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Summary: Com S 633: Randomness in Computation
Lecture 22 Scribe: K.S.Gopalakrishnan
The role of Pseudo Random Generators in derandomization is discussed
in detail in the previous lectures. In this lecture we shall study the conditions
under which PRG exists and some related issues.
De nition: Let f : ! such that 8x; jf(x)j = l(jxj) , where l is a
polynomial bounded function. Moreover assume that jf(x)j > jxj for every
x. The function f : ! is said to be Cryptographic pseudo random
generator if the following holds.
f is polynomial time computable.
For every polynomials p and q, for every n, the distribution f(U n ) is
(p; 1
q
) pseudorandom.
Observe that f(U n ) is a p samplable distribution, and f(U n ) is a dis-
tribution over l(n) .
Now let us discuss under what conditions do such functions exist.
Claim: If P = NP then no Cryptographic pseudo random generator ex-
ist.
We shall prove this statement by contradiction.
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