Abstract
In this paper we consider a description of homomorphic and fully homomorphic cryptographic primitives in the p-adic model. This model describes a wide class of ciphers (including substitution ciphers, substitution ciphers streaming, keystream ciphers in the alphabet of p elements), but certainly not all. Homomorphic and fully homomorphic ciphers are used to ensure the credibility of remote computing, including cloud technology. Within considered p-adic model we describe all homomorphic cryptographic primitives with respect to arithmetic and coordinate-wise logical operations in the ring of p-adic integers \(\mathbb Z_p\). We show that there are no fully homomorphic cryptographic primitives for each pair of the considered set of arithmetic and coordinate-wise logical operations on \(\mathbb Z_p\).
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References
V. Anashin, Uniformly distributed sequences of p-adic integers. Math. Notes 55, 109–133 (1994)
V. Anashin, Ergodic transformations in the space of p-adic integers. P-Adic Numbers Ultrametric Anal. Appl. 4(2), 151–160 (2012)
V. Anashin, A. Khrennikov, Applied Algebraic Dynamics. de Gruyter Expositions in Mathematics, vol. 49 (Walter de Gruyter, Berlin, 2009)
V. Anashin, A. Khrennikov, E. Yurova, T-functions revisited: new criteria for bijectivity/transitivity. Des. Codes Crypt. 7, 383–407 (2014)
D.K. Arrowsmith, F. Vivaldi, Geometry of p-adic Siegel discs. Phys. D: Nonlinear Phenom. 71, 222–236 (1994)
E.Y. Axelsson, On recent results of ergodic property for p-adic dynamical systems. P-Adic Numbers Ultrametric Anal. Appl. 6, 235–257 (2014)
E.Y. Axelsson, A. Khrennikov, Generalization of Hensel lemma: finding of roots of p-adic Lipschitz functions. J. Number Theory 158, 217–233 (2016)
E.Y. Axelsson, A. Khrennikov, Groups of automorphisms of p-adic integers and the problem of the existence of fully homomorphic ciphers. arXiv preprint, arXiv:1805.12537 (2018)
D. Bosio, F. Vivaldi, Round-off errors and p-adic numbers. Nonlinearity 13, 309–322 (2000)
A.-H. Fan, S. Fan, L. Liao, Y. Wang, On minimal decomposition of p-adic homographic dynamical systems. Adv. Math. 257, 92–135 (2014)
M. Haghighat, S. Zonouz, M. Abdel-Mottaleb, CloudID: trustworthy cloud-based and cross-enterprise biometric identification. Expert Sys. Appl. 42(21), 7905–7916 (2015)
S. Katok, P-adic Analysis Compared with Real. Mathematics Advanced Study Semesters. Student Mathematical Library, vol. 37 (American Mathematical Society, Providence, 2007)
A. Khrennikov, E. Yurova, Criteria of measure-preserving for p-adic dynamical systems in terms of the van der Put basis. J. Number Theory 133(2), 484–491 (2013)
A. Khrennikov, E. Yurova, Criteria of ergodicity for p-adic dynamical systems in terms of coordinate functions. Chaos, Solitons Fractals 60, 11–30 (2014)
F.M. Mukhamedov, J.F.F. Mendes, On the chaotic behavior of a generalized logistic p-adic dynamical system. J. Diff. Eqs. 243, 125–145 (2007)
D. Micciancio, A First Glimpse of Cryptography’s Holy Grail (Association for Computing Machinery, New York, 2010), p. 96
P.V. Parmar, S.B. Padhar, S.N. Patel, N.I. Bhatt, R.H. Jhaveri, Survey of various homomorphic encryption algorithms and schemes. Int. J. Comput. Appl. 91(8), 26–32 (2014)
J. Pettigrew, J.A.G. Roberts, F. Vivaldi, Complexity of regular invertible p-adic motions. Chaos 11, 849–857 (2001)
D.K. Rappe, Homomorphic cryptosystems and their applications. Dissertation, Technical University of Dortmund (2006)
W.H. Schikhof, Ultrametric Calculus. An Introduction to p-Adic Analysis (Cambridge University Press, Cambridge, 1984)
M. Van Dijk, C. Gentry, S. Halevi, V. Vaikuntanathan, Fully Homomorphic Encryption over the Integers. Advances in Cryptology Eurocrypt (Springer, Berlin, 2010), pp. 24–43
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Axelsson, E.Y., Khrennikov, A. (2019). Description of (Fully) Homomorphic Cryptographic Primitives Within the p-Adic Model of Encryption. In: Lindahl, K., Lindström, T., Rodino, L., Toft, J., Wahlberg, P. (eds) Analysis, Probability, Applications, and Computation. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-04459-6_23
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DOI: https://doi.org/10.1007/978-3-030-04459-6_23
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