I'm trying to implement an authentication protocol based on elliptic curve over prime field in Matlab. I have already done the point addition and point multiplication successfully, but I'm facing a problem while computing the following steps: (Capital letters are points on the curve, small letters are integer scalars)
((y^-1)T4-T2)(rs^-1) = x1P
((y^-1)T5-x1T3)(rs^-1) = X2
Should I compute
y^-1
as
ModInverse(y , prime)? % the multiplicative inverse of y mod prime
Also, is that how I'm supposed to do the subtraction?
nT2 = T2;
nT2{2} = mod(-nT2{2}, prime);
(Negate the y-coordinate of a point) mod prime, and then do the addition on nT2 and another point
I can help you in finding negative co-ordinate of y. I am explaining with a toy example:
sum = [673 146] % Now to convert 2nd element of sum which is the y coordinate, do the following.
sum(1,2) = -sum(1,2) % this will negate the 2nd element of 1st row of sum and store the result into sum. % now to perform subtraction, do the following. Assume that your addend is N1 = [ 6,5] and augend is the sum, therefore, the difference of these two is...
difference = addell(N1,sum,a,b,p) % where a,b and p are parameters of the ECC. %Thaat is a and b are constants and p is the prime modulus of the ECC. For example of an ECC is y^2=x^3+a*x +b mod p
Hope this will suffice your 2nd question.