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❋♦✉♥❞❛t✐♦♥s ♦❢ ❝r②♣t❛♥❛❧②s✐s✿ ♦♥ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s

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

❖✉t❧✐♥❡

• ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s
• ▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ❛♥❞ ❲❛❧s❤ tr❛♥s❢♦r♠
• ❘❡s✐st❛♥❝❡ t♦ ❞✐✛❡r❡♥t✐❛❧ ❛tt❛❝❦s
• ❋✐♥❞✐♥❣ ❣♦♦❞ ❙❜♦①❡s

✶

❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s

✷

❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ❉❡✜♥✐t✐♦♥✳ ❆ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ♦❢ n ✈❛r✐❛❜❧❡s ✐s ❛ ❢✉♥❝t✐♦♥ ❢r♦♠ Fn

2

✐♥t♦ F2✳ ❚r✉t❤ t❛❜❧❡ ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥✳

x1

✵ ✶ ✵ ✶ ✵ ✶ ✵ ✶

x2

✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶

x3

✵ ✵ ✵ ✵ ✶ ✶ ✶ ✶

f(x1, x2, x3)

✵ ✶ ✵ ✵ ✵ ✶ ✶ ✶ ❱❛❧✉❡ ✈❡❝t♦r ♦❢ f✿ ✇♦r❞ ♦❢ 2n ❜✐ts ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❛❧❧ f(x), x ∈ Fn

2 ✳

✸

❱❡❝t♦r✐❛❧ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ❉❡✜♥✐t✐♦♥✳ ❆ ✈❡❝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)

❊❛❝❤ ❢✉♥❝t✐♦♥

Si : (x1, . . . , xn) − → yi

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

✹

❍❛♠♠✐♥❣ ✇❡✐❣❤t ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ❍❛♠♠✐♥❣ ✇❡✐❣❤t ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❍❛♠♠✐♥❣ ✇❡✐❣❤t ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ f✱ wt(f)✱ ✐s t❤❡ ❍❛♠♠✐♥❣ ✇❡✐❣❤t ♦❢ ✐ts ✈❛❧✉❡ ✈❡❝t♦r✳ ❆ ❢✉♥❝t✐♦♥ ♦❢ n ✈❛r✐❛❜❧❡s ✐s ❜❛❧❛♥❝❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ wt(f) = 2n−1✳ Pr♦♣♦s✐t✐♦♥✳ ❆ ✈❡❝t♦r✐❛❧ ❢✉♥❝t✐♦♥ S ✇✐t❤ n ✐♥♣✉ts ❛♥❞ n ♦✉t♣✉ts ✐s ❛ ♣❡r♠✉t❛t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛♥② ♥♦♥③❡r♦ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ✐ts ❝♦♦r❞✐♥❛t❡s

x − →

n

• i=1

λiSi(x), λ = (λ1, . . . , λn) = 0

✐s ❛ ❜❛❧❛♥❝❡❞ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥✳

✺

❆❧❣❡❜r❛✐❝ ♥♦r♠❛❧ ❢♦r♠ ✭❆◆❋✮ ▼♦♥♦♠✐❛❧s ✐♥ x1, . . . , xn✿

• xu,

u ∈ Fn

2

• ✇❤❡r❡ xu =

n

• i=1

xui

i .

❊①❛♠♣❧❡✿ x1

1x0 2x1 3x1 4 = x1x3x4 = x1011✳

Pr♦♣♦s✐t✐♦♥✳ ❆♥② ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ♦❢ n ✈❛r✐❛❜❧❡s ❤❛s ❛ ✉♥✐q✉❡ ♣♦❧②♥♦♠✐❛❧ r❡♣r❡s❡♥t❛t✐♦♥✿

f(x1, . . . , xn) =

• u∈Fn

2

auxu, au ∈ F2.

