Linear Complexity of Some Binary Sequences Derived from Fermat Quotients

We determined the linear complexity of a family of p2-periodic binary threshold sequences and a family of p2-periodic binary sequences constructed using the Legendre symbol,both of which are derived from Fermat quotients modulo an odd prime p.If 2 is a primitive element modulo p2,the linear complexity equals to p2-p or p2-1,which is very close to the period and it is large enough for cryptographic purpose.

A Multivariate Public Key Cryptographic Scheme

This paper presents a multivariate public key cryptographic scheme over a finite field with odd prime characteristic.The idea of embedding and layering is manifested in its construction.The security of the scheme is analyzed in detail,and this paper indicates that the scheme can withstand the up to date differential cryptanalysis.We give heuristic arguments to show that this scheme resists all known attacks.

On the Diophantine Equation y^2= px（x^2＋ 2）

For any fixed odd prime p, let N（p） denote the number of positive integer solutions （x, y） of the equation y^2 = px（x^2 ＋ 2）. In this paper, using some properties of binary quartic Diophantine equations, we prove that ifp ≡ 5 or 7（mod 8）, then N（p） = 0; ifp ≡ 1（mod 8）, then N（p） 〈 1; if p〉 3 andp ≡ 3（rood 8）, then N（p） ≤ 2.

关于哥德巴赫猜想的证明——关于数论

关于数论首先要正确认识,科学划分归类整数,即整数划分归类为奇数(素数)、和数与偶数三大类.在整数划分归类的基础上方能证明哥德巴赫猜想.猜想证明所得到的公理、结论、公式是数论的基本公理、结论、公式.

哥德巴赫猜想太极模型

为了认识自然与社会的演化规律从复杂性中寻找简单性用简单性把握世界的复杂性,中国上古伏羲氏一画开天创造了"易有太极,是生两仪""叁天两地而倚数"形而上的太极日模型。这是只用一就能知道哥德巴赫猜想的演化及其因果关系的模型。

NOTES ON GLAISHER'S CONGRUENCES

Let p be an odd prime and let n ≥1, k ≥0 and r be integers. Denote by B_k the kth Bernoulli number. It is proved that (i) If r ≥1 is odd and suppose p ≥r + 4, then (ii)If r ≥2 is even and suppose p ≥ r + 3, then (modp^2). (iii)-(2n+1)p (modp^2). This result generalizes the Glaisher’s congruence. As a corollary, a generalization of the Wolstenholme’s theorem is obtained.

Chen's Conjecture and Its Generalization

Let l1,l2,…,lg be even integers and x be a sufficiently large number.In this paper,the authors prove that the number of positive odd integers k≤x such that（k+l1）2,（k+l2）2,…,（k+lg）2 can not be expressed as 2n+pαis at least c（g）x,where p is an odd prime and the constant c（g）depends only on g.

Congruences involving generalized central trinomial coefficients

For integers b and c the generalized central trinomial coefficient Tn(b,c)denotes the coefficient of xnin the expansion of(x2+bx+c)n.Those Tn=Tn(1,1)(n=0,1,2,...)are the usual central trinomial coefficients,and Tn(3,2)coincides with the Delannoy number Dn=n k=0n k n+k k in combinatorics.We investigate congruences involving generalized central trinomial coefficients systematically.Here are some typical results:For each n=1,2,3,...,we have n-1k=0(2k+1)Tk(b,c)2(b2-4c)n-1-k≡0(mod n2)and in particular n2|n-1k=0(2k+1)D2k;if p is an odd prime then p-1k=0T2k≡-1p(mod p)and p-1k=0D2k≡2p(mod p),where(-)denotes the Legendre symbol.We also raise several conjectures some of which involve parameters in the representations of primes by certain binary quadratic forms.

The p-Rank of Tame Kernels of Pure Quintic Fields

Let F be a pure quintic field. In this paper, we present some results for the p-rank of K2OF, where p is an odd prime number. In particular, the 5-rank of K20F is studied by the reflection theorem. Some explicit results on the 5-rank of K20F are given in some special cases.

Permutation polynomials with low differential uniformity over finite fields of odd characteristic

In this paper, we propose a construction of functions with low differential uniformity based on known perfect nonlinear functions over finite fields of odd characteristic. For an odd prime power q, it is proved that the proposed functions over the finite field Fq are permutations if and only if q≡3(mod 4).

On the elliptic curve y^2=x^3-2r Dx and factoring integers

Let D=pq be the product of two distinct odd primes.Assuming the parity conjecture,we construct infinitely many r≥1 such that E2rD:y2=x3-2rDx has conjectural rank one and vp(x([k]Q))≠vq(x([k]Q))for any odd integer k,where Q is the generator of the free part of E(Q).Furthermore,under the generalized Riemann hypothesis,the minimal value of r is less than c log4 D for some absolute constant c.As a corollary,one can factor D by computing the generator Q.

Rotational structure of the odd-proton nuclide ^(171)Tm:A projected shell model study

Deformed odd-mass nuclei are ideal examples where the interplay between single-particle and collective degrees of freedom can be studied. Inspired by the recent experimental high-spin data in the odd-proton nuclide 171 Tm, we perform projected shell model(PSM) calculations to investigate structure of the ground band and other bands based on isomeric states. In addi- tion to the usual quadrupole-quadrupole force in the Hamiltonian, we employ the hexadecapole-hexadecapole(HH) interac- tion, in a self-consistent way with the hexadecapole deformation of the deformed basis. It is found that the known experi- mental data can be well described by the PSM calculation. The effect of the HH force on the quasiparticle isomeric states is discussed.