Advanced Information Security
3 PROJECTIVE COORDINATES
Dr. Turki F. Al-Somani
2017
Module Outlines
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Why Projective Coordinates ?
Which Projective Coordinates ?
Homogeneous,
Jacobian,
Lopez-Dahab,
Mixed,
and Edwards coordinate systems
Summary
Why Projective Coordinates ?
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The group operations in an affine
coordinate system involve finite field inversion, which is a very costly
operation, particularly over prime fields.
Projective coordinate systems are used to reduce the need for performing
inversion to only 1.
Several projective coordinate systems have been proposed:
Homogeneous, Jacobian, Lopez-Dahab, Mixed and Edwards coordinate systems
Which Projective Coordinate
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?
The selection of a projective coordinate is based on the number of arithmetic operations, mainly
multiplications.
This is to be expected due to the sequential nature of these architectures where a single multiplier is used.
For high performance implementations, such
sequential architectures are too slow to meet the demand of increasing number of operations.
One solution for meeting this requirement is to exploit the inherent parallelism within the elliptic curve point operations in projective coordinate
Homogeneous Coordinates
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For the Homogeneous, so called projective, coordinate system, an elliptic curve point P takes the form (x, y) = (X/Z, Y/Z).
Let P1, P2 and P3 be three different points on the elliptic curve over GF(p), where P1=(X1, Y1, Z1), P2=(X2, Y2, Z2=1) and P3=(X3, Y3, Z3).
Point addition with the Homogenous
coordinate systems can be computed as:
A=Y2Z1, B=X2Z1−X1, C=A2Z1−B3−2B2X1, X3=BC, Y3=A(B2X1−C)−B3Y1, Z3=B3Z1.
Homogeneous Coordinates (contd.)
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Point doubling, on the other hand, can be computed as: A=aZ12+3X12, B=Y1Z1,
C=X1Y1B, D=A2−8C, X3=2BD, Y3=A(4C−D)
−8Y12B2, Z3=8B3.
Jacobian Coordinates
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For the Jacobian coordinate system, P takes the form (x, y) = (X/Z2, Y/Z3).
Point addition can be computed as: A=X1, B=X2Z12, C=Y1, D=Y2Z13, E=B−A, F=D−C, X3=F2–(E3+2AE2), Y3=F(AE2−X3)−CE3,
Z3=Z1E.
Point doubling, on the other hand, can be computed as: A=4X1Y12, B=3X12+aZ14,
X3=B2−2A, Y3=B(A−X3)−8Y14, Z3=2Y1Z1.
Lopez-Dahab Coordinates
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Lopez-Dahab coordinate system takes the form (x,y)=(X/Z,Y/Z2).
Very efficient in GF(2m)
Point addition can be computed as:
A0=Y12Z12 , A1=Y1Z22 , B0=X2Z1, B1=X1Z2,
C=A0+A1, D=B0+B1, E=Z1Z2, F=DE, Z3=F2, G=D2(F+aE2), H=CF, X3=C2+H+G,
I=D2B0E+X3, J=D2A0+X3, Y3=HI+Z3J.
Point doubling can be computed as:
Z3=Z12X12, X3=X14+bZ14,
Y3=bZ14Z3+X3(aZ3+Y12+bZ14)
Mixed Coordinates
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The Mixed coordinate system adds two points where one is given in some
coordinate system while the other in another coordinate system.
The coordinate system of the resulting point, may be in a third coordinate
system
Mixed Coordinates (contd.)
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Edwards Coordinates
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Recently, Edwards showed that all
elliptic curves over prime fields could be transformed to the shape: x2 + y2 = c2 (1 + x2y2), with (0, c) as neutral element
and with the surprisingly simple and
symmetric addition law of two points P1
= (x1, y1) and P2 = (x2, y2) as:
Edwards Coordinates (contd.)
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To capture a larger class of elliptic curves over the original field, the notion of Edwards form
have been modified to include all curves x2 + y2
= c2 (1 + dx2y2) where cd(1−dc4) ≠ 0.
Point addition with the Edwards coordinate systems can be computed as: B=Z12Z1, C=X1X2, D=Y1Y2,
E=G–(C+D), F=dCD, G=(X1+Y1)(X2+Y2), X3=Z1E(B–
F), Z3=(B–F)(B+F), Y3=Z1(D–C)(B+F).
Point doubling, on the other hand, can be computed as: A=X1+Y1, B=A2, C=X12, D=Y12, E=C+D, F=B–E, H=Z12, I=2H, J=E–I, X3=FJ, Z3=EJ, Y3=E(C–D).
Inherent Parallelism (2006)
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Inherent Parallelism (2006)
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Inherent Parallelism (2006)
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Inherent Parallelism (2010)
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Inherent Parallelism (2010)
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Inherent Parallelism (2010)
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Inherent Parallelism (2013)
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Inherent Parallelism (2013)
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Inherent Parallelism (2015)
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Inherent Parallelism
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Inherent Parallelism
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Summary
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The projective coordinate systems are used to eliminate the need for
performing inversion.
For elliptic curves, many different forms of formulas are found for point addition and doubling.
The selection of a specific projective coordinate systems depends on:
Time
Inherent parallelism.
THANKS & GOOD LUCK
NEXT IS: 4 FIELD ARITHMETIC
Dr. Turki F. Al-Somani
2017
