EC Cryptography Tutorials - Herong's Tutorial Examples

∟Discrete Logarithm Problem (DLP)

∟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.

Now we understand what is a trapdoor function. Let's review the Exponentiation (or Scalar Multiplication) operation Abelian Groups again. We want to know if it can be used as a trapdoor function, because its reversion operation, the DLP (Discrete Logarithm Problem), seems to be harder to solve.

DLP Example 1: Arithmetic addition over the integer set of ..., -2, -1, 0, 1, 2, ... is an Abelian Group. Its DLP is too easy to solve. 

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

So this Abelian Group is not suitable to build a trapdoor function.

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 not that easy solve. But we have efficient algorithms to get it done. 

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

So this Abelian Group is not suitable to build a trapdoor function.

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

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

So this Abelian Group is not suitable to build a trapdoor function.

DLP Example 4: Modular 11 arithmetic multiplication over the integer set of 1, 2, ..., 10 is an Abelian Group. Its DLP is not that easy solve. But we have efficient algorithms to get it done. 

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

So this Abelian Group is not suitable to build a trapdoor function.

DLP Example 5: Modular 6700417 arithmetic multiplication over the integer set of 1, 2, ..., 6700416 is an Abelian Group. Its DLP is not that easy solve. But we have efficient algorithms to get it done. 

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

So this Abelian Group is not suitable to build a trapdoor function.

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 very hard solve when n large number. There is no known algorithms to get it done efficiently. 

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

So this Abelian Group is a good candidate to build a trapdoor function. In the next tutorial, we will compare the effort of performing the scalar multiplication with the effort of solving the DLP to demonstrate that this is a good trapdoor function.

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