Merge branch 'aaisha' into 'master'

Aaisha

See merge request !7

ECC Over GF(2m)/sage/Point_addition_affine_Mont.txt

+def Pointaddition_affine_Mont(x1,y1,x2,y2,a,PRECISION):
+r=(2^PRECISION).__xor__(n); rpoly = str_to_poly(r)
+	r2poly = rpoly * rpoly; r2 = poly_to_str(r2poly)
+	rinvpoly = rpoly^-1; rinv = poly_to_str(rinvpoly)
+
+x1_poly= str_to_poly(x1)
+y1_poly=str_to_poly(y1)
+x2_poly= str_to_poly(x2)
+y2_poly=str_to_poly(y2)
+
+x1_poly_MD = x1_poly * r2poly * rinvpoly
+y1_poly_MD = y1_poly * r2poly * rinvpoly
+x2_poly_MD = x2_poly * r2poly * rinvpoly
+y2_poly_MD = y2_poly * r2poly * rinvpoly
+
+#print 'x1_poly=',x1_poly
+#print 'y1_poly=',y1_poly
+#print 'x2_poly=',x2_poly
+#print 'y2_poly=',y2_poly
+
+x1poly_MD = (1 * r2poly * rinvpoly)
+	exp = 2^PRECISION-2
+	for i in reversed(xrange(PRECISION)):
+		x1poly_MD = x1poly_MD * x1poly_MD * rinvpoly
+		if(exp.digits(base=2,padto=PRECISION)[i] == 1):
+			x1poly_MD = x1_poly_MD * x1poly_MD * rinvpoly
+	inv_x1_poly_MD = x1poly_MD
+
+x2poly_MD = (1 * r2poly * rinvpoly)
+	exp = 2^PRECISION-2
+	for i in reversed(xrange(PRECISION)):
+		x2poly_MD = x2poly_MD * x2poly_MD * rinvpoly
+		if(exp.digits(base=2,padto=PRECISION)[i] == 1):
+			x2poly_MD = x2_poly_MD * x2poly_MD * rinvpoly
+	inv_x2_poly_MD = x2poly_MD
+
+a_MD = (a * r2poly * rinvpoly)
+1_MD = (1 * r2poly * rinvpoly)
+	c1_MD = y1_poly_MD + y1_poly_MD * rinvpoly
+	c2_MD = inv_x1_poly_MD + inv_x2_poly_MD * rinvpoly
+	c3_MD = c1_MD * c2_MD * rinvpoly
+	c4_MD = c3_MD * c3_MD * rinvpoly
+	c5_MD = c4_MD + c3_MD * rinvpoly
+	c6_MD = c5_MD +x1_poly_MD * rinvpoly
+	c7_MD = c6_MD +x2_poly_MD * rinvpoly
+	x3_poly_MD = c7_MD +a_MD * rinvpoly
+
+	c1_MD = y1_poly_MD + y1_poly_MD * rinvpoly
+	c2_MD = inv_x1_poly_MD + inv_x2_poly_MD * rinvpoly
+	c3_MD = c1_MD * c2_MD * rinvpoly
+	c4_MD = c3_MD * c3_MD * rinvpoly
+	c5_MD = c4_MD * c3_MD * rinvpoly
+	c6_MD = c5_MD + 1_MD * rinvpoly
+	c7_MD = c6_MD + x2_poly_MD * rinvpoly
+	c8_MD = c6_MD + a_MD * rinvpoly
+	c9_MD = c8_MD + c3_MD * rinvpoly
+	c10_MD = c9_MD + x1_poly_MD * rinvpoly
+	c11_MD = c10_MD + x2_poly_MD * rinvpoly
+	c12_MD = c11_MD + y1_poly_MD * rinvpoly
+	y3_poly_MD = c12_MD +a_MD * rinvpoly
+
+
+
+print 'x3_MD=',poly_to_str(x3_poly_MD)
+print 'y3_MD=',poly_to_str(y3_poly_MD)
\ No newline at end of file

