README.md

ecdlp-pollardrho-gmp

This C code is the GMP implementation of Pollard's Rho Attack on Elliptic Curve Discrete Logarithm Problem (ECDLP).

Getting Started

In order to compile this code, a C compiler and GMP Library (The GNU Multiple Precision Arithmetic Library) is required. gcc version 6.3.0 and GMP 6.1.2. have been used for testing.

Magma Computational Algebra System is used as an intermediate step between theory and programming. Also, the inputs of C code are obtained from Magma (See Testing section). You can purchase the software or can use the Online Calculator. Open-source lovers can use SageMath as well but this repository only contains Magma scripts.

Testing

First step is to create elliptic curve and choose the random variables in Magma. The following functions will be needed:

sum := function(P,Q,A,E)

	x1 := P[1];
	x2 := Q[1];
	y1 := P[2];
	y2 := Q[2];

	if P eq E!0 then return Q;
	elif Q eq E!0 then return P;
	elif x1 eq x2 then
		if y1 ne y2 then return E!0;
		elif y2 eq 0 then return E!0;
		else
			x3 := ((3*x1^2 + A )/(2*y1))^2 - 2*x1 ;
			y3 := (3*x1^2 + A )/(2*y1)*(x1-x3) - y1 ;
			return E![x3,y3];
		end if ;
	else
		x3 := ((y1- y2)/(x1-x2))^2 - x1- x2 ;
		y3 := (y1- y2)/(x1-x2) * (x1-x3) -y1 ;
		return E![x3,y3];
	end if ;

end function;

mult := function(k,P,E,A)
	M :=E!0;
	K := IntegerToSequence(k,2);

	for i in [1..#K] do
		if K[i] eq 1 then
			M := sum(M,P,A,E);
		end if;
		P := sum(P,P,A,E);
	end for;
	return M;
end function;

Then, you can obtain the inputs for attack:

p := NextPrime(Random([0..2^64]));
F := FiniteField(p);

repeat
	repeat
		A := Random(F);
		B := Random(F);
	until (4*A^3 +27*B^2) ne 0 ;
	E := EllipticCurve([A,B]);
until IsPrime(#E);

P := Random(E);
n := Order(P);
k := Random([0..n]);
Q := mult(k,P,E,A);

P; Q; n; k;

The full Magma code can be used for the verification of results of the C code.

Note: Magma Online Calculator has time limitation, thus Magma code cannot be run on Online calc. for 64 bit parameters.