Cryptanalytic results on ‘Dual CRT’ and ‘Common Prime’ RSA

In this paper we study weaknesses of two variants of RSA: Dual RSA and Common Prime RSA. Several schemes under the framework of Dual RSA have been proposed by Sun et al. (IEEE Trans Inf Theory 53(8):2922–2933, 2007). We here concentrate on the Dual CRT-RSA scheme and present certain range of parameters where it is insecure. As a corollary of our work, we prove that the Dual Generalized Rebalanced-RSA (Scheme III of Sun et al.) can be efficiently broken for a significant region where the scheme has been claimed to be secure. Next we consider the Common Prime RSA as proposed by Wiener (IEEE Trans. Inf. Theory 36:553–558, 1990). We present new range of parameters in Common Prime RSA where it is not secure. We use lattice based techniques for the attacks.     

‘冠军快跑’运动会

这是一场竞速运动的盛会．参赛选手为跑步、攀援、飞翔等各类运动的获胜者。本次的方格赛道有些特殊。它们必须按规则穿越各自的跑道。经过的格数最少,率先到达终点的就是冠军。精彩的比赛就要开始了!谁会获胜呢？     

Chains and Trees: ‘Strong’–‘Weak’ Order in Job Scheduling

We present a summary of recent NP-hardness and polynomial time solvability results for the distinction between strong and weak precedence for chains and trees in scheduling. We distinguish between chains and proper trees which are not chains, and demonstrate that the strong-weak precedence distinction for chains is not inclusive with regards to NP-hardness, and conjecture that the same holds for strong-weak tree precedence. The objective is to show that different interpretations for chain and tree order relations in scheduling might have far reaching computational implications.     

Auditing Shaked and Shanthikumar’s ‘excess wealth’

A notion called “excess wealth” was introduced by Shaked and Shanthikumar around 1998 (Probab. Eng. Inf. Sci. 12:1–23, 1998). Subsequent to this, much has been written on it, mostly by Shaked and his colleagues; see Sordo (Insur. Math. Econ. 45(3):466–469, 2009) for a recent review. These works have appeared in the literatures of reliability theory and stochastic orderings. Since the term excess wealth connotes a measure of income inequality—much like its dual, poverty—it should have had an impact in economics and the econometric literature. This, it appears is not the case, at least to the extent that it should be. The purpose of this paper is to investigate the above disconnect by looking at the notion of excess wealth more carefully, but keeping in mind the angle of economics and income. Our conclusion is that an alternative definition of excess wealth better encapsulates what one means by a colloquial use of the term. Our motivation for being attracted to this topic arises from two angles. The first is that the stochastics of diagnostic and threat detection tests, in which we have an interest, has a strong bearing on indices of concentration like the Lorenz Curve, the Gini index, and the entropy. Thus the notion of excess wealth, which conveys a sense of income concentration should also be relevant to diagnostics. The second motivation is to honor Moshe Shaked, a prolific researcher and a friend of the first author, by developing a paper based on an idea that is co-attributed to him.     

Brownian motion with killing and reflection and the ‘‘hot–spots’’ problem

We investigate the hot–spots property for the survival time probability of Brownian motion with killing and reflection in planar convex domains whose boundary consists of two curves, one of which is an arc of a circle, intersecting at acute angles. This leads to the hot–spots property for the mixed Dirichlet–Neumann eigenvalue problem in the domain with Neumann conditions on one of the curves and Dirichlet conditions on the other.Supported in part by NSF Grant # 9700585-DMSSupported in part by NSF Grant # 0203961-DMS     

‘智力快车’乘车资格认证

乘坐“智力快车”的感觉怎么样？很过瘾吧？不过，要想得心应手地驾驶功能丰富的“智力快车”可不是件简单的事儿，你得具备各种过硬的素质，才能在快车的引领下，     

The ‘corrected Durfee’s inequality’ for homogeneous complete intersections

We address the conjecture of Durfee (Math Ann 232:85–98, 1978), bounding the singularity genus $p_g$ by a multiple of the Milnor number $\mu $ for an $n$ -dimensional isolated complete intersection singularity. We show that the original conjecture of Durfee, namely $(n +1)!\cdot p_g \le \mu $ , fails whenever the codimension $r$ is greater than one. Moreover, we propose a new inequality $C_{n,r}\cdot p_g \le \mu $ , and we verify it for homogeneous complete intersections. In the homogeneous case the inequality is guided by a ‘combinatorial inequality’, that might have an independent interest.     

