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 the papers by Andrew Yao in 1982 and Shafi Goldwasser and Silvio Micali in 1983.
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
A d v H i , H i + 1 d i s t ( A ) := | Pr [ x ← $ H i : A ( x ) = 1 ] − Pr [ x ← $ H i + 1 : A ( x ) = 1 ] | , {\displaystyle {\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,
A d v D 1 , D 2 d i s t ( A ) ≤ ∑ i = 0 t − 1 A d v H i , H i + 1 d i s t ( A ) . {\displaystyle {\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
A d v H k , H k + 1 d i s t ( A ) ≥ A d v D 1 , D 2 d i s t ( A ) / t ( n ) . {\displaystyle {\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 fixed negligible advantage function ε(n) between distributions Hi and Hi+1 for every i, so in particular,
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