An Algorithm for Constructing Gröbner and Free Schreier Bases in Free Group Algebras

When Schreier transversals grow wild

“…Let us remark that part of the results here apply in a similar form also to Schreier transversals and Schreier bases of right ideals in free group algebras (see [9], [11], [10]).…”

“…The above shows that when we are given a Schreier transversal T and a corresponding Schreier basis B T for H < F then it is possible to obtain algorithmically a "normal form" modulo H for every element of F , i.e. its coset representative in T , and this demonstrates the importance of Schreier generating sets (whose shape and role is similar to those of Gröbner bases for algebras, see [10]). Thus the generalized word problem for H in F is then solvable.…”

Cryptanalysis of Schemes Based on Polynomial Symmetrical Decomposition

“…Other Algebraic attacks are the F4 and F5 algorithms of Faugére [31−33] as well as the GVW algorithm of Gao et al [34] . Barget et al give the total number of operations in F q for F 5 [33] , i.e., the cost of solving a (zero-dimensional, i.e., finite number of solutions) system of n 2 non-linear equations in 2n variables.…”

A general framework for Noetherian well ordered polynomial reductions

“…When such theory was independently rediscovered by Buchberger [16,17,20] under the name of Gröbner basis, the Pandora box was opened: Buchberger Theory and Algorithm introduced for polynomial rings over a field [16,17,20] was extended to polynomial ring over the integers [60], over Euclidean domains [61], over each ring on which ideal membership is testable and syzygies are computable [105], over domains [82] and PIRs [75], to non-commutative rings which satisfy Poincaré-Birkhoff-Witt Theorem [9], Lie algebras [4,5], solvable polynomial rings [62,63], skew polynomial rings [103,37,38,39], multivariate Ore extensions [83,84,23,26], other algebras which satisfy Poincaré-Birkhoff-Witt Theorem [3,65,66], semigroup rings [94,70,71], function rings [88,89], non-commutative free algebras [9], all effective rings [80,27], reduction rings [99,100,101,102,72], involutive bases [85,106,86,42,…”