Hybrid argument (Cryptography)
Formally, to show two distributions D1 and D2 are computationally indistinguishable, we can define a sequence of hybrid distributions D1 := H0, H1, ..., Ht =: D2 where t is polynomial in the security parameter. Define the advantage of any probabilistic efficient (polynomial-bounded time) algorithm A as
where the dollar symbol ($) denotes that we sample an element from the distribution at random.
By triangle inequality, it is clear that for any probabilistic polynomial time algorithm A,
Thus there must exist some k s.t. 0 ≤ k < t and
Since t is polynomial-bounded, for any such algorithm A, if we can show that its advantage to distinguish the distributions Hi and Hi+1 is negligible for every i, then it immediately follows that its advantage to distinguish the distributions D1 = H0 and D2 = Ht must also be negligible. This fact gives rise to the hybrid argument: it suffices to find such a sequence of hybrid distributions and show each pair of them is computationally indistinguishable.
The hybrid argument is extensively used in cryptography. Some simple proofs using hybrid arguments are:
- If one cannot efficiently predict the next bit of the output of some number generator, then this generator is a pseudorandom number generator (PRG).
- We can securely expand a PRG with 1-bit output into a PRG with n-bit output.
- Lemma 3 in Dodis's notes.
- Theorem 1 in Dodis's notes.
- Lemma 80.5, Corollary 81.7 in Pass's notes.