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The goal of this paper is to calculate explicitly the field index of any quintic number field K generated by a complex root α of a monic irreducible trinomial F(x) = x5 + ax + b ∈ ℤ[x]. In such a way we provide a complete answer to the Problem 22 of Narkiewicz [36] for this class of number fields. Namely, for every prime integer p, we evaluate the highest power of p dividing i(K). In particular, we give sufficient conditions on a and b, which guarantee the non-monogenity of K. [ABSTRACT FROM AUTHOR]

In this paper double, quadruple and octuple primes are discussed. Computational and theoretical properties as well as examples of quadruple and octuple primes are presented, where the octuple primes in most cases have primes between the defining quadruples although the existence of a pure quadruple prime has also been demonstrated computationally. The possibility of sixtentuple primes is also briefly discussed. [ABSTRACT FROM AUTHOR]

We discuss various connections between ideal classes, divisors, Picard and Chow groups of one-dimensional noetherian domains. As a result of these, we give a method to compute Chow groups of orders in global fields and show that there are infinitely many number fields which contain orders with trivial Chow groups. [ABSTRACT FROM AUTHOR]

In this paper, for any rational prime p and for a fixed positive integer ν p , we provide infinite families of number fields K satisfying ν p (i (K)) = ν p . As an application, if i (K) ≠ 1 , then K is not monogenic. We illustrate our results by some computational examples. [ABSTRACT FROM AUTHOR]

We say a monic polynomial g(x) ∈ ℤ[x] of degree n is monogenic if g(x) is irreducible over ℚ and {1, θ, ..., θn−1} is a basis for the ring ℤK of integers of number field K = ℚ(θ), where θ is a root of g(x). Let f (x) = x n + c ∑ i = 1 n (a x) n − i ∈ Z [ x ] and F (x) = x n + c ∑ i = 1 n a i − 1 x n − i ∈ Z [ x ] be irreducible polynomials having degree n ≥ 3. In this paper, we provide necessary and sufficient conditions involving only a, c, n for the polynomials f(x) and F(x) to be monogenic. As an application, we also provide a class of polynomials having a non square-free discriminant and Galois group Sn, the symmetric group on n letters. [ABSTRACT FROM AUTHOR]

Let K be a real algebraic number field. A well-known result, due to Meyer, states that the set ℘ K of Pisot numbers generating K is uniformly discrete and relatively dense in the interval [ 1 , ∞). A recent theorem of Dubickas yields that an algebraic integer, generating K, belongs to the set ℘ K - ℘ K if and only if its other conjugates are of modulus less than 2. In the present paper, we show that 1 ∈ ℘ K - ℘ K , and if K is totally real, then the elements of ℘ K - ℘ K are all algebraic integers of K whose images under the action of all embeddings of K into R , other than the identity of K, belong to the interval (- 2 , 2). Also, we prove that set { θ ′ - θ ∣ θ ′ ∈ ℘ K , θ ∈ ℘ K , θ ′ > θ and (θ , θ ′) ∩ ℘ K = ∅ } is finite and contains at least two elements (resp. at least 2 deg (K) - 1 ) elements when K ≠ Q (resp. when K is totally real). [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we calculate the index of any septic number field K generated by a root α of a monic irreducible trinomial F(x) = x7 + ax4 + b ∈ ℤ[x]. Our approach is based on Engstrom’s results and the factorization of any rational prime in K. In such a way we give a complete answer of Problem 22 of Narkiewicz ([28]) for this family of number fields. As an application of our results, if i(K) ≠ 1, then K is not monogenic. Also, we give generators of power integral bases in some cases where i(K) = 1. Our results are illustrated by some computational examples. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic 0, and let f (x) = x 8 + a x 6 + b x 4 + a x 2 + 1 ∈ F [ x ] be irreducible with Galois group Gal (f) . We extend results in the literature to give a complete and elementary classification of Gal (f) , identified as a transitive subgroup of S8 up to conjugacy. The approach involves testing whether a small number of elements in F are perfect squares. As an application, we give one-parameter families of polynomials for each of the six possible Galois groups. [ABSTRACT FROM AUTHOR]

The idea of extending the classical RSA protocol using algebraic number fields was introduced by Takagi and Naito (Construction of RSA cryptosystem over the algebraic field using ideal theory and investigation of its security. Electron Commun Japan Part III Fund Electr Sci. 2000;83:19–29). Recently, Zheng et al. proposed the use of the ring of algebraic integers of an algebraic number field and the lattice theory to present a high-dimensional form of RSA. The authors claim that their proposal is post-quantum and is significant both from the theoretical and practical point of view. In this article, we prove that the security of Zheng et al.’s scheme is still based on the factorization problem, and we present a practical quantum attack on this proposed scheme, our attack is a quantum polynomial time algorithm that employs Shor’s algorithm as a subroutine.

