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❈r②♣t♦❣r❛♣❤✐❝ ❙❜♦①❡s

❆♥♥❡ ❈❛♥t❡❛✉t ❆♥♥❡✳❈❛♥t❡❛✉t❅✐♥r✐❛✳❢r ❤tt♣✿✴✴✇✇✇✲r♦❝q✳✐♥r✐❛✳❢r✴s❡❝r❡t✴❆♥♥❡✳❈❛♥t❡❛✉t✴ ❙✉♠♠❡r ❙❝❤♦♦❧✱ ➆✐❜❡♥✐❦✱ ❏✉♥❡ ✷✵✶✹

❱❡❝t♦r✐❛❧ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ❆ ✈❡❝t♦r✐❛❧ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ✇✐t❤ n ✐♥♣✉ts ❛♥❞ m ♦✉t♣✉ts ✐s ❛ ❢✉♥❝t✐♦♥ ❢r♦♠ Fn

2 ✐♥t♦ Fm 2 ✿

S : Fn

2

− → Fm

2

(x1, . . . , xn) − → (y1, . . . , ym)

❊①❛♠♣❧❡✳ x ✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ❛ ❜ ❝ ❞ ❡ ❢ S(x) ❢ ❡ ❜ ❝ ✻ ❞ ✼ ✽ ✵ ✸ ✾ ❛ ✹ ✷ ✶ ✺ S1(x) ✶ ✵ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✵ ✶ ✶ S2(x) ✶ ✶ ✶ ✵ ✶ ✵ ✶ ✵ ✵ ✶ ✵ ✶ ✵ ✶ ✵ ✵ S3(x) ✶ ✶ ✵ ✶ ✶ ✶ ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✶ S4(x) ✶ ✶ ✶ ✶ ✵ ✶ ✵ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✵ ✵

✶

❘♦✉♥❞ ❢✉♥❝t✐♦♥ ✐♥ ❛ s✉❜st✐t✉t✐♦♥✲♣❡r♠✉t❛t✐♦♥ ♥❡t✇♦r❦

S S S S S S S S S S

❧✐♥❡❛r ❞✐✛✉s✐♦♥

❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄

x(i)

❄ ❄ ✖✕ ✗✔

+

ki

❄ ✲

x(i+1)

✷

❖✉t❧✐♥❡

• ❆❧❣❡❜r❛✐❝ ❞❡❣r❡❡
• ❉✐✛❡r❡♥t✐❛❧ ✉♥✐❢♦r♠✐t②
• ◆♦♥❧✐♥❡❛r✐t②
• ❋✐♥❞✐♥❣ ❣♦♦❞ ❙❜♦①❡s

✸

❉❡❣r❡❡ ♦❢ ❛♥ ❙❜♦①

f(x1, . . . , xn) =

• u∈Fn

2

au

n

• i=1

xui

i ,

au ∈ F2.

❉❡✜♥✐t✐♦♥✳ ❚❤❡ ❞❡❣r❡❡ ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ✐s t❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ❧❛r❣❡st ♠♦♥♦♠✐❛❧ ✐♥ ✐ts ❛❧❣❡❜r❛✐❝ ♥♦r♠❛❧ ❢♦r♠✳ ❚❤❡ ❞❡❣r❡❡ ♦❢ ❛ ✈❡❝t♦r✐❛❧ ❢✉♥❝t✐♦♥ S ✇✐t❤ n ✐♥♣✉ts ❛♥❞ m ♦✉t♣✉ts ✐s t❤❡ ♠❛①✐♠❛❧ ❞❡❣r❡❡ ♦❢ ✐ts ❝♦♦r❞✐♥❛t❡s✳ Pr♦♣♦s✐t✐♦♥✳ ■❢ S ✐s ❛ ♣❡r♠✉t❛t✐♦♥ ♦❢ Fn