▼♦r❡♦✈❡r✱ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ❆◆❋ ❛♥❞ t❤❡ ✈❛❧✉❡s ♦❢ f s❛t✐s❢②✿

au =

• xu

f(x) ❛♥❞ f(u) =

• xu

ax,

✇❤❡r❡ x u ✐❢ ❛♥❞ ♦♥❧② ✐❢ xi ≤ ui ❢♦r ❛❧❧ 1 ≤ i ≤ n✳

✻

❊①❛♠♣❧❡

x1

✵ ✶ ✵ ✶ ✵ ✶ ✵ ✶

x2

✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶

x3

✵ ✵ ✵ ✵ ✶ ✶ ✶ ✶

f(x1, x2, x3)

✵ ✶ ✵ ✵ ✵ ✶ ✶ ✶

a000 = f(000) = 0 a100 = f(100) ⊕ f(000) = 1 a010 = f(010) ⊕ f(000) = 0 a110 = f(110) ⊕ f(010) ⊕ f(100) ⊕ f(000) = 1 a001 = f(001) ⊕ f(000) = 0 a101 = f(101) ⊕ f(001) ⊕ f(100) ⊕ f(000) = 0 a011 = f(011) ⊕ f(001) ⊕ f(010) ⊕ f(000) = 1 a111 =

x∈F3

2 f(x) = wt(f) mod 2 = 0

f = x1 ⊕ x1x2 ⊕ x2x3.

✼

❈♦♠♣✉t✐♥❣ t❤❡ ❆◆❋

n = 3✿

✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼

f(0) f(1) f(2) f(3) f(4) f(5) f(6) f(7) f(0) f(0) ⊕ f(1) f(2) f(2) ⊕ f(3) f(4) f(4) ⊕ f(5) f(6) f(6) ⊕ f(7) f(0) f(0) ⊕ f(1) f(0) ⊕ f(2) f(0) ⊕ f(1) f(4) f(4) ⊕ f(5) f(4) ⊕ f(6) f(4) ⊕ f(5) ⊕f(2) ⊕ f(3) ⊕f(6) ⊕ f(7) f(0) f(0) ⊕ f(1) f(0) ⊕ f(2) f(0) ⊕ f(1) f(0) ⊕ f(4) f(0) ⊕ f(1) f(0) ⊕ f(2) f(0) ⊕ f(1) ⊕f(2) ⊕ f(3) f(4) ⊕ f(5) ⊕f(4) ⊕ f(6) ⊕f(2) ⊕ f(3) ⊕f(4) ⊕ f(5) ⊕f(6) ⊕ f(7)

✜rst st❡♣✿

f(2i + 1) ← f(2i + 1) ⊕ f(2i)

s❡❝♦♥❞ st❡♣✿

f(4i + j + 2) ← f(4i + j + 2) ⊕ f(4i + j), ∀0 ≤ j < 2

t❤✐r❞ st❡♣✿

f(8i + j + 4) ← f(8i + j + 4) ⊕ f(8i + j), ∀0 ≤ j < 4

✽

❈♦♠♣✉t✐♥❣ t❤❡ ❆◆❋ ❲❤❡♥ t❤❡ ✈❛❧✉❡ ✈❡❝t♦r ✐s st♦r❡❞ ❛s ❛ 32✲❜✐t ✐♥t❡❣❡r ①✿ ① ❫❂ ✭① ✫ ✵①✺✺✺✺✺✺✺✺✮ ❁❁ ✶❀ ① ❫❂ ✭① ✫ ✵①✸✸✸✸✸✸✸✸✮ ❁❁ ✷❀ ① ❫❂ ✭① ✫ ✵①✵❢✵❢✵❢✵❢✮ ❁❁ ✹❀ ① ❫❂ ✭① ✫ ✵①✵✵❢❢✵✵❢❢✮ ❁❁ ✽❀ ① ❫❂ ① ❁❁ ✶✻❀

