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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 K1 K2 Kr−1 Kr

◮ K✿ ♠❛st❡r ❦❡②✳ ◮ F✿ r♦✉♥❞ ❢✉♥❝t✐♦♥✳ ◮ Ki✿ r♦✉♥❞ s✉❜✲❦❡②s✳

EK : Fs

2

→ Fs

2

M → C = EK(M) = FKr ◦ · · · ◦ FK1(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 K1 K2 Kr−1 Kr

◮ K✿ ♠❛st❡r ❦❡②✳ ◮ F✿ r♦✉♥❞ ❢✉♥❝t✐♦♥✳ ◮ Ki✿ r♦✉♥❞ s✉❜✲❦❡②s✳

EK : Fs

2

→ Fs

2

M → C = EK(M) = FKr ◦ · · · ◦ FK1(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 K1 K2 Kr−1 Kr

◮ K✿ ♠❛st❡r ❦❡②✳ ◮ F✿ r♦✉♥❞ ❢✉♥❝t✐♦♥✳ ◮ Ki✿ r♦✉♥❞ s✉❜✲❦❡②s✳

EK : Fs

2

→ Fs

2

M → C = EK(M) = FKr ◦ · · · ◦ FK1(M).

❇✳●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❡ ❢♦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♦ ♦♥ ✳ ✳ ✳

M K

✲

F

✲✳ ✳ ✳✲

F

✲ F r−1(M) ✲

F

✲ C

✳ ✳ ✳

✻

K1

✻

Kr−1

✻

Kr

❲r♦♥❣ ❦❡② r❛♥❞♦♠✐③❛t✐♦♥ ❤②♣♦t❤❡s✐s ✭❲✳❑✳❘✳❍✳✮✳

❇✳●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 Kr

◮ ❉❡❝✐♣❤❡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♦ ♦♥ ✳ ✳ ✳

M K

✲

F

✲✳ ✳ ✳✲

F

✲ F r−1(M) ✲

F

✲ C

✳ ✳ ✳

✻

K1

✻

Kr−1

✻

Kr

✲ F −1 ✻

k k = Kr k = Kr

❄ ✲ ✉♥✐❢♦r♠❧②

r❛♥❞♦♠

❲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❡❞ ✐♥✳ ◮ nkey t❤❡ ♥✉♠❜❡r ♦❢ ❜✐ts ♦❢ k∗✳ ◮ Σk ✐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❝❤✳

PS

def

= Pr [k∗ ∈ L] .

◮ ❉❡✜♥✐♥❣ ❛ ❝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✿ PS✳ ◮ 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✐♥❣ PS ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ N ❛♥❞ ℓ✳ ◮ ❊①♣r❡ss✐♥❣ N ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ PS ❛♥❞ 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 Σk ❢♦r ❡❛❝❤ ❝❛♥❞✐❞❛t❡✳ ▼♦❞❡❧ Σk ∼ Bin (N, p∗) ✐❢ k = k∗, Bin (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✿

PrM,K

• π, M ⊕ γ, F r−1

K

(M) = 0

• = 1

2 + ε. p∗ = 1 2 + ε ❛♥❞ p = 1 2.

◮ ❙t❛t✐st✐❝s ❡①tr❛❝t❡❞ ❢r♦♠ N ❦♥♦✇♥ ♣❧❛✐♥t❡①t✴❝✐♣❤❡rt❡①t ♣❛✐rs

(mi, ci)✿ Σk =

N

• i=1

π, mi ⊕ γ, F −1

k (ci). ◮ ❈r✐t❡r✐♦♥ ❢♦r ♦r❞❡r✐♥❣ ❝❛♥❞✐❞❛t❡s✿

• Σk

N − 1 2

• .

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✼ ✴ ✸✻

❆♥❛❧②s✐s

❚②♣✐❝❛❧ ✈❛❧✉❡s ❢♦r ❛ ✻✹✲❜✐t ❝✐♣❤❡r ✭s = 64✮✿ p∗ = 1 2 + 2−32 ❛♥❞ p = 1 2. ■♥ t❤✐s ❞♦♠❛✐♥✱ t❤❡ ●❛✉ss✐❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r t❤❡ ❜✐♥♦♠✐❛❧ ❞✐str✐❜✉t✐♦♥ ✐s t✐❣❤t✳ ■♥ ❬▼❛ts✉✐ ✶✾✾✸❪✿ N = O

• 1/ε2

✳ ■♥ ❬❏✉♥♦❞ ✷✵✵✶❪✿ ❛ ♣r❡❝✐s❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ r❛♥❦ ♦❢ k∗✳ ■♥ ❬❙❡❧ç✉❦ ✷✵✵✽❪✿ PS ≈ Φ

• 2

√ N|ε| + Φ−1

• 1 −

ℓ 2nkey+1

• .

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✽ ✴ ✸✻

❉✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s

◮ ◆♦♥✲✐❞❡❛❧ ❜❡❤❛✈✐♦r✿

PrM,K

• F r−1

K

(M) ⊕ F r−1

K

(M ⊕ δ1) = δ2

• = p∗.

p∗ > 2−s ❛♥❞ p = 1 2s − 1 ≈ 2−s.

