## Section: New Results

### Tensor rank of multiplication over finite fields

Determining the tensor rank of multiplication over finite fields is a problem of great interest in algebraic complexity theory, but it also has practical importance: it allows us to obtain multiplication algorithms with a low bilinear complexity, which are of crucial significance in cryptography. In collaboration with S. Ballet and J. Chaumine [35] , J. Pieltant obtained new asymptotic bounds for the symmetric tensor rank of multiplication in finite extensions of finite fields ${\mathbb{F}}_{q}$. In the more general (not-necessarily-symmetric) case, J. Pieltant and H. Randriam obtained new uniform upper bounds for multiplication in extensions of ${\mathbb{F}}_{q}$. They also gave purely asymptotic bounds substantially improving those coming from uniform bounds, by using a family of Shimura curves defined over ${\mathbb{F}}_{q}$. This work will appear in Mathematics of Computation [15] .