1. Ref. Cryptography: theory and practice Douglas R. Stinson
Shannon’s theory
Ref. Cryptography: theory and practice
Douglas R. Stinson

2. Shannon’s theory
1949, “Communication theory of Secrecy Systems” in Bell Systems Tech. Journal.
Two issues:
What is the concept of perfect secrecy? Does there any cryptosystem provide perfect secrecy?
It is possible when a key is used for only one encryption
How to evaluate a cryptosystem when many plaintexts are encrypted using the same key?

3. Outline Introduction Elementary probability theory Perfect secrecy
One-time pad
Elementary probability theory
Perfect secrecy
Entropy
Spurious keys and unicity distance

4. Categories of cryptosystem (1)
Computational security:
The best algorithm for breaking a cryptosystem requires at least N operations, where N is a very large number
No known practical cryptosystem can be proved to be secure under this definition
Study w.r.t certain types of attacks (ex. exhaustive key search) does not guarantee security against other type of attack

5. Categories of cryptosystem (2)
Provable security
Reduce the security of the cryptosystem to some well-studied problems that is thought to be difficult
Ex. RSA  integer factoring problem
Unconditional security
A cryptosystem cannot be broken, even with infinite computational resources

6. One-Time Pad Unconditional security !!!
Described by Gilbert Vernam in 1917
Use a random key that was truly as long as the message, no repetitions
The One-Time Pad is an evolution of the Vernham cipher, which was invented by Gilbert Vernham in 1918, and used a long tape of random letters to encrypt the message. An Army Signal Corp officer, Joseph Mauborgne, proposed an improvement using a random key that was truly as long as the message, with no repetitions, which thus totally obscures the original message.
Since any plaintext can be mapped to any ciphertext given some key, there is simply no way to determine which plaintext corresponds to a specific instance of ciphertext.
For ciphertext

7. Example: one-time pad
Given ciphertext with Vigenère Cipher: ANKYODKYUREPFJBYOJDSPLREYIUNOFDOIUERFPLUYTS
Decrypt by hacker 1:
Ciphertext: ANKYODKYUREPFJBYOJDSPLREYIUNOFDOIUERFPLUYTS
Key: pxlmvmsydofuyrvzwc tnlebnecvgdupahfzzlmnyih
Plaintext: mr mustard with the candlestick in the hall
Decrypt by hacker 2:
Ciphertext: ANKYODKYUREPFJBYOJDSPLREYIUNOFDOIUERFPLUYTS
Key: pftgpmiydgaxgoufhklllmhsqdqogtewbqfgyovuhwt
Plaintext: miss scarlet with the knife in the library
Which one?

8. a b c d e f g h i j k l m n o p q r s t u v w x y z ?
B C D E F G H I J K L M N O P Q R S T U V W X Y Z ? A
C D E F G H I J K L M N O P Q R S T U V W X Y Z ? A B
D E F G H I J K L M N O P Q R S T U V W X Y Z ? A B C
E F G H I J K L M N O P Q R S T U V W X Y Z ? A B C D
F G H I J K L M N O P Q R S T U V W X Y Z ? A B C D E
G H I J K L M N O P Q R S T U V W X Y Z ? A B C D E F
H I J K L M N O P Q R S T U V W X Y Z ? A B C D E F G
I J K L M N O P Q R S T U V W X Y Z ? A B C D E F G H
J K L M N O P Q R S T U V W X Y Z ? A B C D E F G H I
K L M N O P Q R S T U V W X Y Z ? A B C D E F G H I J
L M N O P Q R S T U V W X Y Z ? A B C D E F G H I J K
M N O P Q R S T U V W X Y Z ? A B C D E F G H I J K L
N O P Q R S T U V W X Y Z ? A B C D E F G H I J K L M
O P Q R S T U V W X Y Z ? A B C D E F G H I J K L M N
P Q R S T U V W X Y Z ? A B C D E F G H I J K L M N O
Q R S T U V W X Y Z ? A B C D E F G H I J K L M N O P
R S T U V W X Y Z ? A B C D E F G H I J K L M N O P Q
S T U V W X Y Z ? A B C D E F G H I J K L M N O P Q R
T U V W X Y Z ? A B C D E F G H I J K L M N O P Q R S
U V W X Y Z ? A B C D E F G H I J K L M N O P Q R S T
V W X Y Z ? A B C D E F G H I J K L M N O P Q R S T U
W X Y Z ? A B C D E F G H I J K L M N O P Q R S T U V
X Y Z ? A B C D E F G H I J K L M N O P Q R S T U V W
Y Z ? A B C D E F G H I J K L M N O P Q R S T U V W X
Z ? A B C D E F G H I J K L M N O P Q R S T U V W X Y
? A B C D E F G H I J K L M N O P Q R S T U V W X Y Z a b c d e f g h i j k l m n o p q r s t u v w x y z ?

9. Problem with one-time pad
Truly random key with arbitrary length?
Distribution and protection of long keys
The key has the same length as the plaintext!
One-time pad was thought to be unbreakable, but there was no mathematical proof until Shannon developed the concept of perfect secrecy 30 years later.

