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Im trying to prove that an inefficient double-lengthening PRG exists.

My current approach is to produce a single random bit $b$ and output $r,b$. After producing the 'random' bit a length extension technique can be applied.

A candidate for such a random bit is $<r,x_r>$ where $<\cdot,\cdot>$ is inner product mod $n$.

Another direction is returning $r,f(r)$ Unfortunately, I am not able to proceed with the proof - will appreciate any hints\directions. Thanks!

**This was already in Inefficient double-lengthening PRG but no answer was provided

CryptoMan Leo
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1 Answers1

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What is $f$? There can't be information theoretic PRGs because an n-bit seed can only have at most n bits of entropy - and a PRG needs longer outputs. But if inefficient means taking time longer than the adversary, you can use a complexity theoretic PRG, e.g. the Nisan-Wigderson PRG https://www.math.ias.edu/~avi/PUBLICATIONS/MYPAPERS/NOAM/HARDNESS/final.pdf. The minimum assumption there is that there exists a hard problem for the adversary, i.e. a problem that is assumed to take more time to solve than the adversary's runtime.

platypusses
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