Probabilistic encryption

Probabilistic encryption is the use of randomness in an encryption algorithm, so that when encrypting the same message several times it will, in general, yield different ciphertexts. To be semantically secure, that is, to hide even partial information about the plaintext, an encryption algorithm must be probabilistic.

Suppose that the adversary knows that the plaintext is either "YES" or "NO", or has a hunch that the plaintext might be "ATTACK AT CALAIS". Using a deterministic encryption algorithm is bad in either of these situations, because the adversary can simply try encrypting each possible message that he suspects the plaintext to be. The encryption algorithm must therefore incorporate randomness, ensuring that each plaintext maps into one of a large number of possible ciphertexts.

The first probabilistic encryption scheme was proposed by Goldwasser and Micali based on the hardness of the quadratic residuosity problem and had a message expansion factor equal to the public key size. More efficient probabilistic encryption algorithms include Elgamal, Paillier and various constructions under the random oracle model, including Optimal Asymmetric Encryption Padding (OAEP).

Intuitively, probabilistic encryption can be thought of as padding the plaintext with a random string before encrypting with a deterministic algorithm. Conversely, decryption involves applying a deterministic algorithm and ignoring the random padding. However, early schemes which applied this naive approach were broken due to limitations in some deterministic encryption schemes. Techniques such as OAEP integrate random padding in a manner that is secure using any trapdoor permutation.

Example of probabilistic encryption using any trapdoor permutation:

x - single bit plaintext
f - trapdoor permutation (deterministic encryption algorithm)
b - hard core predicate of f
r - random string

$Enc(x)=(f(r),x\oplus b(r))$

$Dec(y,z)=b(f^{-1}(y))\oplus z$

This is inefficient because only a single bit is encrypted. In other words, the message expansion factor is equal to the public key size.

Example of probabilistic encryption in the random oracle model:

x - plaintext
f - trapdoor permutation (deterministic encryption algorithm)
h - random oracle (typically implemented using a publicly specified hash function)
r - random string

$Enc(x)=(f(r),x\oplus h(r))$

$Dec(y,z)=h(f^{-1}(y))\oplus z$

See Also