3 research outputs found
An exponential separation between MA and AM proofs of proximity
Interactive proofs of proximity allow a sublinear-time verifier to check that a given input is close to the language, using a small amount of communication with a powerful (but untrusted) prover. In this work we consider two natural minimally interactive variants of such proofs systems, in which the prover only sends a single message, referred to as the proof. The first variant, known as MA-proofs of Proximity (MAP), is fully non-interactive, meaning that the proof is a function of the input only. The second variant, known as AM-proofs of Proximity (AMP), allows the proof to additionally depend on the verifier's (entire) random string. The complexity of both MAPs and AMPs is the total number of bits that the verifier observes - namely, the sum of the proof length and query complexity. Our main result is an exponential separation between the power of MAPs and AMPs. Specifically, we exhibit an explicit and natural property Pi that admits an AMP with complexity O(log n), whereas any MAP for Pi has complexity Omega~(n^{1/4}), where n denotes the length of the input in bits. Our MAP lower bound also yields an alternate proof, which is more general and arguably much simpler, for a recent result of Fischer et al. (ITCS, 2014). Lastly, we also consider the notion of oblivious proofs of proximity, in which the verifier's queries are oblivious to the proof. In this setting we show that AMPs can only be quadratically stronger than MAPs. As an application of this result, we show an exponential separation between the power of public and private coin for oblivious interactive proofs of proximity
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An Exponential Separation Between MA and AM Proofs of Proximity
AbstractInteractive proofs of proximity allow a sublinear-time verifier to
check that a given input is close to the language, using a
small amount of communication with a powerful (but untrusted)
prover. In this work, we consider two natural
minimally interactive variants of such
proofs systems, in which the prover only sends a single message,
referred to as the proof.
The first variant, known as -proofs of
Proximity (), is fully non-interactive, meaning that the
proof is a function of the input only. The second variant,
known as -proofs of Proximity (), allows the proof
to additionally depend on the verifier's (entire) random string. The
complexity of both s and s is the total number of bits
that the verifier observes—namely, the sum of the proof length
and query complexity.
Our main result is an exponential separation between the power of
s and s. Specifically, we exhibit an explicit and
natural property
Π
that admits an with complexity
O
(
log
n
)
, whereas any for
Π
has complexity
Ω
~
(
n
1
/
4
)
, where n denotes the length of the input
in bits. Our lower bound also yields an alternate proof,
which is more general and arguably much simpler, for a
recent result of Fischer et al. (ITCS, 2014). Also, Aaronson (Quantum Information & Computation 2012) has shown
a
Ω
(
n
1
/
6
)
lower bound for the same property
Π
.Lastly, we also consider the notion of oblivious proofs of proximity, in which
the verifier's queries are oblivious to the proof.
In this setting, we show
that s can only be quadratically stronger than s. As an
application of this result, we show an exponential separation
between the power of public and private coin for oblivious
interactive proofs of proximity.</jats:p