Discrete logarithm like problems and linear recurring sequences

Previous Article
A new class of majority-logic decodable codes derived from polarity designs
AMC Home
This Issue
Next Article
Asymptotic lower bound on the algebraic immunity of random balanced multi-output Boolean functions

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

Discrete logarithm like problems and linear recurring sequences

Santos González ^1, , Llorenç Huguet ^2, , Consuelo Martínez ^1, and Hugo Villafañe ^2,

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

Abstract
Related Papers

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.

Keywords: hardness assumptions., public key cryptography, Discrete logarithm problem, linear recurring sequences, Diffie Hellman problem, compact representation.

Mathematics Subject Classification: Primary: 11T71; Secondary: 94A5.

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:

[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. Google Scholar
[2]	A. Brouwer, R. Pellikaan and E. Verheul, Doing more with fewer bits,, in, (1999), 321. Google Scholar
[3]	W. Diffie and M. Hellman, New directions in cryptography,, IEEE Trans. Inform. Theory, 22 (1976), 644. Google Scholar
[4]	C. M. Fiduccia, An efficient formula for linear recurring sequences,, SIAM J. Comput., 14 (1985), 106. Google Scholar
[5]	J. von zur Gathen and J. Gerhard, "Modern Computer Algebra,'', Cambridge Univ. Press, (2003). Google Scholar
[6]	K. Giuliani and G. Gong, Analogues to the Gong-Harn and XTR cryptosystems,, in, (2003). Google Scholar
[7]	K. Giuliani and G. Gong, Efficient key agreement and signature schemes using compact representations in $GF(p$¹⁰$)$,, in, (2004). Google Scholar
[8]	K. Giuliani and G. Gong, New LFSR-based cryptosystems and the trace discrete logarithm problem (Trace-DLP),, in, (2005), 298. Google Scholar
[9]	G. Gong and L. Harn, Public-key cryptosystems based on cubic finite field extensions,, IEEE Trans. Inform. Theory, 45 (1999), 2601. Google Scholar
[10]	K. Karabina, Factor-4 and 6 compression of cyclotomic subgroups of $\mathbb F$^_2^4m and $\mathbb F$^_3^6m,, J. Math. Crypt., 4 (2010), 1. Google Scholar
[11]	A. Lenstra, Using cyclotomic polynomials to construct efficient discrete logarithm cryptosystems over finite fields,, in, (1997), 127. Google Scholar
[12]	A. Lenstra and E. Verheul, The XTR public key system,, in, (2000), 1. 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, (1986), 35. 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. Google Scholar
[16]	C. P. Schnorr, Efficient signature generation by smart cards,, J. Cryptology, 4 (1991), 161. 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. Google Scholar
[18]	P. Smith, LUC public-key encryption,, Dr. Dobb's J., (1994), 44. 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, (1994), 14. Google Scholar
[20]	C. H. Tan, X. Yi and C. K. Siew, Signature schemes based on third order shift registers,, in, (2001), 445. 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. 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. Google Scholar
[23]	E. Verheul, Certificates of recoverability with scalable recovery agent security,, in, (2000), 258. Google Scholar

show all references

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. Google Scholar
[2]	A. Brouwer, R. Pellikaan and E. Verheul, Doing more with fewer bits,, in, (1999), 321. Google Scholar
[3]	W. Diffie and M. Hellman, New directions in cryptography,, IEEE Trans. Inform. Theory, 22 (1976), 644. Google Scholar
[4]	C. M. Fiduccia, An efficient formula for linear recurring sequences,, SIAM J. Comput., 14 (1985), 106. Google Scholar
[5]	J. von zur Gathen and J. Gerhard, "Modern Computer Algebra,'', Cambridge Univ. Press, (2003). Google Scholar
[6]	K. Giuliani and G. Gong, Analogues to the Gong-Harn and XTR cryptosystems,, in, (2003). Google Scholar
[7]	K. Giuliani and G. Gong, Efficient key agreement and signature schemes using compact representations in $GF(p$¹⁰$)$,, in, (2004). Google Scholar
[8]	K. Giuliani and G. Gong, New LFSR-based cryptosystems and the trace discrete logarithm problem (Trace-DLP),, in, (2005), 298. Google Scholar
[9]	G. Gong and L. Harn, Public-key cryptosystems based on cubic finite field extensions,, IEEE Trans. Inform. Theory, 45 (1999), 2601. Google Scholar
[10]	K. Karabina, Factor-4 and 6 compression of cyclotomic subgroups of $\mathbb F$^_2^4m and $\mathbb F$^_3^6m,, J. Math. Crypt., 4 (2010), 1. Google Scholar
[11]	A. Lenstra, Using cyclotomic polynomials to construct efficient discrete logarithm cryptosystems over finite fields,, in, (1997), 127. Google Scholar
[12]	A. Lenstra and E. Verheul, The XTR public key system,, in, (2000), 1. 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, (1986), 35. 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. Google Scholar
[16]	C. P. Schnorr, Efficient signature generation by smart cards,, J. Cryptology, 4 (1991), 161. 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. Google Scholar
[18]	P. Smith, LUC public-key encryption,, Dr. Dobb's J., (1994), 44. 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, (1994), 14. Google Scholar
[20]	C. H. Tan, X. Yi and C. K. Siew, Signature schemes based on third order shift registers,, in, (2001), 445. 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. 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. Google Scholar
[23]	E. Verheul, Certificates of recoverability with scalable recovery agent security,, in, (2000), 258. Google Scholar

