In this paper we first show that Tester for an F-algebra A and multilinear forms (see Testers and their Applications ECCC 2012) is equivalent to multilinearalgorithm for the product of elements in A(see Algebraiccomplexity theory. vol. 315, Springer-Verlag). Ourresult is constructive in deterministic polynomial time. We showthat given a tester of size for an F-algebra Aand multilinear forms of degree d one can in deterministicpolynomial time construct a multilinear algorithm for themultiplication of d elements of the algebra of multilinearcomplexity and vise versa.

This with the constructions in above paper give the first polynomialtime construction of a bilinear algorithm with linear bilinearcomplexity for the multiplication of two elements in any extensionfinite field.

We then study the problem of simulating a substitution of anassignment from an F-algebra A in a degree dmultivariate polynomials with substitution of assignments from theground field F. We give a complete classification of allalgebras for which this can be done and show that this problem isequivalent to constructing symmetric multilinearalgorithms for the product of d elements in A.