Composite Integer Research Articles

Longest run of equal parts in a random integer composition

This paper examines a problem in enumerative and asymptotic combinatorics involving the classical structure of integer compositions. What is sought is an analysis on average and in distribution of the length of the longest run of consecutive equal parts in a composition of size n. The problem was posed by Herbert Wilf at the Analysis of Algorithms conference in July 2009 (see arXiv:0906.5196).

Combinatorial operads from monoids

We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operads obtained from usual monoids such as the additive and multiplicative monoids of integers and cyclic monoids. They involve various familiar combinatorial objects: endofunctions, parking functions, packed words, permutations, planar rooted trees, trees with a fixed arity, Schr\"oder trees, Motzkin words, integer compositions, directed animals, and segmented integer compositions. We also recover some already known (symmetric or not) operads: the magmatic operad, the associative commutative operad, the diassociative operad, and the triassociative operad. We provide presentations by generators and relations of all constructed nonsymmetric operads.

Colored compositions, Invert operator and elegant compositions with the “black tie”

This paper shows how the study of colored compositions of integers reveals some unexpected and original connection with the Invert operator. The Invert operator becomes an important tool to solve the problem of directly counting the number of colored compositions for any coloration. The interesting consequences arising from this relationship also give an immediate and simple criterion to determine whether a sequence of integers counts the number of some colored compositions. Applications to Catalan and Fibonacci numbers naturally emerge, allowing to clearly answer to some open questions. Moreover, the definition of colored compositions with the “black tie” provides straightforward combinatorial proofs to a new identity involving multinomial coefficients and to a new closed formula for the Invert operator. Finally, colored compositions with the “black tie” give rise to a new combinatorial interpretation for the convolution operator, and to a new and easy method to count the number of parts of colored compositions.

Computing prime divisors in an interval

We address the problem of finding a nontrivial divisor of a composite integer when it has a prime divisor in an interval. We show that this problem can be solved in time of the square root of the interval length with a similar amount of storage, by presenting two algorithms; one is probabilistic and the other is its derandomized version.

A statistic on n-color compositions and related sequences

A composition of a positive integer in which a part of size n may be assigned one of n colors is called an n-color composition. Let am denote the number of n-color compositions of the integer m. It is known that am=F2m for all m≥1, where Fm denotes the Fibonacci number defined by Fm=Fm−1+Fm−2 if m≥2, with F0=0 and F1=1. A statistic is studied on the set of n-color compositions of m, thus providing a polynomial generalization of the sequence F2m. The statistic may be described, equivalently, in terms of statistics on linear tilings and lattice paths. The restriction to the set of n-color compositions having a prescribed number of parts is considered and an explicit formula for the distribution is derived. We also provide q-generalizations of relations between am and the number of self-inverse n-compositions of 2m+1 or 2m. Finally, we consider a more general recurrence than that satisfied by the numbers am and note some particular cases.

Cryptanalysis and Improvement of Yanlin and Xiaoping's Signature Scheme based on ECDLP and Factoring

Qin Yanlin and Wu Xiaoping proposed a digital signature scheme based on elliptic curve discrete logarithm problem and factoring a composite integer. They claimed that the security of their scheme depends on solving ECDLP and factoring both. In this paper, it is shown that if anyone can solve ECDLP then he can generate a valid signature without knowledge of private keys. An improved scheme is also proposed in this paper. The proposed scheme requires minimal operations in encryption and decryption algorithms which makes it more efficient. General Terms 2000 AMS Subject Classification No. 94A60

Better Lattice Constructions for Solving Multivariate Linear Equations Modulo Unknown Divisors

At CaLC 2001, Howgrave-Graham proposed the polynomial time algorithm for solving univariate linear equations modulo an unknown divisor of a known composite integer, the so-called partially approximate common divisor problem. So far, two forms of multivariate generalizations of the problem have been considered in the context of cryptanalysis. The first is simultaneous modular univariate linear equations, whose polynomial time algorithm was proposed at ANTS 2012 by Cohn and Heninger. The second is modular multivariate linear equations, whose polynomial time algorithm was proposed at Asiacrypt 2008 by Herrmann and May. Both algorithms cover Howgrave-Graham’s algorithm for univariate cases. On the other hand, both multivariate problems also become identical to Howgrave-Graham’s problem in the asymptotic cases of root bounds. However, former algorithms do not cover Howgrave-Graham’s algorithm in such cases. In this paper, we introduce the strategy for natural algorithm constructions that take into account the sizes of the root bounds. We work out the selection of polynomials in constructing lattices. Our algorithms are superior to all known attacks that solve the multivariate equations and can generalize to the case of arbitrary number of variables. Our algorithms achieve better cryptanalytic bounds for some applications that relate to RSA cryptosystems.

