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❙tr❡❛♠ ❝✐♣❤❡r ❝r②♣t❛♥❛❧②s✐s

❆♥♥❡ ❈❛♥t❡❛✉t ❆♥♥❡✳❈❛♥t❡❛✉t❅✐♥r✐❛✳❢r ❤tt♣✿✴✴✇✇✇✲r♦❝q✳✐♥r✐❛✳❢r✴s❡❝r❡t✴❆♥♥❡✳❈❛♥t❡❛✉t✴ ■❝❡ ❇r❡❛❦ ✷✵✶✸

❑❡②str❡❛♠ ❣❡♥❡r❛t♦r

✫✪ ✬✩

Φ

✫✪ ✬✩

Φ

✐♥✐t✐❛❧✐③❛t✐♦♥

❏ ❏ ❏ ❏ ❏ ❏ ❫ ✡ ✡ ✡ ✡ ✡ ✡ ✢

x1

• ❅

❅ ❄ ❄ ❄ ❄ ❄ ❄ ✲ ✲

x0

• ❅

❅ ❄ ❄ ❄ ❄ ❄ ❄ ✲ ✲ ❄

s❡❝r❡t ❦❡② k ❜✐ts ♣✉❜❧✐❝ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥t❡r♥❛❧ st❛t❡ ✜❧t❡r ❦❡②str❡❛♠

f f s0 s1

· · n ❜✐ts tr❛♥s✐t✐♦♥

✶

❆✈♦✐❞✐♥❣ ❣❡♥❡r✐❝ ❛tt❛❝❦s

• ❚❤❡ ✐♥t❡r♥❛❧ st❛t❡ ♠✉st ❜❡ ❛t ❧❡❛st t✇✐❝❡ ❧❛r❣❡r t❤❛♥ t❤❡ ❦❡②

✭❚✐♠❡✲♠❡♠♦r②✲❞❛t❛ tr❛❞❡✲♦✛ ❬●♦❧✐❝ ✾✺❪❬❇❛❜❜❛❣❡ ✾✺❪✮

• ❚❤❡ ❣❡♥❡r❛t♦r ♠✉st ♣❛ss t❤❡ st❛t✐st✐❝❛❧ t❡sts✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡

✜❧t❡r✐♥❣ ❢✉♥❝t✐♦♥ f ♠✉st ❜❡ ❜❛❧❛♥❝❡❞✳

• ❆t ❧❡❛st ♦♥❡ ❢✉♥❝t✐♦♥ ❛♠♦♥❣ Φ ❛♥❞ f ♠✉st ❜❡ ♥♦♥❧✐♥❡❛r✳
• Φ ❤❛s ♥♦ s❤♦rt ❝②❝❧❡s✳

✷

❈❤♦♦s✐♥❣ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ Φ ❚✇♦ str❛t❡❣✐❡s✿

• ❈❤♦♦s❡ ❛ r❛♥❞♦♠✲❧♦♦❦✐♥❣ ♠❛♣♣✐♥❣✴♣❡r♠✉t❛t✐♦♥ ♦♣❡r❛t✐♥❣ ♦♥ ❛

❧❛r❣❡ ✐♥t❡r♥❛❧ st❛t❡✿ t❤❡ ♣❡r✐♦❞ ♦❢ (xt)t≥0 ✐s ❡①♣❡❝t❡❞ t♦ ❜❡ ❝❧♦s❡ t♦ 2

n 2✳ ❙❤♦rt ❝②❝❧❡s ❡①✐st ❜✉t ❛r❡ ✉♥❧✐❦❡❧② t♦ ♦❝❝✉r✳ ❊❣✿ ❘❈✹✳

• ❈❤♦♦s❡ ❛ ♣❡r♠✉t❛t✐♦♥ ✇✐t❤ s♦♠❡ ❦♥♦✇♥ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦♣❡rt✐❡s

♦♣❡r❛t✐♥❣ ♦♥ ❛ s♠❛❧❧ ✐♥t❡r♥❛❧ st❛t❡✿ t❤❡ ♣❡r✐♦❞ ♦❢ (xt)t≥0 ❝❛♥ ❜❡ ♣r♦✈❡❞ t♦ ❜❡ ❝❧♦s❡ t♦ 2n✳ ❙❤♦rt ❝②❝❧❡s ❛r❡ ❛✈♦✐❞❡❞✳ ❊❣✿ ▲❋❙❘ ✭❚♦r✬s ❧❡❝t✉r❡✮✳

✸

❖✉t❧✐♥❡

• ❙t❛t✐st✐❝❛❧ ❛tt❛❝❦s ❛❣❛✐♥st ✜❧t❡r❡❞ ▲❋❙❘s
• ❆❧❣❡❜r❛✐❝ ❛tt❛❝❦s ❛❣❛✐♥st ✜❧t❡r❡❞ ▲❋❙❘s
• ❈♦rr❡❧❛t✐♦♥ ❛tt❛❝❦s ❛❣❛✐♥st ❝♦♠❜✐♥❛t✐♦♥ ❣❡♥❡r❛t♦rs

