Quasi-subfield polynomials were introduced by Huang et al. together with a new algorithm to solve the Elliptic Curve Discrete Logarithm Problem (ECDLP) over finite fields of small characteristic. In this paper we provide both quasi-subfield polynomial families and theorem limiting their existence. Our results do not allow derive any speedup for ECDLP compared previous approaches.

Suppose F is a finite field with q elements and F is a subfield of L, which as an F -vector space, has dimension d. We say that d is the degree of the field extension L over F , and we write [L : F ] = d. We see that this occurs if M ∈ F [x] is irreducible of degree d and L = F [x]/(M). In this case we identify the field F with the subfield of L consisting of constant polynomials. Suppose that ...

The overstretched NTRU problem, which is the NTRU problem with super-polynomial size q in n, is one of the most important candidates for higher level cryptography. Unfortunately, Albrecht et al. in Crypto 2016 and Cheon et al. in ANTS 2016 proposed so-called subfield attacks which demonstrate that the overstretched NTRU problems with power-of-two cyclotomic modulus are not secure enough with gi...

In recent years, graph representations have been used extensively for modelling complicated structural information, such as circuits, images, molecular structures, biological networks, weblogs, XML documents and so on. As a result, frequent subgraph mining has become an important subfield of graph mining. This paper presents a novel Frequent Pattern Graph Mining algorithm, FP-GraphMiner, that c...

Let $ k be a field, G totally ordered abelian group and \mathbb K = k((G)) the maximal field of generalised power series, endowed with canonical valuation v $. We study \mathrm{-Aut} preserving automorphisms subfield k(G)\subseteq K\subseteq $, where k(G) is fraction ring k[G] Under assumption that satisfies two lifting properties we are able to generalise refine Hofberger's decomposition \math...

Kaplansky proved in 1942 that among ail fields with a valuation having a given divisible value group G, a given algebraically closed residue field R, and a given restriction to the minimal subfield (either the trivial valuation on Qor Fp , or the /?-adic valuation on Q), there is one that is maximal in the strong sensé that every other can be embedded in it. In this paper, we construct this fie...

A set of algebraic numbers has the Northcott property if each of its subsets of bounded Weil height is finite. Northcott’s Theorem, which has many Diophantine applications, states that sets of bounded degree have the Northcott property. Bombieri, Dvornicich and Zannier raised the problem of finding fields of infinite degree with this property. Bombieri and Zannier have shown that Q ab , the max...

A finite group G is called admissible over a given field if there exists a central division algebra that contains a G-Galois field extension as a maximal subfield. We give a definition of embedding problems of division algebras that extends both the notion of embedding problems of fields as in classical Galois theory, and the question which finite groups are admissible over a field. In a recent...