In cryptography, the hybrid argument is a proof technique used to show that two distributions are computationally indistinguishable.
History
editHybrid arguments had their origin in a papers by Andrew Yao in 1982 and Shafi Goldwasser and Silvio Micali in 1983.[1]
Formal description
editFormally, 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 n. 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(n) and
Since t is polynomial-bounded, for any such algorithm A, if we can show that it has a negligible advantage function between distributions Hi and Hi+1 for every i, that is,
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.[2]
Applications
editThe 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).[3]
- We can securely expand a PRG with 1-bit output into a PRG with n-bit output.[4]
See also
editNotes
edit- ^ Bellare, Mihir, and Phillip Rogaway. "Code-based game-playing proofs and the security of triple encryption." Cryptology ePrint Archive (2004)
- ^ Lemma 3 in Dodis's notes.
- ^ Theorem 1 in Dodis's notes.
- ^ Lemma 80.5, Corollary 81.7 in Pass's notes.
References
edit- Dodis, Yevgeniy. "Introduction to Cryptography Lecture 5 notes" (PDF). Archived from the original (PDF) on 2014-12-25.
- Pass, Rafael. "A Course in Cryptography" (PDF).
- Fischlin, Marc; Mittelbach, Arno. "An Overview of the Hybrid Argument" (PDF).