CIS 5371 Cryptography 2. Perfect Secret Encryption

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Presentation transcript:

CIS 5371 Cryptography 2. Perfect Secret Encryption Based on: Jonathan Katz and Yehuda Lindel Introduction to Modern Cryptography

Encryption Plaintext Ciphertext Encryption Decryption encryption key decryption key Encryption Plaintext Ciphertext Decryption

Encryption schemes (ciphers) x

Encryption schemes Definition An encryption scheme (Gen,Enc,Dec) over message space M is perfectly secret if for every probability distribution over M, every message mM, and every ciphertext cC for which Pr[C = c]  0: Pr[M = m | C = c] = Pr[M = m] Convention: We consider only probability distributions over M, C that assign non-zero probabilities to all mM and cC.

Encryption schemes Lemma 1 An encryption scheme (Gen,Enc,Dec) over message space M is perfectly secret if and only if for every probability distribution over M, every message mM, and every ciphertext cC: Pr[C = c | M = m] = Pr[C = c]

Encryption schemes

Encryption schemes An equivalent definition for perfect secrecy

Encryption schemes

Encryption schemes

Shannon’s Theorem Theorem Let (Gen,Enc,Dec) be an encryption scheme over a message space M for which |M|= |K|=|C|. The scheme is perfectly secret if and only if: Every key kK is chosen with equal probability 1/|K| by algorithm Gen. For every mM and every cC there is a unique key kK such that Enck(m) outputs c

Shannon’s Theorem

Shannon’s Theorem Proof. We have Pr[C=c|M=m]=Pr[K=k] where c=mk, for any c,m, since the key k. Since the keys are chosen uniformly at random: Pr[C=c|M=m]=1/|K| for any mM. It follows that: Pr[C=c|M=m1] =Pr[C=c|M=m2], for any m1,m2 M

Encryption algorithms: one-time pad

One-time pad Theorem The one time pad encryption scheme is perfectly secret.

One-time pad Proof (use Lemma 2) For any c C and m1, m2  M we have: Pr[C=c|M=m1]=Pr[k=k1]=1/|K| Pr[C=c|M=m2]=Pr[k=k2]=1/|K| It follows that: Pr[C=c|M=m1]=Pr[C=c|M=m2]

Limitations to perfect secrecy Theorem Let (Gen,Enc,Dec) be a perfectly secret encryption scheme over message space M, and let K be the key space as determined by Gen. Then |K|  |M| .