Let $X\cong {SL}_2\left(ℝ\right)∕{SL}_2\left(\mathbb{Z}\right)$ be the space of unimodular lattices in ${ℝ}^2$, and for any $r\ge 0$ denote by $K_r\subset X$ the set of lattices such that all its nonzero vectors have supremum norm at least $e^{-r}$. These are compact nested subsets of $X$, with $K_0={\bigcap }_r{K}_r$ being the union of two closed horocycles. We use an explicit second moment formula for the Siegel transform of the indicator functions of squares in ${ℝ}^2$ centered at the origin to derive an asymptotic formula for the volume of sets $K_r$ as $r\to 0$. Combined with a zero-one law for the set of the $\psi$-Dirichlet numbers established by Kleinbock and Wadleigh (Proc. Amer. Math. Soc. 146 (2018), 1833–1844), this gives a new dynamical Borel–Cantelli lemma for the geodesic flow on $X$ with respect to the family of shrinking targets $\left\{K_r\right\}$.

We briefly discuss linear decomposition and nonlinear decomposition attacks using polynomial-time deterministic algorithms that recover the secret shared keys from public data in many schemes of algebraic cryptography. We show that in this case, contrary to common opinion, typical computational security assumptions are not very relevant to the security of the schemes; i.e., one can break the schemes without solving the algorithmic problems on which the assumptions are based. Also we present another and in some points similar approach, which was established by Tsaban et al.

Before demonstrating the applicability of these two methods to two well-known noncommutative protocols, we cryptanalyze two new cryptographic schemes that have not yet been analyzed.

Further, we introduce a novel method of construction of systems resistant against attacks via linear algebra. In particular, we propose improved versions of the well-known Diffie–Hellman-type (DH) and Anshel–Anshel–Goldfeld (AAG) algebraic cryptographic key-exchange protocols.

We consider power series with positive completely multiplicative coefficients. We obtain a large family of power series that have the unit circle as natural boundary, as well as new $\mathrm{\Omega }$-theorems for power series with positive completely multiplicative coefficients when the argument tends to roots of unity. Also $\mathrm{\Omega }$-estimates for some partial sums of completely multiplicative functions are given.

The Poupard polynomials are polynomials in one variable with integer coefficients, with some close relationship to Bernoulli and tangent numbers. They also have a combinatorial interpretation. We prove that every Poupard polynomial has all its roots on the unit circle. We also obtain the same property for another sequence of polynomials introduced by Kreweras and related to Genocchi numbers. This is obtained through a general statement about some linear operators acting on palindromic polynomials.

We derive the generating function (g.f.) of the number of colored circular palindromic compositions of $N$ with $K$ parts in terms of the g.f. of an input sequence $a$ that determines how many different colors each part of the composition can have. As a result, we get the g.f. of the number of all colored circular palindromic compositions of $N$. Using the latter formula and the g.f. of the number of colored circular compositions, we may easily derive the g.f. of the number of all colored dihedral compositions of $N$.

An asymptotic estimate for the number of positive primitive roots which are square-full integers in arithmetic progressions is derived. The employed method combines two techniques and is based on the character-sum method involving two characters; one character is to take care of being a primitive root, based on a result of Shapiro, and the other character is to take care of being square-full, based on a result of Munsch.