doung AD (2008) Gray S-box for advanced encryption standard. Int Conf Comput Intell Secur 1:253–256
Webster AF, Tavares S (1986) On the design of S-boxes. In: Advances in cryptology: proceedings of CRYPTO\(\_\)85. Lecture notes in computer science, pp 523–534
Yi X, Cheng SX, You XH, Lam KY (2002) A method for obtaining cryptographically strong 8 \(\times \) 8 S-boxes. Int Conf Inf Netw Appl 2(3):14–20
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jinyun Yuan.
Rights and permissions
About this article
Cite this article
Shah, T., Mehmood, N., de Andrade, A.A. et al. Maximal cyclic subgroups of the groups of units of Galois rings: a computational approach. Comp. Appl. Math. 36, 1273–1297 (2017). https://doi.org/10.1007/s40314-015-0281-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40314-015-0281-9