Introduction Elliptic Curve Cryptography is an exciting and promising ...

daughterinsectAI and Robotics

Nov 21, 2013 (3 years and 4 months ago)

77 views

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 in themselves are no
t
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 two source files you may download with this article 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 shou
ld 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 maths behind the EC's and th
e all important
use of EC's 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 p
robably 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 fund
amental and used quite a lot in the code, I will outline
this here:

Given a finite field F
p

where p is a prime number (or more specifically a
prime power
)
and a, b are elements of F
p
,
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
divid
es
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
(which' inverse we are looking for) and
p
are relative prime
then
we can find an inverse.

This also makes sen
se 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 encrypti
on 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 Dev
-
Cpp
and MinGW, gcc version 3.4.2 but has not been te
sted 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.
Finite
FieldElement.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 m
essage from Alice

w
hich 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.