Introduction to Cryptography with Open-Source Software

250 Cryptography with Open-Source Software

sage: b1 = 1/b; b1

x^6 + x^5 + x^4 + x^2 + 1

Now for the affine transformation:

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

sage: a = y^4+y^3+y^2+y+1

sage: c = y^6+y^5+y+1

sage: q = (a*b1.polynomial().subs(x=y)+c).mod(y^8+1); q

y^7 + y^6 + y^4 + y^2 + 1

This last polynomial corresponds to the byte 11010101 as above.

The S-box transformation can be equivalently described using matrices.

Suppo se b

′

is defined a s above, a nd so

b

′

= b

7

b

6

b

5

b

4

b

3

b

2

b

1

b

0

.

Then the S-box output is the byte z = z

7

z

6

z

5

z

4

z

3

z

2

z

1

z

0

for which

























1 0 0 0 1 1 1 1

1 1 0 0 0 1 1 1

1 1 1 0 0 0 1 1

1 1 1 1 0 0 0 1

1 1 1 1 1 0 0 0

0 1 1 1 1 1 0 0

0 0 1 1 1 1