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

8.4. MULTIPLE ENCIPHERINGS 147
Just as the gcd is the largest number that divides both, the lcm is the
smallest number that both divide. (Obviously word order matters!) Are they
connected in anyway besides word order Consider the following chart:
m n mn gcd(m, n) lcm(m, n)
4 8 32 4 8
4 6 24 2 12
3 8 24 1 24
8 10 80 2 40
5 12 60 1 60
8 9 72 1 72
From the examples it appears that gcd(m, n) × lcm(m, n) = m × n, and this is
correct. Since we have a method for computing gcd’s, it is perhaps better to
write
mn
lcm(m, n) =
gcd(m, n) .
In particular, if m and n are relatively prime then lcm(m, n) is just the product
mn.
Examples: What keylength do the following keys produce
(1) hamburger and FrenchFries give a 9 × 11 = 99 letter keyword
(2) Francisco and California and Oakland. letter keyword.
(3) SanFrancisco and California and Oakland give a 12×10×7
2
= 350 letter
keyword.
(4) SanFrancisco and California and Oakland and FrenchFries. 12
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄
Using multiple keywords thus produces the equivalents of very long keywords
relatively easily. How helpful is this Suppose our goal is to have no more than
20 letters of depth per keyword letter in a Vigenère cipher. There are about
70 characters per line of typed text and about 50 lines of text per page, so
about 3500 characters per page. Since 3500/20 = 180, for each page of message
we would need more than 180 letters worth of keyword. So Francisco and
California and Oakland would safely encipher no more than (9 × 10 × 7 =
630)/180 = 3.5 pages of message.
Suppose we must send 100 pages of messages a day We must have keyword
of about 100×180 = 18, 000 letters (= 6 pages), or a clever selection of 4 words,
with lengths 7, 9, 13, and 17. And these words must be changed everyday.
Modern life is making this problem even more severe: speaking at 200 words
per minute on a cell phone means to have a secure conversation one needs in
12 (2) 9 × 10 × 7 = 630, (4) 12×10×7×11
2 = 3850.