In cryptography, the hybrid argument is a proof technique used to show that two distributions are computationally indistinguishable.
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Formal description
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.
Applications
The hybrid argument is extensively used in cryptography. Some simple proofs using hybrid arguments are: