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❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ❇❡♥♦ît ●ér❛r❞ s✉♣❡r✈✐s❡❞ ❜② ❏❡❛♥✲P✐❡rr❡ ❚✐❧❧✐❝❤ ❚❤❡s✐s ❞❡❢❡♥s❡ ✲ ❉❡❝❡♠❜❡r ✾✱ ✷✵✶✵ ❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✶ ✴ ✸✻

■t❡r❛t✐✈❡ ❜❧♦❝❦ ❝✐♣❤❡rs M ✲ ✲ ✲ ✳ ✳ ✳ ✳ ✳ ✳ ✲ ✲ ✲ C F F F F ✻ ✻ ✻ ✻ ✳ ✳ ✳ ✳ ✳ ✳ K K 1 K 2 K r − 1 K r ◮ K ✿ ♠❛st❡r ❦❡②✳ ◮ F ✿ r♦✉♥❞ ❢✉♥❝t✐♦♥✳ ◮ K i ✿ r♦✉♥❞ s✉❜✲❦❡②s✳ E K : F s F s → 2 2 M �→ C = E K ( M ) = F K r ◦ · · · ◦ F K 1 ( M ) . ❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✷ ✴ ✸✻

■t❡r❛t✐✈❡ ❜❧♦❝❦ ❝✐♣❤❡rs M ✲ ✲ ✲ ✳ ✳ ✳ ✳ ✳ ✳ ✲ ✲ ✲ C F F F F ✻ ✻ ✻ ✻ ✳ ✳ ✳ ✳ ✳ ✳ K K 1 K 2 K r − 1 K r ◮ K ✿ ♠❛st❡r ❦❡②✳ ◮ F ✿ r♦✉♥❞ ❢✉♥❝t✐♦♥✳ ◮ K i ✿ r♦✉♥❞ s✉❜✲❦❡②s✳ E K : F s F s → 2 2 M �→ C = E K ( M ) = F K r ◦ · · · ◦ F K 1 ( M ) . ❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✷ ✴ ✸✻

■t❡r❛t✐✈❡ ❜❧♦❝❦ ❝✐♣❤❡rs M ✲ ✲ ✲ ✳ ✳ ✳ ✳ ✳ ✳ ✲ ✲ ✲ C F F F F ✻ ✻ ✻ ✻ ✳ ✳ ✳ ✳ ✳ ✳ K K 1 K 2 K r − 1 K r ◮ K ✿ ♠❛st❡r ❦❡②✳ ◮ F ✿ r♦✉♥❞ ❢✉♥❝t✐♦♥✳ ◮ K i ✿ r♦✉♥❞ s✉❜✲❦❡②s✳ E K : F s F s → 2 2 M �→ C = E K ( M ) = F K r ◦ · · · ◦ F K 1 ( M ) . ❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✷ ✴ ✸✻

✷✳ ❋♦r ❡✈❡r② ♣♦ss✐❜❧❡ ❝❛♥❞✐❞❛t❡ ❢♦r ❉❡❝✐♣❤❡r ❝✐♣❤❡rt❡①ts ❜② ♦♥❡ r♦✉♥❞ ✉s✐♥❣ ✳ ●❡♥❡r❛t❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ st❛t✐st✐❝ ✭❣❡♥❡r❛❧❧② ❛ ❝♦✉♥t❡r✮✳ ✸✳ ❖r❞❡r t❤❡ ❝❛♥❞✐❞❛t❡s r❡❣❛r❞✐♥❣ t❤❡✐r ❧✐❦❡❧✐❤♦♦❞✳ ✹✳ ❚❡st ❛❧❧ t❤❡ ♠❛st❡r ❦❡②s t❤❛t ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❜❡st ❝❛♥❞✐❞❛t❡ ❛♥❞ s♦ ♦♥ ✳ ✳ ✳ ❲r♦♥❣ ❦❡② r❛♥❞♦♠✐③❛t✐♦♥ ❤②♣♦t❤❡s✐s ✭❲✳❑✳❘✳❍✳✮✳ ▲❛st r♦✉♥❞ ❛tt❛❝❦ ✶✳ ❋✐♥❞ ❛ ♥♦♥✲✐❞❡❛❧ ❜❡❤❛✈✐♦r ♦❢ r − 1 r♦✉♥❞s ♦❢ t❤❡ ❝✐♣❤❡r✳ ✲ ✲ ✳ ✳ ✳ ✲ ✲ F r − 1 ( M ) ✲ ✲ C M F F F ✻ ✻ ✻ ✳ ✳ ✳ K K 1 K r − 1 K r ❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✸ ✴ ✸✻

