Curves and Their Applications to Factoring Polynomials

We present new methods for computing square roots and factorization of polynomials over finite fields. We also describe a method for computing in the Jacobian of a singular hyperelliptic curve. There is a compact representation of an element in the Jacobian of a smooth hyperelliptic curve over any field. This compact representation leads an efficient method for computing in Jacobians which is called Cantor's Algorithm. In one part of the dissertation, we show that an extension of this compact representation and Cantor's Algorithm is possible for singular hyperelliptic curves. This extension lead to the use of singular hyperelliptic curves for factorization of polynomials and computing square roots in finite fields. Our study shows that computing the square root of a number mod p is equivalent to finding any of the particular group elements in the Jacobian of a certain singular hyperelliptic curve. This is also true in the case of polynomial factorizations. Therefore the efficiency of our algorithms depends on only the efficiency of the algorithms for computing in the Jacobian of a singular hyperelliptic curve. The algorithms for computing in Jacobians of hyperelliptic curves are very fast especially for small genus and this makes our algorithms especially computing square roots algorithms competitive with the other well-known algorithms. In this work we also investigate superelliptic curves for factorization of polynomials

LEARNING ARITHMETIC READ-ONCE FORMULAS*

Abstract. A formula is read-once if each variable appears at most once in it. An arithmetic read-once formula is one in which the operators are addition, subtraction, multiplication, and division. We present polynomial time algorithms for exact learning of arithmetic read-once formulas over a field. We present a membership and equivalence query algorithm that identifies arithmetic read-once formulas over an arbitrary field. We present a randomized membership query algorithm (i.e., a randomized black box interpolation algorithm) that identifies such formulas over finite fields with at least 2n + 5 elements (where n is the number of variables) and over infinite fields. We also show the existence of nonuniform deterministic membership query algorithms for arbitrary read-once formulas over fields of characteristic 0, and division-free read-once formulas over fields that have at least 2n + elements. For our algorithms, we assume we are able to perform efficiently arithmetic operations on field elements and compute square roots in the field. It is shown that the ability to compute square roots is necessary in the sense that the problem of computing n square roots in a field can be reduced to the problem of identifying an arithmetic formula over n variables in that field. Our equivalence queries are of a slightly nonstandard form, in which counterexamples are required not to be inputs on which the formula evaluates to 0/0. This assumption is shown to be necessary for fields of size o(n! log n) in the sense that we prove there exists no polynomial time identification algorithm that uses only membership and standard equivalence queries

Square root computation over even extension fields

This paper presents a comprehensive study of the computation of square roots over finite extension fields. We propose two novel algorithms for computing square roots over even field extensions of the form \F_{q^{2}}, with $q=p^n,$ $p$ an odd prime and $n\geq 1$. Both algorithms have an associate computational cost roughly equivalent to one exponentiation in \F_{q^{2}}. The first algorithm is devoted to the case when $q\equiv 1 \bmod 4$, whereas the second one handles the case when $q\equiv 3 \bmod 4$. Numerical comparisons show that the two algorithms presented in this paper are competitive and in some cases more efficient than the square root methods previously known

Low Complexity Bit-Parallel Square Root Computation over GF(2m2^m) for all Trinomials

In this contribution we introduce a low-complexity bit-parallel algorithm for computing square roots over binary extension fields. Our proposed method can be applied for any type of irreducible polynomials. We derive explicit formulae for the space and time complexities associated to the square root operator when working with binary extension fields generated using irreducible trinomials. We show that for those finite fields, it is possible to compute the square root of an arbitrary field element with equal or better hardware efficiency than the one associated to the field squaring operation. Furthermore, a practical application of the square root operator in the domain of field exponentiation computation is presented. It is shown that by using as building blocks squarers, multipliers and square root blocks, a parallel version of the classical square-and-multiply exponentiation algorithm can be obtained. A hardware implementation of that parallel version may provide a speedup of up to 50\% percent when compared with the traditional version

On Taking Square Roots without Quadratic Nonresidues over Finite Fields

We present a novel idea to compute square roots over finite fields, without being given any quadratic nonresidue, and without assuming any unproven hypothesis. The algorithm is deterministic and the proof is elementary. In some cases, the square root algorithm runs in $\tilde{O}(\log^2 q)$ bit operations over finite fields with $q$ elements. As an application, we construct a deterministic primality proving algorithm, which runs in $\tilde{O}(\log^3 N)$ for some integers $N$.Comment: 14 page