Hybrid argument (Cryptography)

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

Formal description

Formally, to show two distributions D₁ and D₂ are computationally indistinguishable, we can define a sequence of hybrid distributions D₁ := H₀, H₁, ..., H_t =: D₂ where t is polynomial in the security parameter n. Define the advantage of any probabilistic efficient (polynomial-bounded time) algorithm A as

${\mathsf{Adv}}_{H_i,H_{i+1}}^{\mathsf{dist}}(\mathbf{A}):=|Pr[x\stackrel{\mathrm{\$}}{\leftarrow }{H}_i:\mathbf{A}(x)=1]-Pr[x\stackrel{\mathrm{\$}}{\leftarrow }{H}_{i+1}:\mathbf{A}(x)=1]|,$

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,

${\mathsf{Adv}}_{D_1,D_2}^{\mathsf{dist}}(\mathbf{A})\le \sum _{i=0}^{t-1}{\mathsf{Adv}}_{H_i,H_{i+1}}^{\mathsf{dist}}(\mathbf{A}).$

Thus there must exist some k s.t. 0 ≤ k < t(n) and

${\mathsf{Adv}}_{H_k,H_{k+1}}^{\mathsf{dist}}(\mathbf{A})\ge {\mathsf{Adv}}_{D_1,D_2}^{\mathsf{dist}}(\mathbf{A})/t(n).$

Since t is polynomial-bounded, for any such algorithm A, if we can show that it has a negligible advantage function between distributions H_i and H_i+1 for every i, that is,

$ϵ(n)\ge {\mathsf{Adv}}_{H_k,H_{k+1}}^{\mathsf{dist}}(\mathbf{A})\ge {\mathsf{Adv}}_{D_1,D_2}^{\mathsf{dist}}(\mathbf{A})/t(n),$

then it immediately follows that its advantage to distinguish the distributions D₁ = H₀ and D₂ = H_t 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:

• 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.

References

• Dodis, Yevgeniy. "Introduction to Cryptography Lecture 5 notes". http://cs.nyu.edu/courses/fall08/G22.3210-001/lect/lecture5.pdf. 
• Pass, Rafael. "A Course in Cryptography". https://www.cs.cornell.edu/courses/cs4830/2010fa/lecnotes.pdf. 
• Fischlin, Marc; Mittelbach, Arno. "An Overview of the Hybrid Argument". https://eprint.iacr.org/2021/088.pdf.