▲❛st r♦✉♥❞ ❛tt❛❝❦ ✶✳ ❋✐♥❞ ❛ ♥♦♥✲✐❞❡❛❧ ❜❡❤❛✈✐♦r ♦❢ r − 1 r♦✉♥❞s ♦❢ t❤❡ ❝✐♣❤❡r✳ ✷✳ ❋♦r ❡✈❡r② ♣♦ss✐❜❧❡ ❝❛♥❞✐❞❛t❡ k ❢♦r K r ◮ ❉❡❝✐♣❤❡r ❝✐♣❤❡rt❡①ts ❜② ♦♥❡ r♦✉♥❞ F ✉s✐♥❣ k ✳ ◮ ●❡♥❡r❛t❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ st❛t✐st✐❝ ✭❣❡♥❡r❛❧❧② ❛ ❝♦✉♥t❡r✮✳ ✸✳ ❖r❞❡r t❤❡ ❝❛♥❞✐❞❛t❡s r❡❣❛r❞✐♥❣ t❤❡✐r ❧✐❦❡❧✐❤♦♦❞✳ ✹✳ ❚❡st ❛❧❧ t❤❡ ♠❛st❡r ❦❡②s t❤❛t ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❜❡st ❝❛♥❞✐❞❛t❡ ❛♥❞ s♦ ♦♥ ✳ ✳ ✳ k = K r ❄ ✲ ✲ ✳ ✳ ✳ ✲ ✲ F r − 1 ( M ) ✲ ✲ C ✲ F − 1 M F F F ✲ ✉♥✐❢♦r♠❧② r❛♥❞♦♠ ✻ ✻ ✻ ✻ k � = K r ✳ ✳ ✳ K K 1 K r − 1 K r k ❲r♦♥❣ ❦❡② r❛♥❞♦♠✐③❛t✐♦♥ ❤②♣♦t❤❡s✐s ✭❲✳❑✳❘✳❍✳✮✳ ❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✸ ✴ ✸✻

❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s✿ ♥♦t❛t✐♦♥ ◮ N ✐s t❤❡ ♥✉♠❜❡r ♦❢ s❛♠♣❧❡s ❛✈❛✐❧❛❜❧❡ t♦ t❤❡ ❛tt❛❝❦❡r✳ ◮ k ∗ ✐s t❤❡ ❝♦rr❡❝t ✈❛❧✉❡ ♦❢ t❤❡ s✉❜❦❡② ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥✳ ◮ n key t❤❡ ♥✉♠❜❡r ♦❢ ❜✐ts ♦❢ k ∗ ✳ ◮ Σ k ✐s t❤❡ ❝♦✉♥t❡r ❡①tr❛❝t❡❞ ❢r♦♠ s❛♠♣❧❡s ❢♦r ❛ ❝❛♥❞✐❞❛t❡ k ✳ ❈♦♥❝❡r♥✐♥❣ t❤❡ t✐♠❡ ❝♦♠♣❧❡①✐t②✳ ◮ ❖♥❧② st♦♣ ✇❤❡♥ t❤❡ ❦❡② ✐s r❡❝♦✈❡r❡❞✳ ◮ ❑❡❡♣✐♥❣ ❛ ❧✐st L ♦❢ t❤❡ ❧✐❦❡❧✐❡st ❝❛♥❞✐❞❛t❡s ❢♦r t❤❡ ✜♥❛❧ s❡❛r❝❤✳ = Pr [ k ∗ ∈ L ] . def P S ◮ ❉❡✜♥✐♥❣ ❛ ❝r✐t❡r✐♦♥ t♦ ❞❡t❡r♠✐♥❡ ❝❛♥❞✐❞❛t❡s t♦ ❦❡❡♣✳ ◮ ❋✐①✐♥❣ t❤❡ s✐③❡ ♦❢ t❤❡ ❧✐st ℓ = |L| ✳ ❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✹ ✴ ✸✻

