Abstract
Trivium is an international standard of lightweight stream ciphers (ISO/IEC 29192-3: 2012). In this paper, the Trivium-like NFSRs, a class of Galois NFSRs generalized from the Galois NFSR of Trivium, are studied from the perspective of Fibonacci NFSRs. It is shown that an n-stage Trivium-like NFSR cannot be equivalent to an n-stage Fibonacci NFSR, which is proved by showing the existence of “collision initial states”. As an intermediate conclusion, a necessary and sufficient condition for a kind of linear degeneracy of a Trivium-like NFSR is obtained from the persepective of interleaved sequences. Moreover, the smallest stage number of a Fibonacci NFSR that can generate all the output sequences of an n-stage Trivium-like NFSR is shown to be greater than n − 7 and this value is no less than 371 = 287 + min{93, 84, 111} specifically for the 288-stage Galois NFSR used in Trivium. These results contradict the existence of a equivalent Fibonacci model of Trivium NFSR of small stage, which implies that Trivium algorithm possesses a fair degree of immunity against “structure attack”.
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References
Courtois N T and Meier W, Algebraic Attacks on Stream Ciphers with Linear Feedback, Springer, Berlin, 2003, 345–359.
Meier W and Staffelbach O, Fast correlation attacks on certain stream ciphers, Journal of Cryptology, 1989, 1(3): 159–176.
De Cannière C and Preneel B T, New Stream Cipher Designs: The eSTREAM Finalists, Springer, Berlin/Heidelberg, 2008.
Ågren M, Hell M, Johansson T, et al., Grain-128a: A new version of grain-128 with optional authentication, International Journal of Wireless and Mobile Computing, 2011, 5(1): 48–59.
Canteaut A, Carpov S, Fontaine C, et al., Stream ciphers: A practical solution for efficient homomorphic-ciphertext compression, Journal of Cryptology, 2018, 31(3): 885–916.
Bernstein D J, Caesar: Cryptographic competition for authenticated encryption: Security, applicability, and robustness, 2014, https://competitions.cr.yp.to.
Hell M, Johansson T, Maximov A, et al., The grain family of stream ciphers, New Stream Cipher Designs, Springer, Berlin/Heidelberg, 2008, 179–190.
Borghoff J, Knudsen L R, and Stolpe M, bivium as a mixed-integer linear programming problem, Cryptography and Coding, Springer, Berlin, 2009, 133–152.
Dinur I and Shamir A, Cube attacks on tweakable black box polynomials, Advances in Cryptology, Springer, Berlin, 2009, 278–299.
Maximov A and Biryukov A, Two trivial attacks on trivium, Selected Areas in Cryptography, Springer, Berlin, 2007, 36–55.
Ye C D, Tian T, and Zeng F Y, The MILP-aided conditional differential attack and its application to trivium, Designs, Codes and Cryptography, 2021, 89(2): 317–339.
Hu H G and Gong G, Periods on two kinds of nonlinear feedback shift registers with time varying feedback functions, International Journal of Foundations of Computer Science, 2011, 22(6): 1317–1329.
Lechtaler A C, Cipriano M, García E, et al., Trivium vs. trivium toy, Proceedings of the 20th Argentinean Congress on Computer Science — III Workshop Computer Security, Buenos Aires, 2014, 161–172.
Zhang S Y and Chen G L, New results on the state cycles of trivium, Designs, Codes and Cryptography, 2019, 87(1): 149–162.
Dubrova E and Hell M E, A stream cipher for 5G wireless communication systems, Cryptography and Communications, 2017, 9(2): 273–289.
Zhang J M and Qi W F, Cryptanalysis of an equivalent model of stream cipher espresso, Journal of Cryptologic Research, 2016, 3(1): 91–100.
Ge Y and Parampalli U, Cryptanalysis of the class of maximum period galois nLFSR-based stream ciphers, Cryptography and Communications, 2021, 13(5): 847–864.
Tian T, Zhang J M, and Qi W F, On the uniqueness of a type of cascade connection representations for NFSRs, Designs, Codes and Cryptography, 2019, 87(10): 2267–2294.
Berbain C, Gilbert H, and Joux A, Algebraic and correlation attacks against linearly filtered non linear feedback shift registers, Selected Areas in Cryptography, Springer, Berlin, 2008, 184–198.
Orumiehchiha M A, Pieprzyk J, Steinfeld R, et al., Security analysis of linearly filtered NLFSRs, Journal of Mathematical Cryptology, 2013, 7(4): 313–332.
Dubrova E, A transformation from the fibonacci to the galois NLFSRs, IEEE Transactions on Information Theory, 2009, 55(11): 5263–5271.
Massey J L and Liu R W, Equivalence of nonlinear shift-registers, IEEE Transactions on Information Theory, 1964, 10(4): 378–379.
Lin Z Q, The transformation from the galois NLFSR to the fibonacci configuration, Proceeding of the Fourth International Conference on Emerging Intelligent Data and Web Technologies, Guiyang, 2013, 335–339.
Golomb S W, Shift Register Sequences, Holden-Dan Inc, San Francisco, 1967.
Lidl R and Niederreiter H, Finite fields, Encyclopaedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 1997.
Herstein I N, Topics in Algebra, John Wiley and Sons, New York, 1991.
Zhong J H, Pan Y Y, Kong W H, et al., Necessary and sufficient conditions for galois NFSRs equivalent to fibonacci ones and their application to stream cipher trivium, Cryptology ePrint Archive, 2021, 928, https://eprint.iacr.org/2021/928.
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This research was supported by the National Natural Science Foundation of China under Grant Nos. 12371526, 61872383, 61802430, and 62202494.
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Wang, H., Zheng, Q. & Qi, W. A Fibonacci View on the Galois NFSR Used in Trivium. J Syst Sci Complex 37, 1326–1350 (2024). https://doi.org/10.1007/s11424-024-2295-0
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DOI: https://doi.org/10.1007/s11424-024-2295-0