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What Is Discrete Logarithm Problem (DLP)
This section describes what is Discrete Logarithm Problem (DLP), which is the reverse operation of an exponentiation (or scalar multiplication) operation in an Abelian group. 2022-10-01, 221🔥, 0💬

Additive Notation of Abelian Group
This section describes the Additive notation of an Abelian Group. The addition sign, +, is used as the operator. Number 0 is used as the identity element. 2022-10-01, 217🔥, 0💬

Set Subgroup Order to Higher Value
This section provides a tutorial example on how to set the subgroup order a value greater than the order of the entire group, like 2 times of the modulo, to ensure correct result of scalar multiplications. 2022-10-01, 217🔥, 0💬

What Is Subgroup in Abelian Group
This section describes Subgroups in a Abelian Group. A subgroup in a Abelian Group is a subset of the Abelian Group that itself is an Abelian Group. The subgroup and its parent group are using the same operation. 2022-10-01, 216🔥, 0💬

Algebraic Solution for Point Doubling
This section provides an algebraic solution for calculating the addition operation of two points at the same location on an elliptic curve. 2022-10-01, 212🔥, 0💬

Scalar Multiplication or Exponentiation
This section describes what is Scalar Multiplication or Exponentiation in Abelian Groups. They are used represent the process of performing Abelian Group operations consecutively n times with the same element. 2022-10-01, 212🔥, 0💬

Reduced Elliptic Curve Group - E97(-1,1)
This section provides an example of a reduced Elliptic Curve group E97(-1,1). Some example points on the curve are is also provided. 2022-10-01, 203🔥, 0💬

EC Curves Supported by Java
This section provides is a list of named curves that are supported or not supported by 'keytool' and Java default library. 2022-10-01, 202🔥, 0💬

DLP And Trapdoor Function
This section exams the difficulty level of the Discrete Logarithm Problem (DLP) in several Abelian Group examples to see if them can be used to build trapdoor functions. 2022-10-01, 200🔥, 0💬

Prove of Elliptic Curve Addition Operation
This section describes how to prove that the addition operation on an elliptic curve can be successfully performed geometrically. 2022-10-01, 197🔥, 0💬

Python Program for Reduced Elliptic Curves
This section provides a Python program that finds all points on a reduced elliptic curve group, Ep(a,b). 2022-10-01, 197🔥, 0💬

Reduced Elliptic Curve Group - E127(-1,3)
This section provides an example of a reduced Elliptic Curve group E127(-1,3). An example of addition operation is also provided. 2022-10-01, 197🔥, 0💬

Terminology
List of terms used in this book. 2022-10-01, 196🔥, 0💬

Elliptic Curves in Integer Space
This section describes the fact that elliptic equations in 2-dimensional integer space can not be used to construct Abelian groups. 2022-10-01, 195🔥, 0💬

Elliptic Curve Operation Summary
This section provides a summary of elliptic curve operations and their properties discussed in this chapter. 2022-10-01, 192🔥, 0💬

Multiplicative Notation of Abelian Group
This section describes the multiplicative notation of an Abelian Group. The multiplication sign, *, is used as the operator. Number 1 is used as the identity element. 2022-10-01, 185🔥, 0💬

Converting Elliptic Curve Groups
This section describes steps on how to convert real number elliptic curve groups to cyclic subgroups of integer elliptic curve groups. 2022-10-01, 182🔥, 0💬

How to Calculate "M**e mod n"
This section discusses the difficulties of calculating 'M**e mod n'. The intermediate result of 'M**e' is too big for most programming languages. 2022-10-04, 181🔥, 0💬

Doubling or Squaring in Abelian Group
This section describes what is Doubling or Squaring in Abelian Groups. When performing the Abelian Group operation on two identical elements, this operation is called Squaring in multiplicative notation or Doubling in additive notation. 2022-10-01, 180🔥, 0💬

Modular Multiplication of 10 - Not Abelian Group
This section demonstrates that the modular arithmetic multiplication of 10 (integer multiplication operation followed by a modular reduction of 10) can not define an Abelian Group. 2022-10-01, 179🔥, 0💬

Integer Points of First Region as Element Set
This section describes how to use all integer points from the first region on a reduced elliptic curve as the element set to try to construct an Abelian group. 2022-10-01, 179🔥, 0💬

Elliptic Curves Reduced by Modular Arithmetic
This section describes elliptic curves reduced by modular arithmetic of prime numbers. We can find lots of more integer points on those reduced elliptic curves. 2022-10-01, 177🔥, 0💬

What Is Reduced Elliptic Curve Group
This section describes Reduced Elliptic Curve Groups or Elliptic Curve over Prime Field GF(p), denoted as Ep(a,b), which uses elliptic curve equations reduced by modular arithmetic of prime number p to define the group element set, and uses point addition operation based on the rule of chord reduced... 2022-10-01, 176🔥, 0💬

Finite Elliptic Curve Group, Eq(a,b), q = p^n
This section describes finite elliptic curve groups constructed with modular arithmetic reduction of prime power numbers, p^n. 2022-10-01, 176🔥, 0💬

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Android ASP Astrology Big5 Bitcoin Blowfish Built-in C# Calendar CD-DVD Cheminformatics Chinese Cryptography DES Email Encoding Ethereum Flash GB2312 General H Herong History HTML Java JavaScript JDBC JSP JVM Linux Molecule Movie Music MySQL Neural Network Perl PHP Physics PKI Publication Python Reference REST SOAP Sorting Swing TV Series UML Unicode VBScript Web Web-Services Windows WSDL XML XMLPad XSD XSL-FO