hal-03832419 https://telecom-paris.hal.science/hal-03832419 https://telecom-paris.hal.science/hal-03832419/document https://telecom-paris.hal.science/hal-03832419/file/tri-sym.pdf doi:10.1007/978-3-030-68869-1_5 [INSTITUT-TELECOM] Institut Mines Télécom [INSMI] CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions [TELECOM-PARISTECH] Télécom Paris [UVSQ] Université de Versailles Saint-Quentin-en-Yvelines [UNIV-PARIS-SACLAY] Université Paris-Saclay [LTCI] Laboratoire Traitement et Communication de l'Information [INFRES] Département Informatique et Réseaux [IP_PARIS] Institut Polytechnique de Paris [INSTITUTS-TELECOM] composantes instituts telecom [UVSQ-UPSACLAY] UVSQ-UPSACLAY [C2] Equipe Cybersécurité et Cryptographie Trisymmetric Multiplication Formulae in Finite Fields Randriambololona, Hugues Rousseau, Édouard [INFO.INFO-SC] Computer Science [cs]/Symbolic Computation [cs.SC] [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] [INFO.INFO-AO] Computer Science [cs]/Computer Arithmetic [INFO.INFO-CC] Computer Science [cs]/Computational Complexity [cs.CC] COMM Multiplication is an expensive arithmetic operation, therefore there has been extensive research to find Karatsuba-like formulae reducing the number of multiplications involved when computing a bilinear map. The minimal number of multiplications in such formulae is called the bilinear complexity, and it is also of theoretical interest to asymptotically understand it. Moreover, when the bilinear maps admit some kind of invariance, it is also desirable to find formulae keeping the same invariance. In this work, we study trisymmetric, hypersymmetric, and Galois invariant multiplication formulae over finite fields, and we give an algorithm to find such formulae. We also generalize the result that the bilinear complexity and symmetric bilinear complexity of the twovariable multiplication in an extension field are linear in the degree of the extension, to trisymmetric bilinear complexity, and to the complexity of t-variable multiplication for any t ≥ 3. 2020-07 2022-10-27 en Springer International Publishing WAIFI 2020 Rennes, France