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❍❛s❤ ❋✉♥❝t✐♦♥s ❇❛s❡❞ ♦♥ ❚❤r❡❡ P❡r♠✉t❛t✐♦♥s✿ ❆ ●❡♥❡r✐❝ ❙❡❝✉r✐t② ❆♥❛❧②s✐s

❇❛rt ▼❡♥♥✐♥❦ ❛♥❞ ❇❛rt Pr❡♥❡❡❧ ❑❯ ▲❡✉✈❡♥

❈❘❨P❚❖ ✷✵✶✷ ✖ ❆✉❣✉st ✷✶✱ ✷✵✶✷

✶ ✴ ✶✽

▼♦t✐✈❛t✐♦♥

• ❍❛s❤ ❢✉♥❝t✐♦♥s ❜❛s❡❞ ♦♥ ❜❧♦❝❦ ❝✐♣❤❡rs
• ❉❛✈✐❡s✲▼❡②❡r ✬✽✹✱ P●❱ ✬✾✸✱ ❚❛♥❞❡♠✲❉▼ ✬✾✷✱ ✳✳✳
• ▼❉✺ ✬✾✷✱ ❙❍❆✲✶ ✬✾✺✱ ❙❍❆✲✷ ✬✵✶✱ ❇❧❛❦❡ ✬✵✽✱ ❙❦❡✐♥ ✬✵✽✱ ✳✳✳

❘❡✲❦❡②✐♥❣ r❡❧❛t❡❞✲❦❡② s❡❝✉r✐t②✱ ❡✣❝✐❡♥❝② ❧♦ss✱ ✳ ✳ ✳ ■♥st❡❛❞ ✉s❡ ✜①❡❞✲❦❡② ❜❧♦❝❦ ❝✐♣❤❡rs✱ ♦r ♣❡r♠✉t❛t✐♦♥s

E

F

✷ ✴ ✶✽

▼♦t✐✈❛t✐♦♥

• ❍❛s❤ ❢✉♥❝t✐♦♥s ❜❛s❡❞ ♦♥ ❜❧♦❝❦ ❝✐♣❤❡rs
• ❉❛✈✐❡s✲▼❡②❡r ✬✽✹✱ P●❱ ✬✾✸✱ ❚❛♥❞❡♠✲❉▼ ✬✾✷✱ ✳✳✳
• ▼❉✺ ✬✾✷✱ ❙❍❆✲✶ ✬✾✺✱ ❙❍❆✲✷ ✬✵✶✱ ❇❧❛❦❡ ✬✵✽✱ ❙❦❡✐♥ ✬✵✽✱ ✳✳✳
• ❘❡✲❦❡②✐♥❣ −

→ r❡❧❛t❡❞✲❦❡② s❡❝✉r✐t②✱ ❡✣❝✐❡♥❝② ❧♦ss✱ ✳ ✳ ✳ ■♥st❡❛❞ ✉s❡ ✜①❡❞✲❦❡② ❜❧♦❝❦ ❝✐♣❤❡rs✱ ♦r ♣❡r♠✉t❛t✐♦♥s

E

F

✷ ✴ ✶✽

▼♦t✐✈❛t✐♦♥

• ❍❛s❤ ❢✉♥❝t✐♦♥s ❜❛s❡❞ ♦♥ ❜❧♦❝❦ ❝✐♣❤❡rs
• ❉❛✈✐❡s✲▼❡②❡r ✬✽✹✱ P●❱ ✬✾✸✱ ❚❛♥❞❡♠✲❉▼ ✬✾✷✱ ✳✳✳
• ▼❉✺ ✬✾✷✱ ❙❍❆✲✶ ✬✾✺✱ ❙❍❆✲✷ ✬✵✶✱ ❇❧❛❦❡ ✬✵✽✱ ❙❦❡✐♥ ✬✵✽✱ ✳✳✳
• ❘❡✲❦❡②✐♥❣ −

→ r❡❧❛t❡❞✲❦❡② s❡❝✉r✐t②✱ ❡✣❝✐❡♥❝② ❧♦ss✱ ✳ ✳ ✳

• ■♥st❡❛❞ ✉s❡ ✜①❡❞✲❦❡② ❜❧♦❝❦ ❝✐♣❤❡rs✱ ♦r ♣❡r♠✉t❛t✐♦♥s

E

F π F

✷ ✴ ✶✽

▼♦t✐✈❛t✐♦♥

• ❇❧❛❝❦✲❈♦❝❤r❛♥✲❙❤r✐♠♣t♦♥ ✬✵✺✿

♥♦ s❡❝✉r❡ 2n✲t♦✲n✲❜✐t ❢✉♥❝t✐♦♥ ✉s✐♥❣ ✶ n✲❜✐t ♣❡r♠✉t❛t✐♦♥ ❝❛❧❧

π F

• ❡♥❡r❛❧✐③❡❞ ❜② ❘♦❣❛✇❛②✲❙t❡✐♥❜❡r❣❡r ✬✵✽✱ ❙t❛♠ ✬✵✽✱ ❙t❡✐♥❜❡r❣❡r ✬✶✵

✲t♦✲ ✲❜✐t ❢✉♥❝t✐♦♥ ✉s✐♥❣ ✲❜✐t ♣❡r♠✉t❛t✐♦♥s✿ ❝♦❧❧✐s✐♦♥s ✐♥ q✉❡r✐❡s ✭❛❧♠♦st ❛❧✇❛②s✮