◮ ❙t❛t✐st✐❝s ❡①tr❛❝t❡❞ ❢r♦♠ N ❝✐♣❤❡rt❡①ts (ci 1, ci 2) ❝♦rr❡s♣♦♥❞✐♥❣

t♦ ❝❤♦s❡♥ ♣❧❛✐♥t❡①ts (mi

1, mi 2) ✇✐t❤ ❞✐✛❡r❡♥❝❡ δ1✿

Σi

k =

1 ✐❢ F −1

k (ci 1) ⊕ F −1 k (ci 2) = δ2

♦t❤❡r✇✐s❡✳

◮ ❈r✐t❡r✐♦♥ ❢♦r ♦r❞❡r✐♥❣ ❝❛♥❞✐❞❛t❡s✿

Σk =

N

• i=1

Σi

k.

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✾ ✴ ✸✻

❆♥❛❧②s✐s

❚②♣✐❝❛❧ ✈❛❧✉❡s ❢♦r s = 64✿ p∗ = 2−60 ❛♥❞ p = 2−64. ■♥ t❤✐s ❞♦♠❛✐♥✱ t❤❡ P♦✐ss♦♥ ❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r t❤❡ ❜✐♥♦♠✐❛❧ ❞✐str✐❜✉t✐♦♥ ✐s t✐❣❤t✳ ❬❇✐❤❛♠✱ ❙❤❛♠✐r ✶✾✾✵❪✿ ❢♦r p∗ s✉✣❝✐❡♥t❧② ❧❛r❣❡r t❤❛♥ 2−s✱ N = O (1/p∗) . ① ■♥ ❬❙❡❧ç✉❦ ✷✵✵✽❪✱ PS ≈ Φ

• Np2

∗/p − Φ−1(1 − ℓ 2nkey )

• 1 + p∗/p
• .

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✶✵ ✴ ✸✻

❚r✉♥❝❛t❡❞ ❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s

◮ ◆♦♥✲✐❞❡❛❧ ❜❡❤❛✈✐♦r✿

PrM,K

• F r−1

K

(M) ⊕ F r−1

K

(M ⊕ δ) ∈ ∆2

• δ ∈ ∆1
• = p∗.

p∗ > |∆2| · 2−s ❛♥❞ p = |∆2| 2s − 1 ≈ |∆2| · 2−s.

◮ ❙t❛t✐st✐❝s ❡①tr❛❝t❡❞ ❢r♦♠ N ❝✐♣❤❡rt❡①ts (ci 1, ci 2) ❝♦rr❡s♣♦♥❞✐♥❣

t♦ ❝❤♦s❡♥ ♣❧❛✐♥t❡①ts (mi

1, mi 2) ✇✐t❤ ❞✐✛❡r❡♥❝❡ ✐♥ ∆1✿

Σi

k =

1 ✐❢ F −1

k (ci 1) ⊕ F −1 k (ci 2) ∈ ∆2

♦t❤❡r✇✐s❡✳

◮ ❈r✐t❡r✐♦♥ ❢♦r ♦r❞❡r✐♥❣ ❝❛♥❞✐❞❛t❡s✿

Σk =

N

• i=1

Σi

k.

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✶✶ ✴ ✸✻

❆♥❛❧②s✐s

◆♦ t②♣✐❝❛❧ ✈❛❧✉❡s ❢♦r ♣r♦❜❛❜✐❧✐t✐❡s s✐♥❝❡ ✐t ❞❡♣❡♥❞s ♦♥ |∆2|✳

• 2−60, 2−64

,

• 2−15.8, 2−16

,

• 0.5 + 2−32, 0.5
• .

❇♦t❤ t❤❡ P♦✐ss♦♥ ❛♥❞ t❤❡ ●❛✉ss✐❛♥ ❛♣♣r♦①✐♠❛t✐♦♥s ♠❛② ♥♦t ❜❡ ✈❛❧✐❞✳

✲✶✷✵ ✲✶✵✵ ✲✽✵ ✲✻✵ ✲✹✵ ✲✷✵ ✵ ✶✵ ✶✺ ✷✵ ✷✺ log (Pr [Σk∗ = x]) x ❇✐♥♦♠✐❛❧

• ❛✉ss✐❛♥

P♦✐ss♦♥ ✲✶✶ ✲✶✵ ✲✾ ✲✽ ✲✼ ✲✻ ✲✺ ✲✹ ✹✸✵✵ ✹✸✷✺ ✹✸✺✵ ✹✸✼✺ x ❇✐♥♦♠✐❛❧

• ❛✉ss✐❛♥

P♦✐ss♦♥

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✶✷ ✴ ✸✻

❆♥❛❧②s✐s

◆♦ t②♣✐❝❛❧ ✈❛❧✉❡s ❢♦r ♣r♦❜❛❜✐❧✐t✐❡s s✐♥❝❡ ✐t ❞❡♣❡♥❞s ♦♥ |∆2|✳

• 2−60, 2−64

,

• 2−15.8, 2−16

,

• 0.5 + 2−32, 0.5
• .

