Computing Sparse Multiples of Polynomials

Efficiently Computing Real Roots of Sparse Polynomials

We propose an efficient algorithm to compute the real roots of a sparse polynomial f ∈ R[x] having k non-zero realvalued coefficients. It is assumed that arbitrarily good approximations of the non-zero coefficients are given by means of a coefficient oracle. For a given positive integer L, our algorithm returns disjoint disks ∆1, . . . ,∆s ⊂ C, with s < 2k, centered at the real axis and of radi...

Sparse squares of polynomials

We answer a question left open in an article of Coppersmith and Davenport which proved the existence of polynomials whose powers are sparse, and in particular polynomials whose squares are sparse (i.e., the square has fewer terms than the original polynomial). They exhibit some polynomials of degree 12 having sparse squares, and ask whether there are any lower degree complete polynomials with t...

Factors of Sparse Polynomials are Sparse

Abstract This paper was removed due to an error in the proof (Claim 4.12 as stated is not true). The authors would like to thank Ilya Volkovich for pointing out a counterexample to this papers main result in positive characteristic: If F is a field with prime characteristic p, then the polynomial xp1 + x p 2 + . . . + x p n has the following factor: (x1 + x2 + . . . + xn) p−1, which has sparsit...

Multiples of Primitive Polynomials and Their Products over GF(2)

A standard model of nonlinear combiner generator for stream cipher system combines the outputs of several independent Linear Feedback Shift Register (LFSR) sequences using a nonlinear Boolean function to produce the key stream. Given such a model, cryptanalytic attacks have been proposed by finding out the sparse multiples of the connection polynomials corresponding to the LFSRs. In this direct...

Probabilistic Algorithms Sparse Polynomials