✹

❙t❛t✐st✐❝❛❧ ❛tt❛❝❦s ❛❣❛✐♥st ✜❧t❡r❡❞ ▲❋❙❘s

✺

Pr✐♥❝✐♣❧❡ f st ut

ut+γ1 ut+γ2ut+γ3 . . . ut+γn

✲ ✻ ✻ ✻ ✻ ✻ ✻ ✲ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ ✻

■❢ f ✐s ❜❛❧❛♥❝❡❞✱ t❤❡♥ t❤❡ ♦✉t♣✉t s❡q✉❡♥❝❡ ❝♦♥t❛✐♥s r♦✉❣❤❧② t❤❡ s❛♠❡ ♥✉♠❜❡rs ♦❢ 0 ❛♥❞ 1✳ ◗✉❡st✐♦♥✿ ❲❤❛t ✐❢ ✇❡ ❝♦♥s✐❞❡r ♣❛✐rs ♦❢ ❜✐ts❄ ❉♦ ✇❡ ❣❡t ❛ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥❄

✻

❊①❛♠♣❧❡

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✍✌ ✎☞

×

✍✌ ✎☞

×

✍✌ ✎☞

✰

✍✌ ✎☞

✰

❍ ❍ ❍ ❨ ✟✟ ✟ ✯ ✟✟ ✟ ✯ ❍ ❍ ❍ ❨ P P P P ✐ ✏✏ ✏ ✶ ✲ ✻

♦✉t♣✉t

✛ ❄ ✲

❣❡♥❡r❛t❡s t❤❡ 127✲❜✐t s❡q✉❡♥❝❡ ✵✶✵✵✵✵✶✶✵✵✵✵✵✶✵✵✵✵✶✶✶✶✶✵✶✶✶✵✶✵✵✵✶✵✵✶✶✶✵✶✶✶✵✶✶✵✵✶✶✵✵✵✶✶✵✵✶✶✶✶ ✵✶✶✵✶✶✶✵✵✶✶✵✵✶✶✵✵✶✵✵✶✵✶✵✵✶✶✵✵✶✵✶✶✶✵✵✶✶✶✵✶✵✶✵✵✶✶✵✶✶✵✵✶✵✵✵✶✶✶✵ ✵✶✶✵✶✵✵

→ ■t ❤❛s 64 ♦♥❡s✳

✼

❊①❛♠♣❧❡

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✍✌ ✎☞

×

✍✌ ✎☞

×

✍✌ ✎☞

✰

✍✌ ✎☞

✰

❍ ❍ ❍ ❨ ✟✟ ✟ ✯ ✟✟ ✟ ✯ ❍ ❍ ❍ ❨ P P P P ✐ ✏✏ ✏ ✶ ✲ ✻

♦✉t♣✉t

✛ ❄ ✲

❣❡♥❡r❛t❡s t❤❡ 127✲❜✐t s❡q✉❡♥❝❡ ✵✶✵✵✵✵✶✶✵✵✵✵✵✶✵✵✵✵✶✶✶✶✶✵✶✶✶✵✶✵✵✵✶✵✵✶✶✶✵✶✶✶✵✶✶✵✵✶✶✵✵✵✶✶✵✵✶✶✶✶ ✵✶✶✵✶✶✶✵✵✶✶✵✵✶✶✵✵✶✵✵✶✵✶✵✵✶✶✵✵✶✵✶✶✶✵✵✶✶✶✵✶✵✶✵✵✶✶✵✶✶✵✵✶✵✵✵✶✶✶✵ ✵✶✶✵✶✵✵

(st, st+1) =

            

(0, 0) 30 t✐♠❡s (0, 1) 32 t✐♠❡s (1, 0) 32 t✐♠❡s (1, 1) 32 t✐♠❡s

✽

❊①❛♠♣❧❡

✲ ✲ ✲ ✲ ✲ ✲ ✲ ✍✌ ✎☞

×

✍✌ ✎☞

×

✍✌ ✎☞

✰

✍✌ ✎☞

✰

❍ ❍ ❍ ❨ ✟✟ ✟ ✯ ✟✟ ✟ ✯ ❍ ❍ ❍ ❨ P P P P ✐ ✏✏ ✏ ✶ ✲ ✻

♦✉t♣✉t

✛ ❄ ✲

❣❡♥❡r❛t❡s t❤❡ 127✲❜✐t s❡q✉❡♥❝❡ ✵✶✵✵✵✵✶✶✵✵✵✵✵✶✵✵✵✵✶✶✶✶✶✵✶✶✶✵✶✵✵✵✶✵✵✶✶✶✵✶✶✶✵✶✶✵✵✶✶✵✵✵✶✶✵✵✶✶✶✶ ✵✶✶✵✶✶✶✵✵✶✶✵✵✶✶✵✵✶✵✵✶✵✶✵✵✶✶✵✵✶✵✶✶✶✵✵✶✶✶✵✶✵✶✵✵✶✶✵✶✶✵✵✶✵✵✵✶✶✶✵ ✵✶✶✵✶✵✵

(st, st+2) =

            

(0, 0) 22 t✐♠❡s (0, 1) 40 t✐♠❡s (1, 0) 39 t✐♠❡s (1, 1) 24 t✐♠❡s

✾

❊①❛♠♣❧❡ ❆t t✐♠❡ t✿ ❆t t✐♠❡ t + 2✿

✲ ✲ c ✲ ✲ b ✲ ✲a ✲ ✍✌ ✎☞

×

✍✌ ✎☞

×

✍✌ ✎☞

✰

✍✌ ✎☞

✰

❍ ❍ ❍ ❨ ✟✟ ✟ ✯ ✟✟ ✟ ✯ ❍ ❍ ❍ ❨ P P P P ✐ ✏✏ ✏ ✶ ✲ ✻

♦✉t♣✉t

✛ ❄ ✲

st = f(c, b, a)