❇♦t❤ t❤❡ P♦✐ss♦♥ ❛♥❞ t❤❡ ●❛✉ss✐❛♥ ❛♣♣r♦①✐♠❛t✐♦♥s ♠❛② ♥♦t ❜❡ ✈❛❧✐❞✳

✲✻✵ ✲✺✵ ✲✹✵ ✲✸✵ ✲✷✵ ✲✶✵ ✵ ✸✷✵ ✸✹✵ ✸✻✵ ✸✽✵ ✹✵✵ ✹✷✵ log (Pr [Σk∗ = x]) x ❇✐♥♦♠✐❛❧

• ❛✉ss✐❛♥

P♦✐ss♦♥

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✶✷ ✴ ✸✻

❆♣♣r♦①✐♠❛t✐♥❣ t❤❡ t❛✐❧s ♦❢ t❤❡ ❜✐♥♦♠✐❛❧ ❞✐str✐❜✉t✐♦♥

▼❛✐♥ t♦♦❧ ✭❢♦❧❦❧♦r❡✮ ❙✉♣♣♦s✐♥❣ t❤❛t Σk ∼ Bin (N, p)✱ t❤❡♥✱ ❢♦r τ < p✱ Pr [Σk ≤ τN] ∼

N→∞

p√1 − τ (p − τ) √ 2πNτ · e−ND(τ||p), ❛♥❞✱ ❢♦r τ > p✱ Pr [Σk ≥ τN] ∼

N→∞

(1 − p)√τ (τ − p)

• 2πN(1 − τ)

· e−ND(τ||p). ❚❤❡ ❑✉❧❧❜❛❝❦✲▲❡✐❜❧❡r ❞✐✈❡r❣❡♥❝❡✿ D (a||b) def = a · ln a b

• + (1 − a) · ln

1 − a 1 − b

• .

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✶✸ ✴ ✸✻

❉❛t❛ ❝♦♠♣❧❡①✐t② ✭✶✴✷✮

Pr [Σk∗ < τ N] ≤ α , Pr [Σk ≥ τ N] ≤ β. τ

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✶✹ ✴ ✸✻

❉❛t❛ ❝♦♠♣❧❡①✐t② ✭✶✴✷✮

Pr [Σk∗ < τ N] ≤ α , Pr [Σk ≥ τ N] ≤ β. τ N ↓

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✶✹ ✴ ✸✻

❉❛t❛ ❝♦♠♣❧❡①✐t② ✭✶✴✷✮

Pr [Σk∗ < τ N] ≤ α , Pr [Σk ≥ τ N] ≤ β. τ N ↑

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✶✹ ✴ ✸✻

❉❛t❛ ❝♦♠♣❧❡①✐t② ✭✶✴✷✮

Pr [Σk∗ < τ N] ≤ α , Pr [Σk ≥ τ N] ≤ β. τ

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✶✹ ✴ ✸✻

❉❛t❛ ❝♦♠♣❧❡①✐t② ✭✷✴✷✮

❊st✐♠❛t❡s ❢♦r ◆ ❬❇❧♦♥❞❡❛✉✱ ●✳ ✷✵✵✾❪ ❚✇♦ ❡st✐♠❛t❡s ❢♦r t❤❡ ❞❛t❛ ❝♦♠♣❧❡①✐t② ♦❢ ❛ s✐♠♣❧❡ st❛t✐st✐❝❛❧ ❝r②♣t✲ ❛♥❛❧②s✐s ✇✐t❤ s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② ❝❧♦s❡ t♦ ✵✳✺ ❛r❡

N ′ def = − 1 D (p∗||p)

• ln
• λ β
• D (p∗||p)
• + 1

2 ln

• − ln
• λ β
• D (p∗||p)
• ,

❛♥❞ N′′ def = −ln (2√πβ) D (p∗||p) . ❇♦✉♥❞s ♦♥ ❡rr♦r ♠❛❞❡ ✉s✐♥❣ N′ ❛♥❞ N′′ ❣✉❛r❛♥t❡❡ t❤❡✐r ❛❝❝✉r❛❝②✳

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✶✺ ✴ ✸✻

❊♠♣✐r✐❝❛❧ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❡st✐♠❛t❡s

▲✐♥❡❛r ❝r②♣t❛♥❛❧②s✐s

✹✺ ✹✻ ✹✼ ✹✽ ✹✾ ✺✵ ✹ ✻ ✽ ✶✵ ✶✷ ✶✹ ✶✻ ✶✽ ✷✵ log2(N) − log2(β) N ❬❙❡❧ç✉❦ ✷✵✵✽❪ N′ N′′

❉✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s

✹✻ ✹✽ ✺✵ ✺✷ ✺✹ ✺✻ ✺✽ ✹ ✻ ✽ ✶✵ ✶✷ ✶✹ ✶✻ ✶✽ ✷✵ − log2(β) N ❬❙❡❧ç✉❦ ✷✵✵✽❪ N′ N′′

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✶✻ ✴ ✸✻

❊♠♣✐r✐❝❛❧ ❛❝❝✉r❛❝② ♦❢ t❤❡ ❡st✐♠❛t❡s

❚r✉♥❝❛t❡❞ ❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s

✷✷ ✷✸ ✷✹ ✷✺ ✷✻ ✻ ✽ ✶✵ ✶✷ ✶✹ ✶✻ ✶✽ ✷✵ − log2(β) N ❬❙❡❧ç✉❦ ✷✵✵✽❪ N′ N′′

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✶✻ ✴ ✸✻

❋✐①✐♥❣ t❤❡ ❧✐st s✐③❡

❲❡ ❣❛✈❡ ❡st✐♠❛t❡s ♦❢ N

◮ ❢♦r PS = 1 − α ≈ 0.5✱ ◮ ❢✉♥❝t✐♦♥ ♦❢ β✿ ♣r♦♣♦rt✐♦♥ ♦❢ ❦❡♣t ❝❛♥❞✐❞❛t❡s✳

◆♦✇✱ ✇❡ ✜① t❤❡ ❧✐st s✐③❡ |L| = ℓ✳

◮ ❊①♣r❡ss✐♥❣ PS ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ N ❛♥❞ ℓ✳

PS =

N

• i=0

Pr [Σk∗ = i] · Bn−ℓ,ℓ(G(i)), ✇❤❡r❡ G ✐s t❤❡ ❝✉♠✉❧❛t✐✈❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ Σk=k∗✳

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✶✼ ✴ ✸✻

❙✉❝❝❡ss Pr♦❜❛❜✐❧✐t② ✭✶✴✷✮

PS =

N

• i=0

Pr [Σk∗ = i] · Bn−ℓ,ℓ(G(i)).

✵ ✵✳✷ ✵✳✹ ✵✳✻ ✵✳✽ ✶ ✵ ✹✵✵ ✽✵✵ ✶✷✵✵ ✶✻✵✵ ✷✵✵✵ i

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✶✽ ✴ ✸✻

❙✉❝❝❡ss Pr♦❜❛❜✐❧✐t② ✭✷✴✷✮

❚❤❡♦r❡♠ ❬❇❧♦♥❞❡❛✉✱ ●✳✱ ❚✐❧❧✐❝❤ ✷✵✵✾❪ ■❢ G−1 ❞❡♥♦t❡s t❤❡ ✐♥✈❡rs❡ ❝✉♠✉❧❛t✐✈❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❝♦✉♥t❡rs Σk ❢♦r k = k∗✱ t❤❡♥✱ PS ≈

N

• i=G−1(t0)

Pr [Σk∗ = i] , ✇✐t❤ t0

def

= 1 −

ℓ−1 2nkey −2✳

❋♦r♠✉❧❛ ✐♥ ❬❙❡❧ç✉❦ ✷✵✵✽❪✿ ∞

Φ−1

w

• 1−

ℓ 2nkey

ϕr(x) dx.

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✶✾ ✴ ✸✻

▼✉❧t✐♣❧❡ ❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s

▼✉❧t✐♣❧❡ ❝r②♣t❛♥❛❧②s❡s✿ ❡①tr❛❝t✐♥❣ ♠♦r❡ ✐♥❢♦r♠❛t✐♦♥ ✉s✐♥❣ s❡✈❡r❛❧ ❝❤❛r❛❝t❡r✐st✐❝s✳ PrM,K

• F r−1

K

(M) ⊕ F r−1

K

(M ⊕ δj

1) = δj 2

• = pj

∗.

❍❡r❡ t❤❡ ❝♦✉♥t❡rs ❛r❡ Σj

k def

= #

• (m1, m2 = m1 ⊕ δj

1, c1, c2), F −1 k (c1) ⊕ F −1 k (c2) = δj 2

• ,

Σk

def

=

• j

Σj

k.

▼❛✐♥ ❞✐✣❝✉❧t② Σj

k ∼ Bin

• N, pj

∗

• ✇✐t❤ pj1

∗ = pj2 ∗ .

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✷✵ ✴ ✸✻

❆♥❛❧②③✐♥❣ ♠✉❧t✐♣❧❡ ❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s

▼❛✐♥ ✐ss✉❡✿ ❡st✐♠❛t✐♥❣ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ❛ s✉♠ ♦❢ ❜✐♥♦♠✐❛❧ ✈❛r✐❛❜❧❡s✳ ① ■♥ ❧✐t❡r❛t✉r❡✱ ❙❡❧ç✉❦✬s ❢♦r♠✉❧❛ ✐s ✉s❡❞✳ ① ❯s✐♥❣ P♦✐ss♦♥ ❛♣♣r♦①✐♠❛t✐♦♥✱ t❤❡ ❜❡❤❛✈✐♦r ♦❢ ❝♦✉♥t❡rs ❢♦r ❧❛r❣❡ ❞❡✈✐❛t✐♦♥s ✐s ♥♦t ❝❛✉❣❤t✳ ❯s❡ ❛♥♦t❤❡r ❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r t❤❡ t❛✐❧s✳ ▼❛✐♥ t♦♦❧

• ❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠✉❧❛ ✉s❡❞ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❜✐♥♦♠✐❛❧ t❛✐❧s ❢♦r

❛♣♣r♦①✐♠❛t✐♥❣ t❤❡ t❛✐❧s ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ❛ s✉♠ ♦❢ ✐✳✐✳❞✳ ✈❛r✐❛❜❧❡s✳

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✷✶ ✴ ✸✻

❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

✵ ✵✳✷ ✵✳✹ ✵✳✻ ✵✳✽ ✶ ✷✽✳✺ ✷✾ ✷✾✳✺ ✸✵ ✸✵✳✺ ✸✶ PS log2(N) ❖✉rs ❙❡❧ç✉❦ P♦✐ss♦♥ ♦♥❧② ❊①♣❡r✐♠❡♥t❛❧

