Algebra of Polynomially Bounded Sequences and Negligible Functions

In this article we formalize negligible functions that play an essential role in cryptology [10], [2]. Generally, a cryptosystem is secure if the probability of succeeding any attacks against the cryptosystem is negligible. First, we formalize the algebra of polynomially bounded sequences [20]. Next, we formalize negligible functions and prove the set of negligible functions is a subset of the algebra of polynomially bounded sequences. Moreover, we then introduce equivalence relation between polynomially bounded sequences, using negligible functions.Shinshu University, Nagano, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Mihir Bellare. A note on negligible functions, 2002.Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433–439, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507–513, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Oded Goldreich. Foundations of Cryptography: Volume 1, Basic Tools. Cambridge University Press, 2001.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1): 35–40, 1990.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841–845, 1990.Artur Korniłowicz. On the real valued functions. Formalized Mathematics, 13(1):181–187, 2005.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273–275, 1990.Richard Krueger, Piotr Rudnicki, and Paul Shelley. Asymptotic notation. Part I: Theory. Formalized Mathematics, 9(1):135–142, 2001.Richard Krueger, Piotr Rudnicki, and Paul Shelley. Asymptotic notation. Part II: Examples and problems. Formalized Mathematics, 9(1):143–154, 2001.Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265–268, 1997.Hiroyuki Okazaki and Yuichi Futa. Polynomially bounded sequences and polynomial sequences. Formalized Mathematics, 23(3):205–213, 2015. doi:10.1515/forma-2015-0017. [Crossref]Henryk Oryszczyszyn and Krzysztof Prażmowski. Real functions spaces. Formalized Mathematics, 1(3):555–561, 1990.Konrad Raczkowski and Andrzej Nędzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213–216, 1991.Konrad Raczkowski and Andrzej Nędzusiak. Series. Formalized Mathematics, 2(4):449–452, 1991.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329–334, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825–829, 2001.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990