Introduction to Cryptography with Open-Source Software

202 Cryptography with Open-Source Software

To get a better understanding of the DDT, consider the element in row 62,

column 5, which is 4. This means that there are four different tuples

(x

1

, x

2

, y

1

, y

2

)

for which x

1

⊕ x

2

= x

′

= 62 = 0b111110, y

1

⊕ y

2

= y

′

= 5 = 0b0101, and

y

1

= S(x

1

), y

2

= S(x

2

). These can be found very easily:

sage: for x1 in range(64):

....: for x2 in range(64):

....: y1 = Sb1(x1)

....: y2 = Sb1(x2)

....: if x1^^x2 == 62 and y1^^y2 == 5:

....: print x1,x2,y1,y2

....:

8 54 2 7

21 43 12 9

43 21 9 12

54 8 7 2

Note that if (x

1

, x

2

, y

1

, y

2

) is a tuple satisfying the requirements, then so is

(x

2

, x

1

, y

2

, y

1

) by the commutativity of the exclusive-or (XOR) ...