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✷✷ ✴ ✸✻

Pr♦♣♦s❡❞ ❛tt❛❝❦ ♦♥ ✶✽✲r♦✉♥❞ P❘❊❙❊◆❚

■♠♣r♦✈❡♠❡♥ts ❢r♦♠ ❬❲❛♥❣ ✷✵✵✽❪

◮ ❯s❡ ♦❢ ❞✐✛❡r❡♥t✐❛❧s ✇✐t❤ ❞✐✛❡r❡♥t ♦✉t♣✉t ❞✐✛❡r❡♥❝❡s✳ ◮ ❇❡tt❡r ❡st✐♠❛t✐♦♥ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ♣r♦❜❛❜✐❧✐t✐❡s✳ ◮ ❙♣❡❝✐✜❝ ❛♥❛❧②s✐s t❤❛t ❞♦ ♥♦t ✉s❡ ●❛✉ss✐❛♥ ❛♣♣r♦①✐♠❛t✐♦♥✳

❉❛t❛ ❚✐♠❡ ❱❡rs✐♦♥ ❘♦✉♥❞s ❚②♣❡ ❬❲❛♥❣✵✽❪ 264.0 264.0 ✽✵ ✶✻ ✭♠✉❧t✐✳✮ ❞✐✛✳ ❬❖❱❚❑✵✾❪ 263.0 2104.0 ✶✷✽ ✶✼ r❡❧❛t❡❞ ❦❡②s s✉❜♠✐tt❡❞ 262.0 275.0 ✽✵ ✶✽ ♠✉❧t✐✳ ❞✐✛✳ ❬❆❧❜❈✐❞✵✾❪ 262.0 2113.0 ✶✷✽ ✶✾ ❛❧❣✳ ❞✐✛✳ ❬❈♦❧❙t❛✵✾❪ 257.0 257.0 ✽✵ ✷✹ st❛t✳ s❛t✳ ❬❈❤♦✶✵❪ 264.0 272.0 ✽✵ ✷✻ ♠✉❧t✐✳ ❧✐♥✳

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✷✸ ✴ ✸✻

❆❞✈❛♥t❛❣❡ ❛♥❞ ❣❛✐♥

◮ Ψ✿ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ r❛♥❦ ♦❢ k∗ ❛♠♦♥❣

t❤❡ 2nkey ❝❛♥❞✐❞❛t❡s✳

◮ ❆❞✈❛♥t❛❣❡✿

a def = − log2 Med(Ψ) 2nkey

• .

◮ ●❛✐♥✿

Γ def = − log2 2 E(Ψ) − 1 2nkey

• .

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✷✹ ✴ ✸✻

❆❞✈❛♥t❛❣❡ ✈s ❣❛✐♥ ✭✶✴✷✮

✵ ✺✵ ✶✵✵ ✶✺✵ ✷✵✵ ✷✺✵ 20 22 24 26 28 210 212 214 216 218 220 222 224 226 228 230 232 ❘❡♣❛rt✐t✐♦♥ ♦❢ ψ✬s

✻

Med(Ψ)

✻

E(Ψ)

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✷✺ ✴ ✸✻

❆❞✈❛♥t❛❣❡ ✈s ❣❛✐♥ ✭✷✴✷✮

• ❛✐♥✿

① ♣r♦✈✐❞❡s ♣❡ss✐♠✐st✐❝ r❡s✉❧ts❀ E(Ψ) ❝❛♥ ❜❡ ❡❛s✐❧② ❡st✐♠❛t❡❞✳ ❆❞✈❛♥t❛❣❡✿ ♣r♦✈✐❞❡s ♥♦♥✲♣❡ss✐♠✐st✐❝ r❡s✉❧ts❀ ① ❡st✐♠❛t✐♥❣ Med(Ψ) ♠❛② ♥♦t ❜❡ ❡❛s②✳ ❘❡♠❛r❦ ♦♥ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡ ❀ ❀ ✳

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✷✻ ✴ ✸✻

❆❞✈❛♥t❛❣❡ ✈s ❣❛✐♥ ✭✷✴✷✮

• ❛✐♥✿

① ♣r♦✈✐❞❡s ♣❡ss✐♠✐st✐❝ r❡s✉❧ts❀ E(Ψ) ❝❛♥ ❜❡ ❡❛s✐❧② ❡st✐♠❛t❡❞✳ ❆❞✈❛♥t❛❣❡✿ ♣r♦✈✐❞❡s ♥♦♥✲♣❡ss✐♠✐st✐❝ r❡s✉❧ts❀ ① ❡st✐♠❛t✐♥❣ Med(Ψ) ♠❛② ♥♦t ❜❡ ❡❛s②✳ ❘❡♠❛r❦ ♦♥ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡

◮ Med(Ψ) = 218.05❀ ◮ E(Ψ) = 219.99❀ ◮ E(log2(Ψ)) = 17.96✳

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✷✻ ✴ ✸✻

❙♦♠❡ ❞❡✜♥✐t✐♦♥

❆♥ ❛❧t❡r♥❛t✐✈❡ q✉❛♥t✐t② t♦ ❧♦♦❦ ❛t ✐s ❡♥tr♦♣②✳ H(X)

def

= EX log2 (Pr [X]) , H(X|Y )

def

= EX,Y log2 (Pr [X|Y ]) ,

◮ Y ✿ t❤❡ ✈❛r✐❛❜❧❡ ❝♦♥t❛✐♥✐♥❣ t❤❡ st❛t✐st✐❝s✳ ◮ K′✿ t❤❡ s✉❜✲❦❡② t♦ r❡❝♦✈❡r✳ ◮ H(K′|Y )✿ q✉❛♥t✐❢② t❤❡ ✉♥❝❡rt❛✐♥t② ♦♥ t❤❡ ❦❡② ❦♥♦✇✐♥❣

s❛♠♣❧❡s✳ ❍❡✉r✐st✐❝ ❚❛❦✐♥❣ ❛ ❧✐st ♦❢ s✐③❡ ℓ = 2H(K′|Y ) ❧❡❛❞s t♦ ❛ s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② ❣r❡❛t❡r t❤❛♥ 0.5✳

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✷✼ ✴ ✸✻

▲✐♥❦s ❜❡t✇❡❡♥ ❡♥tr♦♣② ❛♥❞ ❛❞✈❛♥t❛❣❡

a def = − log2 Med(Ψ) 2nkey

• s✐♠✐❧❛r t♦ ? = − log2
• 2H(K′|Y )

2nkey

• .

❋♦r♠✉❧❛ ❢♦r

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✷✽ ✴ ✸✻

▲✐♥❦s ❜❡t✇❡❡♥ ❡♥tr♦♣② ❛♥❞ ❛❞✈❛♥t❛❣❡

a def = − log2 Med(Ψ) 2nkey

• s✐♠✐❧❛r t♦ I(K′; Y ) = − log2
• 2H(K′|Y )

2nkey

• .

I(K′; Y ) def = H(K′) − H(K′|Y ). ❋♦r♠✉❧❛ ❢♦r I(K′; Y )

I(K′; Y ) =

• k′
• y

Pr [K′ = k′, Y = y] log2

• Pr [K′ = k′, Y = y]

Pr [K′ = k′] Pr [Y = y]

• ,

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✷✽ ✴ ✸✻

❊st✐♠❛t✐♥❣ I(K′; Y )

❇♦✉♥❞✐♥❣ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥ ❜② ❛ s✉♠ ♦❢ q✉❛♥t✐t✐❡s ❡❛s✐❡r t♦ ❝♦♠♣✉t❡✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ❢✉♥❝t✐♦♥ ♦❢ ❛ ✈❛r✐❛❜❧❡ A ✐s g(A)✳ ▼❛✐♥ t♦♦❧ ■❢ g(Y |K′) =

• j

g(Yj|K′

j),

t❤❡♥✱ I(K′; Y ) ≤

• j

I(K′

j; Yj).

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✷✾ ✴ ✸✻

❆♣♣❧✐❝❛t✐♦♥ t♦ ♠✉❧t✐♣❧❡ ❧✐♥❡❛r ❝r②♣t❛♥❛❧②s✐s ✭✶✴✸✮

❍❡r❡ t❤❡ ✈❛r✐❛❜❧❡s ❛r❡ ❞❡❝♦♠♣♦s❡❞ r❡❣❛r❞✐♥❣ ❛♣♣r♦①✐♠❛t✐♦♥s✳ PrM,K [πj, M ⊕ γj, C = κj, K] = 1 2 + εj. ❈♦✉♥t❡rs ❛r❡ Σj

def

=

N

• i=1

πj, mi ⊕ γj, ci. ❚❤❡♥✱ ✇❡ ✉s❡ t❤❡ ❜♦✉♥❞ ✇✐t❤ Yj

def

= N − 2Σj 2Nεj ❛♥❞ K′

j def

= κj, K. Yj = (−1)K′

j + Bj ✇✐t❤ Bj ∼ N

• 0,

1 4Nε2

j

• .

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✸✵ ✴ ✸✻

❆♣♣❧✐❝❛t✐♦♥ t♦ ♠✉❧t✐♣❧❡ ❧✐♥❡❛r ❝r②♣t❛♥❛❧②s✐s ✭✷✴✸✮

g(Y |K′) =

• j

g(Yj|K′

j) ⇐

⇒ ❛♣♣r♦①✐♠❛t✐♦♥s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✳ ❚❤❡♥✱ I(K′

j; Yj) ≤ Cap(σ2 j ),

✇❤❡r❡ Cap(σ2

j ) ✐s t❤❡ ❝❛♣❛❝✐t② ♦❢ t❤❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧ ✇✐t❤ ♥♦✐s❡

✈❛r✐❛♥❝❡ σ2

j def

= 1/4Nε2

j✳

I(K′; Y ) ≤

• j

Cap

• σ2

j

• ≈

2N

j ε2 j

ln 2 + O  

j

ε4

j

  .

