Datasheet

In GF(2
n
) fully reduced polynomials are of degree strictly lower than degree(P[X]). In many cases the
polynomials used in this library are only partially reduced and so have a degree higher or equal than
degree(P[X]), but this degree is maintained strictly lower than (degree(P[X]) + 15).
43.3.7.1.2 Coordinates System
In this implementation, several choices have been made related to the coordinate systems managed by
the elliptic curve primitives.
There are two systems currently managed by the library:
• Affine Coordinates System where each curve point has two coordinates (X,Y)
• Projective Coordinates System where each point is represented with three coordinates (X,Y,Z)
Converting from the affine coordinates system to a projective coordinates system and is performed by
extending its representation having Z = 1:
(X,Y) ⇒ (X,Y, Z= 1)
Converting from a projective coordinate to an affine one is a service offered by the library. The formula to
perform this conversion is:
(X,Y, Z) ⇒ (X ⇒ Z,Y/Z
2
)
43.3.7.1.3 Points Representation in Memory
Depending on the representation (Projective or Affine), points are represented in memory as shown in the
following figure.
Figure 43-15. Point Representation in Memory
In this figure, the modulus is represented as a reference, and to show that coordinates are always to be
provided on the length of the modulus plus one 32-bit word.
Different types of representations are listed here:
Affine representation:  =

 <  × 
15

 <  × 
15
SAM D5x/E5x Family Data Sheet
Public Key Cryptography Controller (PUKCC)
© 2019 Microchip Technology Inc.
Datasheet
DS60001507E-page 1545