10. Preview of perfect secrecy (1)
When we discuss the security of a cryptosystem, we should specify the type of attack that is being considered
Ciphertext-only attack
Unconditional security assumes infinite computational time
Theory of computational complexity ×
Probability theory ˇ

11. Preview of perfect secrecy (2)
Definition: A cryptosystem has perfect secrecy if Pr[x|y] = Pr[x] for all xP, yC
Idea: Oscar can obtain no information about the plaintext by observing the ciphertext
Alice
Bob y x
Oscar

12. Outline Introduction Elementary probability theory Perfect secrecy
One-time pad
Elementary probability theory
Perfect secrecy
Entropy
Spurious keys and unicity distance

13. Discrete random variable (1)
Def: A discrete random variable, say X, consists of a finite set X and a probability distribution defined on X.
The probability that the random variable X takes on the value x is denoted Pr[X=x] or Pr[x]
0≤Pr[x] for all xX,

14. Discrete random variable (2)
Ex. Consider a coin toss to be a random variable defined on {head, tails} , the associated probabilities Pr[head]=Pr[tail]=1/2
Ex. Throw a pair of dice. It is modeled by Z={(1,1), (1,2), …, (2,1), (2,2), …, (6,6)}
where Pr[(i,j)]=1/36 for all i, j.
sum=4 corresponds to {(1,3), (2,2), (3,1)} with probability 3/36

15. Joint and conditional probability
X and Y are random variables defined on finite sets X and Y, respectively.
Def: the joint probability Pr[x, y] is the probability that X=x and Y=y
Def: the conditional probability Pr[x|y] is the probability that X=x given Y=y
Pr[x, y] =Pr[x|y]Pr[y]= Pr[y|x]Pr[x]

16. Bayes’ theorem If Pr[y] > 0, then
Ex. Let X denote the sum of two dice.
Y is a random variable on {D, N}, Y=D if the two dice are the same. (double)

17. Outline Introduction Elementary probability theory Perfect secrecy
One-time pad
Elementary probability theory
Perfect secrecy
Entropy
Spurious keys and unicity distance

18. Definitions
Assume a cryptosystem (P,C,K,E,D) is specified, and a key is used for one encryption
Plaintext is denoted by random variable x
Key is denoted by random variable K
Ciphertext is denoted by random variable y
Plaintext
Ciphertext y x K

19. Perfect secrecy
Definition: A cryptosystem has perfect secrecy if Pr[x|y] = Pr[x] for all xP, yC
Idea: Oscar can obtain no information about the plaintext by observing the ciphertext
Alice
Bob y x
Oscar

20. Relations among x, K, y Ciphertext is a function of x and K
y is the ciphertext, given that x is the plaintext

21. Relations among x, K, y
x is the plaintext, given that y is the ciphertext

22. Ex. Shift cipher has perfect secrecy (1)
Shift cipher: P=C=K=Z26 , encryption is defined as
Ciphertext:

23. Ex. Shift cipher has perfect secrecy (2)
Pr[y|x]
Apply Bayes’ theorem
Perfect secrecy

24. Perfect secrecy when |K|=|C|=|P|
(P,C,K,E,D) is a cryptosystem where |K|=|C|=|P|.
The cryptosystem provides perfect secrecy iff
every keys is used with equal probability 1/|K|
For every xP, yC, there is a unique key K such that
Ex. One-time pad in Z2

25. Outline Introduction Elementary probability theory Perfect secrecy
One-time pad
Elementary probability theory
Perfect secrecy
Entropy
Spurious keys and unicity distance
How about the security when many plaintexts are
encrypted using one key?

26. Preview (1) We want to know:
Plaintext
Ciphertext xn yn K
We want to know:
the average amount of ciphertext required for an opponent to be able to uniquely compute the key, given enough computing time

27. Preview (2) We want to know:
How much information about the key is revealed by the ciphertext
= conditional entropy H(K|Cn)
We need the tools of entropy

28. Entropy (1) Suppose we have a discrete random variable X
What is the information gained by the outcome of an experiment?
Ex. Let X represent the toss of a coin, Pr[head]=Pr[tail]=1/2
For a coin toss, we could encode head by 1, and tail by 0 => i.e. 1 bit of information

29. Entropy (2)
Ex. Random variable X with Pr[x1]=1/2, Pr[x2]=1/4, Pr[x3]=1/4
The most efficient encoding is to encode x1 as 0, x2 as 10, x3 as 11.
Notice: probability 2-n => n bits
p => -log2 p
The average number of bits to encode X

30. Entropy: definition
Suppose X is a discrete random variable which takes on values from a finite set X. Then, the entropy of the random variable X is defined as

31. Entropy : example Let P={a, b}, Pr[a]=1/4, Pr[b]=3/4.
K={K1, K2, K3}, Pr[K1]=1/2, Pr[K2]=Pr[K3]= 1/4.
encryption matrix: a b K1 1 2 K2 3 K3 4
H(P)=
H(K)=1.5, H(C)=1.85

32. Conditional entropy
Known any fixed value y on Y, information about random variable X
Conditional entropy: the average amount of information about X that is revealed by Y
Theorem: H(X,Y)=H(Y)+H(X|Y)

33. Theorem (1) Let (P,C,K,E,D) be a cryptosystem, then
H(K|C) = H(K) + H(P) – H(C)
Proof:
H(K,P,C) = H(C|K,P) + H(K,P)
Since key and plaintext uniquely determine the ciphertext
H(C|K,P) = 0
H(K,P,C) = H(K,P) = H(K) + H(P)
Key and plaintext are independent

34. Theorem (2) We have Similarly, Now, H(K,P,C) = H(K,P) = H(K) + H(P)
H(K,P,C) = H(K,C) = H(K) + H(C)
H(K|C)= H(K,C)-H(C)
= H(K,P,C)-H(C)
= H(K)+H(P)-H(C)