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Shannon ’ s theory

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

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Shannon ’ s theory

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  1. 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 • 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 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 ? 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 allxP, yC • Idea: Oscar can obtain no information about the plaintext by observing the ciphertext Bob Alice y x Oscar

  12. Outline • Introduction • 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 setX 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 • 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 Ciphertext Plaintext y x K

  19. Perfect secrecy • Definition: A cryptosystem has perfect secrecy if Pr[x|y] = Pr[x] for allxP, yC • Idea: Oscar can obtain no information about the plaintext by observing the ciphertext Bob Alice 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 • 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: the average amount of ciphertext required for an opponent to be able to uniquely compute the key, given enough computing time Ciphertext Plaintext yn xn K

  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: 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)

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