Abstract
In this work we analyze the security of cubic cryptographic constructions with respect to rank weakness. We detail how to extend the big field idea from quadratic to cubic, and show that the same rank defect occurs. We extend the min-rank problem and propose an algorithm to solve it in this setting. We show that for fixed small rank, the complexity is even lower than for the quadratic case. However, the rank of a cubic polynomial in n variables can be larger than n, and in this case the algorithm is very inefficient. We show that the rank of the differential is not necessarily smaller, rendering this line of attack useless if the rank is large enough. Similarly, the algebraic attack is exponential in the rank, thus useless for high rank.
D. E. Escudero—Work done whilst at Universidad Nacional de Colombia.
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Notes
- 1.
Notice that we are using an upper bound to estimate the complexity. This is a customary usage for this kind of attacks. In practice, it has been observed [32] that, on average, this bound is not too far from the actual complexity.
- 2.
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Acknowledgements
This work was partially supported by “Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas”, Colciencias (Colombia), Project No. 111865842333 and Contract No. 049-2015.
We would like to thank Jintai Ding for very useful discussions. We would also like to thank the reviewers of PQCrypto 2018 for their constructive reviews and suggestions. And last but not least, we thank the Facultad de Ciencias of the Universidad Nacional de Colombia sede Medellín for granting us access to the Enlace server, where we ran most of the experiments of this paper.
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Baena, J., Cabarcas, D., Escudero, D.E., Khathuria, K., Verbel, J. (2018). Rank Analysis of Cubic Multivariate Cryptosystems. In: Lange, T., Steinwandt, R. (eds) Post-Quantum Cryptography. PQCrypto 2018. Lecture Notes in Computer Science(), vol 10786. Springer, Cham. https://doi.org/10.1007/978-3-319-79063-3_17
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