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A simple C++ implementation of Elliptic Curve Cryptography

, 7 Jan 2008 CPOL
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A finite field EC and simple ECC scheme in C++ to help understand the principles.

Introduction

Elliptic Curve Cryptography is an exciting and promising method of encrypting data which achieves the same, or better, strength with far smaller key lengths than traditional encryption methods such as RSA. Elliptic Curves are themselves not rocket science, but the plethora of articles and mathematical background out there do leave it somewhat as "a non-trivial exercise to the causal reader" to actually see how the scheme can be implemented and used. Alas, I for one do not code for a living anymore and hence I always look for compact, to the point, implementations showing with code exactly how something works.

I hope that the source files you download with this article will provide one such source of compact, easy to understand, material to demystify and indeed realize how Elliptic Curves (notice the capitalization here...) can be coded in C++ and used to encrypt and decrypt messages between the ever present Alice and Bob...

Background

Yes, there is plenty of background. Firstly, you should understand the basics of Elliptic Curves, and I have found no better place to learn about them than here: Certicom's EC tutorial. It explains the math behind the ECs, and the important use of ECs over finite fields, i.e., over integers modulo some other integer (usually chosen to be a prime so that the period of any sequence of integers generated by multiplication or addition becomes "long enough").

Secondly, you should probably take some time to think relatively deeply about how finite fields actually work. Finding the inverse of a number in a finite field, for example, is not immediately trivial (unless you do this sort of thing for a living). And, since that is quite fundamental and used quite a lot in the code, I will outline this here:

Given a finite field Fp where p is a prime number (or more specifically a prime power) and a, b are elements of Fp, a is the multiplicative inverse of b if (and only if):

(a * b) mod p == 1

Which makes sense, as (in "real speak") a is the inverse of b if a*b == 1, i.e., a = 1/b.

To find a, given b and p, requires the use of the "Greatest Common Divisor" (GCD) which returns the largest integer less than (or equal to) a (or) and b that divides a and b evenly.

If this integer is 1, then a and b are relative prime, since only 1 can divide them both evenly. Now, given a and b and p, if b (the inverse of which we are looking for) and p are relative prime, then we can find an inverse. This also makes sense since since if b and p are relative prime, you can always write:

b*u + p*v == 1

since GCD(b,p) == 1 only if b and p have this relation. Now, if p is an actual prime number, then b always has an inverse modulo p...

Since pretty much *all* modern encryption schemes use prime numbers and modulo arithmetic one way or the other, it is a Good Thing to learn the basics.

Using the code

I hope the code is pretty much self explanatory. It was developed using DevCpp, MinGW, and GCC version 3.4.2, but has not been tested on other compilers. Although I do like my C++, I have not gone overboard with anything that could cause "ANSI compliance" issues, but please let me know if you find anything.

To get started, go to look at int main(... at the bottom of main.cpp. FiniteFieldElement.hpp is the header file implementing modular arithmetic using normal integers. Warning: I have had to adjust for the modulus of negative numbers, and I assume (since the ANSI standard doesn't explicitly state anything about it) that it could be different on other compilers. Just be forewarned that if something doesn't look right, that could be the reason!

Also: this is implemented using bog standard machine integers, no special big-integer support here.

The example encrypts a message from Alice which is "1972", so if everything is running alright, you should see that Bob's decrypted message reads just that.

Points of interest

Great fun to implement this, in particular when it worked. I encourage anybody to expand on Elliptic Curve implementations to ensure that the understanding and knowledge of these powerful mathematical entities is spread out as much as possible. Security can't be secure enough.

History

First version written over Christmas in the south of not-so-sunny France.

License

This article, along with any associated source code and files, is licensed under The Code Project Open License (CPOL)

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About the Author

Jarl Ostensen
Program Manager Electronic Arts Inc
United Kingdom United Kingdom
Started working for Scala Multimedia Television A/S back in the early 90's and moved on from there to work for Bullfrog Productions Ltd in the UK, later Electronic Arts Inc, where I've stayed since...