2 ✱ t❤❡♥ deg S ≤ n − 1✳

✹

❊①❛♠♣❧❡ x ✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ❛ ❜ ❝ ❞ ❡ ❢ S1(x) ✶ ✵ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✵ ✶ ✶ S2(x) ✶ ✶ ✶ ✵ ✶ ✵ ✶ ✵ ✵ ✶ ✵ ✶ ✵ ✶ ✵ ✵ S3(x) ✶ ✶ ✵ ✶ ✶ ✶ ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✶ S4(x) ✶ ✶ ✶ ✶ ✵ ✶ ✵ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✵ ✵ S1 = 1 + x1 + x3 + x2x3 + x4 + x2x4 + x3x4 + x1x3x4 + x2x3x4 S2 = 1 + x1x2 + x1x3 + x1x2x3 + x4 + x1x4 + x1x2x4 + x1x3x4 S3 = 1 + x2 + x1x2 + x2x3 + x4 + x2x4 + x1x2x4 + x3x4 + x1x3x4 S4 = 1 + x3 + x1x3 + x4 + x2x4 + x3x4 + x1x3x4 + x2x3x4

✺

❘❡s✐st❛♥❝❡ t♦ ❞✐✛❡r❡♥t✐❛❧ ❛tt❛❝❦s

✻

❉✐✛❡r❡♥❝❡ t❛❜❧❡ ♦❢ ❛♥ ❙❜♦①

a \ b

✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ❛ ❜ ❝ ❞ ❡ ❢ ✶ ✷ ✵ ✹ ✷ ✵ ✷ ✷ ✵ ✵ ✵ ✷ ✵ ✵ ✵ ✷ ✷ ✷ ✷ ✵ ✷ ✹ ✵ ✷ ✵ ✹ ✵ ✵ ✵ ✵ ✵ ✵ ✸ ✷ ✵ ✹ ✵ ✷ ✵ ✵ ✵ ✵ ✻ ✵ ✵ ✵ ✷ ✵ ✹ ✷ ✵ ✷ ✹ ✵ ✵ ✵ ✷ ✷ ✵ ✵ ✷ ✵ ✵ ✷ ✺ ✵ ✹ ✷ ✵ ✵ ✵ ✷ ✷ ✵ ✵ ✹ ✷ ✵ ✵ ✵ ✻ ✹ ✵ ✵ ✵ ✵ ✹ ✵ ✹ ✵ ✵ ✵ ✵ ✹ ✵ ✵ ✼ ✵ ✷ ✵ ✵ ✷ ✷ ✷ ✵ ✷ ✷ ✷ ✵ ✵ ✷ ✵ ✽ ✵ ✹ ✵ ✵ ✵ ✹ ✵ ✵ ✵ ✵ ✵ ✵ ✹ ✵ ✹ ✾ ✷ ✷ ✵ ✷ ✷ ✵ ✵ ✵ ✹ ✵ ✵ ✷ ✵ ✷ ✵ ❛ ✵ ✵ ✷ ✷ ✵ ✷ ✷ ✷ ✵ ✷ ✷ ✵ ✵ ✵ ✷ ❜ ✵ ✵ ✷ ✵ ✹ ✵ ✷ ✷ ✵ ✵ ✵ ✻ ✵ ✵ ✵ ❝ ✵ ✷ ✵ ✵ ✵ ✷ ✵ ✵ ✷ ✷ ✷ ✷ ✵ ✹ ✵ ❞ ✷ ✵ ✵ ✵ ✷ ✵ ✵ ✵ ✵ ✷ ✵ ✵ ✽ ✷ ✵ ❡ ✵ ✵ ✵ ✵ ✵ ✵ ✹ ✵ ✵ ✵ ✹ ✵ ✵ ✹ ✹ ❢ ✵ ✵ ✵ ✹ ✵ ✵ ✵ ✹ ✷ ✷ ✵ ✷ ✵ ✵ ✷