✾

❉❡❣r❡❡ ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ❉❡✜♥✐t✐♦♥✳ ❚❤❡ ❞❡❣r❡❡ ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ✐s t❤❡ ❞❡❣r❡❡ ♦❢ t❤❡ ❧❛r❣❡st ♠♦♥♦✲ ♠✐❛❧ ✐♥ ✐ts ❆◆❋✳ Pr♦♣♦s✐t✐♦♥✳ ❚❤❡ ✇❡✐❣❤t ♦❢ ❛♥ n✲✈❛r✐❛❜❧❡ ❢✉♥❝t✐♦♥ f ✐s ♦❞❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ deg f = n✳ ❉❡✜♥✐t✐♦♥✳ ❚❤❡ ❞❡❣r❡❡ ♦❢ ❛ ✈❡❝t♦r✐❛❧ ❢✉♥❝t✐♦♥ S ✇✐t❤ n ✐♥♣✉ts ❛♥❞ m ♦✉t♣✉ts ✐s t❤❡ ♠❛①✐♠❛❧ ❞❡❣r❡❡ ♦❢ ✐ts ❝♦♦r❞✐♥❛t❡s✳

✶✵

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

✶✶

■❞❡♥t✐❢②✐♥❣ Fn

2 ✇✐t❤ ❛ ✜♥✐t❡ ✜❡❧❞

Fn

2 ✐s ✐❞❡♥t✐✜❡❞ ✇✐t❤ t❤❡ ✜♥✐t❡ ✜❡❧❞ ✇✐t❤ 2n ❡❧❡♠❡♥ts✳

F2n = {0} ∪ {αi, 0 ≤ i ≤ 2n − 2}

✇❤❡r❡ α ✐s ❛ r♦♦t ♦❢ ❛ ♣r✐♠✐t✐✈❡ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ n✳

⇒ ❢♦r ❛♥② i, αi =

n−1

• j=0

λjαj

❊①❛♠♣❧❡ ❢♦r n = 4✿ ♣r✐♠✐t✐✈❡ ♣♦❧②♥♦♠✐❛❧✿ 1 + x + x4✱ α ❛ r♦♦t ♦❢ t❤✐s ♣♦❧②♥♦♠✐❛❧✳

F24

1 α α2 α3 α4 α5 α6 α7

1 α α2 α3 α + 1 α2 + α α3 + α2 α3 + α + 1

F4

2

✵✵✵✵ ✵✵✵✶ ✵✵✶✵ ✵✶✵✵ ✶✵✵✵ ✵✵✶✶ ✵✶✶✵ ✶✶✵✵ ✶✵✶✶ α8 α9 α10 α11 α12 α13 α14

α2 + 1 α3 + α α2 + α + 1 α3 + α2 + α α3 + α2 + α + 1 α3 + α2 + 1 α3 + 1

✵✶✵✶ ✶✵✶✵ ✵✶✶✶ ✶✶✶✵ ✶✶✶✶ ✶✶✵✶ ✶✵✵✶

✶✷

❚❤❡ ✉♥✐✈❛r✐❛t❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❙❜♦①❡s ❆♥② ✈❡❝t♦r✐❛❧ ❢✉♥❝t✐♦♥ ✇✐t❤ n ✐♥♣✉ts ❛♥❞ n ♦✉t♣✉ts ❝❛♥ ❜❡ s❡❡♥ ❛s

S : F2n − → F2n

❚❤❡♥✱

S(X) =

2n−1

• i=0

ciXi , ci ∈ F2n.

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

S(X) = α12 + α2X + α13X2 + α6X3 + α10X4 + αX5 + α10X6 + α2X7 +α9X8 + α4X9 + α7X10 + α7X11 + α5X12 + X13 + α6X14

❘❡♠❛r❦✳ ❚❤❡ ✭♠✉❧t✐✈❛r✐❛t❡✮ ❞❡❣r❡❡ ♦❢ Xi ✐s ❡①❛❝t❧② t❤❡ ♥✉♠❜❡r ♦❢ ♦♥❡s ✐♥ t❤❡ ❜✐♥❛r② ❡①♣❛♥s✐♦♥ ♦❢ i✳

✶✸

▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ ❛ ❢✉♥❝t✐♦♥ ❛♥❞ ❲❛❧s❤ tr❛♥s❢♦r♠

