1.12. AN APPLICATION TO CRYPTOGRAPHY
81
Similarly,
p
divides
a
h
a
. But then
a
h
a
is divisible by both
p
and
q
, and hence by
n
D
pq
D
l
:
c
:
m
:.p; q/
.
n
Let
r
be any natural number relatively prime to
m
, and let
s
be an
inverse of
r
modulo
m
,
rs
1
.
mod
m/
. We can encrypt and decrypt
natural numbers
a
that are relatively prime to
n
by first raising them to the
r
th
power modulo
n
, and then raising the result to the
s
th
power modulo
n
.
Lemma 1.12.2.
For all integers
a
, if
b
a
r
.
mod
n/;
then
b
s
a .
mod
n/:
Proof.
Write
rs
D
1
C
tm
Then
b
s
a
rs
D
a .
mod
n/
, by Lemma
1.12.1
.
n
Here is how these observations can be used to encrypt and decrypt
information: I pick two very large primes
p
and
q
, and I compute the
quantities
n
,
m
,
r
, and
s
. I publish
n
and
r
(or I send them to you privately,
but I don’t worry very much if some snoop intercepts them). I keep
p
,
q
,
m
, and
s
secret.
Any message that you might like to send me is first encoded as a
natural number by a standard procedure; for example, a text message is
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 Fall '08
 EVERAGE
 Algebra, Cryptography, Remainder, Natural number, Prime number, Euclidean algorithm

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