7 research outputs found
INTERLEAVED GROUP PRODUCTS
Let be the special linear group . We show that if
and are sampled uniformly from large
subsets and of then their interleaved product is nearly uniform over . This extends a result of the first
author, which corresponds to the independent case where and are product
sets. We obtain a number of other results. For example, we show that if is
a probability distribution on such that any two coordinates are uniform
in , then a pointwise product of independent copies of is nearly
uniform in , where depends on only. Extensions to other groups are
also discussed.
We obtain closely related results in communication complexity, which is the
setting where some of these questions were first asked by Miles and Viola. For
example, suppose party of parties receives on its
forehead a -tuple of elements from . The parties
are promised that the interleaved product is equal either to the identity or to some
other fixed element , and their goal is to determine which of the two
the product is equal to. We show that for all fixed and all sufficiently
large the communication is , which is tight. Even for
the previous best lower bound was . As an application, we
establish the security of the leakage-resilient circuits studied by Miles and
Viola in the "only computation leaks" model
Mixing in Non-Quasirandom Groups
We initiate a systematic study of mixing in non-quasirandom groups. Let A and B be two independent, high-entropy distributions over a group G. We show that the product distribution AB is statistically close to the distribution F(AB) for several choices of G and F, including:
1) G is the affine group of 2x2 matrices, and F sets the top-right matrix entry to a uniform value,
2) G is the lamplighter group, that is the wreath product of ?? and ?_{n}, and F is multiplication by a certain subgroup,
3) G is H? where H is non-abelian, and F selects a uniform coordinate and takes a uniform conjugate of it.
The obtained bounds for (1) and (2) are tight.
This work is motivated by and applied to problems in communication complexity. We consider the 3-party communication problem of deciding if the product of three group elements multiplies to the identity. We prove lower bounds for the groups above, which are tight for the affine and the lamplighter groups
Simultaneous Multiparty Communication Protocols for Composed Functions
In the Number On the Forehead (NOF) multiparty communication model,
players want to evaluate a function on some input by broadcasting bits according to a
predetermined protocol. The input is distributed in such a way that each player
sees all of it except . In the simultaneous setting, the players
cannot speak to each other but instead send information to a referee. The
referee does not know the players' input, and cannot give any information back.
At the end, the referee must be able to recover from what
she obtained.
A central open question, called the barrier, is to find a function
which is hard to compute for or more players (where the 's
have size ) in the simultaneous NOF model. This has important
applications in circuit complexity, as it could help to separate from
other complexity classes. One of the candidates belongs to the family of
composed functions. The input to these functions is represented by a boolean matrix , whose row is the input and is a
block-width parameter. A symmetric composed function acting on is specified
by two symmetric - and -variate functions and , that output
where is the -th block of width
of . As the majority function is conjectured to be outside of
, Babai et. al. suggested to study , with large
enough.
So far, it was only known that is not enough for to
break the barrier in the simultaneous deterministic NOF model. In this
paper, we extend this result to any constant block-width , by giving a
protocol of cost for any symmetric composed
function when there are players.Comment: 17 pages, 1 figure; v2: improved introduction, better cost analysis
for the 2nd protoco