On Obtaining Pseudorandomness from Error-Correcting Codes

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

Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes

We give an improved explicit construction of highly unbalanced bipartite expander graphs with expansion arbitrarily close to the degree (which is polylogarithmic in the number of vertices). Both the degree and the number of right-hand vertices are polynomially close to optimal, whereas the previous constructions of Ta-Shma et al. [2007] required at least one of these to be quasipolynomial in the optimal. Our expanders have a short and self-contained description and analysis, based on the ideas underlying the recent list-decodable error-correcting codes of Parvaresh and Vardy [2005]. Our expanders can be interpreted as near-optimal “randomness condensers,” that reduce the task of extracting randomness from sources of arbitrary min-entropy rate to extracting randomness from sources of min-entropy rate arbitrarily close to 1, which is a much easier task. Using this connection, we obtain a new, self-contained construction of randomness extractors that is optimal up to constant factors, while being much simpler than the previous construction of Lu et al. [2003] and improving upon it when the error parameter is small (e.g., 1/poly(n)).Engineering and Applied Science

On Obtaining Pseudorandomness from Error-Correcting Codes

Constructing pseudorandom objects based on codes has been the focus of some recent research. These constructions were based on specific algebraic codes and were rather simple in their structure in that a random index into a codeword was picked and $m$ subsequent symbols output. In this work, we explore the question of whether it is possible to extend the scope of application of this paradigm of constructions to larger families of codes. We show in this work that there exist such pseudorandom objects based on cyclic, linear codes that fool linear tests. When restricted to just algebraic codes, our techniques yield constructions that fool low-degree tests. Specifically, our results show that Reed-Solomon codes can be used to obtain pseudorandom objects, albeit in a weakened form. To the best of our knowledge, this is the first instance of Reed-Solomon codes being used to this effect. In the process, we also touch upon one of the holy grails of derandomization. It should come as no surprise that pseudorandom objects that fool low-degree tests are automatically correlated to derandomizing polynomial identity testing. We look at whether our constructions are general enough to answer this important question and while we come up short in our endeavor, we believe our approach adds a new perspective to this problem and hopefully a meaningful opening to solving it.</p

On Obtaining Pseudorandomness from Error-Correcting Codes

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

On obtaining pseudorandomness from error-correcting codes

c ○ 200