✷ ✸ ✹ ✺

✸ ✴ ✶✽

▼♦t✐✈❛t✐♦♥

• ❇❧❛❝❦✲❈♦❝❤r❛♥✲❙❤r✐♠♣t♦♥ ✬✵✺✿

♥♦ s❡❝✉r❡ 2n✲t♦✲n✲❜✐t ❢✉♥❝t✐♦♥ ✉s✐♥❣ ✶ n✲❜✐t ♣❡r♠✉t❛t✐♦♥ ❝❛❧❧

π F

• ●❡♥❡r❛❧✐③❡❞ ❜② ❘♦❣❛✇❛②✲❙t❡✐♥❜❡r❣❡r ✬✵✽✱ ❙t❛♠ ✬✵✽✱ ❙t❡✐♥❜❡r❣❡r ✬✶✵
• mn✲t♦✲rn✲❜✐t ❢✉♥❝t✐♦♥ ✉s✐♥❣ k n✲❜✐t ♣❡r♠✉t❛t✐♦♥s✿ ❝♦❧❧✐s✐♦♥s ✐♥

(2n)1−(m−r+1)/(k+1) q✉❡r✐❡s ✭❛❧♠♦st ❛❧✇❛②s✮

F ✷ π ✸ π ✹ π ✺ π 2n → n 2n/3 2n/2

5 2n → n

2n/6 23n/8 2n/2 4n → 2n 1 2n/4 22n/5 2n/2

✸ ✴ ✶✽

▼♦t✐✈❛t✐♦♥

• ❇❧❛❝❦✲❈♦❝❤r❛♥✲❙❤r✐♠♣t♦♥ ✬✵✺✿

♥♦ s❡❝✉r❡ 2n✲t♦✲n✲❜✐t ❢✉♥❝t✐♦♥ ✉s✐♥❣ ✶ n✲❜✐t ♣❡r♠✉t❛t✐♦♥ ❝❛❧❧

π F

• ●❡♥❡r❛❧✐③❡❞ ❜② ❘♦❣❛✇❛②✲❙t❡✐♥❜❡r❣❡r ✬✵✽✱ ❙t❛♠ ✬✵✽✱ ❙t❡✐♥❜❡r❣❡r ✬✶✵
• mn✲t♦✲rn✲❜✐t ❢✉♥❝t✐♦♥ ✉s✐♥❣ k n✲❜✐t ♣❡r♠✉t❛t✐♦♥s✿ ❝♦❧❧✐s✐♦♥s ✐♥

(2n)1−(m−r+1)/(k+1) q✉❡r✐❡s ✭❛❧♠♦st ❛❧✇❛②s✮

F ✷ π ✸ π ✹ π ✺ π 2n → n 2n/3 2n/2

5 2n → n

2n/6 23n/8 2n/2 4n → 2n 1 2n/4 22n/5 2n/2

✸ ✴ ✶✽

❙❡❝✉r✐t② ▼♦❞❡❧

πi, π−1

i

✻ ❄

q q✉❡r✐❡s

❛❞✈❡rs❛r② A

• ■❞❡❛❧ ♣❡r♠✉t❛t✐♦♥ ♠♦❞❡❧✿ πi✬s r❛♥❞♦♠❧② ❣❡♥❡r❛t❡❞
• ❆❞✈❡rs❛r② q✉❡r② ❛❝❝❡ss t♦ πi✬s

s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t②

✹ ✴ ✶✽

❙❡❝✉r✐t② ▼♦❞❡❧

πi, π−1

i

✻ ❄

q q✉❡r✐❡s

❛❞✈❡rs❛r② A

✲

❞✐st✐♥❝t (x1, x2), (x′

1, x′ 2) s✳t✳

F(x1, x2) = F(x′

1, x′ 2)

• ■❞❡❛❧ ♣❡r♠✉t❛t✐♦♥ ♠♦❞❡❧✿ πi✬s r❛♥❞♦♠❧② ❣❡♥❡r❛t❡❞
• ❆❞✈❡rs❛r② q✉❡r② ❛❝❝❡ss t♦ πi✬s

Advcol

F (q) = max A

s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② A s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t②

✹ ✴ ✶✽

❙❡❝✉r✐t② ▼♦❞❡❧

πi, π−1

i

✻ ❄

q q✉❡r✐❡s

❛❞✈❡rs❛r② A

✲

z ∈ {0, 1}n

✲

(x1, x2) s✳t✳ F(x1, x2) = z

• ■❞❡❛❧ ♣❡r♠✉t❛t✐♦♥ ♠♦❞❡❧✿ πi✬s r❛♥❞♦♠❧② ❣❡♥❡r❛t❡❞
• ❆❞✈❡rs❛r② q✉❡r② ❛❝❝❡ss t♦ πi✬s

Advcol

F (q) = max A

s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② A Advepre

F

(q) = max

A

max

z∈{0,1}n s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② A ✹ ✴ ✶✽

Pr✐♦r ❈♦♥str✉❝t✐♦♥s ✖ ❙❤r✐♠♣t♦♥✲❙t❛♠ ✬✵✽

x1 x2 z

n n n

f1 f2 f3

• 2n✲t♦✲n✲❜✐t ❢✉♥❝t✐♦♥ ✉s✐♥❣ 3 ♦♥❡✲✇❛② ❢✉♥❝t✐♦♥s
• ❖♣t✐♠❛❧ ❝♦❧❧✐s✐♦♥ s❡❝✉r✐t②
• ❈♦❧❧✐s✐♦♥ s❡❝✉r✐t② ✐❢ fi(x) = πi(x) ⊕ x ✭s❤♦✇❡❞ ❜② ❛✉t♦♠❛t❡❞ ❛♥❛❧②s✐s✮

