3,249 research outputs found
Query-to-Communication Lifting for BPP
For any -bit boolean function , we show that the randomized
communication complexity of the composed function , where is an
index gadget, is characterized by the randomized decision tree complexity of
. In particular, this means that many query complexity separations involving
randomized models (e.g., classical vs. quantum) automatically imply analogous
separations in communication complexity.Comment: 21 page
Simulation Theorems via Pseudorandom Properties
We generalize the deterministic simulation theorem of Raz and McKenzie
[RM99], to any gadget which satisfies certain hitting property. We prove that
inner-product and gap-Hamming satisfy this property, and as a corollary we
obtain deterministic simulation theorem for these gadgets, where the gadget's
input-size is logarithmic in the input-size of the outer function. This answers
an open question posed by G\"{o}\"{o}s, Pitassi and Watson [GPW15]. Our result
also implies the previous results for the Indexing gadget, with better
parameters than was previously known. A preliminary version of the results
obtained in this work appeared in [CKL+17]
MaxSAT Resolution and Subcube Sums
We study the MaxRes rule in the context of certifying unsatisfiability. We
show that it can be exponentially more powerful than tree-like resolution, and
when augmented with weakening (the system MaxResW), p-simulates tree-like
resolution. In devising a lower bound technique specific to MaxRes (and not
merely inheriting lower bounds from Res), we define a new proof system called
the SubCubeSums proof system. This system, which p-simulates MaxResW, can be
viewed as a special case of the semialgebraic Sherali-Adams proof system. In
expressivity, it is the integral restriction of conical juntas studied in the
contexts of communication complexity and extension complexity. We show that it
is not simulated by Res. Using a proof technique qualitatively different from
the lower bounds that MaxResW inherits from Res, we show that Tseitin
contradictions on expander graphs are hard to refute in SubCubeSums. We also
establish a lower bound technique via lifting: for formulas requiring large
degree in SubCubeSums, their XOR-ification requires large size in SubCubeSums
Adventures in Monotone Complexity and TFNP
Separations: We introduce a monotone variant of Xor-Sat and show it has exponential monotone circuit complexity. Since Xor-Sat is in NC^2, this improves qualitatively on the monotone vs. non-monotone separation of Tardos (1988). We also show that monotone span programs over R can be exponentially more powerful than over finite fields. These results can be interpreted as separating subclasses of TFNP in communication complexity.
Characterizations: We show that the communication (resp. query) analogue of PPA (subclass of TFNP) captures span programs over F_2 (resp. Nullstellensatz degree over F_2). Previously, it was known that communication FP captures formulas (Karchmer - Wigderson, 1988) and that communication PLS captures circuits (Razborov, 1995)
Block encryption of quantum messages
In modern cryptography, block encryption is a fundamental cryptographic
primitive. However, it is impossible for block encryption to achieve the same
security as one-time pad. Quantum mechanics has changed the modern
cryptography, and lots of researches have shown that quantum cryptography can
outperform the limitation of traditional cryptography.
This article proposes a new constructive mode for private quantum encryption,
named , which is a very simple method to construct quantum
encryption from classical primitive. Based on mode, we
construct a quantum block encryption (QBE) scheme from pseudorandom functions.
If the pseudorandom functions are standard secure, our scheme is
indistinguishable encryption under chosen plaintext attack. If the pseudorandom
functions are permutation on the key space, our scheme can achieve perfect
security. In our scheme, the key can be reused and the randomness cannot, so a
-bit key can be used in an exponential number of encryptions, where the
randomness will be refreshed in each time of encryption. Thus -bit key can
perfectly encrypt qubits, and the perfect secrecy would not be broken
if the -bit key is reused for only exponential times.
Comparing with quantum one-time pad (QOTP), our scheme can be the same secure
as QOTP, and the secret key can be reused (no matter whether the eavesdropping
exists or not). Thus, the limitation of perfectly secure encryption (Shannon's
theory) is broken in the quantum setting. Moreover, our scheme can be viewed as
a positive answer to the open problem in quantum cryptography "how to
unconditionally reuse or recycle the whole key of private-key quantum
encryption". In order to physically implement the QBE scheme, we only need to
implement two kinds of single-qubit gates (Pauli gate and Hadamard gate),
so it is within reach of current quantum technology.Comment: 13 pages, 1 figure. Prior version appears in
eprint.iacr.org(iacr/2017/1247). This version adds some analysis about
multiple-message encryption, and modifies lots of contents. There are no
changes about the fundamental result
Spatial Isolation Implies Zero Knowledge Even in a Quantum World
Zero knowledge plays a central role in cryptography and complexity. The
seminal work of Ben-Or et al. (STOC 1988) shows that zero knowledge can be
achieved unconditionally for any language in NEXP, as long as one is willing to
make a suitable physical assumption: if the provers are spatially isolated,
then they can be assumed to be playing independent strategies. Quantum
mechanics, however, tells us that this assumption is unrealistic, because
spatially-isolated provers could share a quantum entangled state and realize a
non-local correlated strategy. The MIP* model captures this setting. In this
work we study the following question: does spatial isolation still suffice to
unconditionally achieve zero knowledge even in the presence of quantum
entanglement? We answer this question in the affirmative: we prove that every
language in NEXP has a 2-prover zero knowledge interactive proof that is sound
against entangled provers; that is, NEXP \subseteq ZK-MIP*. Our proof consists
of constructing a zero knowledge interactive PCP with a strong algebraic
structure, and then lifting it to the MIP* model. This lifting relies on a new
framework that builds on recent advances in low-degree testing against
entangled strategies, and clearly separates classical and quantum tools. Our
main technical contribution consists of developing new algebraic techniques for
obtaining unconditional zero knowledge; this includes a zero knowledge variant
of the celebrated sumcheck protocol, a key building block in many probabilistic
proof systems. A core component of our sumcheck protocol is a new algebraic
commitment scheme, whose analysis relies on algebraic complexity theory.Comment: 55 pages. arXiv admin note: text overlap with arXiv:1704.0208
Spatial isolation implies zero knowledge even in a quantum world
Zero knowledge plays a central role in cryptography and complexity. The seminal work of Ben-Or et al. (STOC 1988) shows that zero knowledge can be achieved unconditionally for any language in NEXP, as long as one is willing to make a suitable physical assumption: if the provers are spatially isolated, then they can be assumed to be playing independent strategies. Quantum mechanics, however, tells us that this assumption is unrealistic, because spatially-isolated provers could share a quantum entangled state and realize a non-local correlated strategy. The MIP* model captures this setting. In this work we study the following question: does spatial isolation still suffice to unconditionally achieve zero knowledge even in the presence of quantum entanglement? We answer this question in the affirmative: we prove that every language in NEXP has a 2-prover zero knowledge interactive proof that is sound against entangled provers; that is, NEXP ⊆ ZK-MIP*. Our proof consists of constructing a zero knowledge interactive PCP with a strong algebraic structure, and then lifting it to the MIP* model. This lifting relies on a new framework that builds on recent advances in low-degree testing against entangled strategies, and clearly separates classical and quantum tools. Our main technical contribution is the development of new algebraic techniques for obtaining unconditional zero knowledge; this includes a zero knowledge variant of the celebrated sumcheck protocol, a key building block in many probabilistic proof systems. A core component of our sumcheck protocol is a new algebraic commitment scheme, whose analysis relies on algebraic complexity theory
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