✶✵

❊①❛♠♣❧❡ ❆t t✐♠❡ t✿ ❆t t✐♠❡ t + 2✿

✲ ✲ c ✲ ✲ b ✲ ✲a ✲ ✍✌ ✎☞

×

✍✌ ✎☞

×

✍✌ ✎☞

✰

✍✌ ✎☞

✰

❍ ❍ ❍ ❨ ✟✟ ✟ ✯ ✟✟ ✟ ✯ ❍ ❍ ❍ ❨ P P P P ✐ ✏✏ ✏ ✶ ✲ ✻

♦✉t♣✉t

✛ ❄ ✲ ✲ ✲d ✲ ✲ c ✲ ✲ b ✲ ✍✌ ✎☞

×

✍✌ ✎☞

×

✍✌ ✎☞

✰

✍✌ ✎☞

✰

❍ ❍ ❍ ❨ ✟✟ ✟ ✯ ✟✟ ✟ ✯ ❍ ❍ ❍ ❨ P P P P ✐ ✏✏ ✏ ✶ ✲ ✻

♦✉t♣✉t

✛ ❄ ✲

st = f(c, b, a) st+2 = f(d, c, b)

✶✶

❊①❛♠♣❧❡ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ (st, st+2) ✐s ❣✐✈❡♥ ❜② t❤❡ tr✉t❤ t❛❜❧❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

(a, b, c, d) → (f(c, b, a), f(d, c, b)) d c b a y1 y2 d c b a y1 y2

✵ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✶ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✶ ✶ ✶ ✵ ✶ ✵ ✶ ✶ ✶ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✵ ✶ ✵ ✶ ✶ ✵ ✶ ✶ ✵ ✶ ✶ ✶ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✶ ✶ ✵ ✵ ✶ ✵ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✵

✶✷

❊①❛♠♣❧❡ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ (st, st+2) ✐s ❣✐✈❡♥ ❜② t❤❡ tr✉t❤ t❛❜❧❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥

(a, b, c, d) → (f(c, b, a), f(d, c, b)) d c b a y1 y2 d c b a y1 y2

✵ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✶ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✶ ✶ ✶ ✵ ✶ ✵ ✶ ✶ ✶ ✶ ✵ ✶ ✵ ✵ ✶ ✵ ✶ ✶ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✶ ✶ ✵ ✶ ✶ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✶ ✶ ✵ ✵ ✶ ✵ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶

→ Pr[st = st+2] = 5 8 > 1 2

✶✸

❘❡❧❛t❡❞ ❞❡s✐❣♥ ❝r✐t❡r✐❛ ❆✉❣♠❡♥t❡❞ ❢✉♥❝t✐♦♥ ❬❆♥❞❡rs♦♥ ✾✶❪

{0, 1}γn−γ1+m − → {0, 1}m (uγn, uγn+1, . . . , uγ1+m−1) − →

• f(uγn, · · · , uγ1), f(uγn+1, · · · , uγ1+1), . . . ,

f(uγn+m−1, · · · , uγ1+m−1)

• s❤♦✉❧❞ ❜❡ ❜❛❧❛♥❝❡❞ ❢♦r ❛❧❧ m ≤ (γn − γ1)✳

❚❤✐s ❝♦♥❞✐t✐♦♥ ❤♦❧❞s ❢♦r ✐♥st❛♥❝❡ ✐❢ f ✐s ❧✐♥❡❛r ✇✐t❤ r❡s♣❡❝t t♦ ✐ts ✜rst ♦r t♦ ✐ts ❧❛st ✐♥♣✉t ✈❛r✐❛❜❧❡✳ ❈❤♦✐❝❡ ♦❢ t❤❡ s❡❧❡❝t✐♦♥ ❝❡❧❧s✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ♣❛✐rs (i, j) ✇✐t❤ t❤❡ s❛♠❡ γj − γi s❤♦✉❧❞ ❜❡ ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡✳

✶✹

❆❧❣❡❜r❛✐❝ ❛tt❛❝❦s ❛❣❛✐♥st ✜❧t❡r❡❞ ▲❋❙❘s

✶✺

❘❡❝♦✈❡r✐♥❣ t❤❡ ✐♥✐t✐❛❧ st❛t❡ ❢♦r ❛ ❧✐♥❡❛r tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥

• ❚❤❡ ✐♥✐t✐❛❧ st❛t❡ ✐s s❡❝r❡t✿

✐♥✐t✐❛❧ st❛t❡ = k0, . . . , kn−1 ∈ {0, 1}n

• ▲✐♥❡❛r tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ ♦♥ {0, 1}n✿

xt = Lt(k0, . . . , kn−1)

• ❋✐❧t❡r✐♥❣ ❢✉♥❝t✐♦♥ ❢r♦♠ {0, 1}n ✐♥t♦ {0, 1}✿

st = f(xt) = f

• Lt(k0, . . . , kn−1)
• Pr♦❜❧❡♠✳ ❘❡❝♦✈❡r t❤❡ ✐♥✐t✐❛❧ st❛t❡ k0, . . . , kn−1 ❢r♦♠ t❤❡ ❦♥♦✇❧❡❞❣❡ ♦❢

N ❦❡②str❡❛♠ ❜✐ts s0, s1, . . . , sN−1✳

✶✻

❇❛s✐❝ ❛❧❣❡❜r❛✐❝ ❛tt❛❝❦ ❙❡t ✉♣ t❤❡ ❡♥❝✐♣❤❡r✐♥❣ ❡q✉❛t✐♦♥s✿

         s0 = f(k0, . . . , kn−1) s1 = f ◦ L(k0, . . . , kn−1) st = f ◦ Lt(k0, . . . , kn−1)

❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ✇✐t❤ n ✈❛r✐❛❜❧❡s ♦❢ ❞❡❣r❡❡ d = deg(f) ✳ ❚❤❡ ❧❛③② ❝r②♣t♦❣r❛♣❤❡r✿

d

• i=1

n i

• ≃ nd

d! ❦❡②str❡❛♠ ❜✐ts

❚✐♠❡ ❝♦♠♣❧❡①✐t②✿ n3d ♦♣❡r❛t✐♦♥s ✳

✶✼

❇❛s✐❝ ❛❧❣❡❜r❛✐❝ ❛tt❛❝❦ ❙❡t ✉♣ t❤❡ ❡♥❝✐♣❤❡r✐♥❣ ❡q✉❛t✐♦♥s✿

         s0 = f(k0, . . . , kn−1) s1 = f ◦ L(k0, . . . , kn−1) st = f ◦ Lt(k0, . . . , kn−1)

❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ✇✐t❤ n ✈❛r✐❛❜❧❡s ♦❢ ❞❡❣r❡❡ d = deg(f) ✳

• ❚❤❡ ❧❛③② ❝r②♣t♦❣r❛♣❤❡r✿ ❛s❦ ▼❛rt✐♥✦

d

• i=1

n i

• ≃ nd

d! ❦❡②str❡❛♠ ❜✐ts

❚✐♠❡ ❝♦♠♣❧❡①✐t②✿ n3d ♦♣❡r❛t✐♦♥s ✳

✶✽

❇❛s✐❝ ❛❧❣❡❜r❛✐❝ ❛tt❛❝❦ ❙❡t ✉♣ t❤❡ ❡♥❝✐♣❤❡r✐♥❣ ❡q✉❛t✐♦♥s✿

         s0 = f(k0, . . . , kn−1) s1 = f ◦ L(k0, . . . , kn−1) st = f ◦ Lt(k0, . . . , kn−1)

❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ✇✐t❤ n ✈❛r✐❛❜❧❡s ♦❢ ❞❡❣r❡❡ d = deg(f) ✳

• ❚❤❡ st✉♣✐❞ ❝r②♣t♦❣r❛♣❤❡r✿ ❙♦❧✈❡ t❤❡ s②st❡♠ ❜② ❧✐♥❡❛r✐③❛t✐♦♥

d

• i=1

n i

• ≃ nd

d! ❦❡②str❡❛♠ ❜✐ts

❚✐♠❡ ❝♦♠♣❧❡①✐t②✿ n3d ♦♣❡r❛t✐♦♥s ✳

✶✾

❊①❛♠♣❧❡

k9

✲k8 ✲k7 ✲k6 ✲k5 ✲k4

f

✲k3 ✲k2 ✲

k1

✲ ✒✑ ✓✏

✰

❄

k0

✛ ✲ ✻ ✻ ✻ ✻ ✻ ✻ ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ ✟ ❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜ ❜ ✻

✇❤❡r❡ f = x1 + x6 + x2x4 + x4x5 + x2x3x4x5 + x1x2x3x4x5✳ ❲❡ ❣❡t ❛ s②st❡♠ ♦❢ ❞❡❣r❡❡ 5 ✇✐t❤ 10 ✉♥❦♥♦✇♥s✳

✷✵

❆❧❣❡❜r❛✐❝ ❛tt❛❝❦s ❬❈♦✉rt♦✐s✲▼❡✐❡r ✵✸❪ ▲❡t AN(f) = {g, g(x)f(x) = 0 ❢♦r ❛❧❧ x ∈ {0, 1}n} ❜❡ t❤❡ s❡t ♦❢ ❛♥♥✐❤✐❧❛t♦rs ♦❢ f✳ ▲❡t g ∈ AN(f)✱ ✐✳❡✳✱ s✉❝❤ t❤❛t g(x)f(x) = 0 ❢♦r ❛❧❧ x ∈ {0, 1}n✳

g(xt)f(xt) = g(xt)st = 0 = ⇒ g ◦ Lt(k0, . . . , kn−1) = 0 ✐❢ st = 1 .

▲❡t h ∈ AN(1 + f)✱ ✐✳❡✱ s✉❝❤ t❤❛t h(x)(1 + f(x)) = 0 ❢♦r ❛❧❧ x ∈ {0, 1}n✳

h(xt)(1 + f(xt)) = h(xt)(1 + st) = 0 = ⇒ h ◦ Lt(k0, . . . , kn−1) = 0 ✐❢ st = 0 .

❆❧❣❡❜r❛✐❝ s②st❡♠ ✇✐t❤ n ✈❛r✐❛❜❧❡s ♦❢ ❞❡❣r❡❡

d = min{deg(g), g ∈ AN(f) ∪ AN(1 + f), g = 0} .

✷✶

❆❧❣❡❜r❛✐❝ ❛tt❛❝❦s ❬❈♦✉rt♦✐s✲▼❡✐❡r ✵✸❪ ▲❡t AN(f) = {g, g(x)f(x) = 0 ❢♦r ❛❧❧ x ∈ {0, 1}n} ❜❡ t❤❡ s❡t ♦❢ ❛♥♥✐❤✐❧❛t♦rs ♦❢ f✳ ▲❡t g ∈ AN(f)✱ ✐✳❡✳✱ s✉❝❤ t❤❛t g(x)f(x) = 0 ❢♦r ❛❧❧ x ∈ {0, 1}n✳

g(xt)f(xt) = g(xt)st = 0 = ⇒ g ◦ Lt(k0, . . . , kn−1) = 0 ✐❢ st = 1 .