✺ ✴ ✶✽

Pr✐♦r ❈♦♥str✉❝t✐♦♥s ✖ ❘♦❣❛✇❛②✲❙t❡✐♥❜❡r❣❡r ✬✵✽

x1 x2 z

n n n

π1 π2 π3

a11 a12 a21 a22 a23 a31 a32 a33 a34 a41 a42 a43 a44 a45

• 2n✲t♦✲n✲❜✐t ❢✉♥❝t✐♦♥ ✭♦✈❡r F2n✮ ✉s✐♥❣ 3 ♣❡r♠✉t❛t✐♦♥s

❈♦❧❧✐s✐♦♥✴♣r❡✐♠❛❣❡ s❡❝✉r✐t② ✐❢ s❛t✐s❢② ✏✐♥❞❡♣❡♥❞❡♥❝❡ ❝r✐t❡r✐♦♥✑

❊①❝❧✉❞❡s ❜✐♥❛r②

✻ ✴ ✶✽

Pr✐♦r ❈♦♥str✉❝t✐♦♥s ✖ ❘♦❣❛✇❛②✲❙t❡✐♥❜❡r❣❡r ✬✵✽

x1 x2 z

n n n

π1 π2 π3

a11 a12 a21 a22 a23 a31 a32 a33 a34 a41 a42 a43 a44 a45

• 2n✲t♦✲n✲❜✐t ❢✉♥❝t✐♦♥ ✭♦✈❡r F2n✮ ✉s✐♥❣ 3 ♣❡r♠✉t❛t✐♦♥s
• ❈♦❧❧✐s✐♦♥✴♣r❡✐♠❛❣❡ s❡❝✉r✐t② ✐❢ aij s❛t✐s❢② ✏✐♥❞❡♣❡♥❞❡♥❝❡ ❝r✐t❡r✐♦♥✑

− → ❊①❝❧✉❞❡s ❜✐♥❛r② aij

✻ ✴ ✶✽

❖✉r ❈♦♠♣r❡ss✐♦♥ ❋✉♥❝t✐♦♥ ❉❡s✐❣♥

• 2n✲t♦✲n ❝♦♠♣r❡ss✐♦♥ ❢✉♥❝t✐♦♥ ✉s✐♥❣ ♣❡r♠✉t❛t✐♦♥s ❛♥❞ ✲♦♣❡r❛t♦rs

x1 x2 z

n n n

π1 π2 π3

a11 a12 a21 a22 a23 a31 a32 a33 a34 a41 a42 a43 a44 a45 aij ∈ {0, 1}

▼✉❧t✐✲♣❡r♠✉t❛t✐♦♥ s❡tt✐♥❣✿ ✬s ❛❧❧ ❞✐✛❡r❡♥t ❙✐♥❣❧❡✲♣❡r♠✉t❛t✐♦♥ s❡tt✐♥❣✿

✼ ✴ ✶✽

❖✉r ❈♦♠♣r❡ss✐♦♥ ❋✉♥❝t✐♦♥ ❉❡s✐❣♥

• 2n✲t♦✲n ❝♦♠♣r❡ss✐♦♥ ❢✉♥❝t✐♦♥ ✉s✐♥❣ ♣❡r♠✉t❛t✐♦♥s ❛♥❞ ✲♦♣❡r❛t♦rs

x1 x2 z

n n n

π1 π2 π3

a11 a12 a21 a22 a23 a31 a32 a33 a34 a41 a42 a43 a44 a45 aij ∈ {0, 1}

• ▼✉❧t✐✲♣❡r♠✉t❛t✐♦♥ s❡tt✐♥❣✿ πi✬s ❛❧❧ ❞✐✛❡r❡♥t
• ❙✐♥❣❧❡✲♣❡r♠✉t❛t✐♦♥ s❡tt✐♥❣✿ π1 = π2 = π3

✼ ✴ ✶✽

x1 x2 z

n n n

π1 π2 π3

a11 a12 a21 a22 a23 a31 a32 a33 a34 a41 a42 a43 a44 a45

✽ ✴ ✶✽

x1 x2 z

n n n

π1 π2 π3

a11 a12 a21 a22 a23 a31 a32 a33 a34 a41 a42 a43 a44 a45

x1 x2 z

n n n

π1 π2 π3

a12 a11 a22 a21 a23 a32 a31 a33 a34 a42 a41 a43 a44 a45

✽ ✴ ✶✽

x1 x2 z

n n n

π1 π2 π3

a11 a12 a21 a22 a23 a31 a32 a33 a34 a41 a42 a43 a44 a45

x2 x1 z

n n n

π1 π2 π3

a12 a11 a22 a21 a23 a32 a31 a33 a34 a42 a41 a43 a44 a45

✽ ✴ ✶✽

❊q✉✐✈❛❧❡♥❝❡ ❈❧❛ss❡s

❉❡✜♥✐t✐♦♥✿ ❊q✉✐✈❛❧❡♥❝❡ ❈❧❛ss

❈♦♠♣r❡ss✐♦♥ ❢✉♥❝t✐♦♥s F ❛♥❞ F′ ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ ❢♦r ❜♦t❤ ❝♦❧❧✐s✐♦♥ ❛♥❞ ♣r❡✐♠❛❣❡ s❡❝✉r✐t② t❤❡r❡ ❡①✐sts ❛ t✐❣❤t ❜✐✲❞✐r❡❝t✐♦♥❛❧ r❡❞✉❝t✐♦♥