We study the index i (K) of any septic number field K generated by a root of an irreducible trinomial of type F (x) = x 7 + a x 2 + b ∈ Z [ x ] . We show that the unique prime which can divide i (K) is 2 . Moreover, we give necessary and sufficient conditions on a and b so that 2 is a common index divisor of K . Further, we show that i (K) = 2 whenever 2 divides i (K) . In this way, we answer completely Problem 6 and Problem 22 of Narkiewicz [34] for these families of number fields. As an application of our results, if 2 divides i (K) , then the ring O K of integers of K has no power integral basis. We illustrate our results by giving some numerical examples. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to calculate the index of any number field K generated by a root of an irreducible trinomial x 4 + a x + b ∈ Z [ x ] . Our approach is based on Engstrom's results and the factorization of 2 Z K and 3 Z K . In particular, we reformulate the Davis and Spearman's results. Namely, for every prime integer p, we evaluate ν p (i (K)) . The existence of a common index divisor of K guaranties the non monogenity of K. [ABSTRACT FROM AUTHOR]

The reduction algorithm is used to compute the reduced ideals of a number field. However, there are reduced ideals that can never be obtained from this algorithm. In this paper, we will show that these ideals have inverses of larger norms among reduced ones. Especially, for some number fields, we present the conditions of the reduced ideals produced by the reduction algorithm. Additionally, our results partly answer the question by R. Schoof who asks about the number of the reduced ideals that can not be produced by the reduction algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we study the monogenity of any number field defined by a monic irreducible trinomial F (x) = x 12 + a x m + b ∈ Z [ x ] with 1 ≤ m ≤ 11 an integer. For every integer m, we give sufficient conditions on a and b so that the field index i(K) is not trivial. In particular, if i (K) ≠ 1 , then K is not monogenic. For m = 1 , we give necessary and sufficient conditions on a and b, which characterize when a rational prime p divides the index i(K). For every prime divisor p of i(K), we also calculate the highest power p dividing i(K), in such a way we answer the problem 22 of Narkiewicz (Elementary and analytic theory of algebraic numbers, Springer Verlag, Auflag, 2004) for the number fields defined by trinomials x 12 + a x + b . [ABSTRACT FROM AUTHOR]

Given an integer n > 1 , it is a classical Diophantine problem that whether n can be written as a sum of two rational cubes. The study of this problem, considering several special cases of n, has a copious history that can be traced back to the works of Sylvester [14], Satgé [11], Selmer [12] etc., and up to the recent works of Alpöge et al. [1]. In this article, we consider the cube sum problem for cube-free integers n which are coprime to 3 and have exactly two distinct prime factors. [ABSTRACT FROM AUTHOR]

We consider pure quartic relative extensions of the number field Q (i) of type K = Q (a + b i 4) , where a , b ∈ Z and b ≠ 0 , such that a + b i ∈ Z [ i ] is square-free. We describe integral bases of these fields. The index form equation is reduced to a relative cubic Thue equation over Q (i) and some corresponding quadratic form equations. We consider monogenity of K and relative monogenity of K over Q (i) . We shall show how our former method based on the factors of the index form can be used in the relative case to exclude relative monogenity in some cases. [ABSTRACT FROM AUTHOR]

Elliptic curves have become key ingredients for instantiating zero-knowledge proofs and more generally proof systems. Recently, there have been many tailored constructions of these curves that aim at efficiently implementing different kinds of proof systems. In this survey we provide the reader with a comprehensive overview on existing work and revisit the contributions in terms of efficiency and security. We present an overview at three stages of the process: curves to instantiate a SNARK, curves to instantiate a recursive SNARK, and also curves to express an elliptic-curve related statement. We provide new constructions of curves for SNARKs and generalize the state-of-the-art constructions for recursive SNARKs. We also exhaustively document the existing work and open-source implementations. [ABSTRACT FROM AUTHOR]

Let K be a number field generated by a root θ of a monic irreducible trinomial F (x) = x n + a x m + b ∈ ℤ [ x ] . In this paper, we study the problem of monogenity of K. More precisely, we provide some explicit conditions on a, b, n, and m for which K is not monogenic. As applications, we show that there are infinite families of non-monogenic number fields defined by trinomials of degree n = 2r · 3k with r and k two positive integers. We also give infinite families of non-monogenic sextic number fields defined by trinomials. Some illustrating examples are giving at the end of this paper. [ABSTRACT FROM AUTHOR]

In this article, we deal with the problem of monogenity of quintic number fields defined by monic irreducible quadrinomials F (x) = x 5 + a x m + b x + c ∈ Z [ x ] with m = 2, 3, 4. We give sufficient conditions on a, b, c, and m so that the index of the field K is nontrivial and we evaluate ν p (i (K)) in each case. In particular, if i (K) ≠ 1 , then K is not monogenic. Our results are illustrated by computational examples. [ABSTRACT FROM AUTHOR]

For a number field K defined by a trinomial F (x) = x6 + ax + b ∈ ℤ[x], Jakhar and Kumar gave some necessary conditions on a and b, which guarantee the non-monogenity of K [25]. In this paper, for every prime integer p, we characterize when p is a common index divisor of K. In particular, if any one of these conditions holds, then K is not monogenic. In such a way our proposed results extend those of Jakhar and Kumar. [ABSTRACT FROM AUTHOR]

