Difference families from rings

A strong indication about the existence of a (7p, 4, 1) difference family with p ? 7 (mod 12) a prime has been given in [11]. Here, developing some ideas of that paper, we give, much more generally, a strong indication about the existence of a cyclic (pq, 4, 1) difference family whenever p and q are primes congruent to 7 (mod 12) and of a cyclic (pq, 5, 1) difference family whenever p and q are primes congruent to 11 (mod 20). Indeed we give an algorithm for their construction that seems to be always successful and we have checked it works whenever both primes p and q do not exceed 1,000. All our (pq, 4, 1) and (pq, 5, 1) difference families have the nice property of admitting a multiplier of order 3 or 5, respectively, that fixes almost all base blocks. As an intermediate result we also find an optimal (p, 5, 1) optical orthogonal code for every prime p ? 11 (mod 20) not exceeding 10,000.

We prove new existence and nonexistence results for modular Golomb rulers in this paper. We completely determine which modular Golomb rulers of order $k$ exist, for all $k\leq 11$, and we present a general existence result that holds for all $k \geq 3$. We also derive new nonexistence results for infinite classes of modular Golomb rulers and related structures such as difference packings, optical orthogonal codes, cyclic Steiner systems and relative difference families.

Designs from pairs of finite fields. A cyclic unital U(6) and other regular steiner 2-designs

Grid-block difference families and related combinatorial structures

On group automata over finite fields

We consider q-deformations of Witt rings, based on geometric operations on zeta functions of motives over finite fields, and we use these deformations to construct q-analogs of the Bost-Connes quantum statistical mechanical system. We show that the q-deformations obtained in this way can be related to Habiro ring constructions of analytic functions over F_1 and to categorifications of Bost-Connes systems.

Abstruct- Elliptic cuwe cryptosystems have the potential to provide relatively small block size, high-security public key schemes that can be efficiently implemented. As with other known public key schemes, such as RSA and discrete exponentiation in a finite field, some care must be exercised when selecting the parameters involved, in this case the elliptic curve and the underlying field. Specific classes of cuwes that give little or no advantage over previously known schemes are discussed. The main result of the paper is to demonstrate the reduction of the elliptic curve logarithm problem to the logarithm problem in the multiplicative group of an extension of the underlying finite field. For the class of supersingular elliptic curves, the reduction takes probabilistic polynomial time, thus providing a probabilistic subexponential time algorithm for the former problem. Index Tem- Discrete logarithms, elliptic curves, public key CryPtOSraPhY.

This paper presents a variant of Advance Encryption Standard (AES) cipher that is depending on the same structure of the original AES Substitution Permutation Network (SPN) and the same mathematical background represented by Galois Field (GF28). The proposed cipher encrypt and decrypt data with block size of 512-bit and with key size of 128-bit expanded into 512-bits which means encrypt block that four times larger than the block of the original cipher and provide high margin of security with key size 128-bit of key needs 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">512</sup> possibility instead of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">128</sup> , since this cipher is based on two perfect techniques for the expansion and generating key reaching to 512-bit for each round. This cipher depends upon strong mathematical foundations and solid algebraic representation and introduces deeper insights in developing AES.

Explicit Constructions of Perfect Hash Families from Algebraic Curves over Finite Fields

Seven infinite classes of relative difference families with variable block sizes are presented explicitly. In particular, a balanced (gv,g,K,1)-DF with g=Σk∈K[(k2-k)/2] is explicitly given for: (i) K={3,4,5} and every v coprime to 6; (ii) K={3,4,6}, {3,5,6} or {3,4,5,6} and every v coprime to 30. As far as the authors are aware, these difference families can be viewed as the first explicit constructions of infinite classes of optimal variable-weight optical orthogonal codes with more than two weights. It is observed, however, that there are infinitely many values of v for which an optimal (v,W,1,Q) -OOC exists, whatever the set of weights W and the weight distribution sequence Q are.

Two types of permutation polynomials with special forms

The p-ary function f(x) mapping GF(p4k) to GF(p) and given by f(x)=Tr4k(xp3k+p2k-pk+1+x2) is proven to be a weakly regular bent function and the exact value of its Walsh transform coefficients is found. This is the first proven infinite class of nonquadratic generalized bent functions over the fields of an arbitrary odd characteristic. The proof is based on a few new results in the area of exponential sums and polynomials over finite fields that may also be interesting as independent problems.

A note on difference families from cyclotomy

On the support varieties of tilting modules

We consider the boomerang uniformity of an infinite class of (locally-APN) power maps and show that their boomerang uniformity over the finite field $$\mathbb {F}_{2^n}$$ is 2 and 4, when $$n \equiv 0 \pmod 4$$ and $$n \equiv 2 \pmod 4$$ , respectively. As a consequence, we show that for this class of power maps, the differential uniformity is strictly greater than their boomerang uniformity.