δS(a, b) = #{X ∈ Fn

2,

S(X ⊕ a) ⊕ S(X) = b}

✼

❘❡s✐st❛♥❝❡ t♦ ❞✐✛❡r❡♥t✐❛❧ ❛tt❛❝❦s ❬◆②❜❡r❣ ❑♥✉❞s❡♥ ✾✷❪✱❬◆②❜❡r❣ ✾✸❪ ❈r✐t❡r✐♦♥ ♦♥ t❤❡ ❙❜♦①✳ ❆❧❧ ❡♥tr✐❡s ✐♥ t❤❡ ❞✐✛❡r❡♥❝❡ t❛❜❧❡ ♦❢ S s❤♦✉❧❞ ❜❡ s♠❛❧❧✳

δ(S) = max

a,b=0 #{X ∈ Fn 2,

S(X ⊕ a) ⊕ S(X) = b}

♠✉st ❜❡ ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡✳

δ(S) ✐s ❝❛❧❧❡❞ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ✉♥✐❢♦r♠✐t② ♦❢ S ✭❛❧✇❛②s ❡✈❡♥✮✳

❚❤❡♦r❡♠✳ ❋♦r ❛♥② ❙❜♦① S ✇✐t❤ n ✐♥♣✉ts ❛♥❞ n ♦✉t♣✉ts✱

δ(S) ≥ 2 .

❚❤❡ ❢✉♥❝t✐♦♥s ❛❝❤✐❡✈✐♥❣ t❤✐s ❜♦✉♥❞ ❛r❡ ❝❛❧❧❡❞ ❛❧♠♦st ♣❡r❢❡❝t ♥♦♥❧✐♥❡❛r ❢✉♥❝t✐♦♥s ✭❆P◆✮✳

✽

❋♦r ❙P◆ ✉s✐♥❣ S ❊①♣❡❝t❡❞ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ 2✲r♦✉♥❞ ❝❤❛r❛❝t❡r✐st✐❝

≤ δ(S) 2n d

✇❤❡r❡ d ✐s t❤❡ ❜r❛♥❝❤ ♥✉♠❜❡r ♦❢ t❤❡ ❧✐♥❡❛r ❧❛②❡r✳ ❊①♣❡❝t❡❞ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ 2✲r♦✉♥❞ ❞✐✛❡r❡♥t✐❛❧ ❬❉❛❡♠❡♥ ❘✐❥♠❡♥ ✵✷❪

