Divisor Problem Research Articles

Simultaneous diagonalization of incomplete matrices and applications

We consider the problem of recovering the entries of diagonal matrices $\{U_a\}_a$ for $a = 1,\ldots,t$ from multiple incomplete samples $\{W_a\}_a$ of the form $W_a=PU_aQ$, where $P$ and $Q$ are unknown matrices of low rank. We devise practical algorithms for this problem depending on the ranks of $P$ and $Q$. This problem finds its motivation in cryptanalysis: we show how to significantly improve previous algorithms for solving the approximate common divisor problem and breaking CLT13 cryptographic multilinear maps.

A note on Titchmarsh divisor problem under the generalized Riemann hypothesis

Let Λ(n) be the von Mangoldt function and τ(n) the divisor function. We focus on the summation T ( x ) = ∑ 1 < n ≤ x Λ ( n ) τ ( n − 1 ) . Under the Riemann hypothesis related to Dirichlet L-functions, it is proved that T ( x ) = x P 1 ( log x ) + O ( x 1 − 𝜃 ) holds for 𝜃<16×10−4. Here P1(t) is a polynomial of degree 1.

A Bombieri-type theorem for convolution with application on number field

Let $$K$$ be an imaginary quadratic number field and $$\mathcal{O}_K$$ be its ring of integers. We show that, if the arithmetic functions $$f, g\colon\mathcal{O}_K\rightarrow \mathbb{C}$$ both have level of distribution $$\vartheta$$ for some $$0<\vartheta\leq 1/2$$ then the Dirichlet convolution $$f*g$$ also has level of distribution $$\vartheta$$ . As an application we also obtain an analogue of the Titchmarsh divisor problem for product of two primes in imaginary quadratic fields.

Algorithms for CRT-variant of Approximate Greatest Common Divisor Problem

AbstractThe approximate greatest common divisor problem (ACD) and its variants have been used to construct many cryptographic primitives. In particular, the variants of the ACD problem based on Chinese remainder theorem (CRT) are being used in the constructions of a batch fully homomorphic encryption to encrypt multiple messages in one ciphertext. Despite the utility of the CRT-variant scheme, the algorithms that secures its security foundation have not been probed well enough.In this paper, we propose two algorithms and the results of experiments in which the proposed algorithms were used to solve the variant problem. Both algorithms take the same time complexity $\begin{array}{} \displaystyle 2^{\tilde{O}(\frac{\gamma}{(\eta-\rho)^2})} \end{array}$ up to a polynomial factor to solve the variant problem for the bit size of samples γ, secret primes η, and error bound ρ. Our algorithm gives the first parameter condition related to η and γ size. From the results of the experiments, it has been proved that the proposed algorithms work well both in theoretical and experimental terms.

Brill-Noether generality of binary curves

AbstractWe show that the space $G^r_{\underline d}(X)$ of linear series of certain multi-degree $\underline d=(d_1,d_2)$ (including the balanced ones) and rank r on a general genus-g binary curve X has dimension $\rho _{g,r,d}=g-(r+1)(g-d+r)$ if nonempty, where $d=d_1+d_2$ . This generalizes Caporaso’s result from the case $r\leq 2$ to arbitrary rank, and shows that the space of Osserman-limit linear series on a general binary curve has the expected dimension, which was known for $r\leq 2$ . In addition, we show that the space $G^r_{\underline d}(X)$ is still of expected dimension after imposing certain ramification conditions with respect to a sequence of increasing effective divisors supported on two general points $P_i\in Z_i$ , where $i=1,2$ and $Z_1,Z_2$ are the two components of X. Our result also has potential application to the lifting problem of divisors on graphs to divisors on algebraic curves.

Nonnegative Inverse Elementary Divisors Problem for Lists with Nonnegative Real Parts

In this paper, sufficient conditions for the existence and construction of nonnegative matrices with prescribed elementary divisors for a list of complex numbers with nonnegative real part are obtained, and the corresponding nonnegative matrices are constructed. In addition, results of how to perturb complex eigenvalues of a nonnegative matrix while keeping its elementary divisors and its nonnegativity are derived.

Counting primitive subsets and other statistics of the divisor graph of [formula omitted

Let Q(n) denote the count of the primitive subsets of the integers {1,2…,n}. We give a new proof that Q(n)=α(1+o(1))n for some constant α, which allows us to give a good error term and to improve upon the lower bound for the value of α. We also show that the method developed can be applied to many similar problems related to the divisor graph, including other questions about primitive sets, geometric-progression-free sets, and the divisor graph path-cover problem.