▲❡t h ∈ AN(1 + f)✱ ✐✳❡✱ s✉❝❤ t❤❛t h(x)(1 + f(x)) = 0 ❢♦r ❛❧❧ x ∈ {0, 1}n✳

h(xt)(1 + f(xt)) = h(xt)(1 + st) = 0 = ⇒ h ◦ Lt(k0, . . . , kn−1) = 0 ✐❢ st = 0 .

❆❧❣❡❜r❛✐❝ s②st❡♠ ✇✐t❤ n ✈❛r✐❛❜❧❡s ♦❢ ❞❡❣r❡❡

d = min{deg(g), g ∈ AN(f) ∪ AN(1 + f), g = 0} .

✷✷

❊①❛♠♣❧❡

f = x1 + x6 + x2x4 + x4x5 + x2x3x4x5 + x1x2x3x4x5

❱❛❧✉❡ ✈❡❝t♦r ♦❢ f✿ ✵✶✵✶✵✶✵✶✵✶✶✵✵✶✶✵✵✶✵✶✵✶✵✶✶✵✵✶✶✵✶✶✶✵✶✵✶✵✶✵✶✵✵✶✶✵✵✶✶✵✶✵✶✵✶✵✵✶✶✵✵✶✵✵ ✶✵✶✵✶✵✶✵✵✵✵✵✵✵✵✵✶✵✶✵✶✵✶✵✵✵✵✵✵✵✵✵✵✶✵✶✵✶✵✶✵✵✵✵✵✵✵✵✵✶✵✶✵✶✵✶✵✵✵✵✵✵✵✵ → ❚❤❡ ❢✉♥❝t✐♦♥ g ✇✐t❤ t❤❡ ♣r❡✈✐♦✉s ✈❛❧✉❡ ✈❡❝t♦r ❜❡❧♦♥❣s t♦ AN(f)✳

g = 1 + x1 + x4 + x1x4 + x6 + x4x6

■♥ ♣❛rt✐❝✉❧❛r✱ deg g = 2✳

✷✸

❊①❛♠♣❧❡

k9

✲k8 ✲k7 ✲k6 ✲k5 ✲k4

g

✲k3 ✲k2 ✲

k1

✲ ✒✑ ✓✏

✰

❄

k0

✛ ✲ ✻ ✻ ✻ ✻ ✻ ✻ ✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ ✟ ❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜ ❜ ✻

■❢ st = 1✱ t❤❡♥ t❤❡ ♦✉t♣✉t ♦❢ t❤✐s ❣❡♥❡r❛t♦r ✐s 0✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ❦❡②str❡❛♠ ✐s 10110...✳ = 1 + k9 + k3 + k3k9 + k0 + k0k3 = 1 + k11 + k5k11 + k2 + k2k5 = 1 + k1 + k8 + k1k5 + k5k8 + k2 + k2k5 = 1 + k2 + k9 + k2k6 + k6k9 + k3 + k3k6 ❙✐♠✐❧❛r ❝♦♠♣❧❡①✐t② ❛s t❤❡ ❛tt❛❝❦ ❢♦r ❛ ✜❧t❡r✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ ❞❡❣r❡❡ 2✳

✷✹

❈♦♠♣❧❡①✐t② ♦❢ t❤❡ ❛tt❛❝❦

n ❂ s✐③❡ ♦❢ t❤❡ ✐♥t❡r♥❛❧ st❛t❡ AI(f) ❂ ❛❧❣❡❜r❛✐❝ ✐♠♠✉♥✐t② ♦❢ t❤❡ ✜❧t❡r✐♥❣ ❢✉♥❝t✐♦♥ f AI(f) ❂ min{deg(g), g ∈ AN(f) ∪ AN(1 + f), g = 0}✳

◆✉♠❜❡r ♦❢ ♦♣❡r❛t✐♦♥s✿

 

AI(f)

• i=1

n i  

3

≃ n3AI(f) .

✷✺

❙❡❝✉r✐t② ❝r✐t❡r✐♦♥

AI(f) ❂ ❛❧❣❡❜r❛✐❝ ✐♠♠✉♥✐t② ♦❢ t❤❡ ✜❧t❡r✐♥❣ ❢✉♥❝t✐♦♥ f AI(f) ❂ min{deg(g), g ∈ AN(f) ∪ AN(1 + f), g = 0}✳

❈♦♠♣❛r✐s♦♥ ✇✐t❤ ❡①❤❛✉st✐✈❡ s❡❛r❝❤✿ ■❢ n = 2k ✇❤❡r❡ k ✐s t❤❡ ❦❡② s✐③❡✱ ✇❡ ♠✉st ❤❛✈❡ (2k)3AI(f) ≥ 2k ✐✳❡✳✱

AI(f) ≥ k 3(1 + log2 k)

❊①❛♠♣❧❡✳ k = 128 ❜✐ts✱ n = 256 ❜✐ts✳

− → AI(f) ≥ 7 .

