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 multilinear
algorithm for the product of elements in A
(see Algebraic
complexity theory. vol. 315, Springer-Verlag). Our
result is constructive in deterministic polynomial time. We show
that given a tester of size \nu for an F-algebra A
and multilinear forms of degree d one can in deterministic
polynomial time construct a multilinear algorithm for the
multiplication of d elements of the algebra of multilinear
complexity \nu and vise versa.

This with the constructions in above paper give the first polynomial
time construction of a bilinear algorithm with linear bilinear
complexity for the multiplication of two elements in any extension
finite field.

We then study the problem of simulating a substitution of an
assignment from an F-algebra A in a degree d
multivariate polynomials with substitution of assignments from the
ground field F. We give a complete classification of all
algebras for which this can be done and show that this problem is
equivalent to constructing symmetric multilinear
algorithms for the product of d elements in A.