Abstract
We demonstrate an efficient method for computing a Gröbner basis of a zero-dimensional ideal describing the key-recovery problem from a single plaintext/ciphertext pair for the full AES-128. This Gröbner basis is relative to a degree-lexicographical order. We investigate whether the existence of this Gröbner basis has any security implications for the AES.
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Bayer, D., Stillman, M.: On the complexity of computing syzygies. Journal of Symbolic Computation 6(2/3), 135–147 (1988)
Becker, T., Weispfenning, V.: Gröbner Bases – A Computational Approach to Commutative Algebra. Springer, Heidelberg (1991)
Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. PhD thesis, University of Innsbruck, Austria (1965)
Buchberger, B.: A criterion for Detecting Unnecessary Reductions in the Construction of Groebner Bases, London, UK, Johannes Kepler University Linz, vol. 72, pp. 3–21. Springer, Heidelberg (1979)
Buchmann, J., Pyshkin, A., Weinmann, R.-P.: Block Ciphers Sensitive to Gröbner Basis Attacks. In: Pointcheval, D. (ed.) CT-RSA 2006. LNCS, vol. 3860, pp. 313–331. Springer, Heidelberg (2006)
Cid, C., Leurent, G.: An analysis of the XSL Algorithm. In: Roy, B. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 333–352. Springer, Heidelberg (2005)
Collart, S., Kalkbrener, M., Mall, D.: Converting Bases with the Gröbner Walk. Journal of Symbolic Computation 24(3/4), 465–469 (1997)
Courtois, N., Pieprzyk, J.: Cryptanalysis of Block Ciphers with Overdefined Systems of Equations. In: Zheng, Y. (ed.) ASIACRYPT 2002. LNCS, vol. 2501, pp. 267–287. Springer, Heidelberg (2002)
Cox, D.A., Little, J.B., O’Shea, D.: Ideals, Varieties, and Algorithms, 2nd edn., p. 536. Springer, New York (1996)
Faugère, J.-C.: A New Efficient Algorithm for Computing Gröbner bases (F4). Journal of Pure and Applied Algebra 139(1-3), 61–88 (1999)
Faugère, J.-C.: A new efficient algorithm for computing Gröbner bases without reduction to zero (F5). In: Symbolic and Algebraic Computation – ISSAC 2002, pp. 75–83. ACM, New York (2002)
Faugère, J.-C., Gianni, P., Lazard, D., Mora, T.: Efficient Computation of Zero-Dimensional Gröbner Bases by Change of Ordering. Journal of Symbolic Computation 16(4), 329–344 (1993)
Ferguson, N., Schroeppel, R., Whiting, D.: A Simple Algebraic Representation of Rijndael. In: Vaudenay, S., Youssef, A.M. (eds.) SAC 2001. LNCS, vol. 2259, pp. 103–111. Springer, Heidelberg (2001)
Kalkbrener, M.: On the Complexity of Gröbner Bases Conversion. Journal of Symbolic Computation 28(1-2), 265–273 (1999)
Murphy, S., Robshaw, M.: Further Comments on the Structure of Rijndael. AES Comment to NIST (August. 2000)
Murphy, S., Robshaw, M.J.B.: Essential Algebraic Structure within the AES. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 1–16. Springer, Heidelberg (2002)
National Institute of Standards and Technology. FIPS-197: Advanced Encryption Standard (November 2001), Available at, http://csrc.nist.gov/publications/fips/
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Buchmann, J., Pyshkin, A., Weinmann, RP. (2006). A Zero-Dimensional Gröbner Basis for AES-128. In: Robshaw, M. (eds) Fast Software Encryption. FSE 2006. Lecture Notes in Computer Science, vol 4047. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11799313_6
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DOI: https://doi.org/10.1007/11799313_6
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