Inverting square systems algebraically is exponential
نویسندگان
چکیده
منابع مشابه
Inverting square systems algebraically is exponential
In this paper, we prove that the degree of regularity of the family of Square systems, an HFE type of systems, over a prime finite field of odd characteristics q is exactly q, and therefore prove that • inverting Square systems algebraically is exponential, when q = O(n), where n is the number of variables of the system.
متن کاملSuuciency, Exponential Families, and Algebraically Independent Numbers
We construct a continuous, strictly increasing, and bounded function T which maps Lebesgue almost the entire real line onto an algebraically independent set of real numbers. It follows that P n i=1 T (X i) is a uniformly continuous one-dimensional suucient statistic for every IID{model P n = fP n : P 2 Pg such that each P 2 P has a Lebesgue density. It also follows that the one{parameter expone...
متن کاملCertifying solutions to square systems of polynomial-exponential equations
Smale’s α-theory certifies that Newton iterations will converge quadratically to a solution of a square system of analytic functions based on the Newton residual and all higher order derivatives at the given point. Shub and Smale presented a bound for the higher order derivatives of a system of polynomial equations based in part on the degrees of the equations. For a given system of polynomial-...
متن کاملThe square root rank of the correlation polytope is exponential
The square root rank of a nonnegative matrix A is the minimum rank of a matrix B such that A = B ◦B, where ◦ denotes entrywise product. We show that the square root rank of the slack matrix of the correlation polytope is exponential. Our main technique is a way to lower bound the rank of certain matrices under arbitrary sign changes of the entries using properties of the roots of polynomials in...
متن کاملInverting HFE Is Quasipolynomial
In the last ten years, multivariate cryptography has emerged as a possible alternative to public key cryptosystems based on hard computational problems from number theory. Notably, the HFE scheme [17] appears to combine efficiency and resistance to attacks, as expected from any public key scheme. However, its security is not yet completely understood. On one hand, since the security is related ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Finite Fields and Their Applications
سال: 2014
ISSN: 1071-5797
DOI: 10.1016/j.ffa.2013.10.004