Greatest common divisors with moving targets and consequences for linear recurrence sequences

We establish consequences of the moving form of Schmidt’s Subspace Theorem. Indeed, we obtain inequalities that bound the logarithmic greatest common divisor of moving multivariable polynomials evaluated at moving S S -unit arguments. In doing so, we complement recent work of Levin. As an additional application, we obtain results that pertain to the greatest common divisor problem for algebraic linear recurrence sequences. These observations are motivated by previous related works of Corvaja-Zannier, Levin, and others.

A quadratic divisor problem and moments of the Riemann zeta-function

We estimate asymptotically the fourth moment of the Riemann zeta-function twisted by a Dirichlet polynomial of length T^{\frac14 - \varepsilon} . Our work relies crucially on Watt's theorem on averages of Kloosterman fractions. In the context of the twisted fourth moment, Watt's result is an optimal replacement for Selberg's eigenvalue conjecture. Our work extends the previous result of Hughes and Young, where Dirichlet polynomials of length T^{\frac{1}{11}-\varepsilon} were considered. Our result has several applications, among others to the proportion of critical zeros of the Riemann zeta-function, zero spacing and lower bounds for moments. Along the way we obtain an asymptotic formula for a quadratic divisor problem, where the condition am_1m_2 - bn_1n_2 = h is summed with smooth averaging on the variables m_1, m_2, n_1, n_2, h and arbitrary weights in the average on a,b . Using Watt's work allows us to exploit all averages simultaneously. It turns out that averaging over m_1, m_2, n_1, n_2, h right away in the quadratic divisor problem considerably simplifies the combinatorics of the main terms in the twisted fourth moment.

Some results on divisor problems related to cusp forms

Let $$\lambda _{f}(n)$$ be the normalized Fourier coefficients of a holomorphic Hecke cusp form of full level. We study a generalized divisor problem with $$\lambda _{f}(n)$$ over some special sequences. More precisely, for any fixed integer $$k\ge 2$$ and $$j\in \{1,2,3,4\},$$ we are interested in the following sums $$\begin{aligned} S_{k}(x,j):=\sum _{n\le x}\lambda _{k,f}(n^{j})=\sum _{n\le x}\sum _{n=n_{1}n_{2}\cdots n_{k}}\lambda _{f}(n_{1}^{j})\lambda _{f}(n_{2}^{j})\cdots \lambda _{f}(n_{k}^{j}). \end{aligned}$$

The mean square discrepancy in the divisor problem

We study the mean square of the error term, or the mean square discrepancy, in the Dirichlet divisor problem. An analysis of the off-diagonal terms in the mean square of the Voronoi summation formula leads to a precise conjecture for this mean square discrepancy. Surprisingly, the discrepancy contains a slowly oscillating, unbounded term.

Generalized GCD for toric Fano varieties

We study the greatest common divisor problem for torus invariant blowing-up morphisms of nonsingular toric Fano varieties. Our main result applies the theory of Okounkov bodies together with an arithmetic form of Cartan's Second Main theorem, which has been established by Ru and Vojta. It also builds on Silverman's geometric concept of greatest common divisor. As a special case of our results, we deduce a bound for the generalized greatest common divisor of pairs of nonzero algebraic numbers.

A binary quadratic Titchmarsh divisor problem

We consider a binary quadratic variant of the Titchmarsh divisor problem and give an asymptotic formula for $\sum _{p^2+q^2\leq N} \tau (p^2+q^2+1)$, where $p,q$ are primes.

On generalizations of the Titchmarsh divisor problem

On generalizations of the Titchmarsh divisor problem

Fast Additive Partially Homomorphic Encryption From the Approximate Common Divisor Problem

This paper presents two efficient partially homomorphic encryption schemes built upon the approximate common divisor problem, believed to be resistant to quantum computer attacks. Both proposals, named FAHE1 and FAHE2, are additively homomorphic and have a symmetric nature, meaning that they are useful in scenarios where encryption and decryption are performed by the same entity. This is the case, for example, of encrypted databases stored in a public cloud. We also evaluate the performance of our proposals in comparison with two alternatives displaying additive homomorphism: the traditional Paillier asymmetric cryptosystem, which is not quantum-resistant; and the XPIR algorithm, which is both quantum-resistant and symmetric. Our experimental results show that both solutions provide considerable speed-ups when compared to Paillier. Namely, encryption and decryption with FAHE1 are, respectively, 120 and 25 times faster than Paillier’s, while for FAHE2 both operations run more than 1000 times faster. In addition, when compared with a highly optimized XPIR code, our reference implementation remains quite competitive while producing smaller ciphertexts.

