Lecture 4: The mod function
Daniel Frances c 2012
Contents
1
Introduction
2
2
Centrality of mod arithmetic to Internet computations
2
2.1
Modular Addition
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Modular Subtraction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.3
Modular Multiplication
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.4
Modular Division
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.5
Modular Exponentiation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1
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1
Introduction
It turns out that the entire issue of polynomial versus exponential algorithms is fundamental
to Encryption over the Internet. In the early days of the Internet, it was realized that to
support commercial transactions over the Internet, governments needed to ensure that there
were secure means for encrypting information.
It used to be the case that encryption algorithms were shrouded in the highest level of
secrecy. If your enemy knew your encryption algorithm then he would know how to decode it.
Developers of encryption algorithms worked for top secret organizations, and their algorithms
would never see the light of day.
Then the approach changed, encryption algorithms became widely known. In fact currently
used encryption algorithms were the winners of international contests among largely univer
sity academics, to see who could develop the most secure algorithms.
The way these algorithms manage their secrecy, is to use a “secret
key
”. The key is simply
a number. The key becomes
somehow
known to the intended reader of the message, but
remains secret to the intruder.
The algorithms are so good that without the
key
it is
physically impossible to decipher the message. The way this is done is by ensuring that the
algorithms to encode with the key are really fast, but the algorithms to decode with the key
are sufficiently slow to prevent the intruder from trying all possible keys. And what better
way to slow down the decoding than to use an exponential decoding algorithm!!! You want
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 Spring '12
 Frances
 Cryptography, Addition, Modular division, N qa

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