Comments and Discussions

 
QuestionNeed Algorithm Pin
Prashant211112-Mar-15 21:22
memberPrashant211112-Mar-15 21:22 
Questionecc implementation in software Pin
Member 1035328123-Oct-13 9:02
memberMember 1035328123-Oct-13 9:02 
Questionecc Pin
sangamkumar16-May-13 0:01
membersangamkumar16-May-13 0:01 
GeneralHow to generate random Pin
Krishna Veer17-Jun-12 21:13
memberKrishna Veer17-Jun-12 21:13 
QuestionBig Endian vs Little Endian Pin
Member 90770115-Jun-12 9:47
memberMember 90770115-Jun-12 9:47 
Questionbigint? Pin
vixen1623-Feb-11 9:18
membervixen1623-Feb-11 9:18 
AnswerRe: bigint? Pin
Jarl Ostensen31-Mar-11 7:36
memberJarl Ostensen31-Mar-11 7:36 
GeneralRe: please help Pin
Jarl Ostensen16-Oct-10 5:46
memberJarl Ostensen16-Oct-10 5:46 
Generalgreat work sir [modified] Pin
sangamkumar4-Oct-10 20:38
membersangamkumar4-Oct-10 20:38 
GeneralRe: great work sir Pin
Jarl Ostensen4-Oct-10 23:58
memberJarl Ostensen4-Oct-10 23:58 
Generalecc Pin
saauuraabh28-May-10 0:08
membersaauuraabh28-May-10 0:08 
GeneralRe: ecc Pin
Jarl Ostensen29-May-10 1:22
memberJarl Ostensen29-May-10 1:22 
Generalnotices Pin
Member 15130997-Apr-10 4:04
memberMember 15130997-Apr-10 4:04 
GeneralRe: notices Pin
Jarl Ostensen8-Apr-10 2:19
memberJarl Ostensen8-Apr-10 2:19 
GeneralPlease Help! Pin
HabibVaez22-Nov-09 8:26
memberHabibVaez22-Nov-09 8:26 
GeneralRe: Please Help! Pin
Jarl Ostensen8-Apr-10 2:15
memberJarl Ostensen8-Apr-10 2:15 
QuestionC# ECC encrypt/decrypt? Pin
Member 151309928-Oct-09 4:34
memberMember 151309928-Oct-09 4:34 
GeneralRe:Help Pin
astra kevin19-Jul-09 22:04
memberastra kevin19-Jul-09 22:04 
GeneralTo compile with GCC... Pin
Jarl Ostensen2-Apr-09 8:51
memberJarl Ostensen2-Apr-09 8:51 
GeneralRE: to the point, implementation Pin
k0walski2k31-May-08 9:20
memberk0walski2k31-May-08 9:20 
GeneralRe: RE: to the point, implementation Pin
Jarl Ostensen8-Jun-08 23:48
memberJarl Ostensen8-Jun-08 23:48 
GeneralIs it really that secure? [modified] Pin
Rene Pilon9-Jan-08 5:52
memberRene Pilon9-Jan-08 5:52 
GeneralRe: Is it really that secure? Pin
Jarl Ostensen11-Jan-08 6:27
memberJarl Ostensen11-Jan-08 6:27 
GeneralMath Pin
axelriet28-Dec-07 22:30
memberaxelriet28-Dec-07 22:30 
GeneralRe: Math Pin
Jarl Ostensen7-Jan-08 7:28
memberJarl Ostensen7-Jan-08 7:28 
GeneralCompiling code with MS MC Pin
Jarl Ostensen27-Dec-07 11:36
memberJarl Ostensen27-Dec-07 11:36 
GeneralRe: Compiling code with MS MC Pin
Member 15130995-Apr-10 12:21
memberMember 15130995-Apr-10 12:21 
GeneralGood Work Pin
Hatem Mostafa26-Dec-07 5:27
memberHatem Mostafa26-Dec-07 5:27 
GeneralRe: Good Work Pin
Jarl Ostensen27-Dec-07 11:30
memberJarl Ostensen27-Dec-07 11:30 

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