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
https://doi.org/10.22405/2226-8383-2023-24-4-191-205
Abstract
In this paper there are given necessary and sufficient conditions under which a function of fixed variables 𝜓: F^(𝑖+1)_𝑞 → F_𝑞 is bijective, where 𝑖 ∈ N ∪ {0}, F(𝑖+1)_𝑞 is the (𝑖 + 1)-ary Cartesian power of the Galois field F_𝑞 of 𝑞 = 𝑝^𝑘 elements, 𝑝 is an odd prime number and 𝑘 ∈ N. In addition, such conditions of the bijective functions 𝜓 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 Z_𝑝, 𝑝 ̸= 2. In particular, the representation of 1-Lipschitz functions in terms of its coordinate functions on the ring of 2-adic integers 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 Z_𝑝, 𝑝 ̸= 2 is being continued, with special attention to the representation of bijective 1-Lipschitz functions in terms of its coordinate functions on Z_𝑝, 𝑝 ̸= 2.
About the Authors
Perez Aniel’ LopezCuba
Master’s degree in Telematics, Information Security specialist
Justiz Oristela Cuellar
Cuba
graduate student of the Faculty of Mathematics of the Tula State
Lev Tolstoy Pedagogical Institute, master’s degree in Applied Mathematics, doctor of mathematical sciences, professor
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Review
For citations:
Lopez P.A., Cuellar J.O. 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. Chebyshevskii Sbornik. 2023;24(4):191-205. https://doi.org/10.22405/2226-8383-2023-24-4-191-205
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