FreivaldsTrick

LEMMA: Let X be any vector (not all zeros) in Z₂ⁿ. Let R be a random vector over Z₂ⁿ. Then Pr[X.R = 0] ≤ 1/2

PROOF: Fix X. Let i be an index such that X_i≠ 0.

For each vector R, let F(R) denote R with the ith bit flipped. Pair each vector R to its "partner" F(R). Since F(F(R)) = R, the partner relation is symmetric.

Now, for each pair of partners, (R, F(R)), exactly one of the two values {R.X, F(R).X} is 0 and the other is 1. Thus, half the vectors in Z₂ⁿ have dot-product 1 with X, the remaining vectors have dot-product 0 with X.

QED

Note X.R = ∑_i X_i R_i (mod 2).

Exercise: Generalize this to any finite field.

---

References:

• R. Freivalds. Fast probabilistic algorithms. Lecture Notes in Computer Science, 74:57-69, Springer-Verlag, 1979.