238
4. Combinatorics, Probability, and Information Theory
!
"
Pr xk is the first match
!
"
= Pr xk is a match AND x0 , . . . , xk1 are distinct
#
!
"
= Pr xk is a match # x0 , . . . , xk1 are distinct
232
4. Combinatorics, Probability, and Information Theory
logarithm problem. For the finite field Fp , it solves the discrete logarithm
problem (DLP) in approximately p steps.
Of course, the index cal
224
4. Combinatorics, Probability, and Information Theory
Thus the chance of getting exactly one gold coin and exactly one silver coin
is somewhat larger if the coins are not replaced after each pick.
252
4. Combinatorics, Probability, and Information Theory
Theorem 4.58. Every function having Properties H1 , H2 , and H3 is a constant multiple of the function
H(p1 , . . . , pn ) =
n
!
pi log2 pi ,
250
4. Combinatorics, Probability, and Information Theory
reveal significant information about the key. To study this phenomenon, Shannon introduced the concept of entropy in order to quantify the unc
240
4.5.2
4. Combinatorics, Probability, and Information Theory
Discrete logarithms via Pollards method
In this section we describe how to use Pollards method to solve the discrete
logarithm problem
g
Part 2:
1. How does the PRC constitution interpret the history of China?
glorious revolutionary past
one of the longest histories in the world
Dr. Sun Yat-sen destroyed the monarchy and Mao Zedong des
220
4. Combinatorics, Probability, and Information Theory
Pr(F | E c ) = Pr(Output is No | m has property A)N
!
"N
= 1 Pr(Output is Yes | m has property A)
#
$N
1
1
from Property (2) of the Monte Car
4.6. Information theory
249
We sum (4.50) over all c C and divide by #C to obtain
f (k) =
1
1 !
.
f (c) =
#C
#C
cC
This shows that f (k) is constant, independent of the choice of k K, which
is precise
260
4. Combinatorics, Probability, and Information Theory
an assignment of truth values that makes the expression true. Cook proves
that SAT has the following properties:
1. Every N P problem is polyn
222
4. Combinatorics, Probability, and Information Theory
chosen at random without replacement. Let X denote the number of white
balls chosen. Then X is a random variable taking on the integer values
4.6. Information theory
4.6.4
257
The algebra of secrecy systems
We make only a few brief remarks about the algebra of cryptosystems. In [117],
Shannon considers ways of building new cryptosystems by
4.4. Collision algorithms and meet-in-the-middle attacks
4.4
227
Collision algorithms and
meet-in-the-middle attacks
A simple, yet surprisingly powerful, search method is based on the observation
that
4.6. Information theory
247
matched with dierent keys, which shows that there must be at least as many
keys as there are plaintexts.
Given the restriction on the relative sizes of the key, ciphertext,
4.4. Collision algorithms and meet-in-the-middle attacks
233
the list (4.32) may be viewed as selecting n elements from the urn, and we
would like to know the probability of selecting at least one red
244
4. Combinatorics, Probability, and Information Theory
computational resources that may be brought to bear against them. For example, symmetric ciphers such as the simple substitution cipher (Secti
4.4. Collision algorithms and meet-in-the-middle attacks
231
For the first question, Bob uses the reasonably accurate lower bound of
formula (4.29) to set
2
Pr(match) 1 en
/N
=
1
.
2
It is easy to sol
Character Name
Monkey D. Luffy
Nami
Roronoa Zoro
Usopp
Vinsmoke Sanji
Tony Tony Chopper
Nico Robin
Franky
Brook
Donquixote Doflamingo
Trafalgar D. Water Law
Jinbe
Portgas D. Ace
Nefertari Vivi
Crocodi