Abstract
Is it possible that a block cipher apparently immune to classical differential cryptanalysis can be attacked considering a different operation on the message space? Recently Calderini and Sala showed how to effectively compute alternative operations on a vector space which can serve as message space for a block cipher such that the resulting structure is still a vector space. The latter were used to mount a linearisation attack against a toy cipher. Here we investigate how alternative operations interact with the layers of a substitution–permutation network and show how they influence the differential probabilities, when the difference taken into consideration is different from the usual bit-wise addition modulo two. Furthermore, we design a block cipher which appears to be secure with respect to classical differential cryptanalysis, but weaker with respect to our attack which makes use of alternative operations.
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Notes
Note that the distributivity of \(\circ \) over \(+\) does not hold.
Note that only the resistance to differential cryptanalysis is considered and we do not claim any other resistance criteria for the security of this small cipher.
References
Abazari F., Sadeghian B.: Cryptanalysis with ternary difference: applied to block cipher PRESENT. Cryptology ePrint Archive, Report 2011/022, (2011).
Biham E., Anderson R., Knudsen L.: Serpent: A New Block Cipher Proposal. In Fast Software Encryption, pp. 222–238. Springer, New York (1998).
Biham E., Biryukov A., Shamir A.: Cryptanalysis of Skipjack reduced to 31 rounds using impossible differentials. In: International Conference on the Theory and Applications of Cryptographic Techniques, pp. 12–23. Springer, New York (1999).
Borghoff J., Canteaut A., Güneysu T., Kavun E.B., Knezevic M., Knudsen L.R., Leander G., Nikov V., Paar C., Rechberger C., et al.: PRINCE—a low-latency block cipher for pervasive computing applications. In: International Conference on the Theory and Application of Cryptology and Information Security, pp. 208–225. Springer, New York (2012).
Brunetta C., Calderini M., Sala M.: Algorithms and bounds for hidden sums in cryptographic trapdoors. arXiv:1702.08384 (2017).
Berson T.A.: Differential cryptanalysis mod \(2^{\wedge }\) 32 with applications to MD5. In: Eurocrypt, vol. 658, pp. 71–80. Springer, New York (1992).
Blondeau C., Gérard B.: Links between theoretical and effective differential probabilities: experiments on PRESENT. IACR Cryptol. ePrint Arch. 2010, 261 (2010).
Bogdanov A., Knudsen L.R., Leander G., Paar C., Poschmann A., Robshaw M.J.B., Seurin Y., Vikkelsoe C.: PRESENT: an ultra-lightweight block cipher. In: CHES ’07, pp. 450–466. Springer, New York (2007).
Biham E., Shamir A.: Differential cryptanalysis of DES-like cryptosystems. J. Cryptol. 4(1), 3–72 (1991).
Caranti A., Dalla Volta F., Sala M.: Abelian regular subgroups of the affine group and radical rings. Publ. Math. Debrecen 69(3), 297–308 (2006).
Calderini M., Sala M.: Elementary abelian regular subgroups as hidden sums for cryptographic trapdoors. arXiv:1702.00581 (2017).
Daemen J., Rijmen V.: Probability distributions of correlation and differentials in block ciphers. J. Math. Cryptol. 1(3), 221–242 (2007).
Daemen J., Rijmen V.: The Design of Rijndael: AES-the Advanced Encryption Standard. Springer, New York (2013).
Knudsen L.R., Leander G., Poschmann A., Robshaw M.J.B.: PRINTcipher: a block cipher for IC-printing. In: CHES, vol. 6225, pp. 16–32. Springer, New York (2010).
Knudsen L.R.: Truncated and higher order differentials. In: International Workshop on Fast Software Encryption, pp. 196–211. Springer, New York (1994).
Knudsen L.: DEAL—a 128-bit block cipher. In: NIST AES Proposal (1998).
Nyberg K.: Differentially uniform mappings for cryptography. In: Workshop on the Theory and Application of of Cryptographic Techniques, pp. 55–64. Springer, New York (1993).
Acknowledgements
Roberto Civino thankfully acknowledges support by the Department of Mathematics of the University of Trento and by COST Action IC1306. Roberto Civino and Massimiliano Sala are grateful to MIUR-Italy for financial support via PRIN 2015TW9LSR “Group theory and applications”.
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.
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Civino, R., Blondeau, C. & Sala, M. Differential attacks: using alternative operations. Des. Codes Cryptogr. 87, 225–247 (2019). https://doi.org/10.1007/s10623-018-0516-z
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DOI: https://doi.org/10.1007/s10623-018-0516-z