Algorithms Based on *-Algebras, and Their Applications to Isomorphism of Polynomials with One Secret, Group Isomorphism, and Polynomial Identity Testing

Abstract

We consider two basic algorithmic problems concerning tuples of (skew-)symmetric matrices. The first problem asks to decide, given two tuples of (skew-)symmetric matrices (B 1 , . . . , B m ) and (C 1 , . . . , C m ), whether there exists an invertible matrix A such that for every i ∈ {1, . . . , m}, A t B i A = C i . We show that this problem can be solved in randomized polynomial time over finite fields of odd size, the reals, and the complex numbers. The second problem asks to decide, given a tuple of square matrices (B 1 , . . . , B m ), whether there exist invertible matrices A and D, such that for every i ∈ {1, . . . , m}, AB i D is (skew-)symmetric. We show that this problem can be solved in deterministic polynomial time over fields of characteristic not 2. For both problems we exploit the structure of the underlying * -algebras (algebras with an involutive antiautomorphism), and utilize results and methods from the module isomorphism problem.Applications of our results range from multivariate cryptography, group isomorphism, to polynomial identity testing. Specifically, these results imply efficient algorithms for the following problems. (1) Test isomorphism of quadratic forms with one secret over a finite field of odd size. This problem belongs to a family of problems that serves as the security basis of certain authentication schemes proposed by Patarin (Eurocrypt 1996). (2) Test isomorphism of p-groups of class 2 and exponent p (p odd) with order p in time polynomial in the group order, when the commutator subgroup is of order p O( √ ) . (3) Deterministically reveal two families of singularity witnesses caused by the skew-symmetric structure. This represents a natural next step for the polynomial identity testing problem, in the direction set up by the recent resolution of the non-commutative rank problem (Garg-Gurvits-Oliveira-Wigderson, FOCS 2016; Ivanyos-Qiao-Subrahmanyam, ITCS 2017).

From Independent Sets and Vertex Colorings to Isotropic Spaces and Isotropic Decompositions: Another Bridge between Graphs and Alternating Matrix Spaces

“…Fortunately, it turns out that this problem has been studied in computer algebra over finite fields by Brooksbank, Maglione, and Wilson in [20]. Their strategy can be readily applied to R and C, using some ingredients from [59].…”

“…Theorem 5 ([20,Theorem 3.6], [59]). The isotropic 2-decomposition problem can be solved in randomized polynomial time over F q with q odd, and in deterministic polynomial time over R and C. Furthermore, over F q with q odd, the algorithm also outputs the linear bases of the two subspaces in an isotropic 2-decomposition.…”

“…Besides indicating how to handle R and C, we wish to give some reader a flavor of how the so-called * -algebra technique, pioneered by Wilson [111,112] in computer algebra, is applied to this setting. This technique was recently shown to be useful in polynomial identity testing and multivariate cryptography [59].…”

“…Based on this, involutions on simple algebras are also classified [3, Chap. X.4], and explicit lists for F q , R, and C can be found in [59].…”

“…As mentioned in Section 1.3, we give an exposition of the proof of Theorem 5 for F q in [20], using some ingredients from [59] to handle R and C. The main purpose is to give the reader a flavor of how the so-called * -algebra technique is applied in this context. We could not give all the details here, as that would be too long and unnecessary; the interested reader may wish to go to [21,59,111], which contain detailed proofs for using * -algebras to solve several closely related problems.…”