scholarly journals Generalized Group–Subgroup Pair Graphs

2020 ◽  
pp. 169-185
Author(s):  
Kazufumi Kimoto
Keyword(s):  
Zeta Function ◽  
Riemann Hypothesis ◽  
Cayley Graphs ◽  
Hash Functions ◽  
Finite Graph ◽  
Ramanujan Graphs ◽  
Basic Properties ◽  
Ramanujan Graph ◽  
Second Eigenvalue ◽  
Cryptographic Hash Functions

Abstract A regular finite graph is called a Ramanujan graph if its zeta function satisfies an analog of the Riemann Hypothesis. Such a graph has a small second eigenvalue so that it is used to construct cryptographic hash functions. Typically, explicit family of Ramanujan graphs are constructed by using Cayley graphs. In the paper, we introduce a generalization of Cayley graphs called generalized group–subgroup pair graphs, which are a generalization of group–subgroup pair graphs defined by Reyes-Bustos. We study basic properties, especially spectra of them.

scholarly journals Uniform logical cryptanalysis of CubeHash function

2010 ◽  
Vol 23 (3) ◽  
pp. 357-366
Author(s):  
Miodrag Milic ◽  
Vojin Senk
Keyword(s):  
Hash Function ◽  
Hash Functions ◽  
Promising Method ◽  
International Competition ◽  
Great Effort ◽  
Cryptographic Hash Function ◽  
The World ◽  
Cryptographic Hash Functions ◽  
Made In

In this paper we present results of uniform logical cryptanalysis method applied to cryptographic hash function CubeHash. During the last decade, some of the most popular cryptographic hash functions were broken. Therefore, in 2007, National Institute of Standards and Technology (NIST), announced an international competition for a new Hash Standard called SHA-3. Only 14 candidates passed first two selection rounds and CubeHash is one of them. A great effort is made in their analysis and comparison. Uniform logical cryptanalysis presents an interesting method for this purpose. Universal, adjustable to almost any cryptographic hash function, very fast and reliable, it presents a promising method in the world of cryptanalysis.

scholarly journals A (p,v)-Extension of Hurwitz-Lerch Zeta Function and its Properties

2018 ◽  
Author(s):  
Gauhar Rahman ◽  
KS Nisar ◽  
Shahid Mubeen
Keyword(s):  
Zeta Function ◽  
Beta Function ◽  
Integral Representations ◽  
Basic Properties ◽  
Mellin Transformation ◽  
Special Cases ◽  
Lerch Zeta Function ◽  
Derivative Formulas ◽  
Generating Relations

In this paper, we define a (p,v)-extension of Hurwitz-Lerch Zeta function by considering an extension of beta function defined by Parmar et al. [J. Classical Anal. 11 (2017) 81–106]. We obtain its basic properties which include integral representations, Mellin transformation, derivative formulas and certain generating relations. Also, we establish the special cases of the main results.

scholarly journals The Riemann Hypothesis is most likely true

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Complex Numbers ◽  
Natural Numbers ◽  
Chebyshev Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Sum Of Divisors

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. The Riemann hypothesis belongs to the David Hilbert's list of 23 unsolved problems and it is one of the Clay Mathematics Institute's Millennium Prize Problems. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n> 5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1} > e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show that the Riemann hypothesis is most likely true.

scholarly journals On the density of Cayley graphs of R.Thompson’s group F in symmetric generators

2021 ◽  
pp. 1-13
Author(s):  
V. S. Guba
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Finite Graph ◽  
Vertex Degree ◽  
Doubling Property ◽  
Generating Set ◽  
Finite Set ◽  
Standard Set ◽  
Thompson’S Group ◽  
Average Vertex

By the density of a finite graph we mean its average vertex degree. For an [Formula: see text]-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with [Formula: see text] generators is amenable if and only if the density of the corresponding Cayley graph equals [Formula: see text]. A famous problem on the amenability of R. Thompson’s group [Formula: see text] is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators [Formula: see text], is at least [Formula: see text]. This estimate has not been exceeded so far. For the set of symmetric generators [Formula: see text], where [Formula: see text], the same example only gave an estimate of [Formula: see text]. There was a conjecture that for this generating set equality holds. If so, [Formula: see text] would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set [Formula: see text], the inequality [Formula: see text] holds. In this paper, we disprove this conjecture showing that the density of the Cayley graph of [Formula: see text] in symmetric generators [Formula: see text] strictly exceeds [Formula: see text]. Moreover, we show that even larger generating set [Formula: see text] does not have doubling property.

Hash Functions and Their Applications

2018 ◽  
pp. 66-77
Author(s):  
Kannan Balasubramanian
Keyword(s):  
Hash Function ◽  
Hash Functions ◽  
Digital Signatures ◽  
Message Authentication ◽  
Authenticated Encryption ◽  
Authentication Codes ◽  
Message Authentication Codes ◽  
Message Digest ◽  
Cryptographic Hash Functions

Cryptographic Hash Functions are used to achieve a number of Security goals like Message Authentication, Message Integrity, and are also used to implement Digital Signatures (Non-repudiation), and Entity Authentication. This chapter discusses the construction of hash functions and the various attacks on the Hash functions. The Message Authentication Codes are similar to the Hash functions except that they require a key for producing the message digest or hash. Authenticated Encryption is a scheme that combines hashing and Encryption. The Various types of hash functions like one-way hash function, Collision Resistant hash function and Universal hash functions are also discussed in this chapter.

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