Abstract
“Mirror Theory” is the theory that evaluates the number of solutions of affine systems of equalities ( = ) and non equalities ( ≠ ) in finite groups. It is deeply related to the security and attacks of many generic cryptographic secret-key schemes, like random Feistel schemes (balanced or unbalanced), Misty schemes, Xor of two pseudo-random bijections to generate a pseudo-random function etc. In this chapter we will assume that the groups are abelian. Most of the time in cryptography the group is \(((\mathbb{Z}/2\mathbb{Z})^{n},\oplus )\) and this chapter concentrates on these cases. We present here general definitions, some theorems, and many examples and computer simulations.
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References
Hall, M., Jr.: A combinatorial problem on abelian groups. In: Proceedings of the American Mathematical Society, vol. 3(4), pp. 584–587 (1952)
Hall, M., Jr.: A problem in combinatorial group theory. Ars Combinatoria 7, 3–5 (1979)
Patarin, J.: On linear systems of equations with distinct variables and small block size. In: WON, D., Kim, S. (eds.), Information and Communications Security – ICISC ’05, vol. 3935, Lecture Notes in Computer Science, pp. 299–321. Springer, Heidelberg (2005)
Patarin, J.: A Proof of Security in O(2\(^{\mbox{ n}}\)) for the Xor of Two Random Permutations. In: Safavi-Naini, R. (ed.), Information and Communications Security – ICITS 2008, vol. 5155, Lecture Notes in Computer Science, pp. 232–248. Springer, Heidelberg (2008)
Patarin J.: The “Coefficients H” technique. In: Avanzi, R., Keliher, L., Sica, F. (ed.), Selected Areas in Cryptography – SAC ’08, vol. 5381, Lecture Notes in Computer Science, pp. 328–345. Springer, Heidelberg (2009)
Patarin, J.: A Proof of Security in O(2\(^{\mbox{ n}}\)) for the Xor of two random permutations∖∖ - Proof with the H σ technique-, in Cryptology ePrint Archive: Report 2008/010
Patarin, J.: Introduction to mirror theory: analysis of systems of linear equalities and linear non equalities for cryptography, in Cryptology ePrint Archive: Report 2010/287
Patarin, J.: Security of balanced and unbalanced Feistel schemes with linear non equalities, in Cryptology ePrint Archive: Report 2010/293
Patarin, J.: Security in O(2\(^{\mbox{ n}}\)) for the Xor of two random permutations ∖∖ - Proof with the standard H technique -, in Cryptology ePrint Archive: Report 2013/368
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Nachef, V., Patarin, J., Volte, E. (2017). Introduction to Mirror Theory. In: Feistel Ciphers. Springer, Cham. https://doi.org/10.1007/978-3-319-49530-9_14
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DOI: https://doi.org/10.1007/978-3-319-49530-9_14
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