О КОНФЕРЕНЦИИ ПАМЯТИ АНАТОЛИЯ АЛЕКСЕЕВИЧА КАРАЦУБЫ ПО ТЕОРИИ ЧИСЕЛ И ПРИЛОЖЕНИЯМ

In January, 2014, the I’st one-day international “Conference to the Memory of A.A. Karatsuba on Number Theory and Applications” took place in Steklov Mathematical Institute of Russian Academy of sciences. The aims of this conference were presentation of new and important results in different branches of number theory (especially in branches connected with works of A. A. Karatsuba), the exchange by new number-theoretical ideas and insight with new methods and tendencies in number theory. The 2’nd Conference was organized by Steklov Mathematical Institute of Russian Academy of sciences together with Moscow State university in January, 2015. The present paper contains wide annotations of reports of 2’nd Conference.  В январе 2014 г. в Математическом институте им. В. А. Стеклова РАН состоялась первая однодневная международная “Конференция памяти Анатолия Алексеевича Карацубы по теории чисел и приложениям”. Целями этой конференции были представление новых и значимых результатов в различных направлениях теории чисел (особенно в тех, что связаны с творчеством А.А. Карацубы), обмен новыми теоретико-числовыми идеями и ознакомление с новыми методами и тенденциями в теории чисел. Вторая международная Конференция была проведена Математическим институтом им. В. А. Стеклова РАН совместно с Московским Государственным университетом имени М. В. Ломоносова с 30 по 31 января 2015 г. Настоящая статья содержит развёрнутые аннотации докладов, прочитанных на второй Конференции.

Monotone and fast computation of Euler’s constant

Abstract We construct sequences of finite sums ( l ˜ n ) n ≥ 0 $(\tilde{l}_{n})_{n\geq 0}$ and ( u ˜ n ) n ≥ 0 $(\tilde{u}_{n})_{n\geq 0}$ converging increasingly and decreasingly, respectively, to the Euler-Mascheroni constant γ at the geometric rate 1/2. Such sequences are easy to compute and satisfy complete monotonicity-type properties. As a consequence, we obtain an infinite product representation for 2 γ $2^{\gamma }$ converging in a monotone and fast way at the same time. We use a probabilistic approach based on a differentiation formula for the gamma process

Toward verification of the Riemann hypothesis: Application of the Li criterion

We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence \{\lambda_k\}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we determine new bounds for relevant Riemann zeta function sums and the sequence itself. We find that the Riemann hypothesis holds if certain conjectured properties of a sequence \eta_j are valid. The constants \eta_j enter the Laurent expansion of the logarithmic derivative of the zeta function about s=1 and appear to have remarkable characteristics. {\em On our conjecture}, not only does the Riemann hypothesis follow, but an inequality governing the values \lambda_n and inequalities for the sums of reciprocal powers of the nontrivial zeros of the zeta function.Comment: to appear in Math. Physics, Analysis and Geometry; 1 figur

Efficient GF(p m ) Arithmetic Architectures for Cryptographic Applications

Recently, there has been a lot of interest on cryptographic applications based on fields OF(p"), for p > 2. This contribution presents OF(p TM) multipliers architectures, where p is odd. We present designs which trade area for performance based on the number of coefficients that the multiplier processes at one time. Families of irreducible polynomials are introduced to reduce the complexity of the modulo reduction operation and, thus, improved the efficiency of the multiplier. We, then, specialize to fields OF(3 TM) and provide the first cubing architecture pre- sented in the literature. We synthesize our architectures for the special case of OF(397) on the XCV1000-8-FG1156 and XC2VP20-7-FF1156 FPGAs and provide area/performance numbers and comparisons to previous OF(3 TM) and OF(2 TM) implementations. Finally, we provide tables of irreducible polynomials over OF(3) of degree m with 2 _< m _< 255

A variant of NTRU with non-invertible polynomials

Abstract. We introduce a generalization of the NTRU cryptosystem and describe its advantages and disadvantages as compared with the original NTRU protocol. This extension helps to avoid the potential problem of finding “enough ” invertible polynomials within very thin sets of polynomials, as in the original version of NTRU. This generalization also exhibits certain attractive “pseudorandomness ” properties that can be proved rigorously using bounds for exponential sums. 1 A Generalization of NTRU In this generalization of the original NTRU cryptosystem [5, 6], one selects integer parameters (N, p, q) and four sets Lf, Lg, Lϕ, Lm of polynomials in the ring R = Z[X]/(X N − 1) as in the standard version of NTRU. We denote by ⊙ the operation of multiplication in the ring R. The parameters q and p are distinct prime numbers such that gcd(N, q) = 1, and the sets Lf, Lg, Lϕ, Lm are chosen to satisfy the “width condition” �p ϕ ⊙ g + f ⊙ m � <