game.txt

We define following abstract game: 

For some abstract functionality F, adversary A commits to N elements such that it knows to compute F for K elements.
Then it plays two games: 

G1: 
This game uniformly samples n1 pairwise distinct indices from [N], call this set S1. 
If A knows how to compute F for all n1 elements, it wins. 

G2: 
This game uniformly samples n1 pairwise distinct indices from [N]/S1, call this set S2. 
If A doesn't know how to compute F for all n2 elements, it wins.

We can model this games with negative hypergeometric distribution as follows. When game samples index i, 
if A knows how to compute F for for i, we mark this event as 1, and naturally with 0 when A doesn't know how to compute it.

G1: 
A fails to win when any of these events happen: 
e0: 0 
e1: 1, 0 
e2: 1, 1, 0 
e3: 1, 1, 1, 0 
.
.
e_n1: 1, .. 1 {n1 - 1 times}, 0 

so A wins with probability: 1- (sum e_i)

G2: 
A fails to win when any of these events happen: 
e0: 1 
e1: 0, 1 
e2: 0, 0, 1
e3: 0, 0, 0, 1 
.
.
e_n2: 0, .. 0 {n2 - 1 times}, 1

so A wins with probability: 1- (sum e_i)