Cryptography Reference
In-Depth Information
that a sequence (we sometimes refer to this as a string ) of zeros and ones such
as 1101001 does actually represent a binary number . A full explanation of binary
numbers and how they relate to the more familiar decimal numbers is provided
in the Mathematics Appendix. This also includes an explanation of hex , which is
most useful to us as a compact way of representing binary numbers.
XOR
Modern symmetric cryptographic algorithms process binary data by conducting
various different operations on the data. One common operation is to compute
the exclusive or , better known as XOR, of two binary strings (numbers). This
is essentially the equivalent of 'addition' for binary numbers. Thus, every time
that we refer to the binary operation XOR, it is reasonable to interpret this as
'adding' the two binary strings together. When we refer to this operation in text
we use the term XOR, but when we write XOR in mathematical notation we
commonly use the symbol (which itself indicates that we are conducting a type
of 'addition'). The XOR operation is described in more detail in the Mathematics
Appendix.
EXPONENTIATION
A mathematical operation that we will often need to refer to is exponentiation .
This means raising a number to a power , which means multiplying the original
number by itself a certain number of times. Commonly we will need to raise the
number 2 to various powers. We use the conventional notation 2 k tomean 'raising
the number 2 to the power k ', whichmeans multiplying 2 by itself k times. In other
words:
2 k
=
2 × 2 ×···× 2 ,
where there is a total of k occurrences of 2 on the right. As an example
with k = 4:
2 4
=
2 × 2 × 2 × 2 = 16 .
More generally, we use the notation a b to mean raising a to the power b , which
just means multiplying the number a by itself b times. In other words:
a b
=
a × a × a ×···× a ,
where there is a total of b occurrences of a on the right. As an example with a = 3
and b = 5:
3 5
=
3 × 3 × 3 × 3 × 3 = 243 .
Another simple fact that we will need later is that if we raise a to the power b , and
then raise the result to the power c , then this is the same as raising a to the power
b × c . In other words:
( a b ) c
a b × c
=
.
 
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