Understanding Probabilistic Encryption

Probabilistic encryption stands as a cornerstone in modern cryptography, introducing randomness to enhance security and prevent adversaries from exploiting patterns or repetitions in encrypted data. It forms a crucial part of cryptographic protocols ensuring data confidentiality.

Overview of Probabilistic Encryption

Probabilistic encryption algorithms generate varying ciphertexts for the same plaintext, adding an element of randomness during encryption. Unlike deterministic encryption, where identical plaintexts always produce the same ciphertext, probabilistic encryption schemes aim to obscure patterns, thereby bolstering security against various attacks.

Security Assurances and Semantic Security

The fundamental security goal of probabilistic encryption is semantic security. This concept ensures that an adversary, even with access to multiple ciphertexts and knowledge of the encryption algorithm, cannot distinguish between encryptions of the same plaintext. Proving this security property involves rigorous mathematical formulations and proofs.

Notions and Proofs in Probabilistic Encryption

Indistinguishability

Security proofs often revolve around the notion of indistinguishability. The idea is to demonstrate that an adversary, given two ciphertexts resulting from the encryption of the same plaintext, cannot discern which ciphertext corresponds to which plaintext.

Mathematically, this can be expressed as follows:

Let Enc(m) represent the encryption of plaintext m.

For two plaintexts m0 and m1 such that m0≠m1, and their corresponding ciphertexts c0=Enc(m0) and c1=Enc(m1), the encryption scheme is indistinguishable if, for any probabilistic polynomial-time adversary A: |Pr[A(c0)=1]−Pr[A(c1)=1]| is negligible, where Pr[A(c)=1] denotes the probability that the adversary A correctly guesses the bit associated with ciphertext c.

Chosen Plaintext Attack (CPA) Security

Another aspect of security analysis involves resistance against chosen plaintext attacks. Here, the encryption scheme is considered secure if an adversary, even with the ability to choose plaintexts and obtain their corresponding ciphertexts, cannot infer information about other plaintexts.

Mathematically proving CPA security involves demonstrating that the advantage of an adversary in distinguishing ciphertexts is negligible.

AdvCPA(A)=|Pr[A(Enc(μ))=1]−1/2|

where AdvCPA(A) represents the advantage of adversary A in a CPA game, and μ is a randomly chosen plaintext.

Reductions to Computational Problems

Security proofs of probabilistic encryption often rely on reductions to well-established computational problems in cryptography. For instance, schemes might be proven secure under the assumption that breaking them is at least as hard as solving the Decisional Diffie-Hellman problem or the Discrete Logarithm Problem.

Conclusion

Probabilistic encryption, with its incorporation of randomness, stands as a critical tool in securing sensitive information. By providing semantic security assurances and demonstrating resistance against various attacks through rigorous mathematical formulations and proofs, these encryption schemes play a pivotal role in ensuring data confidentiality in modern cryptographic systems.

Refer this paper by Silvio Micali and Shafi Goldwasser - Probabilistic Encryption