✶✹

■❞❡❛ ❆❧❣❡❜r❛✐❝ ❛tt❛❝❦s ✭❛♥❞ ✈❛r✐❛♥ts✮✿ ✉s❡ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡ ✐♥♣✉t ❛♥❞ ♦✉t♣✉t ❜✐ts ♦❢ t❤❡ ❝✐♣❤❡r ✇❤✐❝❤ ❤♦❧❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② 1✳ ❜✉t t❤❡ ❞❡❣r❡❡ ✐s ✉s✉❛❧❧② t♦♦ ❤✐❣❤✦ ▲✐♥❡❛r ✭♦r ❝♦rr❡❧❛t✐♦♥✮ ❛tt❛❝❦s ❬❙✐❡❣❡♥t❤❛❧❡r ✽✺❪❬▼❛ts✉✐ ✾✸❪✿ ✉s❡ ❧✐♥❡❛r r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡ ✐♥♣✉t ❛♥❞ ♦✉t♣✉t ❜✐ts ♦❢ t❤❡ ❝✐♣❤❡r ✇❤✐❝❤ ❤♦❧❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ❧❡ss t❤❛♥ 1✳

✶✺

❊①❛♠♣❧❡ ❈♦♠♣✉t❡

f(x1, x2, x3, x4) = 1 ⊕ x1 ⊕ x4 ⊕ S2(x)

x ✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ❛ ❜ ❝ ❞ ❡ ❢ S1(x) ✶ ✵ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✵ ✶ ✶ ✵①❝✻✻✺ S2(x) ✶ ✶ ✶ ✵ ✶ ✵ ✶ ✵ ✵ ✶ ✵ ✶ ✵ ✶ ✵ ✵ ✵①✷❛✺✼ S3(x) ✶ ✶ ✵ ✶ ✶ ✶ ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✶ ✵①✾✵✼❜ S4(x) ✶ ✶ ✶ ✶ ✵ ✶ ✵ ✶ ✵ ✵ ✶ ✶ ✵ ✵ ✵ ✵ ✵①✵❝❛❢ 1 ⊕ x1 ⊕ x4 = 0xffff + 0xaaaa + 0xff00 = 0xaa55 S2(x) = 0x2a57 f(x) = 0x8002 ❚❤❡ r❡❧❛t✐♦♥ f(x) = 0 ❤♦❧❞s ❢♦r 14 ♦❢ t❤❡ 16 ✈❛❧✉❡s ♦❢ x ∈ F4

2✱

✐✳❡✳✱ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② 14

16 = 7 8✳

✶✻

❈♦♠♣✉t✐♥❣ t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ❛❧❧ ❧✐♥❡❛r r❡❧❛t✐♦♥s ❇✐❛s ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ❋♦r ❛♥② ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ f ♦❢ n ✈❛r✐❛❜❧❡s

E(f) =

• x∈Fn

2

(−1)f(x) = 2n − 2wt(f).

❊q✉✐✈❛❧❡♥t❧②✱

Pr[f(x) = 1] = wt(f) 2n = 1 2

• 1 − E(f)

2n

• .

→ ✇❡ ♥❡❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ ❜✐❛s❡s ♦❢ ❛❧❧ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s x − → b · S(x) ⊕ a · x .

✶✼

▲✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ ❛♥ ❙❜♦①

a \ b

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

Pr

x [a · x · b · S(x) = 1] = 1

2

• 1 − E[a, b]

2n

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

2(1 + 12 16) = 7 8✳

✶✽

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

Fn

2

− → Z a − → E(f + ϕa) =

x∈Fn

2 (−1)f(x)+a·x

✇❤❡r❡ ϕa : x −

→ a · x

❲❛❧s❤ tr❛♥s❢♦r♠ ♦❢ ❛ ✈❡❝t♦r✐❛❧ ❢✉♥❝t✐♦♥ S✿

Fn

2 × Fn 2

− → Z (a, b) − → E(b · S + ϕa) =

x∈Fn

2 (−1)b·S(x)+a·x

✶✾

❈♦♠♣✉t✐♥❣ t❤❡ ❲❛❧s❤ tr❛♥s❢♦r♠

f(x)

✵ ✶ ✵ ✵ ✵ ✶ ✶ ✶

(−1)f(x)

