Cryptology - Unofficial St. Mary's College of California Web Site

56 CHAPTER 4. THE EUCLIDEAN ALGORITHM
Examples:
(1) Encipher multiply with key 7m + 2.
Rather than just adding or multiplying we combine the two.
plaintext m u l t i p l y
plainnumbers 13 21 12 20 9 16 12 25
×7 91 147 84 140 63 112 84 175
+2 93 149 86 142 65 114 86 177
%26 15 19 8 12 13 10 8 21
ciphertext O S H L M J H U
So the ciphertext is OSHLMJHU
(2) Encipher decimate with key 5m + 8.
(3) Encipher conquer with key 9m + 3. 1
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄
4.2 GCD’s and the Euclidean Algorithm
Our Decimation and Linear Ciphers have two problems: the enciphering key
must be chosen “properly” (a term I still haven’t defined) and once it is chosen
the corresponding deciphering key must somehow be determined. It is perhaps
not surprising that the solutions to these two problems are related. However it
is probably surprising that the solutions involve greatest common divisors and
were known to Euclid, some 2500 years ago.
Euclid (c. 350 B.C.E.) textbook, The Elements, is the most successful ever
written, and, with The Bible, one of the most published books of all time,
appearing in over 1000 editions. Euclid taught at the academy in Alexandria,
but this is about all we know about his life. The Elements deals with plane
and solid geometry and number theory, while other books of Euclid cover such
topics as astronomy, mechanics, music, and optics.
The Greatest Common Divisors, or gcd, of two integers is exactly what
the name suggests: it is the largest integer that divides both. For example, 14
is divisible by 1, 2, 7 and 14, while 10 is divisible by 1, 2, 5 and 10. The largest
divisor that 14 and 10 have in common is 2, and so gcd(14, 10) = 2.
For some reason most people seem able to rather automatically compute the
gcd’s of small numbers.
1 (2) BGWAU MDG, (3) DHYZJ VI.