ECC Over GF(2m)/sage/Point_addition_jacobi.txt

+def Pointaddition_Jacobi(x1,y1,z1,x2,y2,z2,PRECISION):
+
+x1_poly = str_to_poly(x1)
+y1_poly = str_to_poly(y1)
+z1_poly = str_to_poly(b1)
+x2_poly = str_to_poly(x2)
+y2_poly = str_to_poly(y2)
+z2_poly = str_to_poly(z2)
+
+print 'x1_poly=',x1_poly
+print 'y1_poly=',y1_poly
+print 'z1_poly=',z1_poly
+
+print 'x2_poly=',x2_poly
+print 'y2_poly=',y2_poly
+print 'z2_poly=',z2_poly
+
+
+o1_poly = z1_poly * z1_poly
+b_poly = x2_poly * o1_poly
+d_poly = y2_poly * o1_poly * z1_poly
+
+e_poly = x1_poly + b_poly
+f_poly = y1_poly + d_poly
+z3_poly = e_poly * z1_poly
+
+h_poly = f_poly * x2_poly + z3_poly * y2_poly
+i_poly = f_poly + z3_poly
+g_poly = e_poly + z1_poly
+
+x3_poly = a_poly * g_poly + f_poly * i_poly + e_poly * e_poly * e_poly
+y3_poly = i_poly * x3_poly  + g_poly * h_poly
+
+
+print 'x3=',poly_to_str(x3_poly)
+print 'y3=',poly_to_str(y3_poly)
+print 'z3=',poly_to_str(z3_poly)
+
+
+#formulas:
+      #O1 = Z12
+      #B = X2*O1
+      #D = Y2*O1*Z1
+      #E = X1+B
+      #F = Y1+D
+      #Z3 = E*Z1
+      #H = F*X2+Z3*Y2
+      #I = F+Z3
+      #G = Z32
+      #X3 = a2*G+F*I+E*E2
+      #Y3 = I*X3+G*H
\ No newline at end of file

ECC Over GF(2m)/sage/Point_addition_jacobi_mont.txt

+def Pointaddition_Jacobi_Mont(x1,y1,z1,x2,y2,z2,PRECISION):
+r=(2^PRECISION).__xor__(n); rpoly = str_to_poly(r)
+	r2poly = rpoly * rpoly; r2 = poly_to_str(r2poly)
+	rinvpoly = rpoly^-1; rinv = poly_to_str(rinvpoly)
+
+x1_poly = str_to_poly(x1)
+y1_poly = str_to_poly(y1)
+z1_poly = str_to_poly(b1)
+x2_poly = str_to_poly(x2)
+y2_poly = str_to_poly(y2)
+z2_poly = str_to_poly(z2)
+
+x1_poly_MD = x1_poly * r2poly * rinvpoly
+y1_poly_MD = y1_poly * r2poly * rinvpoly
+z1_poly_MD = z1_poly * r2poly * rinvpoly
+x2_poly_MD = x2_poly * r2poly * rinvpoly
+y2_poly_MD = y2_poly * r2poly * rinvpoly
+z2_poly_MD = z2_poly * r2poly * rinvpoly
+
+o1_poly_MD = z1_poly_MD * z1_poly_MD * rinvpoly
+b_poly_MD = x2_poly_MD * o1_poly_MD * rinvpoly
+c1_poly_MD = y2_poly_MD * o1_poly_MD * rinvpoly
+d_poly_MD = c1_poly_MD * z1_poly_MD * rinvpoly
+
+e_poly_MD = x1_poly_MD + b_poly_MD * rinvpoly
+f_poly_MD = y1_poly_MD + d_poly_MD * rinvpoly
+z3_poly_MD = e_poly_MD * z1_poly_MD * rinvpoly
+
+c2_poly_MD = f_poly_MD * x2_poly_MD * rinvpoly
+c3_poly_MD = z3_poly_MD * y2_poly_MD * rinvpoly
+h_poly_MD = c2_poly_MD + c3_poly_MD  * rinvpoly
+i_poly_MD = f_poly_MD + z3_poly_MD * rinvpoly
+g_poly_MD = e_poly_MD + z1_poly_MD * rinvpoly
+c4_poly_MD = a_poly_MD * g_poly_MD * rinvpoly
+c5_poly_MD = f_poly_MD * i_poly_MD * rinvpoly
+c6_poly_MD = e_poly_MD * e_poly_MD * rinvpoly
+c9_poly_MD = c4_poly_MD + c5_poly_MD * rinvpoly
+c10_poly_MD = c6_poly_MD * e_poly * rinvpoly
+x3_poly_MD = c9_poly_MD + c10_poly_MD * rinvpoly
+
+c7_poly_MD = i_poly_MD * x3_poly_MD * rinvpoly
+c8_poly_MD = g_poly _MD* h_poly_MD * rinvpoly
+y3_poly_MD = c7_poly_MD + c8_poly_MD * rinvpoly
+
+
+print 'x3_MD=',poly_to_str(x3_poly_MD)
+print 'y3_MD=',poly_to_str(y3_poly_MD)
+print 'z3_MD=',poly_to_str(z3_poly_MD)
+
+
+#formulas:
+      #O1 = Z12
+      #B = X2*O1
+      #D = Y2*O1*Z1
+      #E = X1+B
+      #F = Y1+D
+      #Z3 = E*Z1
+      #H = F*X2+Z3*Y2
+      #I = F+Z3
+      #G = Z32
+      #X3 = a2*G+F*I+E*E2
+      #Y3 = I*X3+G*H
\ No newline at end of file

