From the Riddles of Ancient Egypt to Cryptography in the Renaissance — 3500 Years in the Making - Codes: The Guide to Secrecy from Ancient to Modern Times - page 12

positive integers and 0) in our scheme. (This is called modular arithmetic in
mathematical terms; in this case, modulo 26, and here 26 is called the modulus ).
We perform modular arithmetic in our daily lives when we look at our clocks as
mod 24 arithmetic. Once the 24 hours are done, we begin again to count from
zero to the midnight hour. This is what we will do here modulo 26. We need
a symbol other than = to denote our addition since the outcome will not be
strict equality, but rather equality after throwing away multiples of 26. Since
we might change the value of 26 for some other ciphers, then we need to keep
track of it as well. We do this by writing

25+3 ≡ 2 (mod 26) ,

for instance, in our current example since 25 + 3 = 26 + 2, which is just 2
when the 26 is discarded. Continuing then, we get that the plaintext numerical
equivalents 25 , 4 , 1 , 17 , 0become 2 , 7 , 4 , 20 , 3, and using Table 1.3, the ciphertext
message becomes CHEUD . Once sent, the recipient uses the key 3 to decipher
by first converting the ciphertext to letters via Table 1.3, then calculating, for
instance 2 1 = 25, given that

multiples of 26 are treated as 0and no negative numbers are allowed in our
arithmetic, described above. (In other words,

− 3 ≡ 25 (mod 26), since 2 − 3= − 1=26 −

1 is the same as 25 modulo
26, and we must choose 25 since only the nonnegative numbers less than 26
are allowed.) Similarly, all other numbers are decrypted to yield 25 , 4 , 1 , 17 , 0,
which, via Table 1.2 becomes zebra .
The Caesar cipher is a simple example of more general ciphers called a > ne
ciphers about which we will learn when we revisit the Caesar cipher in Chap-
ter 3. The introduction of the Caesar cipher is an opportunity to solidify our
understanding of ciphers in general. First, we describe it verbally, followed by
an illustration. As we have seen, a cipher not only involves a set of plain-
text/ciphertext pairs ( p, c ), but also a key k used to encipher and decipher.
Moreover, the key has to satisfy certain properties. We want to ensure that
when we encipher a plaintext element using the key, there is only one possible
ciphertext element, and there is only one possible decryption to plaintext possi-
ble. (In mathematical terms each key is called a one-to-one function.) Thus, we
may describe a cipher or cryptosystem as a set (a collection of distinct objects)
of plaintext/ciphertext pairs ( p, c ) together with (one or more) enciphering keys
k , each having a corresponding deciphering key d , called the inverse of k , such
that k ( p )= c and d ( c )= p . In other words, the action of enciphering using
k , denoted by k ( p )= c is “unlocked” by d when d is applied to c , denoted by
d ( c )= p . Hence, the action of k followed by d has the unique result of doing
“nothing” to p , namely,

−

d ( k ( p )) = d ( c )= p.

(In mathematical terms, this action is called an identity function since it iden-
tifies the original object with itself, p in this case.) These properties ensure a
well-defined cryptosystem, a definition that we will be using throughout.