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Two versions of the Ajtai-Szemer\'edi Theorem are considered in the Cartesian square of a finite non-Abelian group $G$. In case $G$ is sufficiently quasirandom, we obtain strong forms of both versions: if $E \subseteq G\times G$ is fairly…
We are concerned with the Hidden Subgroup Problem for finite groups. We present a simplified analysis of a quantum algorithm proposed by Hallgren, Russell and Ta-Shma as well as a detailed proof of a lower bound on the probability of…
For a positive integer $m$, a finite group $G$ is said to admit a tournament $m$-semiregular representation (TmSR for short) if there exists a tournament $\Gamma$ such that the automorphism group of $\Gamma$ is isomorphic to $G$ and acts…
Random access codes are a type of communication task that is widely used in quantum information science. The optimal average success probability that can be achieved through classical strategies is known for any random access code. However,…
We revisit the so-called compressed oracle technique, introduced by Zhandry for analyzing quantum algorithms in the quantum random oracle model (QROM). To start off with, we offer a concise exposition of the technique, which easily extends…
The standard definition of quantum state randomization, which is the quantum analog of the classical one-time pad, consists in applying some transformation to the quantum message conditioned on a classical secret key $k$. We investigate…
It is well known that quantum, randomized and deterministic (sequential) query complexities are polynomially related for total boolean functions. We find that significantly larger separations between the parallel generalizations of these…
The notion of an open collar is generalized to that of a pseudo-collar. Important properties and examples are discussed. The main result gives conditions which guarantee the existence of a pseudo-collar structure on the end of an open…
We consider compact matrix quantum groups whose fundamental corepresentation matrix has entries which are partial isometries with central support. We show that such quantum groups have a simple representation as semi-direct product quantum…
Common random string model is a popular model in classical cryptography. We study a quantum analogue of this model called the common Haar state (CHS) model. In this model, every party participating in the cryptographic system receives many…
A permutoid is a set of partial permutations that contains the identity and is such that partial compositions, when defined, have at most one extension in the set. In 2004 Peter Cameron conjectured that there can exist no algorithm that…
We study the relationship between notions of pseudorandomness in the quantum and classical worlds. Pseudorandom quantum state generator (PRSG), a pseudorandomness notion in the quantum world, is an efficient circuit that produces states…
We prove the security of quantum key distribution against the most general attacks which can be performed on the channel, by an eavesdropper who has unlimited computation abilities, and the full power allowed by the rules of classical and…
A quantum tamper-evident encryption scheme is a non-interactive symmetric-key encryption scheme mapping classical messages to quantum ciphertexts such that an honest recipient of a ciphertext can detect with high probability any meaningful…
We extend the formalism of cluster state quantum secret sharing, as presented in Markham and Sanders [Phys. Rev. A 78, 042309 (2008)] and Keet et al. [Phy. Rev. A 82, 062315 (2010)], to the continuous-variable regime. We show that both…
We prove that finitary isomorphisms with finite expectation exist between Bernoulli shifts over the same free group only if the shifts have the same distribution. This generalizes the integer case result of Schmidt. We provide new proofs of…
We consider the following simple scenario: Alice has one of many possible messages, drawn from a known distribution, and wants to maximize the probability that Bob guesses her message correctly. We prove that if Alice can send only a qudit…
Subsets of F_2^n that are eps-biased, meaning that the parity of any set of bits is even or odd with probability eps close to 1/2, are powerful tools for derandomization. A simple randomized construction shows that such sets exist of size…
In this paper, we extend recent results about the distribution of even and odd gaps of a numerical semigroup. We find that, for any numerical semigroup, the distribution can be computed in terms of the numbers of or the sums of odd and even…
We generalize the Poisson limit theorem to binary functions of random objects whose law is invariant under the action of an amenable group. Examples include stationary random fields, exchangeable sequences, and exchangeable graphs. A…