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✸✶ ✴ ✸✻

❆♣♣❧✐❝❛t✐♦♥ t♦ ♠✉❧t✐♣❧❡ ❧✐♥❡❛r ❝r②♣t❛♥❛❧②s✐s ✭✸✴✸✮

❚❤❡♦r❡♠ ❋♦r ℓ = 1✱ ✐❢

• j

Cap

• σ2

j

• > nkey,

t❤❡♥ t❤❡ s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② t❡♥❞s t♦ ✶ ✇✐t❤ t❤❡ ♥✉♠❜❡r ♦❢ ❛♣♣r♦①✐✲ ♠❛t✐♦♥s✳ ■♥ t❤✐s ❝❛s❡✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡st✐♠❛t❡s ❢♦r N✿ ① ●❛✐♥ → N ≈ nkey + 1

• j ε2

j

✳ ❊♥tr♦♣② → N ≈ nkey 2

j ε2 j

✳

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✸✷ ✴ ✸✻

❆♣♣❧✐❝❛t✐♦♥ t♦ ▼❛ts✉✐✬s ❆❧❣♦r✐t❤♠ ✷

❚❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐s ❞♦♥❡ ❛♠♦♥❣ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ❢♦r k∗✳

I(K′; Y ) ≤

• R+ f 1(y) log2

f 1(y) f(y)

• + (2nkey − 1)f 0(y) log2

f 0(y) f(y)

• dy.

❯s✐♥❣ t❤✐s ❜♦✉♥❞✱ ✇❡ ❝❛♥ ❡①♣❧❛✐♥ ♦❜s❡r✈❛t✐♦♥s ✐♥ ❬❏✉♥♦❞ ✷✵✵✶❪✿ ❊①♣❡r✐♠❡♥t❛❧ t✐♠❡ ❝♦♠♣❧❡①✐t② ✐s 241 ✇❤✐❧❡ t❤❡ t❤❡♦r❡t✐❝❛❧ ❝♦♠♣❧❡①✐t② ♦❜t❛✐♥❡❞ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❡①♣❡❝t❡❞ r❛♥❦ ♦❢ t❤❡ ❦❡② ✐s 243✳ ❆♣♣❧②✐♥❣ t❤❡ ❜♦✉♥❞ ♦♥ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥ ❧❡❛❞s t♦ ❛ t✐♠❡ ❝♦♠♣❧❡①✐t② ♦❢ 241✳

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✸✸ ✴ ✸✻

❆♣♣❧✐❝❛t✐♦♥ t♦ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ❝r②♣t❛♥❛❧②s✐s

❲❡ ❡❛s✐❧② ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❜♦✉♥❞ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ H(Σk∗) t❤❡ ❡♥tr♦♣② ♦❢ t❤❡ ❝♦✉♥t❡r ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❝♦rr❡❝t ❝❛♥❞✐❞❛t❡✱ I(K′; Y ) ≤

N

• j=1

I(K; Yj) ≤ N · (d − H(Σk∗)).

✵ ✷ ✹ ✻ ✽ ✶✵ ✶✷ ✷✸ ✷✹ ✷✺ ✷✻ ✷✼ ✷✽ ❆❞✈❛♥t❛❣❡ log2(N) ❊♠♣✐r✐❝❛❧ [❍❈◆✵✾] ❊♥tr♦♣②

◮ ❆tt❛❝❦ ♣r❡s❡♥t❡❞ ❜②

❍❡r♠❡❧✐♥✱ ◆②❜❡r❣ ❛♥❞ ❈❤♦ ❛t ❋❙❊ ✷✵✵✾✳

◮ ❋♦r ✹ ❜❛s❡

❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ t❤❡ ▲▲❘ ♠❡t❤♦❞✳

◮ ❉❛t❛ ♣r♦✈✐❞❡❞ ❜②

❛✉t❤♦rs✳

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✸✹ ✴ ✸✻

❖t❤❡r ✇♦r❦s ❛♥❞ ♣❡rs♣❡❝t✐✈❡s

❙♦♠❡ ♦t❤❡r ✇♦r❦s✿

◮ ❡①♣❡r✐♠❡♥ts ♦♥ t❤❡ ✉s❡ ♦❢ ❛ ❧✐♥❡❛r ❞❡❝♦❞✐♥❣ ❛❧❣♦r✐t❤♠ ❢♦r

r❡❝♦✈❡r✐♥❣ t❤❡ ❦❡② ✐♥ ♠✉❧t✐♣❧❡ ❧✐♥❡❛r ❝r②♣t❛♥❛❧②s✐s❀

◮ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❛ ♠✉❧t✐♣❧❡ ❧✐♥❡❛r ❝r②♣t❛♥❛❧②s✐s ♦♥ ❉❊❙✳ ◮ ❡①♣❡r✐♠❡♥ts ♦♥ ❞✐✛❡r❡♥t✐❛❧ ❝r②♣t❛♥❛❧②s✐s ❬❇❧♦♥❞❡❛✉✱ ●✳ ✷✵✶✵❪✳

P❡rs♣❡❝t✐✈❡s✿

◮ ❖t❤❡r ✇❛② ❢♦r ❤❛♥❞❧✐♥❣ ♠✉❧t✐♣❧❡ ❛tt❛❝❦s✳ ◮ ❆♣♣❧✐❝❛t✐♦♥ ♦❢ ❡♥tr♦♣② ❛♣♣r♦❛❝❤ t♦ ♦t❤❡r ❝r②♣t❛♥❛❧②s❡s✳ ◮ ❇♦✉♥❞✐♥❣ t❤❡ s✉❝❝❡ss r❛t❡ ✇❤❡♥ t❛❦✐♥❣ ℓ = 2H(K′|Y )✳