[1]	Gerhard Frey. Relations between arithmetic geometry and public key cryptography. Advances in Mathematics of Communications, 2010, 4 (2) : 281-305. doi: 10.3934/amc.2010.4.281
[2]	Gérard Maze, Chris Monico, Joachim Rosenthal. Public key cryptography based on semigroup actions. Advances in Mathematics of Communications, 2007, 1 (4) : 489-507. doi: 10.3934/amc.2007.1.489
[3]	Felipe Cabarcas, Daniel Cabarcas, John Baena. Efficient public-key operation in multivariate schemes. Advances in Mathematics of Communications, 2019, 13 (2) : 343-371. doi: 10.3934/amc.2019023
[4]	Rainer Steinwandt, Adriana Suárez Corona. Cryptanalysis of a 2-party key establishment based on a semigroup action problem. Advances in Mathematics of Communications, 2011, 5 (1) : 87-92. doi: 10.3934/amc.2011.5.87
[5]	Joan-Josep Climent, Juan Antonio López-Ramos. Public key protocols over the ring $E_{p}^{(m)}$. Advances in Mathematics of Communications, 2016, 10 (4) : 861-870. doi: 10.3934/amc.2016046
[6]	Yong-Kum Cho. A quadratic Fourier representation of the Boltzmann collision operator with an application to the stability problem. Kinetic & Related Models, 2012, 5 (3) : 441-458. doi: 10.3934/krm.2012.5.441
[7]	Hongyan Yan, Yun Sun, Yuanguo Zhu. A linear-quadratic control problem of uncertain discrete-time switched systems. Journal of Industrial & Management Optimization, 2017, 13 (1) : 267-282. doi: 10.3934/jimo.2016016
[8]	Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1743-1767. doi: 10.3934/dcdsb.2018235
[9]	Shengbing Deng, Zied Khemiri, Fethi Mahmoudi. On spike solutions for a singularly perturbed problem in a compact riemannian manifold. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2063-2084. doi: 10.3934/cpaa.2018098
[10]	Anton Stolbunov. Constructing public-key cryptographic schemes based on class group action on a set of isogenous elliptic curves. Advances in Mathematics of Communications, 2010, 4 (2) : 215-235. doi: 10.3934/amc.2010.4.215
[11]	Mikko Orispää, Markku Lehtinen. Fortran linear inverse problem solver. Inverse Problems & Imaging, 2010, 4 (3) : 485-503. doi: 10.3934/ipi.2010.4.485
[12]	Marcelo Disconzi. On a linear problem arising in dynamic boundaries. Evolution Equations & Control Theory, 2014, 3 (4) : 627-644. doi: 10.3934/eect.2014.3.627
[13]	V.N. Malozemov, A.V. Omelchenko. On a discrete optimal control problem with an explicit solution. Journal of Industrial & Management Optimization, 2006, 2 (1) : 55-62. doi: 10.3934/jimo.2006.2.55
[14]	Davide Bellandi. On the initial value problem for a class of discrete velocity models. Mathematical Biosciences & Engineering, 2017, 14 (1) : 31-43. doi: 10.3934/mbe.2017003
[15]	Senda Ounaies, Jean-Marc Bonnisseau, Souhail Chebbi, Halil Mete Soner. Merton problem in an infinite horizon and a discrete time with frictions. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1323-1331. doi: 10.3934/jimo.2016.12.1323
[16]	Xiao Lan Zhu, Zhi Guo Feng, Jian Wen Peng. Robust design of sensor fusion problem in discrete time. Journal of Industrial & Management Optimization, 2017, 13 (2) : 825-834. doi: 10.3934/jimo.2016048
[17]	Mathias Staudigl, Jan-Henrik Steg. On repeated games with imperfect public monitoring: From discrete to continuous time. Journal of Dynamics & Games, 2017, 4 (1) : 1-23. doi: 10.3934/jdg.2017001
[18]	Xiangtuan Xiong, Jinmei Li, Jin Wen. Some novel linear regularization methods for a deblurring problem. Inverse Problems & Imaging, 2017, 11 (2) : 403-426. doi: 10.3934/ipi.2017019
[19]	Qiang Du, M. D. Gunzburger, L. S. Hou, J. Lee. Analysis of a linear fluid-structure interaction problem. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 633-650. doi: 10.3934/dcds.2003.9.633
[20]	Shigeaki Koike, Hiroaki Morimoto, Shigeru Sakaguchi. A linear-quadratic control problem with discretionary stopping. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 261-277. doi: 10.3934/dcdsb.2007.8.261

2018 Impact Factor: 0.879

American Institute of Mathematical Sciences