• ■♥t✉✐t✐♦♥✿ F ❛♥❞ F′ ❡q✉✐✈❛❧❡♥t −

→ ❵❡q✉❛❧❧② s❡❝✉r❡✬ ❲❡ ✐❞❡♥t✐❢② ✹ ❡q✉✐✈❛❧❡♥❝❡ r❡❞✉❝t✐♦♥s

❊①❛♠♣❧❡ r❡❞✉❝t✐♦♥ ♦❢ ♣r❡✈✐♦✉s s❧✐❞❡ ✸ ❡①tr❛ r❡❞✉❝t✐♦♥s

❲❡ r❡str✐❝t t♦ ❡q✉✐✈❛❧❡♥❝❡ ✇✳r✳t✳ t❤❡s❡ r❡❞✉❝t✐♦♥s ♦♥❧②

✾ ✴ ✶✽

❊q✉✐✈❛❧❡♥❝❡ ❈❧❛ss❡s

❉❡✜♥✐t✐♦♥✿ ❊q✉✐✈❛❧❡♥❝❡ ❈❧❛ss

❈♦♠♣r❡ss✐♦♥ ❢✉♥❝t✐♦♥s F ❛♥❞ F′ ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ ❢♦r ❜♦t❤ ❝♦❧❧✐s✐♦♥ ❛♥❞ ♣r❡✐♠❛❣❡ s❡❝✉r✐t② t❤❡r❡ ❡①✐sts ❛ t✐❣❤t ❜✐✲❞✐r❡❝t✐♦♥❛❧ r❡❞✉❝t✐♦♥

• ■♥t✉✐t✐♦♥✿ F ❛♥❞ F′ ❡q✉✐✈❛❧❡♥t −

→ ❵❡q✉❛❧❧② s❡❝✉r❡✬

• ❲❡ ✐❞❡♥t✐❢② ✹ ❡q✉✐✈❛❧❡♥❝❡ r❡❞✉❝t✐♦♥s
• ❊①❛♠♣❧❡ r❡❞✉❝t✐♦♥ ♦❢ ♣r❡✈✐♦✉s s❧✐❞❡
• ✸ ❡①tr❛ r❡❞✉❝t✐♦♥s
• ❲❡ r❡str✐❝t t♦ ❡q✉✐✈❛❧❡♥❝❡ ✇✳r✳t✳ t❤❡s❡ r❡❞✉❝t✐♦♥s ♦♥❧②

✾ ✴ ✶✽

▼✉❧t✐✲P❡r♠✉t❛t✐♦♥ ❙❡tt✐♥❣ ✖ ▼❛✐♥ ❘❡s✉❧t

x1 x2 z

n n n

π1 π2 π3

F1

x1 x2 z

n n n

π1 π2 π3

F2

x1 x2 z

n n n

π1 π2 π3

F3

x1 x2 z

n n n

π1 π2 π3

F4

❡q✉✐✈❛❧❡♥t t♦✿ ❝♦❧❧✐s✐♦♥ ♣r❡✐♠❛❣❡ ✦❬❝❪ ✪ ✦❬❝❪ ✦❬❝❪ ✦ ✪ ♥♦♥❡ ♦❢ t❤❡s❡ ✪ ❄

✶✵ ✴ ✶✽

▼✉❧t✐✲P❡r♠✉t❛t✐♦♥ ❙❡tt✐♥❣ ✖ ▼❛✐♥ ❘❡s✉❧t

x1 x2 z

n n n

π1 π2 π3

F1

x1 x2 z

n n n

π1 π2 π3

F2

x1 x2 z

n n n

π1 π2 π3

F3

x1 x2 z

n n n

π1 π2 π3

F4

F ❡q✉✐✈❛❧❡♥t t♦✿ ❝♦❧❧✐s✐♦♥ ♣r❡✐♠❛❣❡ F1, F4 ✦❬❝❪ ✪ F2 ✦❬❝❪ ✦❬❝❪ F3 ✦ ✪ ♥♦♥❡ ♦❢ t❤❡s❡ ✪ ❄

✶✵ ✴ ✶✽

▼✉❧t✐✲P❡r♠✉t❛t✐♦♥ ❙❡tt✐♥❣ ✖ Pr♦♦❢ ■❞❡❛ ✭✶✮

x1 x2 z

n n n

π1 π2 π3

a11 a12 a21 a22 a23 a31 a32 a33 a34 a41 a42 a43 a44 a45

■♥ t♦t❛❧ s❝❤❡♠❡s✱ ❜✉t ♠❛♥② tr✐✈✐❛❧❧② ✐♥s❡❝✉r❡ ❋✉♥❝t✐♦♥ ✐s ✏✈❛❧✐❞✑ ✐❢ ❡❛❝❤ ❣r❡❡♥ s❡t ❝♦♥t❛✐♥s ❛ ❲❡ ❝♦♥s✐❞❡r ✈❛❧✐❞ ❝♦♠♣r❡ss✐♦♥ ❢✉♥❝t✐♦♥s ♦♥❧②

✶✶ ✴ ✶✽

▼✉❧t✐✲P❡r♠✉t❛t✐♦♥ ❙❡tt✐♥❣ ✖ Pr♦♦❢ ■❞❡❛ ✭✶✮

x1 x2 z

n n n

π1 π2 π3

a11 a12 a21 a22 a23 a31 a32 a33 a34 a41 a42 a43 a44 a45

• ■♥ t♦t❛❧ 214 s❝❤❡♠❡s✱ ❜✉t ♠❛♥② tr✐✈✐❛❧❧② ✐♥s❡❝✉r❡
• ❋✉♥❝t✐♦♥ ✐s ✏✈❛❧✐❞✑ ✐❢ ❡❛❝❤ ❣r❡❡♥ s❡t ❝♦♥t❛✐♥s ❛ 1
• ❲❡ ❝♦♥s✐❞❡r ✈❛❧✐❞ ❝♦♠♣r❡ss✐♦♥ ❢✉♥❝t✐♦♥s ♦♥❧②

✶✶ ✴ ✶✽

▼✉❧t✐✲P❡r♠✉t❛t✐♦♥ ❙❡tt✐♥❣ ✖ Pr♦♦❢ ■❞❡❛ ✭✷✮

x1 x2 z

n n n

π1 π2 π3

a11 a12 a21 a22 a23 a31 a32 a33 a34 a41 a42 a43 a44 a45

❆♥② ✈❛❧✐❞ ❡q✉✐✈❛❧❡♥t t♦ s♦♠❡ ✇✐t❤ ❛♥❞ ■t s✉✣❝❡s t♦ ❝♦♥s✐❞❡r t❤❡s❡ ❢✉♥❝t✐♦♥s ♦♥❧②

✶✷ ✴ ✶✽

▼✉❧t✐✲P❡r♠✉t❛t✐♦♥ ❙❡tt✐♥❣ ✖ Pr♦♦❢ ■❞❡❛ ✭✷✮

x1 x2 z

n n n

π1 π2 π3

a11 a12 a21 a22 a23 a31 a32 a33 a34 a41 a42 a43 a44 a45

• ❆♥② ✈❛❧✐❞ F ❡q✉✐✈❛❧❡♥t t♦ s♦♠❡ F′ ✇✐t❤

(a11, a12) = (1, 0) ❛♥❞ (a21, a22, a23) = (0, 1, 0) ■t s✉✣❝❡s t♦ ❝♦♥s✐❞❡r t❤❡s❡ ❢✉♥❝t✐♦♥s ♦♥❧②

✶✷ ✴ ✶✽

▼✉❧t✐✲P❡r♠✉t❛t✐♦♥ ❙❡tt✐♥❣ ✖ Pr♦♦❢ ■❞❡❛ ✭✷✮

x1 x2 z

n n n

π1 π2 π3

a31 a32 a33 a34 a41 a42 a43 a44

• ❆♥② ✈❛❧✐❞ F ❡q✉✐✈❛❧❡♥t t♦ s♦♠❡ F′ ✇✐t❤

(a11, a12) = (1, 0) ❛♥❞ (a21, a22, a23) = (0, 1, 0)

• ■t s✉✣❝❡s t♦ ❝♦♥s✐❞❡r t❤❡s❡ ❢✉♥❝t✐♦♥s ♦♥❧②

✶✷ ✴ ✶✽

▼✉❧t✐✲P❡r♠✉t❛t✐♦♥ ❙❡tt✐♥❣ ✖ Pr♦♦❢ ■❞❡❛ ✭✸✮

x1 x2 z

n n n

π1 π2 π3

a31 a32 a33 a34 a41 a42 a43 a44

❋♦✉r ❣❡♥❡r✐❝ ❛tt❛❝❦s

❝♦❧❧✐s✐♦♥ ✐♥ q✉❡r✐❡s ❝♦❧❧✐s✐♦♥ ✐♥ q✉❡r✐❡s ❝♦❧❧✐s✐♦♥ ✐♥ q✉❡r✐❡s ❝♦❧❧✐s✐♦♥ ✐♥ q✉❡r✐❡s

✐s ❝♦❧❧✐s✐♦♥ s❡❝✉r❡ ♦♥❧② ✐❢ ❡q✉✐✈❛❧❡♥t t♦

✶✸ ✴ ✶✽

▼✉❧t✐✲P❡r♠✉t❛t✐♦♥ ❙❡tt✐♥❣ ✖ Pr♦♦❢ ■❞❡❛ ✭✸✮

x1 x2 z

n n n

π1 π2 π3

a31 a32 a33 a34 a41 a42 a43 a44

• ❋♦✉r ❣❡♥❡r✐❝ ❛tt❛❝❦s

(a31 + a33)(a32 + a34) = 0 = ⇒ ❝♦❧❧✐s✐♦♥ ✐♥ 2n/4 q✉❡r✐❡s ∨4

j=1 a3j = a4j = 0 =

⇒ ❝♦❧❧✐s✐♦♥ ✐♥ 2n/3 q✉❡r✐❡s ∧2

j=1 a3ja4,j+2 = a3,j+2a4j =

⇒ ❝♦❧❧✐s✐♦♥ ✐♥ 2n/3 q✉❡r✐❡s a41 + a42 + a43 + a44 = 1 = ⇒ ❝♦❧❧✐s✐♦♥ ✐♥ 22n/5 q✉❡r✐❡s

