Algebraic complexities and algebraic curves over finite fields
Journal of Complexity
Optimal algorithms for multiplication in certain finite fields using elliptic curves
SIAM Journal on Computing
Algebras Having Linear Multiplicative Complexities
Journal of the ACM (JACM)
On multiplication in finite fields
Journal of Complexity
General secure multi-party computation from any linear secret-sharing scheme
EUROCRYPT'00 Proceedings of the 19th international conference on Theory and application of cryptographic techniques
On the tensor rank of multiplication in any extension of F2
Journal of Complexity
The torsion-limit for algebraic function fields and its application to arithmetic secret sharing
CRYPTO'11 Proceedings of the 31st annual conference on Advances in cryptology
Algebraic geometric secret sharing schemes and secure multi-party computations over small fields
CRYPTO'06 Proceedings of the 26th annual international conference on Advances in Cryptology
Asymptotic bounds on frameproof codes
IEEE Transactions on Information Theory
A new approach to error-correcting codes
IEEE Transactions on Information Theory
Low-Discrepancy Sequences and Global Function Fields with Many Rational Places
Finite Fields and Their Applications
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We give new improvements to the Chudnovsky-Chudnovsky method that provides upper bounds on the bilinear complexity of multiplication in extensions of finite fields through interpolation on algebraic curves. Our approach features three independent key ingredients. *We allow asymmetry in the interpolation procedure. This allows to prove, via the usual cardinality argument, the existence of auxiliary divisors needed for the bounds, up to optimal degree. *We give an alternative proof for the existence of these auxiliary divisors, which is constructive, and works also in the symmetric case, although it requires the curves to have sufficiently many points. *We allow the method to deal not only with extensions of finite fields, but more generally with monogeneous algebras over finite fields. This leads to sharper bounds, and is designed also to combine well with base field descent arguments in case the curves do not have sufficiently many points. As a main application of these techniques, we fix errors in, improve, and generalize, previous works of Shparlinski-Tsfasman-Vladut, Ballet, and Cenk-Ozbudak. Besides, generalities on interpolation systems, as well as on symmetric and asymmetric bilinear complexities, are also discussed.