On bijective functions of fixed variables in the Galois field of~$p^k$~elements and on the ring of $p$-adic integers for an odd prime number~$p$

A. Lopez Perez ; O. Cuellar Justiz

Čebyševskij sbornik, Tome 24 (2023) no. 4, pp. 191-205.

Résumé

In this paper there are given necessary and sufficient conditions under which a function of fixed variables $\psi{:} \mathbb{F}_{q}^{i+1}\to\mathbb{F}_{q}$ is bijective, where $ i\in\mathbb{N}\cup\{0\}$, $\mathbb{F}_{q}^{i+1} $ is the $(i+1)$-ary Cartesian power of the Galois field $\mathbb{F}_{q}$ of $ q=p^k $ elements, $ p $ is an odd prime number and $k\in\mathbb{N}$. In addition, such conditions of the bijective functions $\psi$ of fixed variables are used to write a criterion for the preserving Haar measure of functions from the important class of 1-Lipschitz functions in terms of its coordinate functions on the ring of $p$-adic integers $\mathbb{Z}_p, p\neq2$. In particular, the representation of 1-Lipschitz functions in terms of its coordinate functions on the ring of $2$-adic integers $ \mathbb{Z}_2$ turned out to be a general and useful tool for obtaining mathematical results applied in cryptography. In this work, the research of such representation of 1-Lipschitz functions on the ring of $p$-adic integers $ \mathbb{Z}_p,p\neq2$ is being continued, with special attention to the representation of bijective 1-Lipschitz functions in terms of its coordinate functions on $ \mathbb{Z}_p, p\neq2$.

Détail

Keywords: Galois field, bijective function, $1$-Lipschitz function, Haar measure, Haar measure-preserving function, coordenate function, ergodic function.

Bibliographie

[1] Lausch H., Nöbauer W., Algebra of Polynomials, North-Holl. Publ. Co, American Elsevier Publ. Co, 1973 | MR | Zbl

[2] Mahler K., $ p $-Adic Numbers and Their Functions, 2nd., Cambridge University Press, Cambridge, 1981 | MR

[3] Koblitz N., $ p $-adic numbers, $ p $-adic analysis, and zeta-functions, Graduate Texts in Mathematics, 58, 2nd, Springer, Berlin, 1984 | MR

[4] Schiknof W. H., Ultrametric Calculus, Cambridge University Press, Cambridge, 1984 | MR

[5] Lidl R., Niederreiter H., Finite Fields, Mir, M., 1988

[6] Anashin V. S., “Uniformly Distribueted Sequences of $p$-Adic Integers”, Matematicheskie Zametki, 55:2 (1994), 3–46 | MR | Zbl

[7] Anashin V., Khrennikov A., Applied Algebraic Dynamics, De Gruyter Expositions in Mathematics, 49, Walter de Gruyter, Berlin–New York, 2009 | MR | Zbl

[8] Durand F., Paccaut F., “Minimal polynomial dynamics on the set of 3-adic integers”, Bull. London Math. Soc., 42:2 (2009), 302–314 | DOI | MR

[9] Shafarevich I. R., Remizov A. O., Linear Algebra and Geometry, Fizmatlit, 2009

[10] Khrennikov A., Yurova E., “Criteria of ergodicity for $ p $-adic dynamical systems in terms of coordinate functions”, Chaos, Solitons and Fractals, 60 (2014), 11–30 | DOI | MR | Zbl

[11] Yurova E. A., Khrennikov A. Y., “Generalization of Hensel lemma: finding the roots of p-adic Lipschitz functions”, J. Number Theory, 158 (2016), 217–233 | DOI | MR | Zbl

[12] Memic N., Ergodicity conditions on the group of 3-adic integers, Colloquium Mathematicum, Memic, N., 2016 | MR

[13] Memic N., “Ergodic Polynomials on Subsets of $ p $-Adic Integers”, $ p $-Adic Numbers, Ultrametric Analysis and Applications, 8:2 (2016), 149–159 | MR | Zbl

[14] Yurova E. A., Khrennikov A. Y., “Subcoordinate representation of $ p $-adic functions and generalization of Hensel lemma”, Izvestiya RAN : Ser. Mat., 82:3 (2018), 192–206 | DOI | MR | Zbl

[15] Wang S., Hu B., Liu Y., “The autocorrelation properties of single cycle polynomial T-function”, Des. Codes Cryptogr., 86 (2018), 1527–1540 | DOI | MR | Zbl

[16] Sangtae J., “Measure-preservation and the existence of a root of p-adic 1-Lipschitz functions in Mahlers expansion”, $ p $-Adic Numbers, Ultrametric Analysis and Applications, 10:3 (2018), 192–208 | MR | Zbl

[17] da Silva, D. W., de Araujo, C. P., Chow, E., Barillas, B. S., “A new approach towards fully homomorphic encryption over geometric algebra”, 10th Annual Ubiquitous Computing, Electronics Mobile Communication Conference (UEMCON), IEEE, 2019, 0241–0249

[18] Khrennikov A. Y., Yurova E. A., “Description of (fully) homomorphic cryptographic primitives within the p-adic model of encryption”, Analysis, Probability, Applications, and Computation, Springer, 2019, 241–148 | MR

[19] Khrennikov A. Y., Yurova E. A., “Description of (fully) homomorphic cryptographic primitives within the p-adic model of encryption”, Analysis, Probability, Applications, and Computation, Springer, 2019, 241–248 | MR

[20] Furno J., “Natural extensions for $ p $-adic {$\beta$}-shifts and other scaling maps”, Indagationes Mathematicae, 30 (2019), 1099–1108 | DOI | MR | Zbl

[21] Kim D., Kwon Y., K. Song K., “Minimality of 5-adic polynomial dynamical systems”, Dynamical Systems: An International Journal, 35:4 (2020), 584–596 | DOI | MR | Zbl

[22] da Silva D. W., A Mathematical Framework Towards Efficient Clifford-Based Homomorphic Cryptosystems Using p-Adic Numbers, University of Colorado, Colorado Springs, 2020 | MR

[23] Memić N., “Mahler coefficients of uniformly differentiable functions modulo $ p $”, International Journal of Number Theory, 2020 | MR

[24] Gouvêa F. Q., $ p $-adic Numbers An Introduction, 3rd., Springer, 2020 | MR

[25] da Silva D. W., Harmon L., Delavignette G., Araujo C., Leveled Fully Homomorphic Encryption Schemes with Hensel Codes, Cryptology ePrint Archive, 2021

[26] Anashin V. S., “The $ p $-adic Theory of Automata Functions”, Advances in Non-Archimedean Analysis and Applications, Springer, 2021, 9–113 | MR | Zbl

[27] Memic N., Mahler Coefficients of Some 3-adic Ergodic Functions, Acta Mathematica Vietnamica, 2021 | MR

[28] Zúñiga-Galindo W., Bourama T., Advances in Non-Archimedean Analysis and Applications: The $ p $-adic Methodology in STEAM-H, Springer Nature, 2021 | MR

[29] López A. P., “Measuring Preservation and Ergodicity of 1-Lipschitz Functions on the Ring of 3-Adic Integers in Terms of Coordinate Functions”, Vestnik Moskovskogo Universiteta, Seriya 15, 46:2 (2022), 30–33