✐s ❝♦❧❧✐s✐♦♥ s❡❝✉r❡ ♦♥❧② ✐❢ ❡q✉✐✈❛❧❡♥t t♦

✶✸ ✴ ✶✽

▼✉❧t✐✲P❡r♠✉t❛t✐♦♥ ❙❡tt✐♥❣ ✖ Pr♦♦❢ ■❞❡❛ ✭✸✮

x1 x2 z

n n n

π1 π2 π3

a31 a32 a33 a34 a41 a42 a43 a44

• ❋♦✉r ❣❡♥❡r✐❝ ❛tt❛❝❦s

(a31 + a33)(a32 + a34) = 0 = ⇒ ❝♦❧❧✐s✐♦♥ ✐♥ 2n/4 q✉❡r✐❡s ∨4

j=1 a3j = a4j = 0 =

⇒ ❝♦❧❧✐s✐♦♥ ✐♥ 2n/3 q✉❡r✐❡s ∧2

j=1 a3ja4,j+2 = a3,j+2a4j =

⇒ ❝♦❧❧✐s✐♦♥ ✐♥ 2n/3 q✉❡r✐❡s a41 + a42 + a43 + a44 = 1 = ⇒ ❝♦❧❧✐s✐♦♥ ✐♥ 22n/5 q✉❡r✐❡s

• F ✐s ❝♦❧❧✐s✐♦♥ s❡❝✉r❡ ♦♥❧② ✐❢ ❡q✉✐✈❛❧❡♥t t♦ F1, F2, F3, F4

✶✸ ✴ ✶✽

▼✉❧t✐✲P❡r♠✉t❛t✐♦♥ ❙❡tt✐♥❣ ✖ Pr♦♦❢ ■❞❡❛ ✭✹✮

ts

x1 x2 z

n n n

π1 π2 π3

(only for (only for (only for F3, F4) F1, F2, F3) F2, F3, F4)

• F ✐s ❝♦❧❧✐s✐♦♥ s❡❝✉r❡ ♦♥❧② ✐❢ ✐t ✐s ❡q✉✐✈❛❧❡♥t t♦ F1, F2, F3, F4

❘❡♠❛✐♥s t♦ ♣r♦✈❡✿ ✐❢✲r❡❧❛t✐♦♥ ❛♥❞ ♣r❡✐♠❛❣❡ r❡s✐st❛♥❝❡ ❍❛r❞❡st ❛♥❞ ♠♦st t❡❝❤♥✐❝❛❧ ♣❛rt

❝♦❧❧✐s✐♦♥ r❡s✐st❛♥t ✉♣ t♦ q✉❡r✐❡s t✐❣❤t ✭❛s②♠♣t✳✮ ♣r❡✐♠❛❣❡ r❡s✐st❛♥t ✉♣ t♦ q✉❡r✐❡s t✐❣❤t ✭❛s②♠♣t✳✮ ♣r❡✐♠❛❣❡ r❡s✐st❛♥t ✉♣ t♦ q✉❡r✐❡s t✐❣❤t

✶✹ ✴ ✶✽

▼✉❧t✐✲P❡r♠✉t❛t✐♦♥ ❙❡tt✐♥❣ ✖ Pr♦♦❢ ■❞❡❛ ✭✹✮

ts

x1 x2 z

n n n

π1 π2 π3

(only for (only for (only for F3, F4) F1, F2, F3) F2, F3, F4)

• F ✐s ❝♦❧❧✐s✐♦♥ s❡❝✉r❡ ♦♥❧② ✐❢ ✐t ✐s ❡q✉✐✈❛❧❡♥t t♦ F1, F2, F3, F4
• ❘❡♠❛✐♥s t♦ ♣r♦✈❡✿ ✐❢✲r❡❧❛t✐♦♥ ❛♥❞ ♣r❡✐♠❛❣❡ r❡s✐st❛♥❝❡

❍❛r❞❡st ❛♥❞ ♠♦st t❡❝❤♥✐❝❛❧ ♣❛rt

❝♦❧❧✐s✐♦♥ r❡s✐st❛♥t ✉♣ t♦ q✉❡r✐❡s t✐❣❤t ✭❛s②♠♣t✳✮ ♣r❡✐♠❛❣❡ r❡s✐st❛♥t ✉♣ t♦ q✉❡r✐❡s t✐❣❤t ✭❛s②♠♣t✳✮ ♣r❡✐♠❛❣❡ r❡s✐st❛♥t ✉♣ t♦ q✉❡r✐❡s t✐❣❤t

✶✹ ✴ ✶✽

▼✉❧t✐✲P❡r♠✉t❛t✐♦♥ ❙❡tt✐♥❣ ✖ Pr♦♦❢ ■❞❡❛ ✭✹✮

ts

x1 x2 z

n n n

π1 π2 π3

(only for (only for (only for F3, F4) F1, F2, F3) F2, F3, F4)

