• Title, Summary, Keyword: Frobenius

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Efficient Exponentiation in Extensions of Finite Fields without Fast Frobenius Mappings

  • Nogami, Yasuyuki;Kato, Hidehiro;Nekado, Kenta;Morikawa, Yoshitaka
    • ETRI Journal
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    • v.30 no.6
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    • pp.818-825
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    • 2008
  • This paper proposes an exponentiation method with Frobenius mappings. The main target is an exponentiation in an extension field. This idea can be applied for scalar multiplication of a rational point of an elliptic curve defined over an extension field. The proposed method is closely related to so-called interleaving exponentiation. Unlike interleaving exponentiation methods, it can carry out several exponentiations of the same base at once. This happens in some pairing-based applications. The efficiency of using Frobenius mappings for exponentiation in an extension field was well demonstrated by Avanzi and Mihailescu. Their exponentiation method efficiently decreases the number of multiplications by inversely using many Frobenius mappings. Compared to their method, although the number of multiplications needed for the proposed method increases about 20%, the number of Frobenius mappings becomes small. The proposed method is efficient for cases in which Frobenius mapping cannot be carried out quickly.

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A new class of degenerate Frobenius-Euler-Hermite polynomials

  • Khan, Waseem A.
    • Advanced Studies in Contemporary Mathematics
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    • v.28 no.4
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    • pp.567-576
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    • 2018
  • In this paper, we introduce a new class of degenerate Frobenius-Euler-Hermite polynomials and investigate some identities of these polynomials. Some implicit summation formulae and symmetric identities are also derived by applying the generating functions. These results extend some known summations and identities of generalized degenerate Frobenius-Euler-Hermite polynomials studied by Pathan and Khan.

A NEW CLASS OF GENERALIZED APOSTOL-TYPE FROBENIUS-EULER-HERMITE POLYNOMIALS

  • Pathan, M.A.;Khan, Waseem A.
    • Honam Mathematical Journal
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    • v.42 no.3
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    • pp.477-499
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    • 2020
  • In this paper, we introduce a new class of generalized Apostol-type Frobenius-Euler-Hermite polynomials and derive some explicit and implicit summation formulae and symmetric identities by using different analytical means and applying generating functions. These results extend some known summations and identities of generalized Frobenius-Euler type polynomials and Hermite-based Apostol-Euler and Apostol-Genocchi polynomials studied by Pathan and Khan, Kurt and Simsek.

A NEW CLASS OF q-HERMITE-BASED APOSTOL TYPE FROBENIUS GENOCCHI POLYNOMIALS

  • Kang, Jung Yoog;Khan, Waseem A.
    • Communications of the Korean Mathematical Society
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    • v.35 no.3
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    • pp.759-771
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    • 2020
  • In this article, a hybrid class of the q-Hermite based Apostol type Frobenius-Genocchi polynomials is introduced by means of generating function and series representation. Several important formulas and recurrence relations for these polynomials are derived via different generating function methods. Furthermore, we consider some relationships for q-Hermite based Apostol type Frobenius-Genocchi polynomials of order α associated with q-Apostol Bernoulli polynomials, q-Apostol Euler polynomials and q-Apostol Genocchi polynomials.

FROBENIUS MAP ON THE EXTENSIONS OF T-MODULES

  • Woo, Sung-Sik
    • Communications of the Korean Mathematical Society
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    • v.13 no.4
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    • pp.743-749
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    • 1998
  • On the group of all extensions of elliptic modules by the Carlitz module we define Frobenius map and by using a concrete description of the extension group we give an explicit description of the Frobenius map.

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THE FROBENIUS PROBLEM FOR NUMERICAL SEMIGROUPS GENERATED BY THE THABIT NUMBERS OF THE FIRST, SECOND KIND BASE b AND THE CUNNINGHAM NUMBERS

  • Song, Kyunghwan
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.3
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    • pp.623-647
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    • 2020
  • The greatest integer that does not belong to a numerical semigroup S is called the Frobenius number of S. The Frobenius problem, which is also called the coin problem or the money changing problem, is a mathematical problem of finding the Frobenius number. In this paper, we introduce the Frobenius problem for two kinds of numerical semigroups generated by the Thabit numbers of the first kind, and the second kind base b, and by the Cunningham numbers. We provide detailed proofs for the Thabit numbers of the second kind base b and omit the proofs for the Thabit numbers of the first kind base b and Cunningham numbers.

SOME NEW CHARACTERIZATIONS OF QUASI-FROBENIUS RINGS BY USING PURE-INJECTIVITY

  • Moradzadeh-Dehkordi, Ali
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.2
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    • pp.371-381
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    • 2020
  • A ring R is called right pure-injective if it is injective with respect to pure exact sequences. According to a well known result of L. Melkersson, every commutative Artinian ring is pure-injective, but the converse is not true, even if R is a commutative Noetherian local ring. In this paper, a series of conditions under which right pure-injective rings are either right Artinian rings or quasi-Frobenius rings are given. Also, some of our results extend previously known results for quasi-Frobenius rings.

A REMARK ON THE NUMBER OF FROBENIUS CLASSES GENERATING THE GALOIS GROUP OF THE MAXIMAL UNRAMIFIED EXTENSION

  • Jin, Seokho;Kim, Kwang-Seob
    • Honam Mathematical Journal
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    • v.42 no.2
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    • pp.213-218
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    • 2020
  • Assume that K is a number field and Kur is the maximal unramified extension of it. When Gal(Kur/K) is an infinite group. It is known that Gal(Kur/K) is generated by finitely many Frobenius classes of Gal(Kur/K) by Y. Ihara. In this paper, we will give the explicit number of Frobenius classes which generate whole group Gal(Kur/K).

DING INJECTIVE MODULES OVER FROBENIUS EXTENSIONS

  • Wang, Zhanping;Yang, Pengfei;Zhang, Ruijie
    • Bulletin of the Korean Mathematical Society
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    • v.58 no.1
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    • pp.217-224
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    • 2021
  • In this paper, we study Ding injective modules over Frobenius extensions. Let R ⊂ A be a separable Frobenius extension of rings and M any left A-module, it is proved that M is a Ding injective left A-module if and only if M is a Ding injective left R-module if and only if A ⊗R M (HomR(A, M)) is a Ding injective left A-module.