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Efficient handling of sparse data is a key challenge in Computer Science.Binary convolutions, such as polynomial multiplication or the Walsh Transformare a useful tool in many applications and are efficiently solved.  In the last decade, several problems required efficient solution of sparsebinary convolutions. both randomized and deterministic algorithms weredeveloped for efficiently computing the sparse polynomial multiplication. Thekey operation in all these algorithms was length reduction. The sparse data ismapped into small vectors that preserve the convolution result. The reductionmethod used to-date was the modulo function since it preserves location (of the"1" bits) up to cyclic shift.  To date there is no known efficient algorithm for computing the sparse Walshtransform. Since the modulo function does not preserve the Walsh transform anew method for length reduction is needed. In this paper we present such a newmethod - polynomials. This method enables the development of an efficientalgorithm for computing the binary sparse Walsh transform. To our knowledge,this is the first such algorithm. We also show that this method allows a fasterdeterministic computation of sparse polynomial multiplication than currentlyknown in the literature.

Polynomials: a new tool for length reduction in binary discrete convolutions