 Bounds on Certain Multiplications of Affine Combinations.
 Joan Boyar, Faith Fich, and Kim S. Larsen.
Discrete Applied Mathematics, 52(2):155167, 1994.
Let
A and
B be
n x n matrices the entries of which
are affine combinations
of the variables
a_1, .., a_m, b_1, .., b_m over GF(2).
Suppose that, for each
i,
1 <= i <= m, the term
a_i b_i
is an element of the product matrix
C = A x B.
What is the maximum value that
m can have as a function
of
n?
This question arises from
a recent technique for improving the communication
complexity of zeroknowledge proofs.
The obvious upper bound of n^2 is improved to
n^2 / cubicroot(3) + O(n).
Tighter bounds are obtained for smaller values of n.
The bounds for n = 2, n = 3, and n = 4 are tight.

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