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Summary: Com S 633: Randomness in Computation
Lecture 4 Scribe: K. S. Gopalakrishnan
De nition: Let H be a set of functions from U to T. H is 2-universal if 8x 6= y 2 U; ; 2 T
Pr
h2H
[h(x) = ^ h(y) = ] = 1
jT j 2
Clearly, the set of all functions from U to T is 2-universal. However, the cardinality of this set
is very large. Our goal is to construct a family of 2-universal hash functions whose size is small.
We will see some examples now.
Let U = f0; 1; 2; ::::p 1g, and T = f0; 1; 2; ::::p 1g, where p is a prime.
H = fh ab j a; b 2 Z p g
where
h ab (x) = ax + b( mod p)
F ix x 6= y 2 U; ; 2 T
P r a;b2Zp [ax + b = ^ ax + b = ] =
f(a; b) 2 Z 2
p j ax + b = ^ ay + b = g
jZ p j 2
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