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Licensed Unlicensed Requires Authentication Published by De Gruyter June 5, 2017

On the k-th order lfsr sequence with public key cryptosystems

  • Bariş Bülent Kirlar EMAIL logo and Melek Çİl
From the journal Mathematica Slovaca


In this paper, we propose a novel encryption scheme based on the concepts of the commutative law of the k-th order linear recurrences over the finite field 𝔽q for k > 2. The proposed encryption scheme is an ephemeral-static, which is useful in situations like email where the recipient may not be online. The security of the proposed encryption scheme depends on the difficulty of solving some Linear Feedback Shift Register (LFSR) problems. It has also the property of semantic security. For k = 2, we propose another encryption scheme by using the commutative law of Lucas and Fibonacci sequences having also the same security as proposed for k > 2.

MSC 2010: 11B37; 94A60

E-mail: ,

(Communicated by Stanislav Jakubec)


The authors thank the anonymous referees for their detailed and very helpful comments.


[1] Akleylek, S.—Kirlar, B. B.: New methods for public key cryptosystems based on XTR, Security and Communication Networks 8(18) (2015), 3682–3689.10.1002/sec.1291Search in Google Scholar

[2] Akyildiz, E.—Ashraf, M.: An overview of trace based public key cryptography over finite fields, J. Comput. Appl. Math. 259-B (2014), 599–621.10.1016/ in Google Scholar

[3] Ashraf, M.—Kirlar, B. B.: Message transmission for GH-public key cryptosystem, J. Comput. Appl. Math. 259-B (2014), 578–585.10.1016/ in Google Scholar

[4] Bleichenbacher, D.—Bosma, W.—Lenstra, A. K.: Some remarks on Lucas-based cryptosystems. In: Advances in Cryptology – Crypto’95. Lecture Notes in Comput. Sci. 963, 1995, pp. 386–396.10.1007/3-540-44750-4_31Search in Google Scholar

[5] Catalano, D.—Cramer, R.—Damgard, I.—Crescenzo, G. D.—Pointcheval, D.–Takagi, T.: Contemporary Cryptology. Advanced Courses in Mathematics, 2005.10.1007/3-7643-7394-6Search in Google Scholar

[6] Dolev, D.—Dwork, C.—Naor, M.: Non-malleable cryptography. In: Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, 1991, pp. 542–552.10.1145/103418.103474Search in Google Scholar

[7] Duanmu, Q.—Zhang, X.—Wang, Y.—Zhang, K.: A new public-key cryptosystem based on fifth-order LFSR sequence. The 1st International Conference on Information Science and Engineering (ICISE 2009), 2009, pp. 1557–1560.10.1109/ICISE.2009.101Search in Google Scholar

[8] Giuliani, K.—Gong, G.: Analogues to the Gong-Harn and XTR Cryptosystems. Technical report, University of Waterloo, CORR, 2003.Search in Google Scholar

[9] Giuliani, K.—Gong, G.: Efficient key agreement and signature schemes using compact representations in GF(p10). IEEE International Symposium on Information Theory – ISIT’04.Search in Google Scholar

[10] Giuliani, K.—Gong, G.: New LFSR-based cryptosystems and the trace discrete log problem (trace-DLP). In: Sequences and Their Applications – SETA’04. Lecture Notes in Comput. Sci. 3486, 2005, pp. 298–312.10.1007/11423461_22Search in Google Scholar

[11] Giuliani, K.—Gong, G.: A new algorithm to compute remote terms in special types of characteristic sequences. In: Sequences and Their Applications – SETA’06. Lecture Notes in Comput. Sci. 4086, 2006, pp. 237–247.10.1007/11863854_20Search in Google Scholar

[12] Goldwasser, S.—Micali, S.: Probabilistic encryption and how to play mental poker keeping secret all partial information. In: Proceedings of 14 Annual ACM Symposium on Theory of Computing, 1982, pp. 365–377.10.1145/800070.802212Search in Google Scholar

[13] Goldwasser, S.—Micali, S.: Probabilistic encryption, J. Comput. System Sci. 28 (1984), 270–299.10.1016/0022-0000(84)90070-9Search in Google Scholar

