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On Obtaining Pseudorandomness from Error-Correcting Codes

  • Shankar Kalyanaraman
  • Christopher Umans
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4337)

Abstract

A number of recent results have constructed randomness extractors and pseudorandom generators (PRGs) directly from certain error-correcting codes. The underlying construction in these results amounts to picking a random index into the codeword and outputting m consecutive symbols (the codeword is obtained from the weak random source in the case of extractors, and from a hard function in the case of PRGs).

We study this construction applied to general cyclic error-correcting codes, with the goal of understanding what pseudorandom objects it can produce. We show that every cyclic code with sufficient distance yields extractors that fool all linear tests. Further, we show that every polynomial code with sufficient distance yields extractors that fool all low-degree prediction tests. These are the first results that apply to univariate (rather than multivariate) polynomial codes, hinting that Reed-Solomon codes may yield good randomness extractors.

Our proof technique gives rise to a systematic way of producing unconditional PRGs against restricted classes of tests. In particular, we obtain PRGs fooling all linear tests (which amounts to a construction of ε-biased spaces), and we obtain PRGs fooling all low-degree prediction tests.

Keywords

Success Probability Linear Code Cyclic Code Prediction Test Linear Test 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Shankar Kalyanaraman
    • 1
  • Christopher Umans
    • 1
  1. 1.Dept of Computer ScienceCalifornia Institute of TechnologyPasadena

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