Archive:Prime Curve Standard Projective Coordinates - Revision history

wikademia>Wikademia: moved Prime Curve Standard Projective Coordinates to Archive:Prime Curve Standard Projective Coordinates

2009-06-07T06:02:33Z

moved Prime Curve Standard Projective Coordinates to Archive:Prime Curve Standard Projective Coordinates

← Older revision	Revision as of 06:02, 7 June 2009
(No difference)

wikademia>Wikademia at 18:43, 16 May 2009

2009-05-16T18:43:19Z

← Older revision		Revision as of 18:43, 16 May 2009
Line 3:		Line 3:
	===Introduction===		===Introduction===
	'''Standard Projective Coordinates''' are used to represent elliptic curve points on prime curves y^2 = x^3 + ax + b. Their usage might give a speed benefit over ''Affine Coordinates'' when the cost for field inversions is significantly higher than field multiplications. In ''Standard Projective Coordinates'' the triple (X, Y, Z) represents the affine point (X / Z, Y / Z).		'''Standard Projective Coordinates''' are used to represent elliptic curve points on prime curves y^2 = x^3 + ax + b. Their usage might give a speed benefit over ''Affine Coordinates'' when the cost for field inversions is significantly higher than field multiplications. In ''Standard Projective Coordinates'' the triple (X, Y, Z) represents the affine point (X / Z, Y / Z).
			{{?}}
	===Point Doubling (7M + 5S or 7M + 3S) ===		===Point Doubling (7M + 5S or 7M + 3S) ===
	Let (X, Y, Z) be a point (unequal to the ''point at infinity'') represented in ''Standard Projective Coordinates''. Then its double (X', Y', Z') can be calculated by		Let (X, Y, Z) be a point (unequal to the ''point at infinity'') represented in ''Standard Projective Coordinates''. Then its double (X', Y', Z') can be calculated by

wikademia>Wikademia: 2 revisions

2009-05-11T20:28:57Z

2 revisions

← Older revision	Revision as of 20:28, 11 May 2009
(No difference)

wikademia>Dallas1278 at 04:29, 27 July 2008

2008-07-27T04:29:27Z

← Older revision		Revision as of 04:29, 27 July 2008
Line 44:		Line 44:

	[[Category:Cryptography]]		[[Category:Cryptography]]
			{{alphabetical\|P}}

wikademia>Yann at 10:01, 25 September 2005

2005-09-25T10:01:04Z

New page

<[[Cryptography]]

===Introduction===
'''Standard Projective Coordinates''' are used to represent elliptic curve points on prime curves y^2 = x^3 + ax + b. Their usage might give a speed benefit over ''Affine Coordinates'' when the cost for field inversions is significantly higher than field multiplications. In ''Standard Projective Coordinates'' the triple (X, Y, Z) represents the affine point (X / Z, Y / Z).

===Point Doubling (7M + 5S or 7M + 3S) ===
Let (X, Y, Z) be a point (unequal to the ''point at infinity'') represented in ''Standard Projective Coordinates''. Then its double (X', Y', Z') can be calculated by

if (Y == 0)
return POINT_AT_INFINITY
W = a*Z^2 + 3*X^2
S = Y*Z
B = X*Y*S
H = W^2 - 8*B
X' = 2*H*S
Y' = W*(4*B - H) - 8*Y^2*S^2
Z' = 8*S^3
return (X', Y', Z')

Note: if a = -3, then W can also be calculated as W = 3*(X + Z)*(X - Z), saving 2 field squarings.

===Point Addition (12M + 2S) ===
Let (X1, Y1, Z1) and (X2, Y2, Z2) be two points (both unequal to the ''point at infinity'') represented in ''Standard Projective Coordinates''. Then the sum (X3, Y3, Z3) can be calculated by
U1 = Y2*Z1
U2 = Y1*Z2
V1 = X2*Z1
V2 = X1*Z2
if (V1 == V2)
if (U1 != U2)
return POINT_AT_INFINITY
else
return POINT_DOUBLE(X1, Y1, Z1)
U = U1 - U2
V = V1 - V2
W = Z1*Z2
A = U^2*W - V^3 - 2*V^2*V2
X3 = V*A
Y3 = U*(V^2*V2 - A) - V^3*U2
Z3 = V^3*W
return (X3, Y3, Z3)

=== Mixed Addition (with affine point) (9M + 2S) ===
Let (X1, Y1, Z1) be a point represented in ''Standard Projective Coordinates'' and (X2, Y2) a point in ''Affine Coordinates'' (both unequal to the ''point at infinity''). A formula to add those points can be readily derived from the regular standard projective point addition by replacing each occurance of "Z2" by "1" (and thereby dropping three field multiplications).

[[Category:Cryptography]]