✶ ✲✶ ✶ ✶ ✶ ✲✶ ✲✶ ✲✶ st❡♣ ✶ ✵ ✷ ✷ ✵ ✵ ✷ ✲✷ ✵ st❡♣ ✷ ✷ ✷ ✲✷ ✷ ✲✷ ✷ ✷ ✷

E(f + ϕa)

✵ ✹ ✵ ✹ ✹ ✵ ✲✹ ✵ ✜rst st❡♣✿

S(2i) ← S(2i) + S(2i + 1) S(2i + 1) ← S(2i) − S(2i + 1)

s❡❝♦♥❞ st❡♣✿

S(4i + j) ← S(4i + j) + S(4i + j + 2), ∀0 ≤ j < 2 S(4i + j + 2) ← S(4i + j) − S(4i + j + 2), ∀0 ≤ j < 2

t❤✐r❞ st❡♣✿

S(8i + j) ← S(8i + j) + S(8i + j + 4), ∀0 ≤ j < 4 S(8i + j + 4) ← S(8i + j) − S(8i + j + 4), ∀0 ≤ j < 4

❈♦♠♣❧❡①✐t② ✿ n2n ♦♣❡r❛t✐♦♥s✳

✷✵

❙♦♠❡ ❜❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❲❛❧s❤ tr❛♥s❢♦r♠ ▲❡♠♠❛✿

E(ϕa) =

• x∈Fn

2

(−1)a·x = 2n

✐❢ a = 0 ♦t❤❡r✇✐s❡ . Pr♦♣♦s✐t✐♦♥✳ ❚❤❡ ❲❛❧s❤ tr❛♥s❢♦r♠ ✐s ❛♥ ✐♥✈♦❧✉t✐♦♥ ✭✉♣ t♦ ❛ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❝♦♥st❛♥t✮✿ ❢♦r ❛♥② x ∈ Fn

2 ✱

• a∈Fn

2

E(f + ϕa)(−1)a·x =

• u∈Fn

2

• a∈Fn

2

(−1)f(u)+a·u+a·x =

• u∈Fn

2

(−1)f(u)

a∈Fn

2

(−1)a·(x+u) = 2n(−1)f(x)

✷✶

❙♦♠❡ ❜❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❲❛❧s❤ tr❛♥s❢♦r♠ P❛rs❡✈❛❧ ❡q✉❛❧✐t②✳

• a∈Fn

2

E2(f + ϕa) = 22n.

Pr♦♦❢✳

• a∈Fn

2

E2(f + ϕa) =

• a∈Fn

2

  

• x∈Fn

2

(−1)f(x)+a·x

     

• y∈Fn

2

(−1)f(y)+a·y

  

=

• x∈Fn

2

• y∈Fn

2

(−1)f(x)+f(y)

a∈Fn

2

(−1)a·(x+y)

=

2n

x∈Fn

2

(−1)f(x)+f(x)

=

22n . ❬❈❤❡❝❦ ✐t ♦♥ ❡❛❝❤ ❝♦❧✉♠♥ ♦❢ t❤❡ t❛❜❧❡ ♦♥ ❙❧✐❞❡ ✶✽❪

✷✷

▲✐♥❡❛r✐t② ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ❉❡✜♥✐t✐♦♥✳ ❋♦r ❛♥② ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ f ♦❢ n ✈❛r✐❛❜❧❡s✱

L(f) = max

a

|E(f + ϕa)|

✐s ❝❛❧❧❡❞ t❤❡ ❧✐♥❡❛r✐t② ♦❢ f ✭❤✐❣❤❡st ❜✐❛s ❢♦r ❛♥ ❛✣♥❡ ❛♣♣r♦①✐♠❛t✐♦♥✮✳

N L(f) = 2n−1 − 1 2L(f)

