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.

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.

RELATION BETWEEN A CLASS OF QUASIPERIODIC LATTICES GENERATED BY THE SUBSTITUTION METHOD AND THE GENERALIZED CONTINUED FRACTION EXPANSION

1996 ◽  
Vol 10 (17) ◽  
pp. 2081-2101
Author(s):  
TOSHIO YOSHIKAWA ◽  
KAZUMOTO IGUCHI
Keyword(s):  
Continued Fraction ◽  
Positive Real Number ◽  
Real Numbers ◽  
Continued Fraction Expansion ◽  
Positive Real ◽  
Periodic Sequences ◽  
One Dimensional ◽  
Finite Period ◽  
The One ◽  
Positive Integers

The continued fraction expansion for a positive real number is generalized to that for a set of positive real numbers. For arbitrary integer n≥2, this generalized continued fraction expansion generates (n−1) sequences of positive integers {ak}, {bk}, … , {yk} from a given set of (n−1) positive real numbers α, β, …ψ. The sequences {ak}, {bk}, … ,{yk} determine a sequence of substitutions Sk: A → Aak Bbk…Yyk Z, B → A, C → B,…,Z → Y, which constructs a one-dimensional quasiperiodic lattice with n elements A, B, … , Z. If {ak}, {bk}, … , {yk} are infinite periodic sequences with an identical period, then the ratio between the numbers of n elements A, B, … , Z in the lattice becomes a : β : … : ψ : 1. Thereby the correspondence is established between all the sets of (n−1) positive real numbers represented by a periodic generalized continued fraction expansion and all the one-dimensional quasiperiodic lattices with n elements generated by a sequence of substitutions with a finite period.

scholarly journals Good approximation and characterization of subgroups of R = Z

2001 ◽  
Vol 38 (1-4) ◽  
pp. 97-113 ◽  
Author(s):  
A. Bíró ◽  
J. M. Deshouillers ◽  
Vera T. Sós
Keyword(s):  
Continued Fraction ◽  
Irrational Number ◽  
Explicit Construction ◽  
Continued Fraction Expansion ◽  
Open Problems ◽  
Positive Integers

Let be a real irrational number and A =(xn) be a sequence of positive integers. We call A a characterizing sequence of or of the group Z mod 1 if lim n 2A n !1 k k =0 if and only if 2 Z mod 1. In the present paper we prove the existence of such characterizing sequences, also for more general subgroups of R = Z . Inthespecialcase Z mod 1 we give explicit construction of a characterizing sequence in terms of the continued fraction expansion of. Further, we also prove some results concerning the growth and gap properties of such sequences. Finally, we formulate some open problems.

Some properties of the continued fraction expansion of (m/n) e1/q

1970 ◽  
Vol 67 (1) ◽  
pp. 67-74 ◽  
Author(s):  
K. R. Matthews ◽  
R. F. C. Walters
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Image Position ◽  
Positive Integers

Introduction. Continued fractions of the form are called Hurwitzian if b1, …, bh, are positive integers, ƒ1(x), …, ƒk(x) are polynomials with rational coefficients which take positive integral values for x = 0, 1, 2, …, and at least one of the polynomials is not constant. f1(x), …, fk(x) are said to form a quasi-period.

On the Lower Range of Perron's Modular Function

1969 ◽  
Vol 21 ◽  
pp. 808-816 ◽  
Author(s):  
J. R. Kinney ◽  
T. S. Pitcher
Keyword(s):  
Finite Number ◽  
Continued Fraction ◽  
Modular Function ◽  
Finite Collection ◽  
Continued Fraction Expansion ◽  
Lower Range ◽  
Image Position ◽  
Positive Integers

The modular function Mwas introduced by Perron in (6). M(ξ) (for irrational ξ) is denned by the property that the inequalityis satisfied by an infinity of relatively prime pairs (p, q)for positive d,but by at most a finite number of such pairs for negative d.We will writefor the continued fraction expansion of ξ ∈ (0, 1) and for any finite collection y1,…, ykof positive integers we will writeIt is known (see 6) thatWhere

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.

A NEW POINT IN LAGRANGE SPECTRUM

2012 ◽  
Vol 09 (02) ◽  
pp. 393-403
Author(s):  
K. C. PRASAD ◽  
HRISHIKESH MAHATO ◽  
SUDHIR MISHRA
Keyword(s):  
Continued Fraction ◽  
Infinite Sequence ◽  
Cluster Point ◽  
Continued Fraction Expansion ◽  
Irrational Numbers ◽  
Lagrange Spectrum ◽  
Positive Integers

Let I denote the set of all irrational numbers, θ ∈ I, and simple continued fraction expansion of θ be [a0, a1, …, an, …]. Then a0 is an integer and {an}n≥1 is an infinite sequence of positive integers. Let Mn(θ) = [0, an, an-1, …, a1] + [an+1, an+2, …]. Then the set of numbers { lim sup Mn(θ) ∣ θ ∈ I} is called the Lagrange Spectrum 𝔏. Notably 3 is the first cluster point of 𝔏. Essentially lim inf 𝔏 or [Formula: see text]. Perron [Über die approximation irrationaler Zahlen durch rationale, I, S.-B. Heidelberg Akad. Wiss., Abh. 4 (1921) 17 pp; Über die approximation irrationaler Zahlen durch rationale, II, S.-B. Heidelberg Akad. Wiss., Abh.8 (1921) 12 pp.] has found that lim inf { lim sup Mn(θ) ∣ θ = [a0, a1, a2, …, an, …] and [Formula: see text]. This article forwards the value of lim inf{lim sup Mn(θ) ∣ θ = [a0, a1, …, an, …] and an ≥ 4 frequently}, a long awaited cluster point of Lagrange Spectrum.

scholarly journals A Note on Continued Fractions

1960 ◽  
Vol 12 ◽  
pp. 303-308 ◽  
Author(s):  
A. Oppenheim
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Quadratic Irrational ◽  
Image Position ◽  
Positive Integers ◽  
Periodic Continued Fractions

Any real number y leads to a continued fraction of the type(1)where ai, bi are integers which satisfy the inequalities(2)by means of the algorithm(3)the a's being assigned positive integers. The process terminates for rational y; the last denominator bk satisfying bk ≥ ak + 1. For irrational y, the process does not terminate. For a preassigned set of numerators ai ≥ 1, this C.F. development of y is unique; its value being y.Bankier and Leighton (1) call such fractions (1), which satisfy (2), proper continued fractions. Among other questions, they studied the problem of expanding quadratic surds in periodic continued fractions. They state that “it is well-known that not only does every periodic regular continued fraction represent a quadratic irrational, but the regular continued fraction expansion of a quadratic irrational is periodic.

On the Continued Fraction Expansion of Fixed Period in Finite Fields

2015 ◽  
Vol 58 (4) ◽  
pp. 704-712 ◽  
Author(s):  
Hela Benamar ◽  
Amara Chandoul ◽  
M. Mkaouar
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Finite Fields ◽  
Continued Fraction ◽  
Period Length ◽  
Continued Fraction Expansion ◽  
Positive Integers ◽  
Fixed Period

AbstractThe Chowla conjecture states that if t is any given positive integer, there are infinitely many prime positive integers N such that Per() = t, where Per() is the period length of the continued fraction expansion for . C. Friesen proved that, for any k ∈ ℕ, there are infinitely many square-free integers N, where the continued fraction expansion of has a fixed period. In this paper, we describe all polynomials for which the continued fraction expansion of has a fixed period. We also give a lower bound of the number of monic, non-squares polynomials Q such that deg Q = 2d and Per =t.

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