• F ✐s ❝♦❧❧✐s✐♦♥ s❡❝✉r❡ ♦♥❧② ✐❢ ✐t ✐s ❡q✉✐✈❛❧❡♥t t♦ F1, F2, F3, F4
• ❘❡♠❛✐♥s t♦ ♣r♦✈❡✿ ✐❢✲r❡❧❛t✐♦♥ ❛♥❞ ♣r❡✐♠❛❣❡ r❡s✐st❛♥❝❡
• ❍❛r❞❡st ❛♥❞ ♠♦st t❡❝❤♥✐❝❛❧ ♣❛rt
• F1, . . . , F4 ❝♦❧❧✐s✐♦♥ r❡s✐st❛♥t ✉♣ t♦ 2n/2 q✉❡r✐❡s t✐❣❤t ✭❛s②♠♣t✳✮
• F2 ♣r❡✐♠❛❣❡ r❡s✐st❛♥t ✉♣ t♦ 22n/3 q✉❡r✐❡s t✐❣❤t ✭❛s②♠♣t✳✮
• F1, F3, F4 ♣r❡✐♠❛❣❡ r❡s✐st❛♥t ✉♣ t♦ 2n/2 q✉❡r✐❡s t✐❣❤t

✶✹ ✴ ✶✽

▼✉❧t✐✲P❡r♠✉t❛t✐♦♥ ❙❡tt✐♥❣ ✖ ❈♦♥❥❡❝t✉r❡

Z : s❡t ♦❢ q r❛♥❞♦♠ ❡❧❡♠❡♥ts ❢r♦♠ {0, 1}n ✭❞✉♣❧✐❝❛t❡s ♠❛② ♦❝❝✉r✮ X, Y : ❛♥② t✇♦ s❡ts ♦❢ q ❡❧❡♠❡♥ts ❢r♦♠ {0, 1}n ✭♥♦ ❞✉♣❧✐❝❛t❡s✮

❈♦♥❥❡❝t✉r❡

❲✐t❤ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t②✱ t❤❡r❡ ❡①✐st O(q log q) t✉♣❧❡s (x, y, z) ∈ X × Y × Z s✉❝❤ t❤❛t x ⊕ y = z

• ❈♦♥❥❡❝t✉r❡ r❡❧❛t❡s t♦ ❛r❡❛ ♦❢ ❡①tr❡♠❛❧ ❣r❛♣❤ t❤❡♦r②
• ❙✐♠✐❧❛r t♦ ✭❜✉t ♠♦r❡ ❝♦♠♣❧❡① t❤❛♥✮ ❛ ❧♦♥❣st❛♥❞✐♥❣ ♣r♦❜❧❡♠ ♦❢

❩❛r❛♥❦✐❡✇✐❝③ ❢r♦♠ ✶✾✺✶

• ❉❡t❛✐❧❡❞ ❤❡✉r✐st✐❝❛❧ ❛r❣✉♠❡♥t ✐♥ ♣❛♣❡r

✶✺ ✴ ✶✽

❙✐♥❣❧❡✲P❡r♠✉t❛t✐♦♥ ❙❡tt✐♥❣ ✖ ▼❛✐♥ ❘❡s✉❧t

x1 x2 z

n n n

π π π

a11 a12 a21 a22 a23 a31 a32 a33 a34 a41 a42 a43 a44 a45 aij ∈ {0, 1}

❚❤❡♦r❡♠

❋♦r ❛♥② ❝♦♠♣r❡ss✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ t❤✐s ❢♦r♠✱ ❝♦❧❧✐s✐♦♥s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ 22n/5 q✉❡r✐❡s ✭♣r♦♦❢ ✐s s✐♠✐❧❛r✮

✶✻ ✴ ✶✽

❙✐♥❣❧❡✲P❡r♠✉t❛t✐♦♥ ❙❡tt✐♥❣ ✖ ▼❛✐♥ ❘❡s✉❧t

x1 x2 z

n n n

π π π

a11 a12 a21 a22 a23 a31 a32 a33 a34 a41 a42 a43 a44 a45 b1 b2 b3 b4 aij ∈ {0, 1} bi ∈ {0, 1}n

❚❤❡♦r❡♠

❋♦r ❛♥② ❝♦♠♣r❡ss✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ t❤✐s ❢♦r♠✱ ❝♦❧❧✐s✐♦♥s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ 22n/5 q✉❡r✐❡s ✭♣r♦♦❢ ✐s s✐♠✐❧❛r✮

✶✼ ✴ ✶✽

❈♦♥❝❧✉s✐♦♥s

❈♦♠♣❧❡t❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ 2n✲t♦✲n✲❜✐t ❝♦♠♣r❡ss✐♦♥ ❢✉♥❝t✐♦♥s s♦❧❡❧② ❜❛s❡❞ ♦♥ t❤r❡❡ ♣❡r♠✉t❛t✐♦♥s ❛♥❞ ✲♦♣❡r❛t♦rs

• ▼✉❧t✐✲♣❡r♠✉t❛t✐♦♥ s❡tt✐♥❣✿ ❛♥❛❧②s✐s ♦❢ 214 ❢✉♥❝t✐♦♥s
• 216 ❢✉♥❝t✐♦♥s ♦♣t✐♠❛❧❧② ❝♦❧❧✐s✐♦♥ s❡❝✉r❡
• 48 ♦❢ ✇❤✐❝❤ ♦♣t✐♠❛❧❧② ♣r❡✐♠❛❣❡ s❡❝✉r❡
• ❙✐♥❣❧❡✲♣❡r♠✉t❛t✐♦♥ s❡tt✐♥❣✿ ♥♦♥✲❡①✐st❡♥❝❡ ♦❢ ❝♦❧❧✐s✐♦♥ s❡❝✉r❡ F
• ❆tt❛❝❦ ♦♥ 214 ✭♦r ✐♥ ❢❛❝t 24n214✮ ❢✉♥❝t✐♦♥s