✷✻

❋✐♥❞✐♥❣ ❛♥♥✐❤✐❧❛t♦rs ❊①♣❡♥s✐✈❡ t❡❝❤♥✐q✉❡✿ ❈♦♠♣✉t❡ t❤❡ ❆◆❋ ♦❢ ❛❧❧ ❢✉♥❝t✐♦♥s g s✉❝❤ t❤❛t g(x) = 0 ❢♦r ❛❧❧ x s✉❝❤ t❤❛t f(x) = 1✳ ✵✶✵✶✵✶✵✶✵✶✶✵✵✶✶✵✵✶✵✶✵✶✵✶✶✵✵✶✶✵✶✶✶✵✶✵✶✵✶✵✶✵✵✶✶✵✵✶✶✵✶✵✶✵✶✵✵✶✶✵✵✶✵✵ ❄✵❄✵❄✵❄✵❄✵✵❄❄✵✵❄❄✵❄✵❄✵❄✵✵❄❄✵✵❄✵✵✵❄✵❄✵❄✵❄✵❄❄✵✵❄❄✵✵❄✵❄✵❄✵❄❄✵✵❄❄✵❄❄ ■❢ f ✐s ❛ ❜❛❧❛♥❝❡❞ ❢✉♥❝t✐♦♥ ♦❢ n ✈❛r✐❛❜❧❡s✱ 2n−1 ❞✐✛❡r❡♥t ❛♥♥✐❤✐❧❛t♦rs s❤♦✉❧❞ ❜❡ ❡①❛♠✐♥❡❞✳

✷✼

❋✐♥❞✐♥❣ ❛♥♥✐❤✐❧❛t♦rs

x s✉❝❤ t❤❛t f(x) = 1 [wt(f)] 1 RMf(d, n)

❛❧❧ ♠♦♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ≤ d

d

i=0

n

i

• x1

✳ ✳ ✳

xn x1x2

✳ ✳ ✳

xn−1xn

Pr♦♣♦s✐t✐♦♥✳ ❚❤❡r❡ ❡①✐sts g = 0 ✐♥ AN(f) ✇✐t❤ deg g ≤ d ✇❤❡♥

wt(f) <

d

• i=0

n i

• .

❈♦r♦❧❧❛r②✳ ❋♦r ❛♥② f ♦❢ n ✈❛r✐❛❜❧❡s✱

AI(f) ≤ n 2

• .

✷✽

❈♦rr❡❧❛t✐♦♥ ❛tt❛❝❦s ❛❣❛✐♥st ❝♦♠❜✐♥❛t✐♦♥ ❣❡♥❡r❛t♦rs

✷✾

❚❤❡ ❝♦♠❜✐♥❛t✐♦♥ ❣❡♥❡r❛t♦r

✲ ❄ ❄ ❄ ✲ ❄ ❄ ❄ ✲ ❄ ❄ ✲ ✲ ✲ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✲

✳ ✳ ✳ u1

t

u2

t

un

t

f st

st = f(u1

t, u2 t, . . . , un t )

✸✵

❈♦rr❡❧❛t✐♦♥ ❛tt❛❝❦ ❬❙✐❡❣❡♥t❤❛❧❡r ✽✺❪ t❛r❣❡t ▲❋❙❘ t❛r❣❡t ▲❋❙❘ ❝♦rr❡❧❛t✐♦♥

✲ ✲

st

❦❡②str❡❛♠

σt

✇❤❡r❡ p = Pr[st = σt] = 1

2 .

❉✐✈✐❞❡✲❛♥❞✲❝♦♥q✉❡r ❛tt❛❝❦✿ ❘❡❝♦✈❡r t❤❡ ✐♥✐t✐❛❧ st❛t❡ ♦❢ t❤❡ t❛r❣❡t ▲❋❙❘ ❢r♦♠ t❤❡ ❦♥♦✇❧❡❞❣❡ ♦❢ s♦♠❡ ❦❡②str❡❛♠ ❜✐ts✳

✸✶

❇❛s✐❝ ❝♦rr❡❧❛t✐♦♥ ❛tt❛❝❦ ❬❙✐❡❣❡♥t❤❛❧❡r ✽✺❪ ❆❧❣♦r✐t❤♠✿ ❋♦r ❡❛❝❤ ♣♦ss✐❜❧❡ ✐♥✐t✐❛❧✐③❛t✐♦♥ ♦❢ t❤❡ t❛r❣❡t ▲❋❙❘✿ ❈♦♠♣✉t❡ t❤❡ s❡q✉❡♥❝❡ σ ❣❡♥❡r❛t❡❞ ❜② t❤❡ t❛r❣❡t ▲❋❙❘ ❈❤❡❝❦ ✇❤❡t❤❡r σ ✐s ❝♦rr❡❧❛t❡❞ t♦ t❤❡ ❦❡②str❡❛♠ s ✭♦r t♦ ✐ts ❜✐t✇✐s❡ ❝♦♠♣❧❡♠❡♥t✮ ❜② ❝♦♠♣✉t✐♥❣

C =

N−1

• t=0

(−1)st+σt

✸✷

❊①❛♠♣❧❡✿ ●❡✛❡ ❣❡♥❡r❛t♦r

x1x2x3

✵✵✵ ✶✵✵ ✵✶✵ ✶✶✵ ✵✵✶ ✶✵✶ ✵✶✶ ✶✶✶

f(x1, x2, x3)

✵ ✶ ✵ ✵ ✵ ✶ ✶ ✶

f(x1, x2, x3) = x1❄

① ① ① ① ① ①

p = Pr[st = σt] = Pr[f(x1, x2, x3) = x1] = 1 2

• 1 + E(f + x1)

23

• = 3

4 .

