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D.4. RANDOMNESS EXTRACTORS in .NET Writer Code39 in .NET D.4. RANDOMNESS EXTRACTORS




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D.4. RANDOMNESS EXTRACTORS use vs .net code39 generator toadd barcode 39 on .net Microsoft Office Excel Website of this generator provides Code39 for .NET a black-box procedure for computing the underlying predicate when given oracle access to any potential distinguisher. Speci cally, in the proofs of Theorems 7.

19 and 8.18 (which holds for any = 2 (d ) ),13 this black-box procedure was implemented by a relatively small circuit (which depends on the underlying predicate). Hence, this procedure contains relatively little information (regarding the underlying predicate), on top of the observed -bit long output of the extractor/generator.

Speci cally, for some xed polynomial p, the amount of information encoded in the procedure (and def thus available to it) is upper-bounded by b = p( ), while the procedure is supposed to compute the underlying predicate correctly on each input. That is, b bits of information are supposed to fully determined the underlying predicate, which in turn is identical to the n-bit long source. However, if the source has min-entropy exceeding b, then it cannot be fully determined using only b bits of information.

It follows that the foregoing construction constitutes a (b + O(1), 1/6)-extractor (outputting = b (1) bits), where the constant 1/6 is the one used in the proof of Theorem 8.18 (and the argument holds provided that b = n (1) ). Note that this extractor uses a seed of length d = O(d ) = O(log n).

The argument can be extended to obtain (k, poly(1/k))-extractors that output k (1) bits using a seed of length d = O(log n), provided that k = n (1) . We note that the foregoing description has only referred to two abstract properties of the Nisan-Wigderson Generator: (1) the fact that this generator uses any worst-case hard predicate as a black-box, and (2) the fact that its analysis uses any distinguisher as a black-box. In particular, we viewed the ampli cation of worst-case hardness to inapproximability (performed in Theorem 7.

19) as part of the construction of the pseudorandom generator. An alternative presentation, which is more self-contained, replaces the ampli cation step of Theorem 7.19 with a direct argument in the current (information-theoretic) context and plugs the resulting predicate directly into Construction 8.

17. The advantages of this alternative include using a simpler ampli cation (since ampli cation is simpler in the information-theoretic setting than in the computational setting), and deriving transparent construction and analysis (which mirror Construction 8.17 and Theorem 8.

18, respectively). The alternative presentation. The foregoing analysis transforms a generic distinguisher into a procedure that computes the underlying predicate correctly on each input, which fully determines this predicate.

Hence, an upper bound on the information available to this procedure yields an upper bound on the number of possible outcomes of the source that are bad for the extractor. In the alternative presentation, we transform a generic distinguisher into a procedure that only approximates the underlying predicate; that is, the procedure yields a function that is relatively close to the underlying predicate. If the potential underlying predicates are far apart, then this yields the desired bound (on the number of bad source-outcomes that correspond to such predicates).

Thus, the idea is to encode the n-bit long source by an error-correcting code of length n = poly(n) and relative distance 0.5 (1/n)2 , and use the resulting codeword as a truth table of a predicate for Construction 8.17.

14 Such codes (coupled with ef cient encoding algorithms) do exist (see E.1.2.

5), and the bene t in using them is that each n -bit long string (determined by the information available to the aforementioned approximation procedure) may be. 13 14. Recalling that n = 2d , the .NET Code 3/9 restriction = 2 (d ) implies = n (1) . Indeed, the use of this error-correcting code replaces the hardness-ampli cation step of Theorem 7.

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