ECC Over GF(2m)/sage/point_addition_affine_short_weierstrass.txt

+def Pointaddition_affine(x1,x2,y1,y2,a,PRECISION):
+
+x1_poly= str_to_poly(x1)
+y1_poly=str_to_poly(y1)
+x2_poly= str_to_poly(x2)
+y2_poly=str_to_poly(y2)
+
+
+print 'x1_poly=',x1_poly
+print 'y1_poly=',y1_poly
+
+print 'x2_poly=',x2_poly
+print 'y2_poly=',y2_poly
+
+xpoly = (1)
+	expp = 2^PRECISION-2
+	for i in reversed(xrange(PRECISION)):
+			xpoly = xpoly * xpoly
+			if(expp.digits(base = 2,padto=PRECISION)[i] == 1):
+					xpoly = x1_poly* xpoly
+	inv_x1_poly = xpoly
+xpoly = (2)
+	expp = 2^PRECISION-2
+	for i in reversed(xrange(PRECISION)):
+			xpoly = xpoly * xpoly
+			if(expp.digits(base = 2,padto=PRECISION)[i] == 1):
+					xpoly = x2_poly* xpoly
+	inv_x2_poly = xpoly
+
+#Calculation of resultant coordinates
+x3_poly = ( (y1_poly + y2_poly) *(inv_x1_poly + inv_x2_poly) ) ** 2 + ( (y1_poly + y2_poly) * (inv_x1_poly + inv_x2_poly) ) + x1_poly + x2_poly + a
+
+y3_poly = ( (y1_poly + y2_poly) *(inv_x1_poly + inv_x2_poly) ) ** 3 + (x2_poly + a + 1 ) * (y1_poly +  y2_poly *(inv_x1_poly + inv_x2_poly) ) + x1_poly + x2_poly + a + y1_poly
+
+print 'x3=',poly_to_str(x3_poly)
+print 'y3=',poly_to_str(y3_poly)
+
+ #Affine addition formulas: 
+ #(x1,y1)+(x2,y2)=(x3,y3) 
+    
+ #where
+     # x3 = ((y1+y2)/(x1+x2))2+((y1+y2)/(x1+x2))+x1+x2+a2
+     # y3 = ((y1+y2)/(x1+x2))3+(x2+a2+1)*((y1+y2)/(x1+x2))+x1+x2+a2+y1
\ No newline at end of file

README.md

@@ -28,13 +28,45 @@ Within this project work a Linux device driver should be extended by following E
 
 Functions:
 
-Preparation, Montgomery Transformation, Affine-to-Jacobi Transformation,
-Point Doubling, Point Addition, Jacobi-to-Affine Transformation,
+Preparation, Montgomery Transformation, Affine-to-Jacobi Transformation, Point Addition,
+Point Doubling,  Jacobi-to-Affine Transformation,
 Montgomery Back-transformation, Point Validation.
 By using a Linux User Space Application the correct functionality should be verified by performing Point Validations in GF(2m) for the supported ECC precision widths. Based on a
 given Point and elliptic curve equation a statement must be made whether the Point is on the curve or not.For the validation of the calculation inside of the CryptoCore the open-source mathematics
 software SageMath should be used.In order to be able to illustrate the time required for ECC Point Validation with different precision widths the Real Time Library support (-lrt) should be included.
 
+Point Addition:
+
+With 2 distinct points, P and Q, addition is defined as the negation of the point resulting from the intersection of the curve, E, and the straight line defined by the points P and Q, giving the point, R.
+P+Q=R
+(x1,y1)+(x2,y2)=(x3,y3)
+
+An elliptic curve in short Weierstrass form has parameters a2 a6 and coordinates x y satisfying the following equations:
+                   y^2+x*y=x^3+a2*x^2+a6
+
+
+Affine addition formulas: (x1,y1)+(x2,y2)=(x3,y3) where
+  x3 = ((y1+y2)/(x1+x2))^2+((y1+y2)/(x1+x2))+x1+x2+a2
+  y3 = ((y1+y2)/(x1+x2))^3+(x2+a2+1)*((y1+y2)/(x1+x2))+x1+x2+a2+y1
+
+Jacobian coordinates represent x y as X Y Z satisfying the following equations:
+
+  x=X/Z^2
+  y=Y/Z^3
+
+Assumptions: Z2=1.
+Explicit formulas:
+      O1 = Z1^2
+      B = X2*O1
+      D = Y2*O1*Z1
+      E = X1+B
+      F = Y1+D
+      Z3 = E*Z1
+      H = F*X2+Z3*Y2
+      I = F+Z3
+      G = Z3^2
+      X3 = a2*G+F*I+E*E^S2
+      Y3 = I*X3+G*H
 
 Contributors:
 
@@ -42,7 +74,7 @@ Contributors:
 
 2)Harshal Likhar
 
-3)Aisha
+3)Aaisha Ghodekar (Developer)
 
 4)Hamza