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Let ζ5be a primitive fifth root of unity and d≠1be a quadratic fundamental discriminant not divisible by 5. For the 5-dual cyclic quartic field M=Q((ζ5-ζ5-1)d)of the quadratic fields k1=Q(d)and k2=Q(5d)in the sense of the quintic reflection theorem, the possibilities for the isomophism type of the Galois group G5(2)M=Gal(M5(2)/M)of the second Hilbert 5-class field M5(2)of Mare investigated, when the 5-class group Cl5(M)is elementary bicyclic of rank two. Usually, the maximal unramified pro-5-extension M5(∞)of Mcoincides with M5(2)already. The precise length ℓ5Mof the 5-class tower of Mis determined, when G5(2)Mis of order less than or equal to 55. Theoretical results are underpinned by the actual computation of all 83, respectively 93, cases in the range 0

Let pand qbe two primes and ℓbe an odd positive square-free integer such that ℓ∉{1,p,q,pq}. In this paper we give the unit group of some number fields of the form Q(2,p,q,-ℓ).

In RFID technology, communication is based on random numbers, and the numbers used there are pseudo-random too (PRN). As for the PRN, it is generated by the computational tool that creates a sequence of numbers that are generally not related. In cryptography, we usually need to generate the encrypted and decrypted keys, so that we can use the genetic algorithm (GA) to find and present those keys. In this paper, the authors use the GA to find the random keys based on GA operators. The results of this generation attempt are tested through five statistical tests by which they try to determine the keys that are mostly responsible for message-encryption.

We consider the real cyclic quartic number field K=Q(nε0ℓ), where ℓ=2or ℓ≡1(mod4)is a prime, nis a square-free positive integer relatively prime to ℓand ε0the fundamental unit of its quadratic subfield k=Q(ℓ). In this article, assuming the 2-class group CK,2of Kis nontrivial and cyclic, we prove that the order of CK,2is exactly 2.

Let N=K(D3)be a cubic Kummer extension of the cyclotomic field K=Q(ζ3),containing a primitive cube root of unity ζ3,with cube free integer radicand D>1.Denote by fthe conductor of the abelian extension N/K,  and by N∗the relative genus field of N/K. The aim of the present work is to find out all positive integers Dand conductors fsuch that the genus group GalN∗/N≅Z/3Z×Z/3Zis elementary bicyclic.

We study the capitulation of 2-ideal classes of an infinite family of imaginary biquadratic number fields consisting of fields 𝕜=ℚ(pq1q2,i) $\mathbb {k} =\mathbb {Q}(\sqrt {pq_{1}q_{2}}, i)$, where i=−1 $i=\sqrt {-1}$and q1≡q2≡−p≡−1 (mod 4) are different primes. For each of the three quadratic extensions K/𝕜 $\mathbb {K}/\mathbb {k}$inside the absolute genus field 𝕜(∗)of 𝕜, we compute the capitulation kernel of K/𝕜 $\mathbb {K}/\mathbb {k}$. Then we deduce that each strongly ambiguous class of 𝕜/ℚ(i) $\mathbb {k}/\mathbb {Q}(i)$capitulates already in 𝕜(∗).

Let $$p_1$$ p1 and $$p_2$$ p2 be primes such that $$p_1\equiv p_2\equiv 5 \pmod 8$$ p1≡p2≡5mod8 , $$i=\sqrt{-1}$$ i=−1 , $$d=2p_1p_2$$ d=2p1p2 , $$\mathbb K =\mathbb Q (\sqrt{d},i)$$ K=Q(d,i) , $$\mathbb K _2^{(1)}$$ K2(1) be the Hilbert 2-class field of $$\mathbb K $$ K , $$\mathbb K _2^{(2)}$$ K2(2) be the Hilbert 2-class field of $$\mathbb K _2^{(1)}$$ K2(1) , $$G$$ G be the Galois group of $$\mathbb K _2^{(2)}/\mathbb K $$ K2(2)/K and $$\mathbb K ^{(*)}=\mathbb Q (\sqrt{p_1},\sqrt{p_2},\sqrt{2}, i)$$ K(∗)=Q(p1,p2,2,i) be the genus field of $$\mathbb K $$ K . The 2-part $$\mathbf C _{\mathbb{K },2}$$ CK,2 of the class group of $$\mathbb K $$ K is of type $$(2, 2, 2)$$ (2,2,2) . Our goal is to study the 2-class field tower of $$\mathbb K $$ K and to calculate the order of $$G$$ G .

Pour certains corps quartiques cycliques $\mathbb{K}$, on va étudier le problème de capitulation des 2-classes d’idéaux de $\mathbb{K}$et nous déterminons la structure du groupe de Galois du deuxième 2-corps de classes de Hilbert de $\mathbb{K}$sur $\mathbb{K}$.

The smart cards are increasingly used in several fields with critical data that require security. We cite, as example, the medical field and payment shopping with smart card. Therefore, the hardware and software security of smart cards is one of the key elements of the security of sensitive information handled. Currently, several scientific researchers are interested in studying and enhancing the smart cards security. The study of vulnerabilities is a prerequisite for building security guarantees of this type of devices. Indeed, each vulnerability can easily lead to an attack. In this paper, we generate vulnerability test cases based on models of Europay-MasterCard and Visa (EMV) specifications. We used SysML language to model the EMV transaction through machine state diagram. Then, we generated Event-B model that we exploit it to generate vulnerability and robustness test cases. This work aims to ensure the safety of EMV cards against attacks exploiting the vulnerability of the system.

Let d ε N, i = √−1, k = Q (√d, i), k1 be the 2-Hilbert class field of k, k2 the 2-Hilbert class field of k1, Ck, 2 the 2-component of the class group of k and G2 the Galois group of k2/k. We suppose that the group Ck, 2 is of type (2, 2); then k1 contains three extensions Fi/k, i = 1, 2, 3. The aim of this Note is to study the problem of capitulation of the 2-ideal classes of k in Fi, i = 1, 2, 3, and to determine the structure of G2.