Open Problems Column

2021 ◽  
Vol 52 (3) ◽  
pp. 25-25
Author(s):  
William Gasarch
Keyword(s):  
21St Century ◽  
Open Problem ◽  
Open Problems

This issue's Open Problem Column is by Lance Fortnow and its titled Worlds to Die For: Open Oracle Questions for the 21st Century.

scholarly journals The design of statements and the (re)formulation and resolution of open problems that address issues of social relevance with the use of digital technologies in the initial formation of Mathematics teachers

2019 ◽  
Vol 21 (2) ◽  
Author(s):  
Fabiane Fischer Figueiredo ◽  
Claudia Lisete Oliveira Groenwald
Keyword(s):  
Mathematics Teachers ◽  
Open Problem ◽  
Basic Education ◽  
Digital Technologies ◽  
Pedagogical Practice ◽  
Pedagogical Workshop ◽  
Social Relevance ◽  
Open Problems ◽  
Initial Formation ◽  
Design Activities

This paper presents the results obtained with an investigation, in which a pair of Mathematics graduates, participants of an Extension Course, carried out the design of a statement of open problems. This investigation also addressed a theme of social relevance, in which technologies were used so that these problems were (re)formulated and solved, with the use of technological resources, by students of Basic Education. In order for this objective to be achieved, the training teachers carried out the design activities of problems with the use of digital technologies, planning of the pedagogical practice, in which these problems would be proposed, and of execution of this practice, which occurred through a pedagogical workshop. After completing these activities, the graduates had the opportunity to discuss and reflect on the experiences of designer and teacher, so that they contributed to producing knowledge about the design of open problem statements that address issues of social relevance, (re)formulation and resolution using digital technologies, and how to propose such problems.

scholarly journals The Exact Security of PMAC with Two Powering-Up Masks

2019 ◽  
pp. 125-145 ◽  
Author(s):  
Yusuke Naito
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Open Problem ◽  
Random Function ◽  
Block Cipher ◽  
Random Permutation ◽  
Authentication Code ◽  
Open Problems ◽  
Main Research ◽  
Main Research Topic

PMAC is a rate-1, parallelizable, block-cipher-based message authentication code (MAC), proposed by Black and Rogaway (EUROCRYPT 2002). Improving the security bound is a main research topic for PMAC. In particular, showing a tight bound is the primary goal of the research, since Luykx et al.’s paper (EUROCRYPT 2016). Regarding the pseudo-random-function (PRF) security of PMAC, a collision of the hash function, or the difference between a random permutation and a random function offers the lower bound Ω(q2/2n) for q queries and the block cipher size n. Regarding the MAC security (unforgeability), a hash collision for MAC queries, or guessing a tag offers the lower bound Ω(q2m /2n + qv/2n) for qm MAC queries and qv verification queries (forgery attempts). The tight upper bound of the PRF-security O(q2/2n) of PMAC was given by Gaži et el. (ToSC 2017, Issue 1), but their proof requires a 4-wise independent masking scheme that uses 4 n-bit random values. Open problems from their work are: (1) find a masking scheme with three or less random values with which PMAC has the tight upper bound for PRF-security; (2) find a masking scheme with which PMAC has the tight upper bound for MAC-security.In this paper, we consider PMAC with two powering-up masks that uses two random values for the masking scheme. Using the structure of the powering-up masking scheme, we show that the PMAC has the tight upper bound O(q2/2n) for PRF-security, which answers the open problem (1), and the tight upper bound O(q2m /2n + qv/2n) for MAC-security, which answers the open problem (2). Note that these results deal with two-key PMACs, thus showing tight upper bounds of PMACs with single-key and/or with one powering-up mask are open problems.