❇✳●ér❛r❞ ❙t❛t✐st✐❝❛❧ ❝r②♣t❛♥❛❧②s❡s ♦❢ s②♠♠❡tr✐❝✲❦❡② ❛❧❣♦r✐t❤♠s ✸✺ ✴ ✸✻

❊♥tr♦♣② ✐♥ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ❝r②♣t❛♥❛❧②s✐s ✭✶✴✷✮

❋♦r d ❜❛s❡ ❛♣♣r♦①✐♠❛t✐♦♥s PrM,K [πj, M ⊕ γj, C = 0] = 1 2 + εj. Y i

k def

=    π1, mi ⊕ γ1, F −1

k (ci)

✳ ✳ ✳ πd, mi ⊕ γd, F −1

k (ci)

   .

◮ ❋♦r k = k∗✱ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ Y i k∗ ✐s p∗✳ ◮ ❋♦r k = k∗✱ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ Y i k∗ ✐s ✉♥✐❢♦r♠ ♦♥ Fs 2✳

I(K′; Y ) ≤

N

• i=1
• k∈F

nkey 2

I(K′; Y i

k).

❊♥tr♦♣② ✐♥ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ❝r②♣t❛♥❛❧②s✐s ✭✷✴✷✮

I(K′; Y ) ≤

N

• i=1
• k∈F

nkey 2

I(K′; Y i

k).

I(K′; Y i

k) = H(Y i k) − H(Y i k|K′).

❋✐♥❛❧ r❡s✉❧t I(K′; Y ) ≤ N · (d − H(p∗)).

❊♥tr♦♣② ✐♥ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ❝r②♣t❛♥❛❧②s✐s ✭✷✴✷✮

I(K′; Y ) ≤

N

• i=1
• k∈F

nkey 2

H(Y i

k) − H(Y i k|K′).

• k∈F

nkey 2

H(Y i

k|K′) = H(p∗) + (2nkey − 1) · d.

• k∈F

nkey 2

H(Y i

k) = 2nkey · d.

❋✐♥❛❧ r❡s✉❧t I(K′; Y ) ≤ N · (d − H(p∗)).

❆♣♣❧✐❝❛t✐♦♥ t♦ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ❝r②♣t❛♥❛❧②s✐s

❋♦r♠✉❧❛ ✉s❡❞ ❢r♦♠ ❬❍❡r♠❡❧✐♥✱ ❈❤♦✱ ◆②❜❡r❣ ✷✵✵✾❪✿ aLLR ≈ NC(p) 2 − m.

✵ ✷ ✹ ✻ ✽ ✶✵ ✶✷ ✷✸ ✷✹ ✷✺ ✷✻ ✷✼ ✷✽ ❆❞✈❛♥t❛❣❡ log2(N) ❊♠♣✐r✐❝❛❧ [❍❈◆✵✾] ❊♥tr♦♣②

❆♣♣❧✐❝❛t✐♦♥ t♦ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ❝r②♣t❛♥❛❧②s✐s

❋♦r♠✉❧❛ ✉s❡❞ ❢r♦♠ ❬❍❡r♠❡❧✐♥✱ ❈❤♦✱ ◆②❜❡r❣ ✷✵✵✾❪✿ aLLR ≈ NC(p) 2 ln(2) − m.

✵ ✷ ✹ ✻ ✽ ✶✵ ✶✷ ✷✸ ✷✹ ✷✺ ✷✻ ✷✼ ✷✽ ❆❞✈❛♥t❛❣❡ log2(N) ❊♠♣✐r✐❝❛❧ [❍❈◆✵✾] ❊♥tr♦♣②

❆♣♣❧✐❝❛t✐♦♥ t♦ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ❝r②♣t❛♥❛❧②s✐s

❋♦r♠✉❧❛ ✉s❡❞ ❢r♦♠ ❬❍❡r♠❡❧✐♥✱ ❈❤♦✱ ◆②❜❡r❣ ✷✵✵✾❪✿ aLLR ≈ − log2 Φ

• −
• NC(p)
• − m.

✵ ✷ ✹ ✻ ✽ ✶✵ ✶✷ ✷✸ ✷✹ ✷✺ ✷✻ ✷✼ ✷✽ ❆❞✈❛♥t❛❣❡ log2(N) ❊♠♣✐r✐❝❛❧ [❍❈◆✵✾] ❊♥tr♦♣②

❆♣♣❧✐❝❛t✐♦♥ t♦ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r ❝r②♣t❛♥❛❧②s✐s

❋♦r♠✉❧❛ ✉s❡❞ ❢r♦♠ ❬❍❡r♠❡❧✐♥✱ ❈❤♦✱ ◆②❜❡r❣ ✷✵✵✾❪✿ aLLR ≈ − log2

• 1 − Φ
• NC(p)

2m .

✵ ✷ ✹ ✻ ✽ ✶✵ ✶✷ ✷✸ ✷✹ ✷✺ ✷✻ ✷✼ ✷✽ ❆❞✈❛♥t❛❣❡ log2(N) ❊♠♣✐r✐❝❛❧ [❍❈◆✵✾] ❊♥tr♦♣②