❘❡s❡❛r❝❤ ❞✐r❡❝t✐♦♥s✿

• ❡♥❡r❛❧✐③❡ t♦ ❧❛r❣❡r

✬s✱ ❛♥❞ ✇✐t❤ ❞✐✛❡r❡♥t ♣r✐♠✐t✐✈❡s

• ❡♥❡r❛❧✐③❡ ✐♠♣♦ss✐❜✐❧✐t② r❡s✉❧t ✐♥ s✐♥❣❧❡✲♣❡r♠✉t❛t✐♦♥ s❡tt✐♥❣

❈♦♥❥❡❝t✉r❡

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

✶✽ ✴ ✶✽

❈♦♥❝❧✉s✐♦♥s

❈♦♠♣❧❡t❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ 2n✲t♦✲n✲❜✐t ❝♦♠♣r❡ss✐♦♥ ❢✉♥❝t✐♦♥s s♦❧❡❧② ❜❛s❡❞ ♦♥ t❤r❡❡ ♣❡r♠✉t❛t✐♦♥s ❛♥❞ ✲♦♣❡r❛t♦rs

• ▼✉❧t✐✲♣❡r♠✉t❛t✐♦♥ s❡tt✐♥❣✿ ❛♥❛❧②s✐s ♦❢ 214 ❢✉♥❝t✐♦♥s
• 216 ❢✉♥❝t✐♦♥s ♦♣t✐♠❛❧❧② ❝♦❧❧✐s✐♦♥ s❡❝✉r❡
• 48 ♦❢ ✇❤✐❝❤ ♦♣t✐♠❛❧❧② ♣r❡✐♠❛❣❡ s❡❝✉r❡
• ❙✐♥❣❧❡✲♣❡r♠✉t❛t✐♦♥ s❡tt✐♥❣✿ ♥♦♥✲❡①✐st❡♥❝❡ ♦❢ ❝♦❧❧✐s✐♦♥ s❡❝✉r❡ F
• ❆tt❛❝❦ ♦♥ 214 ✭♦r ✐♥ ❢❛❝t 24n214✮ ❢✉♥❝t✐♦♥s
• ❘❡s❡❛r❝❤ ❞✐r❡❝t✐♦♥s✿
• ●❡♥❡r❛❧✐③❡ t♦ ❧❛r❣❡r F✬s✱ ❛♥❞ ✇✐t❤ ❞✐✛❡r❡♥t ♣r✐♠✐t✐✈❡s
• ●❡♥❡r❛❧✐③❡ ✐♠♣♦ss✐❜✐❧✐t② r❡s✉❧t ✐♥ s✐♥❣❧❡✲♣❡r♠✉t❛t✐♦♥ s❡tt✐♥❣
• ❈♦♥❥❡❝t✉r❡

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

✶✽ ✴ ✶✽

❙✉♣♣♦rt✐♥❣ s❧✐❞❡s

✶✾ ✴ ✶✽

x1 x2 z

n n n

π1 π2 π3

a11 a12 a21 a22 a31 a32 a33 a34 a41 a42 a43 a44 a45

✷✵ ✴ ✶✽

x1 x2 z

n n n

π1 π2 π3

a11 a12 a21 a22 a31 a32 a33 a34 a41 a42 a43 a44 a45

x1 x2 z

n n n

π1 π2 π3

a21 a22 a11 a12 a31 a32 a34 a33 a41 a42 a44 a43 a45

✷✵ ✴ ✶✽

x1 x2 z

n n n

π1 π2 π3

a11 a12 a21 a22 a31 a32 a33 a34 a41 a42 a43 a44 a45

x1 x2 z

n n n

π2 π1 π3

a21 a22 a11 a12 a31 a32 a34 a33 a41 a42 a44 a43 a45

✷✵ ✴ ✶✽

❙✉♠♠❛r② ♦❢ ❖✉r ❘❡s✉❧ts

ts

x1 x2 z

n n n

π1 π2 π3

(only for (only for (only for F3, F4) F1, F2, F3) F2, F3, F4) ❝♦❧❧✐s✐♦♥ ♣r❡✐♠❛❣❡ F ❡q✉✐✈❛❧❡♥t t♦✿ s❡❝✉r✐t② ❛tt❛❝❦ s❡❝✉r✐t② ❛tt❛❝❦ F1, F4 2n/2 ❬❝❪ 2n/2 2n/2❬❝❪ 2n/2 F2 2n/2 ❬❝❪ 2n/2 22n/3 ❬❝❪ 22n/3 F3 2n/2❬❝❪ 2n/2 2n/2❬❝❪ 2n/2 ♥♦♥❡ ♦❢ t❤❡s❡ ❄❬❝❪ 22n/5 ❄❬❝❪ ❄ ❛♥② F ✐♥ ❙P✲s❡tt✐♥❣ ❄❬❝❪ 22n/5 ❄❬❝❪ ❄

✷✶ ✴ ✶✽