On a recursive sequence

Kybernetes ◽  
2007 ◽  
Vol 36 (1) ◽  
pp. 98-115
Author(s):  
Mehdi Dehghan ◽  
Reza Mazrooei‐Sebdani
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Open Problem ◽  
Design Methodology ◽  
Initial Conditions ◽  
Open Problems ◽  
Content Type ◽  
Rational Difference Equations ◽  
Global Behaviour ◽  
Recursive Sequence

PurposeThe aim in this paper is to investigate the dynamics of difference equation yn+1=(pyn+yn−k)/(qyn+yn−k), n=0,1,2,… where k∈{1,2,3,…}, the initial conditions y−k, … ,y−1,y0 and the parameters p and q are non‐negative.Design/methodology/approachThe paper studies characteristics such as the character of semicycles, periodicity and the global stability of the above mentioned difference equation.FindingsIn particular, the results solve the open problem introduced by Kulenovic and Ladas in their monograph, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures.Originality/valueThe global behaviour of the solutions of equation yn+1=(pyn+yn−k)/(qyn+yn−k), n=0,1,2,… were investigated providing valuable conclusions on practical data.

scholarly journals On the Maximum Order Complexity of Thue–Morse and Rudin–Shapiro Sequences along Polynomial Values

2020 ◽  
Vol 15 (2) ◽  
pp. 9-22
Author(s):  
Pierre Popoli
Keyword(s):  
Open Problem ◽  
General Pattern ◽  
Correlation Measure ◽  
Open Problems ◽  
Maximum Order ◽  
Large Maximum ◽  
Maximum Order Complexity

AbstractBoth the Thue–Morse and Rudin–Shapiro sequences are not suitable sequences for cryptography since their expansion complexity is small and their correlation measure of order 2 is large. These facts imply that these sequences are highly predictable despite the fact that they have a large maximum order complexity. Sun and Winterhof (2019) showed that the Thue–Morse sequence along squares keeps a large maximum order complexity. Since, by Christol’s theorem, the expansion complexity of this rarefied sequence is no longer bounded, this provides a potentially better candidate for cryptographic applications. Similar results are known for the Rudin–Shapiro sequence and more general pattern sequences. In this paper we generalize these results to any polynomial subsequence (instead of squares) and thereby answer an open problem of Sun and Winterhof. We conclude this paper by some open problems.

scholarly journals AN AFFIRMATIVE ANSWER TO QUASI-CONTRACTION'S OPEN PROBLEM UNDER SOME LOCAL CONSTRAINTS IN JS-METRIC SPACES

2019 ◽  
Vol 24 (3) ◽  
pp. 445-456
Author(s):  
Sanaz Pourrazi ◽  
Farshid Khojasteh ◽  
Mojgan Javahernia ◽  
Hasan Khandani
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Open Problem ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Affirmative Answer ◽  
Point Property ◽  
Open Problems ◽  
Local Constraints ◽  
Strict Fixed Point

In this work, we first present JS-Pompeiu-Hausdorff metric in JS metric spaces and then introduce well-behaved quasi-contraction in order to find an affirmative answer to quasi-contractions’ open problem under some local constraints in JS-metric spaces. In the literature, this problem solved when the constant modules α ∈ [0,1/2] and when α ∈ (1/2,1], finding conditions by which the set of all fixed points be non-empty, has remained open yet. Moreover, we support our result by a notable example. Finally, by taking into account the approximate strict fixed point property we present some worthwhile open problems in these spaces.

ON SOME PROPERTIES OF QUASI-DISTANCE-BALANCED GRAPHS

2018 ◽  
Vol 97 (2) ◽  
pp. 177-184
Author(s):  
ADEMIR HUJDUROVIĆ
Keyword(s):  
Open Problem ◽  
Open Problems ◽  
Balanced Graphs

For an edge$uv$in a graph$G$,$W_{u,v}^{G}$denotes the set of all vertices of$G$that are closer to$u$than to$v$. A graph$G$is said to bequasi-distance-balancedif there exists a constant$\unicode[STIX]{x1D706}>1$such that$|W_{u,v}^{G}|=\unicode[STIX]{x1D706}^{\pm 1}|W_{v,u}^{G}|$for every pair of adjacent vertices$u$and$v$. The existence of nonbipartite quasi-distance-balanced graphs is an open problem. In this paper we investigate the possible structure of cycles in quasi-distance-balanced graphs and generalise the previously known result that every quasi-distance-balanced graph is triangle-free. We also prove that a connected quasi-distance-balanced graph admitting a bridge is isomorphic to a star. Several open problems are posed.