Some comments on Wolfe's ‘away step’

We give a detailed proof, under slightly weaker conditions on the objective function, that a modified Frank-Wolfe algorithm based on Wolfe's away step strategy can achieve geometric convergence, provided a strict complementarity assumption holds.Research supported by FCAC (Québec) and NSERC (Canada).     

‘Minesweeper’ and spectrum of discrete Laplacians

The article is devoted to a problem inspired by the ‘Minesweeper’ computer game. It is shown that certain configurations of open cells guarantee the existence and the uniqueness of solution. Mathematically the problem is reduced to some spectral properties of discrete differential operators. It is shown how the uniqueness can be used to create a new game which preserves the spirit of ‘Minesweeper’ but does not require a computer.     

谈谈‘9’的乘积巧算

多位数9乘任何数，运算起来并不难，但不论是‘加1乘’，还是‘跟踪乘’，拨珠次数都较烦琐。我在工作中摸索出一种巧算方法，也可称它为‘规律法’。现奉献给广大珠算卫作者和珠算爱好者。     

Robert Patterson: American ‘revolutionary’ mathematician

You will not see Robert Patterson’s name mentioned in many mathematics books. While his mathematical works survive, his name is more likely to appear in American history books dealing with the Colonial period, given his associations with the most influential men of that time. In this article, we will examine his mathematical work, as well as his contributions to a newly-formed nation. Most of what we know about Robert Patterson’s ancestors and life is due to his grandson, William Ewing DuBois, who wrote a family history in 1847. For other information, I have drawn upon diaries and a great many letters. All spelling and syntax are copied exactly as they appear.     

Efficient cuts in Lagrangean ‘Relax-and-cut’ schemes

Lagrangean relaxation produces bounds on the optimal value of (mixed) integer programming problems. These bounds, together with integer feasible solution values, provide intervals bracketing the optimal value of the original problem. When the residual gap, i.e., the relative size of the interval, is too large for the approximations to be deemed satisfactory, it is desirable to ‘strengthen’ the Lagrangean bounds. One possible strengthening technique consists of identifying cuts which are violated by the current Lagrangean solution, and dualizing them. Unfortunately not every valid inequality that is currently violated will improve the Lagrangean relaxation bound when dualized. This paper investigates what makes a violated cut ‘efficient’ in improving bounds. It also provides examples of efficient cuts for several (mixed) integer programming problems.     

A Significant ‘False Perception’ of Wittgenstein’s Draft on Mind’s Eye

If we read the Tractatus logico-philosophicus according to the decimal numbering of its propositions, we may understand, finally, the section about the self and the limits of language and world. Proposition 5.64 follows 5.63 (not 5.634); 5.634 follows 5.633 (not 5.6331); and so on. Thus, it becomes clear that the picture of the visual field (TLP 5.6331) cannot be what scholars have always quoted and discussed, i.e. a draft of an eye inside its field of sight. Actually, Wittgenstein’s original drafts depict (to criticise it) the ordinary way of representing the visual field. Following him, the field of vision is ‘without limits’; it does not have a form that implies the existence of an eye, as far as the phenomenological experience does not have a form that implies the existence of an ‘I’. As a result, the current reproductions of the 5.6331 image must be rectified, and all references to this similitude should be radically amended.     

由一道课本例题引出的结论——“‘角平分线’+‘平行线’”构成等腰三角形

题目(人教版《义务教育课程标准实验教科书》八年级·数学上册,12.3.1等腰三角形,例2)求证:如果三角形一个外角的平分线平行于三角形的一边,那么这个三角形是等腰三角形.     

On Jackson's product form with ‘jump-over’ blocking

It is shown that Jackson's product form is retained for queueing networks with capacity constraints under the assumption of a ‘jump-over’ blocking protocol.     

“金猴戏‘珠’”摆件赏析

这只＂金猴戏‘珠’＂摆件原来是中国珠算博物馆（原名＂中国南通珠算博物馆＂）正式开馆的纪念品现在则已经成为收藏界炙手可热的珍罕之物。