■ss✉❡s ❆♥❛❧②③✐♥❣ t❤❡ ❡✣❝✐❡♥❝② ♦❢ ❛ st❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s✐s✳ ◮ ❞❛t❛ ❝♦♠♣❧❡①✐t②✿ N ✳ ◮ s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t②✿ P S ✳ ◮ t✐♠❡ ❝♦♠♣❧❡①✐t②✿ r❡❧❛t❡❞ t♦ ℓ ✳ ◮ ❊❛❝❤ q✉❛♥t✐t② ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ t✇♦ ♦t❤❡rs✳ ❖♥❡ ✇♦✉❧❞ ❧✐❦❡ t♦ q✉❛♥t✐❢② t❤❡ tr❛❞❡♦✛ ❜❡t✇❡❡♥ t❤❡♠ ✐✳❡✳ ◮ ❊①♣r❡ss✐♥❣ P S ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ N ❛♥❞ ℓ ✳ ◮ ❊①♣r❡ss✐♥❣ N ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ P S ❛♥❞ ℓ ✳ ❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✺ ✴ ✸✻

❙✉♠♠❛r② ❇❛s✐❝s ♦❢ st❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s✐s ❙✐♠♣❧❡ st❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ❙♦♠❡ ❦♥♦✇♥ r❡s✉❧ts ❉❛t❛ ❝♦♠♣❧❡①✐t② ❙✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② ▼✉❧t✐♣❧❡ ❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❊♥tr♦♣② ❛s ❛ t♦♦❧ ❢♦r ❛♥❛❧②③✐♥❣ st❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ❆❞✈❛♥t❛❣❡ ✈s ❣❛✐♥ ❊♥tr♦♣②✿ ❛♥ ❛❧t❡r♥❛t✐✈❡ t♦ ❛❞✈❛♥t❛❣❡ ❙♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ❖t❤❡r ✇♦r❦s ❛♥❞ ♣❡rs♣❡❝t✐✈❡s

▼♦❞❡❧ ◮ ❆ ♥♦♥✲✐❞❡❛❧ st❛t✐st✐❝❛❧ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❝✐♣❤❡r ❤❛s ❜❡❡♥ ❢♦✉♥❞✿ st❛t✐st✐❝❛❧ ❝❤❛r❛❝t❡r✐st✐❝✳ ◮ ❋r♦♠ t❤✐s ❝❤❛r❛❝t❡r✐st✐❝ ❛♥❞ t❤❡ s❛♠♣❧❡s✱ ♦♥❡ ✐s ❛❜❧❡ t♦ ❝♦♠♣✉t❡ ❛ ❝♦✉♥t❡r Σ k ❢♦r ❡❛❝❤ ❝❛♥❞✐❞❛t❡✳ ▼♦❞❡❧ � B in ( N, p ∗ ) ✐❢ k = k ∗ , Σ k ∼ B in ( N, p ) ♦t❤❡r✇✐s❡✳ ❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✻ ✴ ✸✻

▲✐♥❡❛r ❝r②♣t❛♥❛❧②s✐s✿ ▼❛ts✉✐✬s ❆❧❣♦r✐t❤♠ ✷ ◮ ◆♦♥✲✐❞❡❛❧ ❜❡❤❛✈✐♦r✿ = 1 � π, M � ⊕ � γ, F r − 1 � � Pr M,K ( M ) � = 0 2 + ε. K p ∗ = 1 p = 1 2 + ε ❛♥❞ 2 . ◮ ❙t❛t✐st✐❝s ❡①tr❛❝t❡❞ ❢r♦♠ N ❦♥♦✇♥ ♣❧❛✐♥t❡①t✴❝✐♣❤❡rt❡①t ♣❛✐rs ( m i , c i ) ✿ N � � π, m i � ⊕ � γ, F − 1 k ( c i ) � . Σ k = i =1 ◮ ❈r✐t❡r✐♦♥ ❢♦r ♦r❞❡r✐♥❣ ❝❛♥❞✐❞❛t❡s✿ � � Σ k N − 1 � � � . � � 2 � ❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✼ ✴ ✸✻