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

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 591
Author(s):  
Michel L. Lapidus ◽  
Machiel van Frankenhuijsen ◽  
Edward K. Voskanian
Keyword(s):  
Approximation Algorithm ◽  
Continued Fraction ◽  
Practical Method ◽  
Lll Algorithm ◽  
Fractal Strings ◽  
Complex Dimensions ◽  
Diophantine Approximations ◽  
Self Similar ◽  
Simultaneous Diophantine Approximations ◽  
Fractal String

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.

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

2021 ◽  
pp. 2150144
Author(s):  
Saidu Isah Abubakar ◽  
Sadiq Shehu
Keyword(s):  
Polynomial Time ◽  
Continued Fraction ◽  
Prime Power ◽  
New Techniques ◽  
Lll Algorithm ◽  
Generalized System ◽  
Key Equation ◽  
Diophantine Approximations ◽  
Simultaneous Diophantine Approximations ◽  
Fraction Method

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].

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

2021 ◽  
pp. 74-86
Author(s):  
Sadiq Shehu ◽  
Abdullahi Hussaini ◽  
Zahriya Lawal
Keyword(s):  
Information Security ◽  
Polynomial Time ◽  
Continued Fractions ◽  
Prime Power ◽  
Electronic Information ◽  
Lll Algorithm ◽  
Key Equation ◽  
Public Keys ◽  
Diophantine Approximations ◽  
Simultaneous Diophantine Approximations

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.

scholarly journals Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings

2018 ◽  
Vol 2 (4) ◽  
pp. 26 ◽  
Author(s):  
Michel Lapidus ◽  
Hùng Lũ’ ◽  
Machiel Frankenhuijsen
Keyword(s):  
Counting Function ◽  
Minkowski Dimension ◽  
Volume Formula ◽  
Fractal Strings ◽  
Complex Dimensions ◽  
Abscissa Of Convergence ◽  
Tubular Neighborhoods ◽  
Fractal String ◽  
Tube Formulas ◽  
Asymptotic Growth Rate

The theory of complex dimensions describes the oscillations in the geometry (spectra and dynamics) of fractal strings. Such geometric oscillations can be seen most clearly in the explicit volume formula for the tubular neighborhoods of a p-adic fractal string L p , expressed in terms of the underlying complex dimensions. The general fractal tube formula obtained in this paper is illustrated by several examples, including the nonarchimedean Cantor and Euler strings. Moreover, we show that the Minkowski dimension of a p-adic fractal string coincides with the abscissa of convergence of the geometric zeta function associated with the string, as well as with the asymptotic growth rate of the corresponding geometric counting function. The proof of this new result can be applied to both real and p-adic fractal strings and hence, yields a unifying explanation of a key result in the theory of complex dimensions for fractal strings, even in the archimedean (or real) case.

scholarly journals Towards quantized number theory: spectral operators and an asymmetric criterion for the Riemann hypothesis

2015 ◽  
Vol 373 (2047) ◽  
pp. 20140240 ◽  
Author(s):  
Michel L. Lapidus
Keyword(s):  
Riemann Hypothesis ◽  
Spectral Operator ◽  
Joint Work ◽  
Fractal Strings ◽  
Complex Dimensions ◽  
The Family ◽  
Expository Article ◽  
Riemann Zeta ◽  
Fractal String ◽  
The Right

This research expository article not only contains a survey of earlier work but also contains a main new result, which we first describe. Given c ≥0, the spectral operator can be thought of intuitively as the operator which sends the geometry onto the spectrum of a fractal string of dimension not exceeding c . Rigorously, it turns out to coincide with a suitable quantization of the Riemann zeta function ζ = ζ ( s ): , where ∂=∂ c is the infinitesimal shift of the real line acting on the weighted Hilbert space . In this paper, we establish a new asymmetric criterion for the Riemann hypothesis (RH), expressed in terms of the invertibility of the spectral operator for all values of the dimension parameter (i.e. for all c in the left half of the critical interval (0,1)). This corresponds (conditionally) to a mathematical (and perhaps also, physical) ‘phase transition’ occurring in the midfractal case when . Both the universality and the non-universality of ζ = ζ ( s ) in the right (resp., left) critical strip (resp., ) play a key role in this context. These new results are presented here. We also briefly discuss earlier joint work on the complex dimensions of fractal strings, and we survey earlier related work of the author with Maier and with Herichi, respectively, in which were established symmetric criteria for the RH, expressed, respectively, in terms of a family of natural inverse spectral problems for fractal strings of Minkowski dimension D ∈(0,1), with , and of the quasi-invertibility of the family of spectral operators (with ).

Complex Dimensions of Self-Similar Fractal Strings

2000 ◽  
pp. 23-54 ◽  
Author(s):  
Michel L. Lapidus ◽  
Machiel van Frankenhuysen
Keyword(s):  
Fractal Strings ◽  
Complex Dimensions ◽  
Self Similar

scholarly journals New Cryptanalytic Attack on RSA Modulus N=pq Using Small Prime Difference Method

Cryptography ◽  
2018 ◽  
Vol 3 (1) ◽  
pp. 2 ◽  
Author(s):  
Muhammad Ariffin ◽  
Saidu Abubakar ◽  
Faridah Yunos ◽  
Muhammad Asbullah
Keyword(s):  
Polynomial Time ◽  
Difference Method ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Basis Reduction ◽  
Diophantine Approximations ◽  
Reduction Methods ◽  
Simultaneous Diophantine Approximations ◽  
Positive Integers ◽  
Better Than

This paper presents new short decryption exponent attacks on RSA, which successfully leads to the factorization of RSA modulus N = p q in polynomial time. The paper has two parts. In the first part, we report the usage of the small prime difference method of the form | b 2 p - a 2 q | < N γ where the ratio of q p is close to b 2 a 2 , which yields a bound d < 3 2 N 3 4 - γ from the convergents of the continued fraction expansion of e N - ⌈ a 2 + b 2 a b N ⌉ + 1 . The second part of the paper reports four cryptanalytic attacks on t instances of RSA moduli N s = p s q s for s = 1 , 2 , … , t where we use N - ⌈ a 2 + b 2 a b N ⌉ + 1 as an approximation of ϕ ( N ) satisfying generalized key equations of the shape e s d - k s ϕ ( N s ) = 1 , e s d s - k ϕ ( N s ) = 1 , e s d - k s ϕ ( N s ) = z s , and e s d s - k ϕ ( N s ) = z s for unknown positive integers d , k s , d s , k s , and z s , where we establish that t RSA moduli can be simultaneously factored in polynomial time using combinations of simultaneous Diophantine approximations and lattice basis reduction methods. In all the reported attacks, we have found an improved short secret exponent bound, which is considered to be better than some bounds as reported in the literature.

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