MEDP2 ≤ δ(S) 2n d−1

❊✳❣✳✱ ❢♦r t❤❡ 4✲r♦✉♥❞ ❆❊❙✱

MEDP4 ≤

• 2−616

❘❡✜♥❡♠❡♥ts ✐♥✈♦❧✈✐♥❣ t❤❡ ✇❤♦❧❡ ❞✐✛❡r❡♥❝❡ t❛❜❧❡ ❬P❛r❦ ❡t ❛❧✳ ✵✸❪✳

✾

❘❡s✐st❛♥❝❡ t♦ ❧✐♥❡❛r ❛tt❛❝❦s

✶✵

▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ ❛♥ ❙❜♦①

a \ b

✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ❛ ❜ ❝ ❞ ❡ ❢ ✶ ✲✹ ✳ ✹ ✳ ✲✹ ✽ ✲✹ ✹ ✽ ✹ ✳ ✲✹ ✳ ✹ ✳ ✷ ✹ ✲✹ ✳ ✲✹ ✳ ✳ ✹ ✹ ✽ ✳ ✹ ✽ ✲✹ ✲✹ ✳ ✸ ✽ ✹ ✹ ✲✹ ✹ ✳ ✳ ✳ ✳ ✹ ✲✹ ✲✹ ✲✹ ✳ ✽ ✹ ✳ ✲✹ ✹ ✹ ✲✹ ✳ ✳ ✲✽ ✳ ✹ ✹ ✹ ✹ ✳ ✽ ✺ ✲✹ ✹ ✳ ✹ ✽ ✳ ✹ ✲✹ ✽ ✳ ✲✹ ✳ ✹ ✲✹ ✳ ✻ ✲✹ ✳ ✹ ✳ ✹ ✽ ✹ ✹ ✲✽ ✹ ✳ ✹ ✳ ✲✹ ✳ ✼ ✳ ✳ ✳ ✽ ✳ ✲✽ ✳ ✳ ✳ ✳ ✽ ✳ ✽ ✳ ✳ ✽ ✳ ✲✹ ✹ ✲✽ ✳ ✹ ✹ ✲✽ ✳ ✲✹ ✲✹ ✳ ✳ ✹ ✲✹ ✾ ✲✹ ✲✶✷ ✳ ✳ ✹ ✲✹ ✳ ✹ ✳ ✳ ✲✹ ✲✹ ✳ ✳ ✹ ❛ ✲✹ ✳ ✲✶✷ ✲✹ ✳ ✹ ✳ ✲✹ ✳ ✹ ✳ ✳ ✲✹ ✳ ✹ ❜ ✳ ✳ ✳ ✹ ✲✹ ✹ ✲✹ ✳ ✳ ✲✽ ✲✽ ✹ ✲✹ ✲✹ ✹ ❝ ✳ ✳ ✳ ✲✹ ✲✹ ✲✹ ✲✹ ✳ ✳ ✽ ✲✽ ✹ ✹ ✲✹ ✲✹ ❞ ✲✹ ✳ ✹ ✹ ✳ ✲✹ ✳ ✲✹ ✳ ✹ ✳ ✳ ✲✶✷ ✳ ✲✹ ❡ ✹ ✲✹ ✳ ✳ ✹ ✹ ✲✽ ✲✹ ✳ ✳ ✹ ✲✹ ✳ ✲✽ ✲✹ ❢ ✲✽ ✹ ✹ ✲✽ ✳ ✲✹ ✲✹ ✳ ✳ ✲✹ ✹ ✳ ✳ ✲✹ ✹

Pr[a · x + b · S(x) = 0] = 1 2

• 1 + W[a, b]

2n

• ❋♦r ✐♥st❛♥❝❡✱ ❢♦r a = 0x9 ❛♥❞ b = 0x2✱ ✇❡ ❤❛✈❡ p = 1

2(1 − 12 16) = 1 8✳

✶✶

❲❛❧s❤ tr❛♥s❢♦r♠ ♦❢ ❛♥ ❙❜♦① ❲❛❧s❤ tr❛♥s❢♦r♠ ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ f ♦❢ n ✈❛r✐❛❜❧❡s

Fn

2

− → Z a − → Wf(a) =

x∈Fn

2 (−1)f(x)+a·x

❲❛❧s❤ tr❛♥s❢♦r♠ ♦❢ ❛♥ ❙❜♦① S✿

Fn

2 × Fm 2

− → Z (a, b) − → WS(a, b) =

x∈Fn

2 (−1)b·S(x)+a·x = Wb·S(a)

✶✷

▲✐♥❡❛r✐t② ♦❢ ❛♥ ❙❜♦① ❈r✐t❡r✐♦♥ ♦♥ t❤❡ ❙❜♦①✳ ❆❧❧ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ S s❤♦✉❧❞ ❤❛✈❡ ❛ s♠❛❧❧ ❜✐❛s✱ ✐✳❡✳✱

L(S) = max

a∈Fn

2 , b∈Fn 2 ,b=0 |WS(a, b)|

♠✉st ❜❡ ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡✳ P❛rs❡✈❛❧✬s ❡q✉❛❧✐t②✿ ❢♦r ❛♥② ♦✉t♣✉t ♠❛s❦ b✱

• a∈Fn

2

W2

S(a, b) = 22n .

✶✸

❋♦r ❙P◆ ✉s✐♥❣ S ❊①♣❡❝t❡❞ sq✉❛r❡ ❝♦rr❡❧❛t✐♦♥ ♦❢ ❛ 2✲r♦✉♥❞ ❧✐♥❡❛r tr❛✐❧

≤ L(S) 2n 2d′

✇❤❡r❡ d′ ✐s t❤❡ ❧✐♥❡❛r ❜r❛♥❝❤ ♥✉♠❜❡r ♦❢ t❤❡ ❧✐♥❡❛r ❧❛②❡r✳ ❊①♣❡❝t❡❞ sq✉❛r❡ ❝♦rr❡❧❛t✐♦♥ ♦❢ ❛ 2✲r♦✉♥❞ ❧✐♥❡❛r ♠❛s❦ ❬❉❛❡♠❡♥ ❘✐❥♠❡♥ ✵✷❪

MELP2 ≤ L(S) 2n 2(d′−1)

❘❡✜♥❡♠❡♥ts ✐♥✈♦❧✈✐♥❣ t❤❡ ✇❤♦❧❡ sq✉❛r❡ ❝♦rr❡❧❛t✐♦♥ t❛❜❧❡ ❬P❛r❦ ❡t ❛❧✳ ✵✸❪✳

✶✹

▲✐♥❦ ❜❡t✇❡❡♥ t❤❡ ❞✐✛❡r❡♥❝❡ ❛♥❞ sq✉❛r❡ ❝♦rr❡❧❛t✐♦♥ t❛❜❧❡s ❚❤❡♦r❡♠✳ ❬❈❤❛❜❛✉❞ ❱❛✉❞❡♥❛② ✾✹❪❬❇❧♦♥❞❡❛✉ ◆②❜❡r❣ ✶✸❪ ❚❤❡r❡ ✐s ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞✐✛❡r❡♥❝❡ t❛❜❧❡

δ(a, b), a ∈ Fn

2, b ∈ Fn 2

❛♥❞ t❤❡ sq✉❛r❡ ❝♦rr❡❧❛t✐♦♥ t❛❜❧❡

W2(a, b), a, b ∈ Fn

2, b ∈ Fn 2

W2(u, v) =

• a,b∈Fn

2

(−1)a·u+b·vδ(a, b) δ(a, b) = 2−2n

• u,v∈Fn

2

(−1)a·u+b·vW2(u, v)

❚❤❡r❡ ✐s ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❙❜♦① ❛♥❞ t❤❡ ❝♦rr❡❧❛t✐♦♥ t❛❜❧❡✳ ❇✉t s❡✈❡r❛❧ ❙❜♦①❡s ♠❛② ❤❛✈❡ t❤❡ s❛♠❡ sq✉❛r❡ ❝♦rr❡❧❛t✐♦♥ t❛❜❧❡✳

✶✺

❋✐♥❞✐♥❣ ❣♦♦❞ ❙❜♦①❡s ✇✳r✳t✳ t❤❡ ♣r❡✈✐♦✉s ❝r✐t❡r✐❛

✶✻

❊q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ ❙❜♦①❡s ❆✣♥❡ ❡q✉✐✈❛❧❡♥❝❡

S2 = A2 ◦ S1 ◦ A1

✇❤❡r❡ A1 ❛♥❞ A2 ❛r❡ t✇♦ ❛✣♥❡ ♣❡r♠✉t❛t✐♦♥s ♦❢ Fn

2 ✳

❈❈❩ ❡q✉✐✈❛❧❡♥❝❡ ❬❈❛r❧❡t ❈❤❛r♣✐♥ ❩✐♥♦✈✐❡✈ ✾✽❪

(x′, S2(x′)) = A (x, S1(x))

✇❤❡r❡ A ✐s ❛♥ ❛✣♥❡ ♣❡r♠✉t❛t✐♦♥ ♦❢ F2n

2 ✳

✶✼

P❡r♠✉t❛t✐♦♥s ♦❢ F4

2

δ(S) ≥ 4 ❛♥❞ L(S) ≥ 8 16 ❝❧❛ss❡s ♦❢ ♦♣t✐♠❛❧ ❙❜♦①❡s ❬▲❡❛♥❞❡r✲P♦s❝❤♠❛♥♥ ✵✼❪ 8 ♦❢ t❤❡♠ ❤❛✈❡ ❛❧❧ x → b · S(x) ♦❢ ❞❡❣r❡❡ 3✳

✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ❛ ❜ ❝ ❞ ❡ ❢

G0

✵ ✶ ✷ ✶✸ ✹ ✼ ✶✺ ✻ ✽ ✶✶ ✶✷ ✾ ✸ ✶✹ ✶✵ ✺

G1

✵ ✶ ✷ ✶✸ ✹ ✼ ✶✺ ✻ ✽ ✶✶ ✶✹ ✸ ✺ ✾ ✶✵ ✶✷

G2

✵ ✶ ✷ ✶✸ ✹ ✼ ✶✺ ✻ ✽ ✶✶ ✶✹ ✸ ✶✵ ✶✷ ✺ ✾

G3

✵ ✶ ✷ ✶✸ ✹ ✼ ✶✺ ✻ ✽ ✶✷ ✺ ✸ ✶✵ ✶✹ ✶✶ ✾

G4

✵ ✶ ✷ ✶✸ ✹ ✼ ✶✺ ✻ ✽ ✶✷ ✾ ✶✶ ✶✵ ✶✹ ✺ ✸

G5

✵ ✶ ✷ ✶✸ ✹ ✼ ✶✺ ✻ ✽ ✶✷ ✶✶ ✾ ✶✵ ✶✹ ✸ ✺

G6

✵ ✶ ✷ ✶✸ ✹ ✼ ✶✺ ✻ ✽ ✶✷ ✶✶ ✾ ✶✵ ✶✹ ✺ ✸

G7

✵ ✶ ✷ ✶✸ ✹ ✼ ✶✺ ✻ ✽ ✶✷ ✶✹ ✶✶ ✶✵ ✾ ✸ ✺

G8

✵ ✶ ✷ ✶✸ ✹ ✼ ✶✺ ✻ ✽ ✶✹ ✾ ✺ ✶✵ ✶✶ ✸ ✶✷

G9

✵ ✶ ✷ ✶✸ ✹ ✼ ✶✺ ✻ ✽ ✶✹ ✶✶ ✸ ✺ ✾ ✶✵ ✶✷

G10

✵ ✶ ✷ ✶✸ ✹ ✼ ✶✺ ✻ ✽ ✶✹ ✶✶ ✺ ✶✵ ✾ ✸ ✶✷

G11

✵ ✶ ✷ ✶✸ ✹ ✼ ✶✺ ✻ ✽ ✶✹ ✶✶ ✶✵ ✺ ✾ ✶✷ ✸

G12

✵ ✶ ✷ ✶✸ ✹ ✼ ✶✺ ✻ ✽ ✶✹ ✶✶ ✶✵ ✾ ✸ ✶✷ ✺

G13

✵ ✶ ✷ ✶✸ ✹ ✼ ✶✺ ✻ ✽ ✶✹ ✶✷ ✾ ✺ ✶✶ ✶✵ ✸

G14

✵ ✶ ✷ ✶✸ ✹ ✼ ✶✺ ✻ ✽ ✶✹ ✶✷ ✶✶ ✸ ✾ ✺ ✶✵

G15

✵ ✶ ✷ ✶✸ ✹ ✼ ✶✺ ✻ ✽ ✶✹ ✶✷ ✶✶ ✾ ✸ ✶✵ ✺ ✶✽

P❡r♠✉t❛t✐♦♥s ♦❢ Fn

2 ✱ n ♦❞❞

❚❤❡♦r❡♠✳ ❬❈❤❛❜❛✉❞ ❱❛✉❞❡♥❛② ✾✹❪ ❋♦r ❛♥② ❢✉♥❝t✐♦♥ S ✇✐t❤ n ✐♥♣✉ts ❛♥❞ n ♦✉♣✉ts✱

L(S) ≥ 2

n+1 2

✇✐t❤ ❡q✉❛❧✐t② ❢♦r ♦❞❞ n ♦♥❧②✳ ❚❤❡ ❢✉♥❝t✐♦♥s ❛❝❤✐❡✈✐♥❣ t❤✐s ❜♦✉♥❞ ❛r❡ ❝❛❧❧❡❞ ❛❧♠♦st ❜❡♥t ❢✉♥❝t✐♦♥s✳