✐s ❝❛❧❧❡❞ t❤❡ ♥♦♥❧✐♥❡❛r✐t② ♦❢ f ✭❞✐st❛♥❝❡ ♦❢ f t♦ t❤❡ ❛✣♥❡ ❢✉♥❝t✐♦♥s✮✳

✷✸

❈❛♥ ✇❡ s❛② s♦♠❡t❤✐♥❣ ❛❜♦✉t L(f)❄

L(f) = max

a

|E(f + ϕa)|

❚❤❡♦r❡♠✳ ❬❘♦t❤❛✉s ✼✻❪ ❋♦r ❛♥② ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ♦❢ n ✈❛r✐❛❜❧❡s✱

L(f) ≥ 2

n 2 ,

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

22n =

• a∈Fn

2

E2(f + ϕa) ≤ max

a∈Fn

2

E2(f + ϕa) × 2n = 2nL2(f)

✇✐t❤ ❡q✉❛❧✐t② ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛❧❧ E2(f + ϕa) ❛r❡ ❡q✉❛❧✳ ❚❤❡♥✱ L(f) ≥ 2

n 2 ✇✐t❤ ❡q✉❛❧✐t② ✐❢ ❛♥❞ ♦♥❧② ✐❢

E(f + ϕa) = ±2

n 2, ∀a ∈ Fn

2 .

■♥ ♣❛rt✐❝✉❧❛r✱ ♥♦♥❡ ♦❢ t❤❡ f + ϕa ✐s ❜❛❧❛♥❝❡❞✳

✷✹

❈❛♥ ✇❡ s❛② s♦♠❡t❤✐♥❣ ❛❜♦✉t L(f)❄ ❲❤❛t ✐s t❤❡ ❧♦✇❡st ♣♦ss✐❜❧❡ ✈❛❧✉❡ ❢♦r L(f) ✇❤❡♥ n ✐s ♦❞❞❄ ❲❤❡♥ f ✐s ❜❛❧❛♥❝❡❞❄ ❋✉♥❝t✐♦♥s ♦❢ ❞❡❣r❡❡ 2✳ ❋♦r n ♦❞❞✱ n = 2t + 1

x1x2 ⊕ x3x4 ⊕ . . . ⊕ x2t−1x2t ⊕ x2t+1

s❛t✐s✜❡s L(f) = 2

n+1 2 ✳ ▼♦r❡♦✈❡r✱ f ✐s ❜❛❧❛♥❝❡❞ ❛♥❞

∀a ∈ Fn

2,

E(f + ϕa) ∈ {0, ±2

n+1 2 }.

❚❤❡♦r❡♠✳

2

n 2 ≤

min

f∈Booln

L(f) ≤ 2

n+1 2

✷✺

❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ✇✐t❤ ❛ ❧♦✇ ❧✐♥❡❛r✐t②

n minf∈Booln L(f)

✺ ✽ ❬❇❡r❧❡❦❛♠♣✲❲❡❧❝❤ ✼✷❪ ✼ ✶✻ ❬▼②❦❦❡❧✈❡✐t ✽✵❪ ✾ ✷✹✱ ✷✻✱ ✷✽✱ ✸✵ ❬❑❛✈✉t✲▼❛✐tr❛✲❨ü❝❡❧ ✵✻❪ ✶✶ ✹✻✲✻✵ ✶✸ ✾✷✲✶✷✵ ✶✺ ✶✽✷✲✷✶✻ ❬P❛t❡rs♦♥✲❲✐❡❞❡♠❛♥♥ ✽✸❪ ❖♣❡♥ ♣r♦❜❧❡♠✳ ❋✐♥❞ t❤❡ ❧♦✇❡st ♣♦ss✐❜❧❡ ❧✐♥❡❛r✐t② ❢♦r ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ♦❢ n ✈❛r✐❛❜❧❡s✱ ✇❤❡r❡ n ✐s ♦❞❞ ❛♥❞ n ≥ 9✳

✷✻

❇❛❧❛♥❝❡❞ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ✇✐t❤ ❛ ❧♦✇ ❧✐♥❡❛r✐t②

n minf∈Baℓn L(f)

