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

2. Encryption Plaintext Ciphertext Encryption Decryption encryption key
decryption key
Encryption
Plaintext
Ciphertext
Decryption

3. Encryption schemes (ciphers)x

4. 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.

5. 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]

6. Encryption schemes

7. Encryption schemes
An equivalent definition for perfect secrecy

8. Encryption schemes

9. Encryption schemes

10. 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

11. Shannon’s Theorem

12. 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

13. Encryption algorithms: one-time pad

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

15. 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]

16. 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| .