Lecture 12 - The Pohlig-Hellman
attack and Chinese Remainder
Theorem
In the last lecture we saw how discrete logarithms could be efficiently solved if p-1 was a power of 2. Today we generalize this
to the case when p-1 is smooth, meaning that it is a prod
5 i %vs#aw P v#pCv#up1gpeY1u y
t D 9 D " 9 D D ( (4 5
" 9
#@xp y BuHr1d y o y vQpf#Hj%
D kT T
( 4 "
k1 y $ ( " 4 5 k T T
5
#1pf1r y y wau hy r1 y
kT
k
f xy 111 y xy Hr11 y f11 y Uxy Hrr y fr
k T
kTT
kTT
kT
k T
k T T
kTT
kTTT
y @v
MIME and Base64 encoding
Math 348, Spring 2008
Many e-mail messages are now sent using the MIME format. MIME stands for Multipurpose Internet Mail Extensions. A typical e-mail message with an attachment will have
the two lines
MIME-Version: 1.0
Content-Ty
Recap of the first day' s lecture on
Substitution Ciphers
At its core, cryptography is about making encryption systems that can keep secrets away from smart and wise attackers who try to
crack them. A famous NSA maxim states that attacks never get worse,
Recap of the lecture on the Civil
War Vigenere discovery
Recently a Civil War encrypted message was discovered (see http:/www.aolnews.com/2010/12/25/civil-war-message-in-a-bottleopened-decoded/). We will now decode it, using just a little knowledge of the
Lecture 4 - Vigenere Cipher and
the Kasiski Attack
Let us review the Vigenere encryption method that we discussed in the last class. Recall that it is a polyalphabetic substitution
cipher, in that it does not necessarily always encrypt the same letter the
Lecture 5 - Vigenere Cipher, Index
of Coincidence, and the Friedman
Attack
In the last lecture we discussed the Kasiski attack, which finds the key length of a Vigenere text based on the pattern of repeated
trigraphs (that is, 3 letter combinations). Toda
Lecture 14 - The Miller - Rabin
primality test
Recall last time that we spoke about Fermat ' s primality test,
which rules out primes rather than showing numbers are primes. If n is a prime,
then Fermat ' s little theorem asserts that an is congruent to a
Lecture 17 - Pollards factoring
algorithms
This lecture actually 2 in class concern two clever factoring algorithms
introduced by J. Pollard. The first, Pollard ' s algorithm for integer
factorization as distinct from his algorithm for discrete logarithms
Homework #1, due February 5
Exercise 1. Decrypt the following message which was created using the Caesar cipher:
LORYHWKHQDPHRIKRQRU
Exercise 2. Eve has intercepted the following ciphertext which was created by using a shift
cipher:
CNMNBYQLIHANLIOMYLM
De
2
Cryptology course packet
Wesley Pegden
Version: January 19, 2010
4
CHAPTER 1. CLASSICAL CRYPTOLOGY
When Caesar used the cipher, he always shifted by 3, but theres no reason
for us to stick with this convention. For example, we could have encrypted the
m
56
digit, 0 and 1, since using any more would be redundant. (For example, even
the number 2 can be represented as 10 in binary).
If we want to represent a decimal number, say 35, in binary, we subtract the
largest power of 2 less than 35, which is 32, giv
84
CHAPTER 3. PUBLIC-CHANNEL CRYPTOGRAPHY
B
H
G
I
A
Chapter 3
D
E
F
It is the same graph because vertices which were adjacent in the rst graph are
still adjacent, while vertices not adjacent in the rst graph are not adjacent here
either.
It can be surpris
