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 Maximum order complexity of the sum of digits function in Zeckendorf base and polynomial subsequences

2021 ◽  
Author(s):  
Damien Jamet ◽  
Pierre Popoli ◽  
Thomas Stoll
Keyword(s):  
Natural Generalization ◽  
Correlation Measure ◽  
Pseudorandom Sequences ◽  
Maximum Order ◽  
Automatic Sequences ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
Large Maximum ◽  
Maximum Order Complexity ◽  
Subword Complexity

AbstractAutomatic sequences are not suitable sequences for cryptographic applications since both their subword complexity and their expansion complexity are small, and their correlation measure of order 2 is large. These sequences are highly predictable despite having a large maximum order complexity. However, recent results show that polynomial subsequences of automatic sequences, such as the Thue–Morse sequence, are better candidates for pseudorandom sequences. A natural generalization of automatic sequences are morphic sequences, given by a fixed point of a prolongeable morphism that is not necessarily uniform. In this paper we prove a lower bound for the maximum order complexity of the sum of digits function in Zeckendorf base which is an example of a morphic sequence. We also prove that the polynomial subsequences of this sequence keep large maximum order complexity, such as the Thue–Morse sequence.

Correlation measure, linear complexity and maximum order complexity for families of binary sequences

2022 ◽  
Vol 78 ◽  
pp. 101977
Author(s):  
Zhixiong Chen ◽  
Ana I. Gómez ◽  
Domingo Gómez-Pérez ◽  
Andrew Tirkel
Keyword(s):  
Linear Complexity ◽  
Binary Sequences ◽  
Correlation Measure ◽  
Maximum Order ◽  
Maximum Order Complexity

scholarly journals On the Maximum Order Complexity of the Thue-Morse and Rudin-Shapiro Sequence

2019 ◽  
Vol 14 (2) ◽  
pp. 33-42
Author(s):  
Zhimin Sun ◽  
Arne Winterhof
Keyword(s):  
Linear Complexity ◽  
Expected Value ◽  
Quality Measure ◽  
Explicit Formulas ◽  
Maximum Order ◽  
Order Of Magnitude ◽  
Large Maximum ◽  
Maximum Order Complexity

AbstractExpansion complexity and maximum order complexity are both finer measures of pseudorandomness than the linear complexity which is the most prominent quality measure for cryptographic sequences. The expected value of the Nth maximum order complexity is of order of magnitude log N whereas it is easy to find families of sequences with Nth expansion complexity exponential in log N. This might lead to the conjecture that the maximum order complexity is a finer measure than the expansion complexity. However, in this paper we provide two examples, the Thue-Morse sequence and the Rudin-Shapiro sequence with very small expansion complexity but very large maximum order complexity. More precisely, we prove explicit formulas for their N th maximum order complexity which are both of the largest possible order of magnitude N. We present the result on the Rudin-Shapiro sequence in a more general form as a formula for the maximum order complexity of certain pattern sequences.

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.

scholarly journals Maximum order-index of matrices over commutative inclines: an answer to an open problem

2007 ◽  
Vol 420 (1) ◽  
pp. 228-234
Author(s):  
Song-Chol Han ◽  
Hong-Xing Li
Keyword(s):  
Open Problem ◽  
Maximum Order

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 Spanning Trees with Many Leaves and Average Distance

10.37236/757 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ermelinda DeLaViña ◽  
Bill Waller
Keyword(s):  
Lower Bounds ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Average Distance ◽  
Independence Number ◽  
Local Independence ◽  
Open Problems ◽  
Maximum Order ◽  
Bipartite Subgraph ◽  
Number Of Leaves

In this paper we prove several new lower bounds on the maximum number of leaves of a spanning tree of a graph related to its order, independence number, local independence number, and the maximum order of a bipartite subgraph. These new lower bounds were conjectured by the program Graffiti.pc, a variant of the program Graffiti. We use two of these results to give two partial resolutions of conjecture 747 of Graffiti (circa 1992), which states that the average distance of a graph is not more than half the maximum order of an induced bipartite subgraph. If correct, this conjecture would generalize conjecture number 2 of Graffiti, which states that the average distance is not more than the independence number. Conjecture number 2 was first proved by F. Chung. In particular, we show that the average distance is less than half the maximum order of a bipartite subgraph, plus one-half; we also show that if the local independence number is at least five, then the average distance is less than half the maximum order of a bipartite subgraph. In conclusion, we give some open problems related to average distance or the maximum number of leaves of a spanning tree.

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