A faster pseudo-primality test

We propose a pseudo-primality test using cyclic extensions of $\mathbb Z/n \mathbb Z$. For every positive integer $k \leq \log n$, this test achieves the security of $k$ Miller-Rabin tests at the cost of $k^{1/2+o(1)}$ Miller-Rabin tests.Comment: Published in Rendiconti del Circolo Matematico di Palermo Journal, Springe

On Kedlaya type inequalities for weighted means

In 2016 we proved that for every symmetric, repetition invariant and Jensen concave mean $\mathscr{M}$ the Kedlaya-type inequality $$\mathscr{A}\big(x_1,\mathscr{M}(x_1,x_2),\ldots,\mathscr{M}(x_1,\ldots,x_n)\big)\le \mathscr{M} \big(x_1, \mathscr{A}(x_1,x_2),\ldots,\mathscr{A}(x_1,\ldots,x_n)\big)$$ holds for an arbitrary $(x_n)$ ($\mathscr{A}$ stands for the arithmetic mean). We are going to prove the weighted counterpart of this inequality. More precisely, if $(x_n)$ is a vector with corresponding (non-normalized) weights $(\lambda_n)$ and $\mathscr{M}_{i=1}^n(x_i,\lambda_i)$ denotes the weighted mean then, under analogous conditions on $\mathscr{M}$, the inequality $$\mathscr{A}_{i=1}^n \big(\mathscr{M}_{j=1}^i (x_j,\lambda_j),\:\lambda_i\big) \le \mathscr{M}_{i=1}^n \big(\mathscr{A}_{j=1}^i (x_j,\lambda_j),\:\lambda_i\big)$$ holds for every $(x_n)$ and $(\lambda_n)$ such that the sequence $(\frac{\lambda_k}{\lambda_1+\cdots+\lambda_k})$ is decreasing.Comment: J. Inequal. Appl. (2018

On the relationship between continuous- and discrete-time quantum walk

Quantum walk is one of the main tools for quantum algorithms. Defined by analogy to classical random walk, a quantum walk is a time-homogeneous quantum process on a graph. Both random and quantum walks can be defined either in continuous or discrete time. But whereas a continuous-time random walk can be obtained as the limit of a sequence of discrete-time random walks, the two types of quantum walk appear fundamentally different, owing to the need for extra degrees of freedom in the discrete-time case. In this article, I describe a precise correspondence between continuous- and discrete-time quantum walks on arbitrary graphs. Using this correspondence, I show that continuous-time quantum walk can be obtained as an appropriate limit of discrete-time quantum walks. The correspondence also leads to a new technique for simulating Hamiltonian dynamics, giving efficient simulations even in cases where the Hamiltonian is not sparse. The complexity of the simulation is linear in the total evolution time, an improvement over simulations based on high-order approximations of the Lie product formula. As applications, I describe a continuous-time quantum walk algorithm for element distinctness and show how to optimally simulate continuous-time query algorithms of a certain form in the conventional quantum query model. Finally, I discuss limitations of the method for simulating Hamiltonians with negative matrix elements, and present two problems that motivate attempting to circumvent these limitations.Comment: 22 pages. v2: improved presentation, new section on Hamiltonian oracles; v3: published version, with improved analysis of phase estimatio

Do Israelis understand the Hebrew bible?

The Hebrew Bible should be taught like a foreign language in Israel too, argues Ghil'ad Zuckermann, inter alia endorsing Avraham Ahuvia’s recently-launched translation of the Old Testament into what Zuckermann calls high-register 'Israeli'. According to Zuckermann, Tanakh RAM fulfills the mission of 'red 'el ha'am' not only in its Hebrew meaning (Go down to the people) but also – more importantly – in its Yiddish meaning ('red' meaning 'speak!', as opposed to its colorful communist sense). Ahuvia's translation is most useful and dignified. Given its high register, however, Zuckermann predicts that the future promises consequent translations into more colloquial forms of Israeli, a beautifully multi-layered and intricately multi-sourced language, of which to be proud.Ghil'ad Zuckerman

Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms

We develop a theory of Tannakian Galois groups for t-motives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given t-motive is equal to the dimension of its Galois group. Using this result we prove that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent.Comment: 39 page

Actions infinit\'esimales dans la correspondance de Langlands locale p-adique

Let V be a two-dimensional absolutely irreducible p-adic Galois representation and let Pi be the p-adic Banach space representation associated to V via Colmez's p-adic Langlands correspondence. We establish a link between the infinitesimal action of GL_2(Q_p) on the locally analytic vectors of Pi, the differential equation associated to V via the theory of Fontaine and Berger, and the Sen polynomial of V. This answers a question of Harris and gives a new proof of a theorem of Colmez: Pi has nonzero locally algebraic vectors if and only if V is potentially semi-stable with distinct Hodge-Tate weights.Comment: Completely revised version, to appear in Math. Annale

Two grumpy giants and a baby

Pollard's rho algorithm, along with parallelized, vectorized, and negating variants, is the standard method to compute discrete logarithms in generic prime-order groups. This paper presents two reasons that Pollard's rho algorithm is farther from optimality than generally believed. First, ``higher-degree local anti-collisions'' make the rho walk less random than the predictions made by the conventional Brent--Pollard heuristic. Second, even a truly random walk is suboptimal, because it suffers from ``global anti-collisions'' that can at least partially be avoided. For example, after (1.5+o(1))\sqrt(l) additions in a group of order l (without fast negation), the baby-step-giant-step method has probability 0.5625+o(1) of finding a uniform random discrete logarithm; a truly random walk would have probability 0.6753\ldots+o(1); and this paper's new two-grumpy-giants-and-a-baby method has probability 0.71875+o(1). Keywords: Pollard rho, baby-step giant-step, discrete logarithms, complexit