In this paper, for any septic number field K generated by a root α of a monic irreducible trinomial F (x) = x 7 + a x 3 + b ∈ Z [ x ] , we describe all prime power divisors of the index of K answering Problem 22 of Narkiewicz [ 26]. In particular, if i (K) ≠ 1 , then K is not mongenic. We illustrate our results by some computational examples. [ABSTRACT FROM AUTHOR]

We construct an efficient dynamic group signature (or more generally an accountable ring signature) from isogeny and lattice assumptions. Our group signature is based on a simple generic construction that can be instantiated by cryptographically hard group actions such as the CSIDH group action or an MLWE-based group action. The signature is of size O (log N) , where N is the number of users in the group. Our idea builds on the recent efficient OR-proof by Beullens, Katsumata, and Pintore (Asiacrypt'20), where we efficiently add a proof of valid ciphertext to their OR-proof and further show that the resulting non-interactive zero-knowledge proof system is online extractable. Our group signatures satisfy more ideal security properties compared to previously known constructions, while simultaneously having an attractive signature size. The signature size of our isogeny-based construction is an order of magnitude smaller than all previously known post-quantum group signatures (e.g., 6.6 KB for 64 members). In comparison, our lattice-based construction has a larger signature size (e.g., either 126 KB or 89 KB for 64 members depending on the satisfied security property). However, since the O (·) -notation hides a very small constant factor, it remains small even for very large group sizes, say 2 20 . [ABSTRACT FROM AUTHOR]

We describe a method to compute mod ℓ Galois representations contained in the H e ´ t 2 of surfaces. We apply this method to the case of a representation with values in GL 3 (F 9) attached to an eigenform over a congruence subgroup of SL 3 . We obtain, in particular, a polynomial with Galois group isomorphic to the simple group PSU 3 (F 9) and ramified at 2 and 3 only. [ABSTRACT FROM AUTHOR]

We show how our p-adic method to compute Galois representations occurring in the torsion of Jacobians of algebraic curves can be adapted to modular curves. The main ingredient is the use of "moduli-friendly" Eisenstein series introduced by Makdisi, which allow us to evaluate modular forms at p-adic points of modular curves and dispenses us of the need for equations of modular curves and for q-expansion computations in the construction of models of modular Jacobians. The resulting algorithm compares very favourably to our complex-analytic method. [ABSTRACT FROM AUTHOR]

A prime p is a Wolstenholme prime if 2 p p ≡ 2 mod p 4 , or, equivalently, if p divides the numerator of the Bernoulli number B p - 3 ; a Vandiver primep is one that divides the Euler number E p - 3 . Only two Wolstenholme primes and eight Vandiver primes are known. We increase the search range in the first case by a factor of ten, and show that no additional Wolstenholme primes exist up to 10 11 , and in the second case by a factor of twenty, proving that no additional Vandiver primes occur up to this same bound. To facilitate this, we develop a number of new congruences for Bernoulli and Euler numbers mod p that are favorable for computation, and we implement some highly parallel searches using GPUs. [ABSTRACT FROM AUTHOR]

Let f(x) ∈ ℤ[x] be monic and irreducible over ℚ, with deg(f) = n. Let K = ℚ(θ), where f(θ) = 0, and let ℤK denote the ring of integers of K. We say f(x) is monogenic if {1, θ, θ2, ..., θn−1} is a basis for ℤK. Otherwise, f(x) is called non-monogenic. In this article, we give necessary and sufficient conditions for a certain class of polynomials to be monogenic. Using these conditions allows us to generate infinite families of non-monogenic polynomials. In particular, for quadrinomials our results show that there exist infinitely many primes p ≥ 3, and integers t ≥ 1 coprime to p, such that f(x) = xp − 2ptxp−1 + p2t2xp−2 + 1 is non-monogenic. Finally, we illustrate this situation with an explicit example. [ABSTRACT FROM AUTHOR]

Let p ≥ 3 be a prime number. In this article, we study the canonical expansion of the primitive p n -th root of unity ζ p n in p-adic Mal'cev–Neumann field L p for n ≥ 1 . More precisely, we give the explicit formula for the first ℵ 0 terms of the expansion of ζ p n , and as an application, we use it to construct a uniformizer of K 2 , m = Q p ζ p 2 , p 1 / p m with m ≥ 1 . [ABSTRACT FROM AUTHOR]

This paper presents an efficient algorithm for computing 11th-power residue symbols in the cyclo-tomic field ℚ(ζ11),$ \mathbb{Q}\left( {{\zeta }_{11}} \right), $where 11 is a primitive 11th root of unity. It extends an earlier algorithm due to Caranay and Scheidler (Int. J. Number Theory, 2010) for the 7th-power residue symbol. The new algorithm finds applications in the implementation of certain cryptographic schemes.