Combinatorial identities and Titchmarsh’s divisor problem for multiplicative functions

Given a multiplicative function $f$ which is periodic over the primes, we obtain a full asymptotic expansion for the shifted convolution sum $\sum_{|h|<n\leq x} f(n) \tau(n-h)$, where $\tau$ denotes the divisor function and $h\in\mathbb{Z}\setminus\{0\}$. We consider in particular the special cases where $f$ is the generalized divisor function $\tau_z$ with $z\in\mathbb{C}$, and the characteristic function of sums of two squares (or more generally, ideal norms of abelian extensions). As another application, we deduce a full asymptotic expansion in the generalized Titchmarsh divisor problem $\sum_{|h|<n\leq x,\,\omega(n)=k} \tau(n-h)$, where $\omega(n)$ counts the number of distinct prime divisors of $n$, thus extending a result of Fouvry and Bombieri-Friedlander-Iwaniec. We present two different proofs: The first relies on an effective combinatorial formula of Heath-Brown's type for the divisor function $\tau_\alpha$ with $\alpha\in\mathbb{Q}$, and an interpolation argument in the $z$-variable for weighted mean values of $\tau_z$. The second is based on an identity of Linnik type for $\tau_z$ and the well-factorability of friable numbers.

Constants in Titchmarsh divisor problems for elliptic curves

Inspired by the analogy between the group of units $$\mathbb {F}_p^{\times }$$ of the finite field with p elements and the group of points $$E(\mathbb {F}_p)$$ of an elliptic curve $$E/\mathbb {F}_p$$, E. Kowalski, A. Akbary & D. Ghioca, and T. Freiberg & P. Kurlberg investigated the asymptotic behaviour of elliptic curve sums analogous to the Titchmarsh divisor sum $$\sum \nolimits _{p \le x} \tau (p + a) \sim C x$$. In this paper, we present a comprehensive study of the constants C(E) emerging in the asymptotic study of these elliptic curve divisor sums in place of the constant C above. Specifically, by analyzing the division fields of an elliptic curve $$E/\mathbb {Q}$$, we prove bounds for the constants C(E) and, in the generic case of a Serre curve, we prove explicit closed formulae for C(E) amenable to concrete computations. Moreover, we compute the moments of the constants C(E) over two-parameter families of elliptic curves $$E/\mathbb {Q}$$. Our methods and results complement recent studies of average constants occurring in other conjectures about reductions of elliptic curves by addressing not only the average behaviour, but also the individual behaviour of these constants, and by providing explicit tools towards the computational verifications of the expected asymptotics.

Uniform bounds of the Piltz divisor problem over number fields

We consider the upper bound of Piltz divisor problem over number fields. Piltz divisor problem is known as a generalization of the Dirichlet divisor problem. We deal with this problem over number fields and improve the error term of this function for many cases. Our proof uses the estimate of exponential sums. We also show uniform results for ideal counting function and relatively $r$-prime lattice points as one of applications.

Order-preserving encryption using approximate common divisors

Order-preservation is a highly desirable property for encrypted databases as it allows range queries over ciphertexts. Order-preserving encryption (OPE) is used in the encrypted database systems CryptDB and Cipherbase. The former has been adopted by several commercial organisations and the latter was developed as an extension of Microsoftâs SQLServer. We present two novel, but simple, randomised OPE schemes based on the general approximate common divisor problem (GACDP) and decisional polynomial approximate common divisor problem (DPolyACDP) respectively. These appear to be the first OPE schemes to be based on a computational hardness primitive, rather than a security game. Our GACDP based scheme is very efficient, requiring only O(1) arithmetic operations for encryption and decryption. Our DPolyACDP based scheme is similarly efficient. We show that these schemes have near optimal information leakage. We demonstrate how our OPE schemes can be integrated into a secure distributed computing system which computes over encrypted data. We report on an extensive evaluation of our GACDP-based algorithms in such a scenario, a MapReduce computation over encrypted data. The results clearly demonstrate extremely favourable execution times in comparison with existing OPE schemes.

Computing Approximate Greatest Common Right Divisors of Differential Polynomials

Differential (Ore) type polynomials with “approximate” polynomial coefficients are introduced. These provide an effective notion of approximate differential operators, with a strong algebraic structure. We introduce the approximate greatest common right divisor problem (GCRD) of differential polynomials, as a non-commutative generalization of the well-studied approximate GCD problem. Given two differential polynomials, we present an algorithm to find nearby differential polynomials with a non-trivial GCRD, where nearby is defined with respect to a suitable coefficient norm. Intuitively, given two linear differential polynomials as input, the (approximate) GCRD problem corresponds to finding the (approximate) differential polynomial whose solution space is the intersection of the solution spaces of the two inputs. The approximate GCRD problem is proven to be locally well posed. A method based on the singular value decomposition of a differential Sylvester matrix is developed to produce an initial approximation of the GCRD. With a sufficiently good initial approximation, Newton iteration is shown to converge quadratically to an optimal solution. Finally, sufficient conditions for existence of a solution to the global problem are presented along with examples demonstrating that no solution exists when these conditions are not satisfied.