82
1. ALGEBRAIC THEMES
the message. What foils the snoop and makes the RSA method useful for
secure communications is that it is at present computationally difﬁcult to
factor very large integers.
12
On the other hand, it is computationally easy to ﬁnd large prime num
bers and to do the computations necessary for encryption and decryption.
So if my primes
p
and
q
are sufﬁciently large, you and I will still be able
to do our computations quickly and inexpensively, but the snoop will not
be able to factor
n
to ﬁnd
p
and
q
and decrypt our secret message.
Example 1.12.3.
I ﬁnd two (randomly chosen) 50–digit prime numbers:
p
D
4588423984 0513596008 9179371668 8547296304 3161712479
and
q
D
8303066083 0407235737 6288737707 9465758615 4960341401:
Their product is
n
D
3809798755658743385477098607864681010895851155818383
984810724595108122710478296711610558197642043079:
I choose a small prime
r
D
55589
, and I send you
n
and
r
by email. I
secretly compute
m
D
19048993778293716927385493039323405054479255779091
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Integers

Click to edit the document details