Limits on the Stretch of Non-adaptive Constructions of Pseudo-Random Generators

The standard approach for constructing a large-stretch pseudo-random generator given a one-way permutation or given a smaller-stretch pseudo-random generator involves repeatedly composing the given primitive with itself. In this paper, we consider whether this approach is necessary, that is, whether there are constructions that do not involve composition. More formally, we consider black-box constructions of pseudo-random generators from pseudo-random generators of smaller stretch or from one-way permutations, where the constructions make only non- adaptive queries to the given object. We consider three classes of such constructions, and for each class, we give a black-box impossibility result that demonstrates a contrast between the stretch that can be achieved by adaptive and non-adaptive black-box constructions.

We first consider constructions that make constantly-many non-adaptive queries to a given pseudo-random generator, where the seed length of the construction is at most

O

(

log

n

) bits longer than the length

n

of each oracle query. We show that such constructions cannot achieve stretch that is even a single bit greater than the stretch of the given pseudo-random generator.

We then consider constructions with arbitrarily long seeds, but where oracle queries are collectively chosen in a manner that depends only on a portion of the seed whose length is at most

O

(

log

n

) bits longer than the length

n

of each query. We show that such constructions making constantly-many non-adaptive queries cannot achieve stretch that is

ω

(

log

n

) bits greater than the stretch of the given pseudo-random generator.

Finally, we consider a class of constructions motivated by streaming computation. Specifically, we consider constructions where the computation of each individual output bit depends only on the seed and on the response to a single query to a one-way permutation. We allow the seed to have a public portion that is arbitrarily long but must always be included in the output, and a non-public portion that is at most

O

(

log

n

) bits longer than the length

n

of each oracle query. We show that such constructions whose queries are chosen non-adaptively based only on the non-public portion of the seed cannot achieve linear stretch.