Fast Bilinear Algorithms for Symmetric Tensor Contractions

In matrix-vector multiplication, matrix symmetry does not permit a straightforward reduction in computational cost. More generally, in contractions of symmetric tensors, the symmetries are not preserved in the usual algebraic form of contraction algorithms. We introduce an algorithm that reduces the bilinear complexity (number of computed elementwise products) for most types of symmetric tensor contractions. In particular, it lowers the bilinear complexity of symmetrized contractions of symmetric tensors of order {s+v} and {v+t} by a factor of {\frac{(s+t+v)!}{s!t!v!}} to leading order. The algorithm computes a symmetric tensor of bilinear products, then subtracts unwanted parts of its partial sums. Special cases of this algorithm provide improvements to the bilinear complexity of the multiplication of a symmetric matrix and a vector, the symmetrized vector outer product, and the symmetrized product of symmetric matrices. While the algorithm requires more additions for each elementwise product, the total number of operations is in some cases less than classical algorithms, for tensors of any size. We provide a round-off error analysis of the algorithm and demonstrate that the error is not too large in practice. Finally, we provide an optimized implementation for one variant of the symmetry-preserving algorithm, which achieves speedups of up to 4.58\times for a particular tensor contraction, relative to a classical approach that casts the problem as a matrix-matrix multiplication.

On the construction of elliptic Chudnovsky-type algorithms for multiplication in large extensions of finite fields

We indicate a strategy in order to construct bilinear multiplication algorithms of type Chudnovsky in large extensions of any finite field. In particular, using the symmetric version of the generalization of Randriambololona specialized on the elliptic curves, we show that it is possible to construct such algorithms with low bilinear complexity. More precisely, if we only consider the Chudnovsky-type algorithms of type symmetric elliptic, we show that the symmetric bilinear complexity of these algorithms is in [Formula: see text] where n corresponds to the extension degree, and [Formula: see text] is the iterated logarithm. Moreover, we show that the construction of such algorithms can be done in time polynomial in n. Finally, applying this method we present the effective construction, step by step, of such an algorithm of multiplication in the finite field 𝔽357.