→❚❛r❣❡t ▲❋❙❘✿ ▲❋❙❘ ✶

✸✸

❙t❛t✐st✐❝❛❧ t❡st

• ■❢ σ ✐s ✐♥❞❡♣❡♥❞❡♥t ❢r♦♠ s✱

Pr[C = x] ≃ 1 √ 2π exp

• −x2

2

• ■❢ σ ✐s t❤❡ ♦✉t♣✉t ♦❢ t❤❡ t❛r❣❡t ▲❋❙❘✱

Pr[C = x] ≃ 1 √ 2π exp

• −(x − N(2p − 1))2

8Np(1 − p)

• ❋♦r p = 3

4 ❛♥❞ N = 50

✸✹

❈♦♠♣❧❡①✐t② ❉❛t❛ ❝♦♠♣❧❡①✐t②✿ ❋♦r ❛ t❛r❣❡t ▲❋❙❘ ♦❢ ❧❡♥❣t❤ L✱ ✇❡ ❝❤♦♦s❡ ❛ ❢❛❧s❡ ❛❧❛r♠ ♣r♦❜❛❜✐❧✐t②

α ≃ 2−L✳ ❚❤❡♥✱ ✇❡ ♥❡❡❞ N ≃ L

• p − 1

2

2

❚✐♠❡ ❝♦♠♣❧❡①✐t②✿

2LN ≃ L2L

• p − 1

2

2

✸✺

❈♦rr❡❧❛t✐♦♥ ❛tt❛❝❦s ✇✐t❤ ❛ ❋❛st ❋♦✉r✐❡r ❚r❛♥s❢♦r♠ ❊❛❝❤ ❜✐t ♣r♦❞✉❝❡❞ ❜② t❤❡ t❛r❣❡t ▲❋❙❘ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ❜✐ts ♦❢ t❤❡ ✐♥✐t✐❛❧ st❛t❡ k✿

σt =

L−1

• i=0

αt,iki = αt · k .

❋♦r t❤❡ ▲❋❙❘ ♦❢ ❧❡♥❣t❤ 4 ✇✐t❤ ❢❡❡❞❜❛❝❦ ♣♦❧②♥♦♠✐❛❧

1 + X + X4✿ σ4 = k3 ⊕ k0 → α4 = (1, 0, 0, 1) σ5 = k3 ⊕ k1 ⊕ k0 → α5 = (1, 1, 0, 1) σ6 = k3 ⊕ k2 ⊕ k1 ⊕ k0 → α6 = (1, 1, 1, 1) σ7 = k2 ⊕ k1 ⊕ k0 → α7 = (1, 1, 1, 0)

❲❡ ♥❡❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ 2L ❝♦rr❡❧❛t✐♦♥s

C(k) =

N−1

• t=0

(−1)st+σt =

N−1

• t=0

(−1)st(−1)αt·k .

✸✻

❈♦rr❡❧❛t✐♦♥ ❛tt❛❝❦s ✇✐t❤ ❛ ❋❛st ❋♦✉r✐❡r ❚r❛♥s❢♦r♠

C(k) =

N−1

• t=0

(−1)st+σt =

N−1

• t=0

(−1)st(−1)αt·k .

❉✐s❝r❡t❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠✿

{0, 1}L − → Z k − →

• F (k) =

x∈{0,1}L F (x)(−1)x·k

❍❡r❡✱

C(k) = F (k)

✇✐t❤ F (x) =

• (−1)st

✐❢ x = αt ✐❢ t❤❡r❡ ✐s ♥♦ s✉❝❤ t ∈ {0, . . . , N − 1}

✸✼

❈♦rr❡❧❛t✐♦♥ ❛tt❛❝❦s ✇✐t❤ ❛ ❋❛st ❋♦✉r✐❡r ❚r❛♥s❢♦r♠

α0 = 1000, α1 = 0100, α2 = 0010, α3 = 0001, α4 = 1001, α5 = 1101, α6 = 1111, α7 = 1110

❖❜s❡r✈❡❞ ❦❡②str❡❛♠✿ s0 . . . s7 ❂ ✵✵✶✵✶✶✵✵

x1

✵ ✶ ✵ ✶ ✵ ✶ ✵ ✶ ✵ ✶ ✵ ✶ ✵ ✶ ✵ ✶

x2

✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶

x3

✵ ✵ ✵ ✵ ✶ ✶ ✶ ✶ ✵ ✵ ✵ ✵ ✶ ✶ ✶ ✶

x4

✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶

t

✵ ✶ ✷ ✼ ✸ ✹ ✺ ✻

F

✵ ✶ ✶ ✵ ✲✶ ✵ ✵ ✶ ✶ ✲✶ ✵ ✲✶ ✵ ✵ ✵ ✶ ✶ ✵ ✶ ✲✶ ✲✶ ✵ ✵ ✷ ✲✶ ✷ ✶ ✶ ✲✶ ✵ ✵ ✵ ✵ ✵ ✶ ✶ ✷ ✵ ✶ ✲✸ ✲✷ ✷ ✶ ✶ ✵ ✷ ✶ ✶ ✶ ✶ ✲✶ ✲✶ ✸ ✲✸ ✶ ✸ ✲✶ ✸ ✲✸ ✶ ✶ ✸ ✲✶ ✶

