A new method for lattice reduction using directional and hyperplanar shearing

A geometric method of lattice reduction based on cycles of directional and hyperplanar shears is presented. The deviation from cubicity at each step of the reduction is evaluated by a parameter called `basis rhombicity' which is the sum of the absolute values of the elements of the metric tensor associated with the basis. The levels of reduction are quite similar to those obtained with the Lenstra–Lenstra–Lovász (LLL) algorithm, at least up to the moderate dimensions that have been tested (lower than 20). The method can be used to reduce unit cells attached to given hyperplanes.

Exploiting the security of N = prqs through approximation of ϕ(N)

This paper reports new techniques that exploit the security of the prime power moduli [Formula: see text] using continued fraction method. Our study shows that the key equation [Formula: see text] can be exploited using [Formula: see text] as good approximation of [Formula: see text]. This enables us to get [Formula: see text] from the convergents of the continued fractions expansion of [Formula: see text] where the bound of the private exponent is [Formula: see text] which leads to the polynomial time factorization of the moduli [Formula: see text]. We further report the polynomial time attacks that can break the security of the generalized prime power moduli [Formula: see text] using generalized system of equation of the form [Formula: see text] and [Formula: see text] by applying simultaneous Diophantine approximations and LLL algorithm techniques where [Formula: see text] and [Formula: see text].

Quasiperiodic Patterns of the Complex Dimensions of Nonlattice Self-Similar Strings, via the LLL Algorithm

The Lattice String Approximation algorithm (or LSA algorithm) of M. L. Lapidus and M. van Frankenhuijsen is a procedure that approximates the complex dimensions of a nonlattice self-similar fractal string by the complex dimensions of a lattice self-similar fractal string. The implication of this procedure is that the set of complex dimensions of a nonlattice string has a quasiperiodic pattern. Using the LSA algorithm, together with the multiprecision polynomial solver MPSolve which is due to D. A. Bini, G. Fiorentino and L. Robol, we give a new and significantly more powerful presentation of the quasiperiodic patterns of the sets of complex dimensions of nonlattice self-similar fractal strings. The implementation of this algorithm requires a practical method for generating simultaneous Diophantine approximations, which in some cases we can accomplish by the continued fraction process. Otherwise, as was suggested by Lapidus and van Frankenhuijsen, we use the LLL algorithm of A. K. Lenstra, H. W. Lenstra, and L. Lovász.

Strategic Way of Factoring Modulus ``\(N = p^r q\)

Cryptography is fundamental to the provision of a wider notion of information security. Electronic information can easily be transmitted and stored in relatively insecure environments. This research was present to factor the prime power modulus \(N = p^r q\) for \(r \geq 2\) using the RSA key equation, if \(\frac{y}{x}\) is a convergents of the continued fractions expansions of \(\frac{e}{N - \left(2^{\frac{2r+1}{r+1}} N^{\frac{r}{r+1}} - 2^{\frac{r-1}{r+1}} N^{\frac{r-1}{r+1}}\right)}\). We furthered our analysis on \(n\) prime power moduli \(N_i = p_i^r q_i\) by transforming the generalized key equations into Simultaneous Diophantine approximations and using the LLL algorithm on \(n\) prime power public keys \((N_i,e_i)\) we were able to factorize the \(n\) prime power moduli \(N_i = p_i^r q_i\), for \(i = 1,....,n\) simultaneously in polynomial time.

Four-loop anomalous dimension of the third and fourth moments of the nonsinglet twist-2 operator in QCD

We present the result for the third and fourth moments of the nonsinglet four-loop anomalous dimension of Wilson twist-2 operators in QCD with full color and flavor structures. We also give general expressions for some contributions to the full four-loop anomalous dimension obtained by means of the method, based on LLL-algorithm, which was proposed by us earlier for the reconstruction of a general form of the anomalous dimension from the fixed values.

Dynamic keys generation for internet of things

<span>In several aspects, interest in IoT has become considerable by researchers and academics in recent years. Data security becomes one of the important challenges facing development of IoT environment. Many algorithms were proposed to secure the IoT applications. The traditional public key cryptographic are inappropriate because it requires high computational. Therefore, lattice-based public-key cryptosystem (LB-PKC) is a favorable technique for IoT security. NTRU is one of a LB-PKC that based on truncated polynomial ring, it has good features, which make it to be an effective alternative to the RSA and ECC algorithms. But, there is LLL algorithm can success to attack it under certain conditions. This paper proposes modifications to NTRU public key cryptosystem to be secure against the lattice-based attack by using LLL algorithm, as well as a method for generating a new keys sequence dynamically. The results from simulations show that the performance of these modifications gives more secure from NTRU. </span>