FreivaldsTrick

ClassS04CS141 | recent changes | Preferences

Difference (from prior major revision) (no other diffs)

Added: 0a1,5
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


Changed: 2,6c7

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


PROOF: Fix X. Let i be an index such that Xi\neq 0.
PROOF: Fix X. Let i be an index such that Xi≠ 0.

Changed: 14c15
{R\cdot X, F(R)\cdot X}
{R.X, F(R).X}

Added: 21a23,24
Note X.R = i Xi Ri (mod 2).


Removed: 28,30d30




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 Xi≠ 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 Xi Ri (mod 2).

Exercise: Generalize this to any finite field.


References:


ClassS04CS141 | recent changes | Preferences
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Last edited February 10, 2004 8:27 pm by NealYoung (diff)
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