Corrections to the Results Derived in 'A Unified Approach to Algorithms Generating Unrestricted and Restricted Integer Compositions and Integer Partitions'; And a Comparison of Four Restricted Integer Composition Generation Algorithms

In this note, I discuss results on integer compositions/partitions given in the paper A Unified Approach to Algorithms Generating Unrestricted and Restricted Integer Compositions and Integer Partitions. I also experiment with four different generation algorithms for restricted integer compositions and find the algorithm designed in the named paper to be pretty slow, comparatively. Some of my comments may be subjective.

GENERALIZED CULLEN NUMBERS WITH THE LEHMER PROPERTY

We say a positive integer n satisfies the Lehmer property if <TEX>${\phi}(n)$</TEX> divides n - 1, where <TEX>${\phi}(n)$</TEX> is the Euler's totient function. Clearly, every prime satisfies the Lehmer property. No composite integer satisfying the Lehmer property is known. In this article, we show that every composite integer of the form <TEX>$D_{p,n}=np^n+1$</TEX>, for a prime p and a positive integer n, or of the form <TEX>${\alpha}2^{\beta}+1$</TEX> for <TEX>${\alpha}{\leq}{\beta}$</TEX> does not satisfy the Lehmer property.

Recursive construction of non-cyclic pandiagonal Latin squares

The pandiagonal Latin squares constructed using Hedayat’s method are cyclic. During the last decades several authors have described methods for constructing pandiagonal Latin squares which are semi-cyclic. In this paper we propose a recursive method for constructing non-cyclic pandiagonal Latin squares of any given order n, where n is a positive composite integer not divisible by 2 or 3. We also investigate the orthogonality properties of the constructed squares and extend our method to construct non-cyclic pandiagonal Sudoku.

PARALLEL ALGORITHM FOR SECOND–ORDER RESTRICTED WEAK INTEGER COMPOSITION GENERATION FOR SHARED MEMORY MACHINES

In 2012, Page presented a sequential combinatorial generation algorithm for generalized types of restricted weak integer compositions called second–order restricted weak integer compositions. Second–order restricted weak integer compositions cover various types of restricted weak integer compositions of n parts such as integer compositions, bounded compositions, and part–wise integer compositions. In this paper, we present a parallel algorithm that derives from our parallelization of Page's sequential algorithm with a focus on load balancing for shared memory machines.

RADICALLY WEAKENING THE LEHMER AND CARMICHAEL CONDITIONS

Lehmer's totient problem asks if there exist composite integers n satisfying the condition φ(n)|(n - 1) (where φ is the Euler-phi function), while Carmichael numbers satisfy the weaker condition λ(n)|(n - 1) (where λ is the Carmichael universal exponent function). We weaken the condition further, looking at those composite n where each prime divisor of φ(n) also divides n - 1. ( So rad (φ(n))|(n - 1).) While these numbers appear to be far more numerous than the Carmichael numbers, we show that their distribution has the same rough upper bound as that of the Carmichael numbers, a bound which is heuristically tight.

Multiplicity of summands in the random partitions of an integer

In this paper, we prove a conjecture of Yakubovich regarding limit shapes of ‘slices’ of two-dimensional (2D) integer partitions and compositions of n when the number of summands m ~Anα for some A > 0 and \(\alpha < \frac{1}{2}\). We prove that the probability that there is a summand of multiplicity j in any randomly chosen partition or composition of an integer n goes to zero asymptotically with n provided j is larger than a critical value. As a corollary, we strengthen a result due to Erdos and Lehner (Duke Math. J.8 (1941) 335–345) that concerns the relation between the number of integer partitions and compositions when \(\alpha = \frac{1}{3}\).

A Security and Efficiency of Authenticated Key Exchange Protocol for Wireless Mobile Ad Hoc Networks

Mobile Ad Hoc Networks (MANETs) are infrastructure-free, self-configuring and stand alone wireless networks. Lack of efficient computations and secure based point authentication, the security and efficiency of MANETs have been the biggest challenges in its wide application. Many researchers have applied RSA and ECC cryptography algorithms in building secure ID and key exchange agreement; however, they also have difficult to face the challenges of factoring large composite integers and computing discrete logarithms. Generally, public key infrastructures are assumed to be unavailable in MANETs. The key exchange problem for this type of network has now become important. In this article, we propose a new NTRU-based authenticated key exchange protocol for MANETs. We take advantage of NTRU cryptosystem of the inherent efficiency and security in this type of wireless networks - without any public key infrastructure - to defend message exchange against the threat of session key attacks, the man-in-the-middle attacks and the brute force attacks.

A trio of Bernoulli relations, their implications for the Ramanujan polynomials and the special values of the Riemann zeta function

We study the interplay between recurrences for zeta related functions at integer values, `Minor Corner Lattice' Toeplitz determinants and integer composition based sums. Our investigations touch on functional identities due to Ramanujan and Grosswald, the transcendence of the zeta function at odd integer values, the Li Criterion for the Riemann Hypothesis and pseudo-characteristic polynomials for zeta related functions. We begin with a recent result for \zeta(2s) and some seemingly new Bernoulli relations, which we use to obtain a generalised Ramanujan polynomial and properties thereof.

