Ref. Cryptography: theory and practice Douglas R. Stinson 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? 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?

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

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

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

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

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?

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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.

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

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

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

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,

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

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]

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)

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

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

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

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

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

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

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

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

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?

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

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

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

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

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

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

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)

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

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)