There are infinitely many Carmichael numbers

We extend our previous computations to show that there are 1401644 Carmichael numbers up to 10 18 . As before, the numbers were generated by a back-tracking search for possible prime factorisations together with a "large prime variation". We present further statistics on the distribution of Carmichael numbers.

/pdf/the-carmichael-numbers-up-to-10-15-1x0eqzjpp6.pdf

The Carmichael Numbers up to 10 15

This large and impressive text-book provides a comprehensive account of the theory of algebraic structures. " The author believes that a methodological approach is to set out from generality and to proceed to the particular. A systematization of this kind was attempted but such efforts had to be kept within bounds, as the author wanted to write a text-book ". Nevertheless it is clear that the overall motivation of this book is to consider algebra from a structural point of view (which is not the same as considering it from an abstract point of view). The first short chapter contains an informal account of set theory, including proofs of the equivalence of the axiom of choice, Zorn's lemma, and the well-ordering theorem. Halmos' book Naive set theory has made us all a bit more sophisticated about set theory; this account is, I think, too naive, and the remarks about the axiom of choice could be misleading. Chapter I I is on structures, and is the longest chapter of the book— nearly two hundred pages. First, an (associative) algebraic structure is defined, and the concept is exemplified by semi-groups, groups, rings, fields and skew-fields. The author then develops the structural theory both in the general case and in the special cases provided by the examples. Thus when considering extensions of structures, he deals with Schreier group extensions and Everett ring extensions. Chapter I I I continues in the same vein, considering algebraic structures with operators, such as groups with operators and modules over a ring. The terminology which Redei uses is frequently confusing, as it does not follow normal practice. For example an abelian group is called a module, although the term left R-module is used in the usual sense; a structure preserving map between left R-modules is called a linear mapping, and those left R-modules which can be expressed as the direct sum of copies of R are called vector spaces.

Algebra, Volume 1. By L. Rédei. Pp. vxiii. 819, £7. 1967. (Pergamon.)

The HKM / HFX cryptosystem is proposed for standardization at the ITU Telecommunication Standardization Sector Study Group 8. It is designed to provide authenticity and confidentiality of FAX messages at a commercial level of security. In addition, the HKM / HFX cryptosystem is designed for unrestricted export.

/pdf/attacks-on-the-hkm-hfx-cryptosystem-3mm55t7kce.pdf

Attacks on the HKM/HFX Cryptosystem

Applications of Exponential Sums in Communications Theory.- Some Applications of Bounds for Designs to the Cryptography.- Further Results on the Relation Between Nonlinearity and Resiliency for Boolean Functions.- Combinatorial Structure of Finite Fields with Two Dimensional Modulo Metrics?.- A New Method for Generating Sets of Orthogonal Sequences for a Synchronous CDMA System.- New Self-Dual Codes over GF(5).- Designs, Intersecting Families, and Weight of Boolean Functions.- Coding Applications in Satellite Communication Systems [Invited Paper].- A Unified Code.- Enhanced Image Coding for Noisy Channels.- Perfectly Secure Authorization and Passive Identification for an Error Tolerant Biometric System.- An Encoding Scheme for Dual Level Access to Broadcasting Networks.- Photograph Signatures for the Protection of Identification Documents.- An Overview of the Isoperimetric Method in Coding Theory (Extended Abstract) [Invited Paper].- Rectangular Basis of a Linear Code.- Graph Decoding of Array Error-Correcting Codes.- Catastrophicity Test for Time-Varying Convolutional Encoders.- Low Complexity Soft-Decision Sequential Decoding Using Hybrid Permutation for Reed-Solomon Codes.- On Efficient Decoding of Alternant Codes over a Commutative Ring?.- Reduced Complexity Sliding Window BCJR Decoding Algorithms for Turbo Codes.- Advanced Encryption Standard (AES) - An Update [Invited Paper].- The Piling-Up Lemma and Dependent Random Variables.- A Cryptographic Application of Weil Descent.- Edit Probability Correlation Attack on the Bilateral Stop/Go Generator.- Look-Up Table Based Large Finite Field Multiplication in Memory Constrained Cryptosystems (Extended Abstract).- On the Combined Fermat/Lucas Probable Prime Test?.- On the Cryptanalysis of Nonlinear Sequences [Invited Paper].- Securing Aeronautical Telecommunications [Invited Paper].- Tensor-Based Trapdoors for CVP and Their Application to Public Key Cryptography (Extended Abstract).- Delegated Decryption.- Fast and Space-Efficient Adaptive Arithmetic Coding?.- Robust Protocol for Generating Shared RSA Parameters.- Some Soft-Decision Decoding Algorithms for Reed-Solomon Codes.- Weaknesses in Shared RSA Key Generation Protocols.- Digital Signature with Message Recovery and Authenticated Encryption (Signcryption) - A Comparison.

Cryptography and Coding

The paper gives a general definition for the concept of strong Dickson pseudoprimes which contains as special cases the Carmichael numbers and the strong Fibonacci pseudoprimes. Furthermore, we give necessary and sufficient conditions for two important classes of strong Dickson pseudoprimes and deduce some properties for their elements. A suggestion of how to improve a primality test by Baillie&Wagstaff concludes the paper.

/pdf/dickson-pseudoprimes-and-primality-testing-15gbxstllp.pdf

Dickson pseudoprimes and primality testing

Several mathematical procedures require the provision of large “random” prime numbers. Fermat’s Little Theorem plays an important role in many primality tests and in motivating the concept of pseudoprimes. It tells us that for a prime number n and an integer b with gcd(b, n) = 1 the congruence b n −1 ≡ 1 mod n holds. An alternative formulation of Fermat’s Little Theorem states that for any prime n and an arbitrary integer b one has 

$$ {b^n} \equiv b\;\bmod \,n. $$

(1)

Generalized Fibonacci Pseudoprimes and Probable Primes

On formal integration in composition rings of polynomials

Necessary and sufficient conditions are given for an odd composite integer n to be a Fibonacci pseudoprime of the m th kind for all m~7Z. One consequence of this characterization is that any such pseudoprime has to be a Carmichael number.

Some Remarks on Strong Fibonacci Pseudoprimes

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Alan Oswald

Author Tools

Chat about Author

Papers

Dickson pseudoprimes and primality testing

Generalized Fibonacci Pseudoprimes and Probable Primes

On formal integration in composition rings of polynomials

Some Remarks on Strong Fibonacci Pseudoprimes