Open problems in the theory of completely positive and copositive matrices

2015 ◽  
Vol 29 ◽  
pp. 46-58 ◽  
Author(s):  
Avi Berman ◽  
Mirjam Dur ◽  
Naomi Shaked-Monderer
Keyword(s):  
Open Problem ◽  
State Of The Art ◽  
The State ◽  
Open Problems ◽  
Completely Positive ◽  
Copositive Matrices ◽  
Completely Positive Matrices ◽  
Positive Matrices

We describe the main open problems which are currently of interest in the theory of copositive and completely positive matrices. We give motivation as to why these questions are relevant and provide a brief description of the state of the art in each open problem.

scholarly journals Universality and Almost Decidability

2017 ◽  
Author(s):  
Cristian S. Calude ◽  
Damien Desfontaines
Keyword(s):  
Open Problem ◽  
Universal Functions ◽  
Open Problems ◽  
Converse Implication ◽  
Density Zero ◽  
Natural Density ◽  
Positive Integers

We present and study new definitions of universal and programmable universal unary functions and consider a new simplicity criterion: almost decidability of the halting set. A set of positive integers S is almost decidable if there exists a decidable and generic (i.e. a set of natural density one) set whose intersection with S is decidable. Every decidable set is almost decidable, but the converse implication is false. We prove the existence of infinitely many universal functions whose halting sets are generic (negligible, i.e. have density zero) and (not) almost decidable. One result—namely, the existence of infinitely many universal functions whose halting sets are generic (negligible) and not almost decidable—solves an open problem in [9]. We conclude with some open problems.

scholarly journals Constructing Isostatic Frameworks for the $\ell^1$ and $\ell^\infty$ Plane

10.37236/8196 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Katie Clinch ◽  
Derek Kitson
Keyword(s):  
Open Problem ◽  
Spanning Trees ◽  
Constructive Method ◽  
Normed Spaces ◽  
Open Problems

We use a new coloured multi-graph constructive method to prove that if the edge-set of a graph $G=(V,E)$ has a partition into two spanning trees $T_1$ and $T_2$ then there is a map $p:V\to \mathbb{R}^2$, $p(v)=(p_1(v),p_2(v))$, such that $|p_i(u)-p_i(v)| \geqslant |p_{3-i}(u)-p_{3-i}(v)|$ for every edge $uv$ in $T_i$ ($i=1,2$). As a consequence, we solve an open problem on the realisability of minimally rigid bar-joint frameworks in the $\ell^1$ or $\ell^\infty$-plane. We also show how to adapt this technique to incorporate symmetry and indicate several related open problems on rigidity, redundant rigidity and forced symmetric rigidity in normed spaces.

scholarly journals Solution to open problems on fuzzy filters in logical algebras and secure communication encoding scheme on filters

Filomat ◽  
2018 ◽  
Vol 32 (5) ◽  
pp. 1679-1686 ◽  
Author(s):  
Wang Wei
Keyword(s):  
Distributive Lattice ◽  
Open Problem ◽  
Secure Communication ◽  
Chinese Remainder Theorem ◽  
Open Problems ◽  
Encoding Scheme ◽  
Equivalent Relation ◽  
Fuzzy Filters

We characterize fuzzy Boolean and implicative filters in pseudo BCK algebras, then get the essential equivalent relation between the two fuzzy filters. Based on this, we solve an open problem in pseudo BCK algebras. By constructing the Chinese remainder theorem with respect to filters in distributive lattice of polynomials, a new kind of communication encoding scheme is obtained, and we analyze the security of the scheme.

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