Hybrid argument (cryptography)
In cryptography, the hybrid argument is a proof technique used to show that two distributions are computationally indistinguishable.
History
Hybrid arguments had their origin in a papers by Andrew Yao in 1982 and Shafi Goldwasser and Silvio Micali in 1983.[1]
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 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 β€ 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
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).[3]
- We can securely expand a PRG with 1-bit output into a PRG with n-bit output.[4]
See also
Notes
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References
- β Bellare, Mihir, and Phillip Rogaway. "Code-based game-playing proofs and the security of triple encryption." Cryptology ePrint Archive (2004)
- β Lemma 3 in Dodis's notes.
- β Theorem 1 in Dodis's notes.
- β Lemma 80.5, Corollary 81.7 in Pass's notes.