Abstract
<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula> be a prime and <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{F}_p $\end{document}</tex-math></inline-formula> the finite field with <inline-formula><tex-math id="M3">\begin{document}$ p $\end{document}</tex-math></inline-formula> elements. We show how, when given an superelliptic curve <inline-formula><tex-math id="M4">\begin{document}$ Y^n+f(X) \in \mathbb{F}_p[X,Y] $\end{document}</tex-math></inline-formula> and an approximation to <inline-formula><tex-math id="M5">\begin{document}$ (v_0,v_1) \in \mathbb{F}_p^2 $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M6">\begin{document}$ v_1^n = -f(v_0) $\end{document}</tex-math></inline-formula>, one can recover <inline-formula><tex-math id="M7">\begin{document}$ (v_0,v_1) $\end{document}</tex-math></inline-formula> efficiently, if the approximation is good enough. As consequence we provide an upper bound on the number of roots of such bivariate polynomials where the roots have certain restrictions. The results has been motivated by the predictability problem for non-linear pseudorandom number generators and, other potential applications to cryptography.</p>
Publisher
American Institute of Mathematical Sciences (AIMS)
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics,Computer Networks and Communications,Algebra and Number Theory,Applied Mathematics,Discrete Mathematics and Combinatorics,Computer Networks and Communications,Algebra and Number Theory
Reference25 articles.
1. L. Babai.On Lovász' lattice reduction and the nearest lattice point problem, Combinatorica, 6 (1986), 1-13.
2. R. C. Baker, M. Munsch and I. E. Shparlinski, Additive energy and a large sieve inequality for sparse sequences, preprint, arXiv: 2103.12659.
3. S. R. Blackburn, D. Gomez-Perez, J. Gutierrez, I. E. Shparlinski.Predicting nonlinear pseudorandom number generators, Math. Comp., 74 (2005), 1471-1494.
4. S. R. Blackburn, D. Gomez-Perez, J. Gutierrez, I. E. Shparlinski.Reconstructing noisy polynomial evaluation in residue rings, J. Algorithms, 61 (2006), 47-59.
5. J. Blömer and A. May, A tool kit for finding small roots of bivariate polynomials over the integers, in Advances in Cryptology–Eurocrypt 2005, Lecture Notes in Comput. Sci., 3494, Springer, Berlin, 2005,251–267.
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