Hybrid argument (Cryptography)

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In cryptography, the hybrid argument is a proof technique used to show that two distributions are computationally indistinguishable.

Formal description[edit]

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.[1]


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).[2]
  • We can securely expand a PRG with 1-bit output into a PRG with n-bit output.[3]


  1. ^ Lemma 3 in Dodis's notes.
  2. ^ Theorem 1 in Dodis's notes.
  3. ^ Lemma 80.5, Corollary 81.7 in Pass's notes.


  • Dodis, Yevgeniy. "Introduction to Cryptography Lecture 5 notes" (PDF).
  • Pass, Rafael. "A Course in Cryptography" (PDF).