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

${\mathsf {Adv}}_{H_{i},H_{i+1}}^{\mathsf {dist}}(\mathbf {A} ):=\left|\Pr[x{\stackrel {\$}{\gets }}H_{i}:\mathbf {A} (x)=1]-\Pr[x{\stackrel {\$}{\gets }}H_{i+1}:\mathbf {A} (x)=1]\right|,$

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,

${\mathsf {Adv}}_{D_{1},D_{2}}^{\mathsf {dist}}(\mathbf {A} )\leq \sum _{i=0}^{t-1}{\mathsf {Adv}}_{H_{i},H_{i+1}}^{\mathsf {dist}}(\mathbf {A} ).$

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

${\mathsf {Adv}}_{H_{k},H_{k+1}}^{\mathsf {dist}}(\mathbf {A} )\geq {\mathsf {Adv}}_{D_{1},D_{2}}^{\mathsf {dist}}(\mathbf {A} )/t(n).$

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,

$\epsilon (n)\geq {\mathsf {Adv}}_{H_{k},H_{k+1}}^{\mathsf {dist}}(\mathbf {A} )\geq {\mathsf {Adv}}_{D_{1},D_{2}}^{\mathsf {dist}}(\mathbf {A} )/t(n),$

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" (PDF).
• Pass, Rafael. "A Course in Cryptography" (PDF).
• Fischlin, Marc; Mittelbach, Arno. "An Overview of the Hybrid Argument" (PDF).