# American Institute of Mathematical Sciences

May  2013, 7(2): 187-195. doi: 10.3934/amc.2013.7.187

## Discrete logarithm like problems and linear recurring sequences

 1 Departamento de Matemáticas, Universidad de Oviedo, C/ Calvo Sotelo s/n, 33007 Oviedo, Spain, Spain 2 Departament de Ciènces Matemàtiques i Informàtica, Universitat de les Illes Balears, Ctra. de Valldemossa km 7, 5, 07122 Palma de Mallorca, Spain, Spain

Received  October 2012 Published  May 2013

In this paper we study the hardness of some discrete logarithm like problems defined in linear recurring sequences over finite fields from a point of view as general as possible. The intractability of these problems plays a key role in the security of the class of public key cryptographic constructions based on linear recurring sequences. We define new discrete logarithm, Diffie-Hellman and decisional Diffie-Hellman problems for any nontrivial linear recurring sequence in any finite field whose minimal polynomial is irreducible. Then, we prove that these problems are polynomially equivalent to the discrete logarithm, Diffie-Hellman and decisional Diffie-Hellman problems in the subgroup generated by any root of the minimal polynomial of the sequence.
Citation: Santos González, Llorenç Huguet, Consuelo Martínez, Hugo Villafañe. Discrete logarithm like problems and linear recurring sequences. Advances in Mathematics of Communications, 2013, 7 (2) : 187-195. doi: 10.3934/amc.2013.7.187
##### References:
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##### References:
 [1] A. Agnew, R. C. Mullin, I. M. Onyszchuk and S. A. Vanstone, An implementation for a fast public-key cryptosystem, J. Cryptology, 3 (1991), 63-79.  Google Scholar [2] A. Brouwer, R. Pellikaan and E. Verheul, Doing more with fewer bits, in "Advances in Cryptology - ASIACRYPT'99,'' Springer-Verlag, (1999), 321-332. Google Scholar [3] W. Diffie and M. Hellman, New directions in cryptography, IEEE Trans. Inform. Theory, 22 (1976), 644-654.  Google Scholar [4] C. M. Fiduccia, An efficient formula for linear recurring sequences, SIAM J. Comput., 14 (1985), 106-112.  Google Scholar [5] J. von zur Gathen and J. Gerhard, "Modern Computer Algebra,'' Cambridge Univ. Press, Cambridge, 2003.  Google Scholar [6] K. Giuliani and G. Gong, Analogues to the Gong-Harn and XTR cryptosystems, in "Combinatorics and Optimization Research Report,'' Univ. Waterloo, 2003. Google Scholar [7] K. Giuliani and G. Gong, Efficient key agreement and signature schemes using compact representations in $GF(p$10$)$, in "Proceedings of the 2004 IEEE International Symposium on Information Theory - ISIT'04,'' Chicago, (2004), 13. Google Scholar [8] K. Giuliani and G. Gong, New LFSR-based cryptosystems and the trace discrete logarithm problem (Trace-DLP), in "Sequences and Their Applications'04,'' Springer-Verlag, (2005), 298-312. Google Scholar [9] G. Gong and L. Harn, Public-key cryptosystems based on cubic finite field extensions, IEEE Trans. Inform. Theory, 45 (1999), 2601-2605.  Google Scholar [10] K. Karabina, Factor-4 and 6 compression of cyclotomic subgroups of $\mathbb F$*24m and $\mathbb F$*36m, J. Math. Crypt., 4 (2010), 1-42.  Google Scholar [11] A. Lenstra, Using cyclotomic polynomials to construct efficient discrete logarithm cryptosystems over finite fields, in "Proceedings of ACISP'97,'' Springer-Verlag, (1997), 127-138. Google Scholar [12] A. Lenstra and E. Verheul, The XTR public key system, in "Advances in Cryptology - CRYPTO'00,'' Springer-Verlag, (2000), 1-19.  Google Scholar [13] R. Lidl and H. Niederreiter, "Finite Fields,'' Addison-Wesley, 1983.  Google Scholar [14] H. Niederreiter, A public-key cryptosystem based on shift-register sequences, in "Advances in Cryptology - EUROCRYPT'85,'' Springer-Verlag, (1986), 35-39. doi: 10.1007/3-540-39805-8_4.  Google Scholar [15] H. Niederreiter, Some new cryptosystems based on feedback shift register sequences, Math. J. Okayama Univ., 30 (1988), 121-149.  Google Scholar [16] C. P. Schnorr, Efficient signature generation by smart cards, J. Cryptology, 4 (1991), 161-174. Google Scholar [17] M. Shirase, D. Han, Y. Hibin, H. Kim and T. Takagi, A more compact representation of XTR cryptosystem, IEICE T. Fund. Electr., E91-A (2008), 2843-2850. Google Scholar [18] P. Smith, LUC public-key encryption, Dr. Dobb's J., (1994), 44-49. Google Scholar [19] P. Smith and C. Skinner, A public-key cryptosystem and a digital signature system based on the Lucas function analogue to discrete logarithms, in "Advances in Cryptology - ASIACRYPT'94,'' Springer-Verlag, (1994), 14-21. Google Scholar [20] C. H. Tan, X. Yi and C. K. Siew, Signature schemes based on third order shift registers, in "Information Security and Privacy,'' Springer-Verlag, (2001), 445-459. Google Scholar [21] C. H. Tan, X. Yi and C. K. Siew, On the n-th order shift register based discrete logarithm, IECE Trans. Fund., E86-A (2003), 1213-1216. Google Scholar [22] C. H. Tan, X. Yi and C. K. Siew, On Diffie-Hellman problems in 3rd order shift register, IECE Trans. Fund., E87-A (2004), 1206-1208. Google Scholar [23] E. Verheul, Certificates of recoverability with scalable recovery agent security, in "Proceedings of PKC 2000,'' Springer-Verlag, (2000), 258-275.  Google Scholar
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