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On Pseudorandom Generators with Linear Stretch in NC 0

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Cryptography in Constant Parallel Time

Part of the book series: Information Security and Cryptography ((ISC))

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Notes

  1. 1.

    Recall that an NC 0 LPRG can be composed with itself a constant number of times to yield an NC 0 PRG with an arbitrary linear stretch. See Remark 4.3.

  2. 2.

    In Chap. 5 we showed that there is an NC 0 construction of a commitment scheme from an arbitrary PRG including one with sublinear stretch (see Corollary 5.2). However, this construction makes a non-black-box use of the underlying PRG, and is thus quite inefficient. The only known parallel construction that makes a black-box use of the PRG is Naor’s original construction, which requires the PRG to have linear stretch.

  3. 3.

    Indeed, some impossibility results regarding randomness-efficient NC 0 sampling of the error distribution have recently appeared in [140].

  4. 4.

    Our assumption is essentially the same as Alekhnovich’s. The main difference between the two assumptions is that the noise vector e in [5] is a random vector of weight exactly \(\left \lceil \mu m \right \rceil \), as opposed to our noise vector whose entries are chosen to be 1 independently with probability μ. In Sect. 7.4.5 we show that our assumption is implied by Alekhnovich’s assumption. Intuitively, the implication follows from the fact that our noise vectors can be viewed as a convex combination of noise vectors of fixed weight. We do not know whether the converse implication holds. Indeed, a distribution D which can be described as a convex combination of distributions D 1,…,D n may be pseudorandom even if each of the distributions D i is not pseudorandom.

  5. 5.

    In fact, cn can be slightly superlinear.

  6. 6.

    The original formulation asserts that, for a random matrix \(M \leftarrow \mathcal {M}_{m(n)=O(n),n,3}\), the pair (M,y=f M,Q (U n )) is indistinguishable from the pair (M,y′) where y′ is a perturbed version of y in which a single bit in a random location is flipped. This implies that the distribution f M,Q (U n ) is weakly unpredictable.

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  1. School of Electrical Engineering, Tel Aviv University, Tel Aviv, Israel

    Benny Applebaum

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  1. Benny Applebaum
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Applebaum, B. (2014). On Pseudorandom Generators with Linear Stretch in NC 0 . In: Cryptography in Constant Parallel Time. Information Security and Cryptography. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17367-7_7

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  • DOI: https://doi.org/10.1007/978-3-642-17367-7_7

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