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Engineering LibreTexts

15.3: ElGamal Encryption

  • Page ID
    7413
  • ElGamal encryption is a public-key encryption scheme that is based on DHKA

    construction 15.6: ElGamal

    The public parameters are a choice of cyclic group ? with n elements and generator g.

    Figure15-5.jpg

    The scheme satisfies correctness, since for all M:

    Figure15-6.jpg

    Security

    Imagine an adversary who is interested in attacking an ElGamal scheme. This adversary sees the public key A = ga and a ciphertext (gb,M gab) go by. Intuitively, the Decisional Diffie-Hellman assumption says that the value gab looks random, even to someone who has seen gaand gb. Thus, the message M is masked with a pseudorandom group element — as we’ve seen before, this is a lot like masking the message with a random pad as in one-time pad. The only change here is that instead of the XOR operation, we are using the group operation in ?.

    More formally, we can prove the security of ElGamal under the DDH assumption:

    Claim 15.7

    If the DDH assumption in group ? is true, then ElGamal in group ? is CPA$-secure.

    Proof

    It suffices to show that ElGamal has pseudorandom ciphertexts when the calling program sees only a single ciphertext. In other words, we will show that ℒpk-ots$-real ≋ ℒpk-ots$-rand, where these libraries are the ℒpk-cpa$-* libraries from Definition 15.2 but with the singleciphertext restriction used in Definition 15.4. It is left as an exercise to show that ℒpk-ots$-real ≋ ℒpk-ots$-rand implies CPA$ security (which in turn implies CPA security); the proof is very similar to that of Claim 15.5.

    The sequence of hybrid libraries is given below:

    Figure15-7.jpg

    The starting point is the ℒpk-ots$-real library, shown here with the details of ElGamal filled in.

    Figure15-8.jpg

    The main body of QUERY computes some intermediate values B and Ab . But since those lines are only reachable one time, it does not change anything to pre-compute them at initialization time.

    Figure15-9.jpg

    We can factor out the generation of A,B,C in terms of the ℒdh-real library from the Decisional Diffie-Hellman security definition (Definition 14.5).

    Figure15-10.jpg

    Applying the security of DDH, we can replace ℒdh-real with ℒdh-rand.

    Figure15-11.jpg

    The call to DHQUERY has been inlined.

    Figure15-12.jpg

    As before, since the main body of QUERY is only reachable once, we can move the choice of B and C into that subroutine instead of at initialization time.

    Figure15-13.jpg

    When b is sampled uniformly from ℤn, the expression B = gb is a uniformly distributed element of ?. Also recall that when C is a uniformly distributed element of ?, then M · C is uniformly distributed — this is analogous to the one-time pad property (see Exercise 2.1). Applying this change gives the library to the left.

    In the final hybrid, the response to QUERY is a pair of uniformly distributed group elements (B,X). Hence that library is exactly ℒpk-ots$-rand, as desired.