EC Cryptography Tutorials - Herong's Tutorial Examples

∟Discrete Logarithm Problem (DLP)

∟Examples of Discrete Logarithm Problem (DLP)

This section describes the Discrete Logarithm Problem (DLP) in several Abelian Group examples, including elliptic curve groups.

Let's now look at some examples of the Discrete Logarithm Problem (DLP).

DLP Example 1: Arithmetic addition over the integer set of ..., -2, -1, 0, 1, 2, ... is an Abelian Group. Its DLP is defined as: 

Given P and (Q = nP), find the smallest n as:
   n = Q/P

This DLP is very easy to solve. For example: 

   If P = 2, Q = 10, then n = 10/2 = 5
   If P = -7, Q = -7, then n = (-7)/(-7) = 1
   If P = 6700417, Q = 6700417, then n = 6700417/6700417 = 1
   ...

DLP Example 2: Arithmetic multiplication over the rational number set of ..., -1, -1/2, -1/3, 0, 1/3, 1/2, 1, ... is an Abelian Group. Its DLP is defined as: 

Given P and (Q = Pn), find the smallest n as:
   n = logP(Q)

This DLP is not so easy to solve, for certain values of P and Q. For example: 

   If P = 2, Q = 16, then n = log2(16) = 4
   If P = 7, Q = 13841287201, then n = log7(13841287201) = ?
   If P = 1/3, Q = 1/27, then n = log1/3(1/27) = 3
   If P = 1/11, Q = 1/1771561, then n = log1/11(1/1771561) = ?
   ...

DLP Example 3: Modular 10 arithmetic addition over the integer set of 0, 1, 2, ..., 9 is an Abelian Group. Its DLP is defined as: 

Given P and (Q = nP), find smallest n as:
   n = Q/P (mod 10)

This DLP is still easy to solve. For example: 

   If P = 2, Q = 2, then n = 2/2 = 1 (mod 10)
   If P = 7, Q = 3, then n = 3/7 = 9 (mod 10)
   ...

DLP Example 4: Modular 11 arithmetic multiplication over the integer set of 1, 2, ..., 10 is an Abelian Group. Its DLP is defined as: 

Given P and (Q = Pn), find the smallest n as:
   n = logP(Q) (mod 11)

This DLP is not so easy to solve, for some values of P. For example: 

   If P = 2, Q = 6, then n = log2(6) = 4  (mod 11)
   If P = 7, Q = 1, then n = log7(1) = ?  (mod 11)
   ...

DLP Example 5: Modular 6700417 arithmetic multiplication over the integer set of 1, 2, ..., 6700416 is an Abelian Group. Its DLP is defined as: 

Given P and (Q = Pn), find the smallest n as:
   n = logP(Q) (mod 6700417)

This DLP is hard to solve now, for large values of P. For example: 

   If P = 131071, Q = 524287, then n = log131071(6) = ?

DLP Example 6: Points on the elliptic curve y2 = x3 - 7x + 10 with the addition operation defined by the rule of chord is an Abelian Group. Its DLP is defined as: 

Given P and Q on the curve, find the smallest n, such that:
   Q = nP

This DLP is hard to solve. For example: 

   Given P = (1,2), Q = (-1,-4), and Q = nP, what is n?

Table of Contents

 About This Book

 Geometric Introduction to Elliptic Curves

 Algebraic Introduction to Elliptic Curves

 Abelian Group and Elliptic Curves

►Discrete Logarithm Problem (DLP)

 Doubling or Squaring in Abelian Group

 Scalar Multiplication or Exponentiation

 What Is Discrete Logarithm Problem (DLP)

►Examples of Discrete Logarithm Problem (DLP)

 What Is Trapdoor Function

 DLP and Trapdoor Function

 Scalar Multiplication on Elliptic Curve as Trapdoor Function

 Finite Fields

 Generators and Cyclic Subgroups

 Reduced Elliptic Curve Groups

 Elliptic Curve Subgroups

 tinyec - Python Library for ECC

 EC (Elliptic Curve) Key Pair

 ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

 ECDSA (Elliptic Curve Digital Signature Algorithm)

 ECES (Elliptic Curve Encryption Scheme)

 EC Cryptography in Java

 Standard Elliptic Curves

 Terminology

 References

 Full Version in PDF/EPUB