Introduction to Cryptography with Open-Source Software

260 Cryptography with Open-Source Software

Also as with Rijndael, this S-box is generated by an affine transformation of

inverses in the field. Suppose that n is a given nybble, which corresponds to

an element of F . Define

n

′

=



0 if n = 0

n

−1

otherwise.

Then

s

n

= an

′

+ b mod (y

4

+ 1)

where

a = y

3

+ y

2

+ 1, b = y

3

+ 1

and both a and b are considered as elements of the ring Z

2

[y]/(y

4

+ 1). This

is not a field as not all elements are invertible.

For example, consider the nybble 4 = 0 100, which corresponds to polyno-

mial x

2

. In F its inverse is x

3

+ x

2

+ 1:

sage: F.<x> = GF(16,name=’x’)

sage: p = 1/x^2; p

x^3 + x^2 +1

Then:

sage: R.<y> = PolynomialRing(GF(2))

sage: a = y^3+y^2+1 ...