• ❆♥② ❆❇ ❢✉♥❝t✐♦♥ ✐s ❆P◆✳

L(S) = 2

n+1 2

= ⇒ δ(S) = 2

• ❚❤❡ ❝♦♥✈❡rs❡ ❤♦❧❞s ❢♦r s♦♠❡ ❝❛s❡s ♦♥❧②✱ ❢♦r ✐♥st❛♥❝❡ ❢♦r ❆P◆

❙❜♦①❡s ♦❢ ❞❡❣r❡❡ 2 ❬❈❛r❧❡t ❈❤❛r♣✐♥ ❩✐♥♦✈✐❡✈ ✾✽❪

✶✾

❑♥♦✇♥ ❆❇ ♣❡r♠✉t❛t✐♦♥s ♦❢ Fn

2 ✱ n ♦❞❞

▼♦♥♦♠✐❛❧s ♣❡r♠✉t❛t✐♦♥s S(x) = xs ♦✈❡r F2n, n = 2t + 1✳ q✉❛❞r❛t✐❝

2i + 1 ✇✐t❤ gcd(i, n) = 1✱

❬●♦❧❞ ✻✽❪✱❬◆②❜❡r❣ ✾✸❪

1 ≤ i ≤ t

❑❛s❛♠✐

22i − 2i + 1 ✇✐t❤ gcd(i, n) = 1

❬❑❛s❛♠✐ ✼✶❪

2 ≤ i ≤ t

❲❡❧❝❤

2t + 3

❬❉♦❜❜❡rt✐♥ ✾✽❪ ❬❈✳✲❈❤❛r♣✐♥✲❉♦❜❜❡rt✐♥ ✵✵❪ ◆✐❤♦

2t + 2

t 2 − 1 ✐❢ t ✐s ❡✈❡♥

❬❉♦❜❜❡rt✐♥ ✾✽❪

2t + 2

3t+1 2

− 1 ✐❢ t ✐s ♦❞❞

❬❳✐❛♥❣✲❍♦❧❧♠❛♥♥ ✵✶❪ ◆♦♥✲♠♦♥♦♠✐❛❧ ♣❡r♠✉t❛t✐♦♥s✳❬❇✉❞❛❣❤②❛♥✲❈❛r❧❡t✲▲❡❛♥❞❡r✵✽❪ ❋♦r n ♦❞❞✱ ❞✐✈✐s✐❜❧❡ ❜② 3 ❛♥❞ ♥♦t ❜② 9✳

