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Some effectivity questions for plane Cremona transformations in the context of symmetric key cryptography

Part of: Arithmetic problems. Diophantine geometry Birational geometry

Published online by Cambridge University Press:  01 March 2021

N. I. Shepherd-Barron
N. I. Shepherd-Barron*
Affiliation:
Department of Mathematics, King's College London, Strand, LondonWC2R 2LS, UK ([email protected])
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Abstract

An effective lower bound on the entropy of some explicit quadratic plane Cremona transformations is given. The motivation is that such transformations (Hénon maps, or Feistel ciphers) are used in symmetric key cryptography. Moreover, a hyperbolic plane Cremona transformation g is rigid, in the sense of [5], and under further explicit conditions some power of g is tight.

MSC classification

Primary: 14E07: Birational automorphisms, Cremona group and generalizations
Secondary: 14G50: Applications to coding theory and cryptography
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 1 , February 2021 , pp. 1 - 28
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Copyright
Copyright © The Authors, 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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References

Blanc, J. and Cantat, S., Dynamical degrees of birational transformations of projective surfaces, J. Amer. Math. Soc. 29 (2016), 415–471.Google Scholar
Blanc, J. and Furter, J.-P., Topologies and structures of the Cremona group, Ann. Math. 178 (2013), 1173–1198.CrossRefGoogle Scholar
Bourbaki, N., Groupes et algèbres de Lie. IV–VI (Hermann, Paris, 1968).Google Scholar
Cantat, S., Elliptic subgroups of the Cremona group with a large normalizer, appendix to T. Delzant and P. Py, Kähler groups, real hyperbolic spaces and the Cremona group, Composito Math. 148 (2012), 153–184.Google Scholar
Cantat, S. and Lamy, S., Normal subgroups of the Cremona group, Acta Math. 210 (2013), 31–94.CrossRefGoogle Scholar
Delzant, T., Sous-groupes distingués et quotients des groupes hyperboliques, Duke Math. J. 83 (1996), 661–682.Google Scholar
Diller, J. and Favre, C., Dynamics of meromorphic maps of surfaces, American J. Math. 123 (2001), 1135–1169.Google Scholar
Gromov, M., On the entropy of holomorphic maps, L'Enseignement Math. 49 (2003), 217–235.Google Scholar
Lonjou, A., Non simplicité du groupe de Cremona sur tout corps, Ann. Inst. Fourier (Grenoble) 66 (2016), 2021–2046.CrossRefGoogle Scholar
Manin, Y., Cubic forms (North Holland, 1970).Google Scholar
McMullen, C., Coxeter groups, Salem numbers and the Hilbert metric, Publ. Math. IHES 95 (2002), 151–183.CrossRefGoogle Scholar
McMullen, C., Dynamics on blow-ups of the projective plane, Publ. Math. IHES 105 (2007), 49–89.Google Scholar
Steinberg, R., Finite reflection groups, Trans. AMS 91 (1959), 493–504.CrossRefGoogle Scholar