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### Paper:

TR22-153 | 8th November 2022 18:40

#### Hardness against Linear Branching Programs and More

TR22-153
Publication: 13th November 2022 11:55
Keywords:

Abstract:

In a recent work, Gryaznov, Pudlák and Talebanfard (CCC '22) introduced a linear variant of read-once
branching programs, with motivations from circuit and proof complexity. Such a read-once linear branching program is
a branching program where each node is allowed to make \$\mathbb{F}_2\$-linear queries, and are read-once in the sense
that the queries on each path is linearly independent. As their main result, they constructed an explicit
function with average-case complexity \$2^{n/3-o(n)}\$ against a slightly restricted model, which they call
strongly read-once linear branching programs. The main tool in their lower bound result is a new type of extractor,
called directional affine extractors, that they introduced.

Our main result is an explicit function with \$2^{n-o(n)}\$ average-case complexity against the strongly read-once
linear branching program model, which is almost optimal. This result is based on a new connection from this problem
to sumset extractors, which is a randomness extractor model introduced by Chattopadhyay and Li (STOC '16) as a
generalization of many other well-studied models including two-source extractors, affine extractors and small-space extractors.
With this new connection, our lower bound naturally follows from a recent construction of sumset extractors
by Chattopadhyay and Liao (STOC '22). In addition, we show that directional affine extractors imply
sumset extractors in a restricted setting. We observe that such restricted sumset sources are enough to derive lower bounds, and obtain an arguably more modular proof of the lower bound by Gryaznov, Pudlák and Talebanfard.

We also initiate a study of pseudorandomness against linear branching programs. Our main result here
is a hitting set generator construction against regular linear branching programs with constant width. We derive this result
based on a connection to Kakeya sets over finite fields.

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