Pseudorandom Generators for Group Products

We prove that the pseudorandom generator introduced by Impagliazzo et al. in [INW94] with proper choice of parameters fools group products of a given finite group G. The seed length is O(log n(|G|O(1)+log 1 δ )), where n is the length of the word and δ is the allowed error. The result implies that the pseudorandom generator with seed length O(log n(2 logw) + log 1 δ )) fools read-once permutation branching programs of width w. As an application of the pseudorandom generator one obtains small-bias spaces for products over all finite groups [MZ09]. Institute of Mathematics, Academy of Sciences, Prague, e-mail: [email protected]. Partially supported by GA ČR P202/10/0854, project No. 1M0021620808 of MŠMT ČR, Institutional Research Plan No. AV0Z10190503 and grant IAA100190902 of GA AV ČR. Chennai Mathematical Institute, Chennai, India. e-mail: [email protected]. The work was done when the author was a student at The Institute of Mathematical Sciences, Chennai, India. Part of the work was done while visiting Institute of Mathematics, Academy of Sciences, Prague supported by project No. 1M0021620808 of MŠMT ČR. Institute of Mathematics, Academy of Sciences, Prague and Institute of Theoretical Computer Science, Prague, e-mail: [email protected]. Partially supported by Institutional Research Plan No. AV0Z10190503, project No. 1M0021620808 of MŠMT ČR and grant IAA100190902 of GA AV ČR.