• F

✷ ✵ ✲✷ ✵ ✵ ✻ ✹ ✲✷ ✷ ✲✹ ✲✷ ✲✹ ✹ ✲✷ ✵ ✲✷

→ ❚✐♠❡ ❝♦♠♣❧❡①✐t② L2L ✐♥st❡❛❞ ♦❢ N2L✳

✸✽

❈♦rr❡❧❛t✐♦♥ ❛tt❛❝❦ ❛s ❛ ❞❡❝♦❞✐♥❣ ♣r♦❜❧❡♠ ❬▼❡✐❡r✲❙t❛✛❡❧❜❛❝❤ ✽✽❪

✛ ✲ ✲ ✲

t❛r❣❡t ▲❋❙❘

L P σ

❜✐♥❛r② s②♠♠❡tr✐❝ ❝❤❛♥♥❡❧

1 1

1 − p 1 − p p p

✲ ✲

• ✒

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❘ ✲

s ✭❦❡②str❡❛♠✮

❊rr♦r ♣r♦❜❛❜✐❧✐t②✿

p = P r[st = σt] < 1 2 (σt)t<N ❜❡❧♦♥❣s t♦ t❤❡ [N, L]✲❧✐♥❡❛r ❝♦❞❡ ❞❡✜♥❡❞ ❜② P ✳

✸✾

▲❋❙❘ ❝♦❞❡ ▲✐♥❡❛r ❝♦❞❡ ♦❢ ❧❡♥❣t❤ N ❛♥❞ ❞✐♠❡♥s✐♦♥ L ✿

(σ0, . . . , σL−1)      1 0 . . . 0 cL . . . g0,t . . . 0 1 . . . 0 cL−1 . . . g1,t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 . . . 1 c1 . . . gL−1,t . . .      = (σ0, . . . . . . σN−1)

✇❤❡r❡ t❤❡ t✲t❤ ❝♦❧✉♠♥ ✐s ❣✐✈❡♥ ❜②

L−1

• i=0

gi,tXi = Xt mod P ⋆(X) ✇✐t❤ P ⋆(X) =

L

• i=0

cL−iXi .

✹✵

Pr♦♣♦s❡❞ ❞❡❝♦❞✐♥❣ t❡❝❤♥✐q✉❡s

• ▼▲✲❞❡❝♦❞✐♥❣ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ▲❋❙❘✲❝♦❞❡ ❬❙✐❡❣❡♥t❤❛❧❡r ✽✺❪❀
• ❈♦♥✈♦❧✉t✐♦♥♥❛❧ ❝♦❞❡ ❛♥❞ ❱✐t❡r❜✐ ❛❧❣♦r✐t❤♠

❬❏♦❤❛♥ss♦♥✲❏ö♥ss♦♥ ✾✾❪❀

• ❚✉r❜♦✲❝♦❞❡ ❬❏♦❤❛♥ss♦♥✲❏ö♥ss♦♥ ✾✾❪❀
• ▲♦✇✲❉❡♥s✐t② P❛r✐t②✲❈❤❡❝❦ ❝♦❞❡s ❛♥❞ ✐t❡r❛t✐✈❡ ❞❡❝♦❞✐♥❣

❬▼❡✐❡r✲❙t❛✛❡❧❜❛❝❤ ✽✽❪✱ ❬❈❛♥t❡❛✉t✲❚r❛❜❜✐❛ ✵✵❪✱ ❬▼✐❤❛❧❥❡✈✐❝ ✲ ❋♦ss♦r✐❡r✲ ■♠❛✐ ✵✵❪❀

• ▼▲✲❞❡❝♦❞✐♥❣ ♦❢ ❛ ❞❡r✐✈❡❞ ❝♦❞❡ ✇✐t❤ s♠❛❧❧❡r ❞✐♠❡♥s✐♦♥

❬❈❤❡♣②s❤♦✈ ✲ ❏♦❤❛♥ss♦♥ ✲ ❙♠❡❡ts ✵✵❪❀

• ❙✉❞❛♥✬s ❛❧❣♦r✐t❤♠ ❢♦r r❡❝♦♥str✉❝t✐♥❣ ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧

❬❏♦❤❛♥ss♦♥✲❏ö♥ss♦♥ ✵✵❪✳

✹✶

❈♦rr❡❧❛t✐♦♥✲✐♠♠✉♥❡ ❝♦♠❜✐♥✐♥❣ ❢✉♥❝t✐♦♥ ❙❡❝✉r✐t② ❝r✐t❡r✐♦♥✿ ❢♦r ❡❛❝❤ ✐♥♣✉t ✈❛r✐❛❜❧❡ xi✱ 1 ≤ i ≤ n✱

p = Pr[f(x1, . . . , xn) = xi] = 1 2 . f ✐s s❛✐❞ t♦ ❜❡ ❝♦rr❡❧❛t✐♦♥✲✐♠♠✉♥❡✳

❊q✉✐✈❛❧❡♥t❧②✱ E(f + ϕa) = 0 ❢♦r ❛❧❧ a ✇✐t❤ wt(a) = 1✳ ❈♦rr❡❧❛t✐♦♥ ❛tt❛❝❦ ✐♥✈♦❧✈✐♥❣ s❡✈❡r❛❧ ▲❋❙❘s✿ ▲❋❙❘ ✶ ▲❋❙❘ ✷ ▲❋❙❘ n f

❆ ❆ ❆ ❆ ❯ ✲ ✁ ✁ ✁ ✁ ✕ ✲

✳ ✳ ✳ s ▲❋❙❘ ik ▲❋❙❘ i1 g σ

✲ ❍ ❍ ❥

• ✒

✳ ✳ ✳ ❝♦rr❡❧❛t✐♦♥

✹✷

• r❛✐♥ ✈✶ ❬❍❡❧❧ ❏♦❤❛♥ss♦♥ ▼❡✐❡r❪

✹✸