[14] Gong, G.—Harn, L.: Public-key cryptosystems based on cubic finite field extensions, IEEE Trans. Inform. Theory 45 (1999), 2601–2605.10.1109/18.796413Search in Google Scholar

[15] Gong, G.—Harn, L.—Wu, H.: The GH public-key cryptosystem. In: SAC’01, Lecture Notes in Comput. Sci. 2259, 2001, pp. 284–300.10.1007/3-540-45537-X_22Search in Google Scholar

[16] Joye, M.—Quisquater, J.-J.: Efficient computation of full Lucas sequences, Electronics Letters 32 (1996), 537–538.10.1049/el:19960359Search in Google Scholar

[17] Lenstra, A. K.—Verheul, E. R.: The XTR public key system. In: Advances in Cryptology – Crypto’00, Lecture Notes in Comput. Sci. 1880, 2000, pp. 1–19.10.1007/3-540-44598-6_1Search in Google Scholar

[18] Lidl, R.—Niederreiter, H.: Finite Fields, Addison-Wesley Publishing Company, Reading, MA, 1983.Search in Google Scholar

[19] Muratovic-Ribic, A.—Wang, Q.: Partitions and compositions over finite fields, Electron. J. Combin. 20 (2013), 1–14.10.37236/2678Search in Google Scholar

[20] Samet, S.—Miri, A.: Privacy preserving ID3 using Gini Index over horizontally partitioned data. In: Proceedings of the 2008 IEEE/ACS International Conference on Computer Systems and Applications, 2008, pp. 645–651.10.1109/AICCSA.2008.4493598Search in Google Scholar

[21] Shirase, M.—Han, D.—Hibin, Y.—Kim, H.—Takagi, T.: A more compact representation of XTR cryptosystem, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E91-A (2008), 2843–2850.10.1093/ietfec/e91-a.10.2843Search in Google Scholar

[22] Smith, P. J.—Lennon, M. J. J.: LUC: A new public key system. In: Proceedings of the IFIP TC11 Ninth International Conference on Information Security IFIP/Sec’93, 1993, pp. 103–117.Search in Google Scholar

[23] Smith, P.—Skinner, C.: A public-key cryptosystem and a digital signature system based on the Lucas function analogue to discrete logarithms. In: Advances in Cryptology – Asiacrypt’94. Lecture Notes in Comput. Sci. 917, 1995, pp. 357–364.10.1007/BFb0000447Search in Google Scholar

[24] Stam, M.—Lenstra, A. K.: Speeding up XTR. In: Advances in Cryptology – Asiacrypt’01. Lecture Notes in Comput. Sci. 2248, 2001, pp. 125–143.10.1007/3-540-45682-1_8Search in Google Scholar

[25] Tan, C.-H.—Yi, X.—Siew, C.-K.: On the n-th Order Shift Register Based Discrete Logarithm, IEICE Trans. Fundamentals E86-A (2003), 1213–1216.Search in Google Scholar

[26] Yen, S.-M.—Laih, C.-S.: Fast algorithms for LUC digital signature computation, IEE Proc. Comput. Tech. 142 (1995), 165–169.10.1049/ip-cdt:19951788Search in Google Scholar

[27] Yu, C.-H.—Chow, S. S. M.—Chung, K.-M.—Liu, F.-H.: Efficient secure two-party exponentiation. In: Topics in Cryptology – CT-RSA 2011. Lecture Notes in Comput. Sci. 6558, 2011, pp. 17–32.10.1007/978-3-642-19074-2_2Search in Google Scholar

[28] Yucas, J. L.: Irreducible polynomials over finite fields with prescribed trace/prescribed constant term, Finite Fields Appl. 12 (2006), 211–221.10.1016/j.ffa.2005.04.006Search in Google Scholar

Received: 2015-1-16
Accepted: 2015-10-13
Published Online: 2017-6-5
Published in Print: 2017-6-27

© 2017 Mathematical Institute Slovak Academy of Sciences

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