S(x) = x2i+1 + ux2jn

3 +2(3−j)n 3 +i

✇✐t❤ gcd(i, n) = 1 ❛♥❞ j = in

3 mod 3

✷✵

P❡r♠✉t❛t✐♦♥s ♦❢ Fn

2 ✱ n ❡✈❡♥

❚❤❡r❡ ❡①✐st ❙❜♦①❡s ✇✐t❤

L(S) = 2

n+2 2

❜✉t ✇❡ ❞♦ ♥♦t ❦♥♦✇♥ ✐❢ t❤✐s ✈❛❧✉❡ ✐s ♠✐♥✐♠❛❧✳ ❆P◆ ♣♦✇❡r ❢✉♥❝t✐♦♥s ♦✈❡r Fn

2 ✱ n ❡✈❡♥✱ ❛r❡ ♥♦t ♣❡r♠✉t❛t✐♦♥s✳

❉♦ t❤❡r❡ ❡①✐st ❆P◆ ♣❡r♠✉t❛t✐♦♥s ❢♦r n ❡✈❡♥❄

✷✶

❑♥♦✇♥ ❆P◆ ♣❡r♠✉t❛t✐♦♥s ♦❢ Fn

2 ✱ n ❡✈❡♥

❋♦r n = 6✳

δ(S) ≥ 2 ❛♥❞ L(S) ≥ 12

❙❂ ④✵✱ ✺✹✱ ✹✽✱ ✶✸✱ ✶✺✱ ✶✽✱ ✺✸✱ ✸✺✱ ✷✺✱ ✻✸✱ ✹✺✱ ✺✷✱ ✸✱ ✷✵✱ ✹✶✱ ✸✸✱ ✺✾✱ ✸✻✱ ✷✱ ✸✹✱ ✶✵✱ ✽✱ ✺✼✱ ✸✼✱ ✻✵✱ ✶✾✱ ✹✷✱ ✶✹✱ ✺✵✱ ✷✻✱ ✺✽✱ ✷✹✱ ✸✾✱ ✷✼✱ ✷✶✱ ✶✼✱ ✶✻✱ ✷✾✱ ✶✱ ✻✷✱ ✹✼✱ ✹✵✱ ✺✶✱ ✺✻✱ ✼✱ ✹✸✱ ✹✹✱ ✸✽✱ ✸✶✱ ✶✶✱ ✹✱ ✷✽✱ ✻✶✱ ✹✻✱ ✺✱ ✹✾✱ ✾✱ ✻✱ ✷✸✱ ✸✷✱ ✸✵✱ ✶✷✱ ✺✺✱ ✷✷⑥❀ s❛t✐s✜❡s

δ(S) = 2 ✱ deg S = 4 ❛♥❞ L(S) = 16 ❬❉✐❧❧♦♥ ✵✾❪

❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✉♥✐✈❛r✐❛t❡ ♣♦❧②♥♦♠✐❛❧ ♦✈❡r F26 ❝♦♥t❛✐♥s 52 ♥♦♥③❡r♦ ♠♦♥♦♠✐❛❧s ✭♦✉t ♦❢ 56 ♣♦ss✐❜❧❡ ♠♦♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❛t ♠♦st 4✮✳ ❚❤✐s ✐s t❤❡ ♦♥❧② ❦♥♦✇♥ ❆P◆ ♣❡r♠✉t❛t✐♦♥ ✇✐t❤ ❛♥ ❡✈❡♥ ♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s✳

✷✷

• ♦♦❞ ♣❡r♠✉t❛t✐♦♥s ♦❢ Fn

2✱ n ❡✈❡♥

❯s✉❛❧❧②✱ ✇❡ s❡❛r❝❤ ❢♦r ♣❡r♠✉t❛t✐♦♥s S ✇✐t❤

δ(S) = 4 ❛♥❞ L(S) = 2

n+2 2

.

▼♦♥♦♠✐❛❧s ♣❡r♠✉t❛t✐♦♥s S(x) = xs ♦✈❡r F2n✳

2i + 1✱ gcd(i, n) = 2 n ≡ 2 mod 4

❬●♦❧❞ ✻✽❪

22i − 2i + 1✱ gcd(i, n) = 2 n ≡ 2 mod 4

❬❑❛s❛♠✐ ✼✶❪

2n − 2

❬▲❛❝❤❛✉❞✲❲♦❧❢♠❛♥♥ ✾✵❪ ❚❤❡ ❧❛st ♦♥❡ ✐s ❛✣♥❡ ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❆❊❙ ❙❜♦①✳

✷✸

❙♦♠❡ ❝♦♥❝❧✉s✐♦♥s

• ▼❛♥② ♦t❤❡r ♣r♦♣❡rt✐❡s ♦❢ ❙❜♦①❡s ❝❛♥ ❜❡ ❡①♣❧♦✐t❡❞ ❜② ❛♥ ❛tt❛❝❦❡r❀
• ❆ str♦♥❣ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡ ♠❛② ✐♥tr♦❞✉❝❡ ✇❡❛❦♥❡ss❡s❀
• ❉♦♥✬t ❢♦r❣❡t ✐♠♣❧❡♠❡♥t❛t✐♦♥✦✦✦

✷✹