9 research outputs found

    The parity of the number of irreducible factors for some pentanomials

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    AbstractIt is well known that the Stickelberger–Swan theorem is very important for determining the reducibility of polynomials over a binary field. Using this theorem the parity of the number of irreducible factors for some kinds of polynomials over a binary field, for instance, trinomials, tetranomials, self-reciprocal polynomials and so on was determined. We discuss this problem for Type II pentanomials, namely xm+xn+2+xn+1+xn+1∈F2[x] for even m. Such pentanomials can be used for the efficient implementation of multiplication in finite fields of characteristic two. Based on the computation of the discriminant of these pentanomials with integer coefficients, we will characterize the parity of the number of irreducible factors over F2 and establish necessary conditions for the existence of this kind of irreducible pentanomials.Our results have been obtained in an experimental way by computing a significant number of values with Mathematica and extracting the relevant properties

    A Swan-like note for a family of binary pentanomials

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    In this note, we employ the techniques of Swan (Pacific J. Math. 12(3): 1099-1106, 1962) with the purpose of studying the parity of the number of the irreducible factors of the penatomial Xn+X3s+X2s+Xs+1∈F2[X]X^n+X^{3s}+X^{2s}+X^{s}+1\in\mathbb{F}_2[X], where ss is even and n>3sn>3s. Our results imply that if n≢±1(mod8)n \not\equiv \pm 1 \pmod{8}, then the polynomial in question is reducible

    On the parity of the number of irreducible factors of self-reciprocal polynomials over finite fields

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    AbstractUsing the Stickelberger–Swan theorem, the parity of the number of irreducible factors of a self-reciprocal even-degree polynomial over a finite field will be hereby characterized. It will be shown that in the case of binary fields such a characterization can be presented in terms of the exponents of the monomials of the self-reciprocal polynomial

    On Space-Time Trade-Off for Montgomery Multipliers over Finite Fields

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    La multiplication dans le corps de Galois à 2^m éléments (i.e. GF(2^m)) est une opérations très importante pour les applications de la théorie des correcteurs et de la cryptographie. Dans ce mémoire, nous nous intéressons aux réalisations parallèles de multiplicateurs dans GF(2^m) lorsque ce dernier est généré par des trinômes irréductibles. Notre point de départ est le multiplicateur de Montgomery qui calcule A(x)B(x)x^(-u) efficacement, étant donné A(x), B(x) in GF(2^m) pour u choisi judicieusement. Nous étudions ensuite l'algorithme diviser pour régner PCHS qui permet de partitionner les multiplicandes d'un produit dans GF(2^m) lorsque m est impair. Nous l'appliquons pour la partitionnement de A(x) et de B(x) dans la multiplication de Montgomery A(x)B(x)x^(-u) pour GF(2^m) même si m est pair. Basé sur cette nouvelle approche, nous construisons un multiplicateur dans GF(2^m) généré par des trinôme irréductibles. Une nouvelle astuce de réutilisation des résultats intermédiaires nous permet d'éliminer plusieurs portes XOR redondantes. Les complexités de temps (i.e. le délais) et d'espace (i.e. le nombre de portes logiques) du nouveau multiplicateur sont ensuite analysées: 1. Le nouveau multiplicateur demande environ 25% moins de portes logiques que les multiplicateurs de Montgomery et de Mastrovito lorsque GF(2^m) est généré par des trinômes irréductible et m est suffisamment grand. Le nombre de portes du nouveau multiplicateur est presque identique à celui du multiplicateur de Karatsuba proposé par Elia. 2. Le délai de calcul du nouveau multiplicateur excède celui des meilleurs multiplicateurs d'au plus deux évaluations de portes XOR. 3. Nous determinons le délai et le nombre de portes logiques du nouveau multiplicateur sur les deux corps de Galois recommandés par le National Institute of Standards and Technology (NIST). Nous montrons que notre multiplicateurs contient 15% moins de portes logiques que les multiplicateurs de Montgomery et de Mastrovito au coût d'un délai d'au plus une porte XOR supplémentaire. De plus, notre multiplicateur a un délai d'une porte XOR moindre que celui du multiplicateur d'Elia au coût d'une augmentation de moins de 1% du nombre total de portes logiques.The multiplication in a Galois field with 2^m elements (i.e. GF(2^m)) is an important arithmetic operation in coding theory and cryptography. In this thesis, we focus on the bit- parallel multipliers over the Galois fields generated by trinomials. We start by introducing the GF(2^m) Montgomery multiplication, which calculates A(x)B(x)x^{-u} in GF(2^m) with two polynomials A(x), B(x) in GF(2^m) and a properly chosen u. Then, we investigate the rule for multiplicand partition used by a divide-and-conquer algorithm PCHS originally proposed for the multiplication over GF(2^m) with odd m. By adopting similar rules for splitting A(x) and B(x) in A(x)B(x)x^{-u}, we develop new Montgomery multiplication formulae for GF(2^m) with m either odd or even. Based on this new approach, we develop the corresponding bit-parallel Montgomery multipliers for the Galois fields generated by trinomials. A new bit-reusing trick is applied to eliminate redundant XOR gates from the new multiplier. The time complexity (i.e. the delay) and the space complexity (i.e. the logic gate number) of the new multiplier are explicitly analysed: 1. This new multiplier is about 25% more efficient in the number of logic gates than the previous trinomial-based Montgomery multipliers or trinomial-based Mastrovito multipliers on GF(2^m) with m big enough. It has a number of logic gates very close to that of the Karatsuba multiplier proposed by Elia. 2. While having a significantly smaller number of logic gates, this new multiplier is at most two T_X larger in the total delay than the fastest bit-parallel multiplier on GF(2^m), where T_X is the XOR gate delay. 3. We determine the space and time complexities of our multiplier on the two fields recommended by the National Institute of Standards and Technology (NIST). Having at most one more T_X in the total delay, our multiplier has a more-than-15% reduced logic gate number compared with the other Montgomery or Mastrovito multipliers. Moreover, our multiplier is one T_X smaller in delay than the Elia's multiplier at the cost of a less-than-1% increase in the logic gate number

    XOR-counts and lightweight multiplication with fixed elements in binary finite fields

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    XOR-metrics measure the efficiency of certain arithmetic operations in binary finite fields. We prove some new results about two different XOR-metrics that have been used in the past. In particular, we disprove an existing conjecture about those XOR-metrics. We consider implementations of multiplication with one fixed element in a binary finite field. Here we achieve a complete characterization of all elements whose multiplication matrix can be implemented using exactly 2 XOR-operations. Further, we provide new results and examples in more general cases, showing that significant improvements in implementations are possible

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    Mathematical aspects of the design and security of block ciphers

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    Block ciphers constitute a major part of modern symmetric cryptography. A mathematical analysis is necessary to ensure the security of the cipher. In this thesis, I develop several new contributions for the analysis of block ciphers. I determine cryptographic properties of several special cryptographically interesting mappings like almost perfect nonlinear functions. I also give some new results both on the resistance of functions against differential-linear attacks as well as on the efficiency of implementation of certain block ciphers
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