In this paper we argue that hand-writing a program generator generator has a number of advantages compared to generating a program generator generator by self-application of a partial evaluator. We show the basic principles of how to construct a program generator generator by presenting a program generator generator for a skeletal language, and we argue that it is not more difficult to use the ...

it is shown that there are exactly seventy-eight 3-generator 2-groups of order $2^{11}$ with trivial schur multiplier. we then give 3-generator, 3-relation presentations for forty-eight of them proving that these groups have deficiency zero.

De, Trevisan and Tulsiani [CRYPTO 2010] show that every distribution over n-bit strings which has constant statistical distance to uniform (e.g., the output of a pseudorandom generator mapping n− 1 to n bit strings), can be distinguished from the uniform distribution with advantage by a circuit of size O(2 2). We generalize this result, showing that a distribution which has less than k bits of ...

Today we will study some conditions under which a very powerful pseudorandom generator can be shown to exist, and also some consequences of the existence of such a pseudorandom generator. We will start by assuming the existence of a permutation p : {0, 1}n → {0, 1}n which is computable in poly(n) time and which, for some constant δ > 0, is (2δn, 2−δn)-one way. (This is an extremely strong assum...

Q: Why are we using circuit lower bounds here, as opposed to a claim such as E ̸⊆ P for example? A: The proof of the Nisan-Wigderson pseudorandom generator relies on nonuniformity, by showing that distinguishing a pseudorandom generator implies a circuit for solving a hard problem – this reduction involves hardwiring advice into a circuit in order to solve the hard problem. A contradiction requi...

We postulate that a distribution is pseudorandom if it cannot be told apart from the uniform distribution by any eecient procedure. This yields a robust deenition of pseudorandom generators as eecient deterministic programs stretching short random seeds into longer pseudorandom sequences. Thus, pseudorandom generators can be used to reduce the randomness-complexity in any eecient procedure. Pse...

AC 0 circuits of size n log n augmented with log n MODm gates, for every odd integer m and any sufficiently small . As a consequence, for every odd integer m, we obtain a pseudorandom generator, based on the MOD2 function, for circuits of size S containing log S MODm gates. Our results are based on recent bounds of exponential sums that were previously introduced for proving lower bounds for MA...

We consider constant depth circuits augmented with few modular, or more generally, arbitrary symmetric gates. We prove that circuits augmented with o(log n) symmetric gates must have size n n) to compute a certain (complicated) function in ACC. This function is also hard on the average for circuits of size n logn augmented with o(log n) symmetric gates, and as a consequence we can get a pseudor...