Part-products of S-restricted integer compositions

If S is a cofinite set of positive integers, an "S-restricted composition of n" is a sequence of elements of S, denoted ?? = (?1, ?2, ... ), whose sum is n. For uniform random S-restricted compositions, the random variable B(??) = ?i ?i is asymptotically lognormal. (A precise statement of the theorem includes an error term to bound the rate of convergence.) The proof is based upon a combinatorial technique for decomposing a composition into a sequence of smaller compositions.

A fast algorithm to computeL(1/2,f×χq)

Let f be a fixed (holomorphic or Maass) modular cusp form. Let χ q be a Dirichlet character mod q . We describe a fast algorithm that computes the value L ( 1 / 2 , f × χ q ) up to any specified precision. In the case when q is smooth or highly composite integer, the time complexity of the algorithm is given by O ( 1 + | q | 5 / 6 + o ( 1 ) ) .

Lattice point generating functions and symmetric cones

We show that a recent identity of Beck-Gessel-Lee-Savage on the generating function of symmetrically contrained compositions of integers generalizes naturally to a family of convex polyhedral cones that are invariant under the action of a finite reflection group. We obtain general expressions for the multivariate generating functions of such cones, and work out the specific cases of a symmetry group of type A (previously known) and types B and D (new). We obtain several applications of the special cases in type B, including identities involving permutation statistics and lecture hall partitions.

Tight Markov chains and random compositions

For an ergodic Markov chain $\{X(t)\}$ on $\mathbb{N}$, with a stationary distribution $\pi$, let $T_{n}>0$ denote a hitting time for $[n]^{c}$, and let $X_{n}=X(T_{n})$. Around 2005 Guy Louchard popularized a conjecture that, for $n\to\infty$, $T_{n}$ is almost $\operatorname{Geometric}(p)$, $p=\pi([n]^{c})$, $X_{n}$ is almost stationarily distributed on $[n]^{c}$ and that $X_{n}$ and $T_{n}$ are almost independent, if $p(n):=\sup_{i}p(i,[n]^{c})\to0$ exponentially fast. For the chains with $p(n)\to0$, however slowly, and with $\sup_{i,j}\|p(i,\cdot)-p(j,\cdot)\|_{\mathrm{TV}}<1$, we show that Louchard’s conjecture is indeed true, even for the hits of an arbitrary $S_{n}\subset\mathbb{N}$ with $\pi(S_{n})\to0$. More precisely, a sequence of $k$ consecutive hit locations paired with the time elapsed since a previous hit (for the first hit, since the starting moment) is approximated, within a total variation distance of order $k\sup_{i}p(i,S_{n})$, by a $k$-long sequence of independent copies of $(\ell_{n},t_{n})$, where $t_{n}=\operatorname{Geometric}(\pi(S_{n}))$, $\ell_{n}$ is distributed stationarily on $S_{n}$ and $\ell_{n}$ is independent of $t_{n}$. The two conditions are easily met by the Markov chains that arose in Louchard’s studies as likely sharp approximations of two random compositions of a large integer $\nu$, a column-convex animal (cca) composition and a Carlitz (C) composition. We show that this approximation is indeed very sharp for each of the random compositions, read from left to right, for as long as the sum of the remaining parts stays above $\ln^{2}\nu$. Combining the two approximations, a composition—by its chain, and, for $S_{n}=[n]^{c}$, the sequence of hit locations paired each with a time elapsed from the previous hit—by the independent copies of $(\ell_{n},t_{n})$, enables us to determine the limiting distributions of $\mu=o(\ln\nu)$ and $\mu=o(\nu^{1/2})$ largest parts of the random cca-composition and the random C-composition, respectively. (Submitted to Annals of Probability in June 2009.)

On reducing factorization to the discrete logarithm problem modulo a composite

The discrete logarithm problem modulo a composite—abbreviate it as DLPC—is the following: given a (possibly) composite integer n ≥ 1 and elements $${a, b \in \mathbb{Z}_n^*}$$, determine an $${x \in \mathbb{N}}$$ satisfying a x = b if one exists. The question whether integer factoring can be reduced in deterministic polynomial time to the DLPC remains open. In this paper we consider the problem $${{\rm DLPC}_\varepsilon}$$ obtained by adding in the DLPC the constraint $${x\le (1-\varepsilon)n}$$, where $${\varepsilon}$$ is an arbitrary fixed number, $${0 < \varepsilon\le\frac{1}{2}}$$. We prove that factoring n reduces in deterministic subexponential time to the $${{\rm DLPC}_\varepsilon}$$ with $${O_\varepsilon((\ln n)^2)}$$ queries for moduli less or equal to n.