A remark on flat ternary cyclotomic polynomials

Let \(\Phi_n(x)\) be the \(n\)-th cyclotomic polynomial. In this paper, for odd primes \(p\lt q \lt r\) with \(q\equiv \pm1\pmod p\) and \(8r\equiv \pm1\pmod {pq}\), we prove that the coefficients of \(\Phi_{pqr}(x)\) do not exceed \(1\) in modulus if and only if (i) \(p=3\), \(q\geq 19\) and \(q\equiv 1\pmod 3\) or (ii) \(p=7\), \(q\geq83\) and \(q\equiv -1\pmod 7\).

Investigation of the expediency of using AVX512 for the implementation of modern algorithms for electronic signatures

Development and investigation of electronic signatures on algebraic lattices is one of the promising directions in post-quantum cryptography. Cryptosystems CRYSTALS-Dilithium and Falcon represent lattice cryptography in the category of electronic signatures in the NIST PQC open competition among the finalists. Most operations in these cryptosystems are reduced to addition and multiplication of polynomials in a finite field with a generating cyclotomic polynomial xN + 1. Using such a field allows the use of a number-theoretic transformation (NTT) to create fast and reliable software implementations. In practice, vectorized set (SIMD) instructions are used to achieve good performance. AVX2 instructions are most often used among existing implementations. At the same time, the possibility of using AVX512 instructions remains little explored. The purpose of this work is to investigate the feasibility of applying AVX512 instructions to optimization of the NTT, used in modern EPs on algebraic lattices. In particular, the paper presents a method for implementing a number-theoretic transformation using AVX512 for CRYSTALS-Dilithium and Falcon. An increase in performance is shown in comparison with the reference optimized author's implementations.

A FAMILY OF -SUPERCONGRUENCES MODULO THE CUBE OF A CYCLOTOMIC POLYNOMIAL

We establish a family of q-supercongruences modulo the cube of a cyclotomic polynomial for truncated basic hypergeometric series. This confirms a weaker form of a conjecture of the present authors. Our proof employs a very-well-poised Karlsson–Minton type summation due to Gasper, together with the ‘creative microscoping’ method introduced by the first author in recent joint work with Zudilin.

Some q-supercongruences modulo the square and cube of a cyclotomic polynomial

Two q-supercongruences of truncated basic hypergeometric series containing two free parameters are established by employing specific identities for basic hypergeometric series. The results partly extend two q-supercongruences that were earlier conjectured by the same authors and involve q-supercongruences modulo the square and the cube of a cyclotomic polynomial. One of the newly proved q-supercongruences is even conjectured to hold modulo the fourth power of a cyclotomic polynomial.

On the condition number of the Vandermonde matrix of the nth cyclotomic polynomial

Recently, Blanco-Chacón proved the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for some families of cyclotomic number fields by giving some upper bounds for the condition number Cond(Vn) of the Vandermonde matrix Vn associated to the nth cyclotomic polynomial. We prove some results on the singular values of Vn and, in particular, we determine Cond(Vn) for n = 2kpℓ, where k, ℓ ≥ 0 are integers and p is an odd prime number.

A further q-analogue of Van Hamme’s (H.2) supercongruence for primes p ≡ 3(mod4)

Long and Ramakrishna [Some supercongruences occurring in truncated hyper- geometric series, Adv. Math. 290 (2016) 773–808] generalized the (H.2) supercongruence of Van Hamme to the modulus [Formula: see text] case. In this paper, we give a [Formula: see text]-analogue of Long and Ramakrishna’s result for [Formula: see text]. A [Formula: see text]-congruence modulo the fourth power of a cyclotomic polynomial, which is a deeper [Formula: see text]-analogue of the (A.2) supercongruence of Van Hamme for [Formula: see text], is also formulated.