✹ ✽ ✺ ✽ ✻ ✶✷ ✼ ✶✻ ✽ ✷✵✱ ✷✹ ✾ ✷✹✱ ✷✽✱ ✸✷ ✶✵ ✸✻✱ ✹✵ ❖♣❡♥ ♣r♦❜❧❡♠✳ ❋✐♥❞ t❤❡ ❧♦✇❡st ♣♦ss✐❜❧❡ ❧✐♥❡❛r✐t② ❢♦r ❛ ❜❛❧❛♥❝❡❞ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ♦❢ n ✈❛r✐❛❜❧❡s✱ ✇❤❡♥ n ≥ 8✳ Pr♦♣♦s✐t✐♦♥✳ ❬❑❛t③ ✼✶❪ ■❢ f ✐s ❜❛❧❛♥❝❡❞✱ ❛❧❧ ✈❛❧✉❡s E(f + ϕa) ❛r❡ ❞✐✈✐s✐❜❧❡ ❜② 2⌈ n−1

deg f ⌉+1✱ ✐✳❡✳✱ ❛t ❧❡❛st ❜② 4 ✭❛♥❞ ❜② 8 ✐❢ deg f < n − 1✮✳

✷✼

▲✐♥❡❛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 |E (b · S + ϕa)| = max

b=0 L(b · S)

♠✉st ❜❡ ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡✳

N L(S) = 2n−1 − 1 2L(S)

✐s ❝❛❧❧❡❞ t❤❡ ♥♦♥❧✐♥❡❛r✐t② ♦❢ S✳

✷✽

❙❜♦①❡s ✇✐t❤ ❛ ❧♦✇ ❧✐♥❡❛r✐t② ❲❤❛t ✐s t❤❡ ❧♦✇❡st ♣♦ss✐❜❧❡ ✈❛❧✉❡ ❢♦r L(S) ✇❤❡♥ S ✐s ❛ ✈❡❝t♦r✐❛❧ ❢✉♥❝t✐♦♥ ✇✐t❤ n ✐♥♣✉ts ❛♥❞ n ♦✉t♣✉ts❄ ❚❤❡♦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✳ ❋♦r n ❡✈❡♥✳ ❚❤❡r❡ ❡①✐st ❙❜♦①❡s ✇✐t❤

L(S) = 2

n+2 2

❜✉t ✇❡ ❞♦ ♥♦t ❦♥♦✇♥ ✐❢ t❤✐s ✈❛❧✉❡ ✐s ♠✐♥✐♠❛❧✳

✷✾

❘❡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 ❈r✐t❡r✐♦♥ ♦♥ t❤❡ ❙❜♦①✳❬◆②❜❡r❣✲❑♥✉❞s❡♥ ✾✷❪ ❆❧❧ ❡♥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◆✮✳

✸✷

❋✐♥❞✐♥❣ ❣♦♦❞ ❙❜♦①❡s

✸✸

❆✣♥❡ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ ❙❜♦①❡s

S1 ❛♥❞ S2 ❛r❡ ❛✣♥❡❧② ❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ❡①✐st t✇♦ ❛✣♥❡ ♣❡r♠✉t❛t✐♦♥s A1 ❛♥❞ A2✱ s✉❝❤ t❤❛t S2 = A2 ◦ S1 ◦ A1

❚❤❡♥✱

δ(S2) = δ(S1) ❛♥❞ L(S2) = L(S1)

✸✹

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 ♦❞❞

L(S) ≥ 2

n+1 2

❛♥❞ δ(S) ≥ 2

• ❆♥② ❆❇ ❙❜♦① ✭✐✳❡✳✱ ✇✐t❤ L(S) = 2

n+1 2 ✮ ✐s ❆P◆ ❬❈❤❛❜❛✉❞✲❱❛✉❞❡♥❛② ✾✹❪✳

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

❙❜♦①❡s ❬❈❛r❧❡t✲❈❤❛r♣✐♥✲❩✐♥♦✈✐❡✈ ✾✽❪✳

• ❆❇ ❙❜♦①❡s ♦✈❡r Fn

2 ❤❛✈❡ ❞❡❣r❡❡ ❛t ♠♦st n+1 2 ✳

✸✻

❑♥♦✇♥ ❆❇ ♣❡r♠✉t❛t✐♦♥s ♦❢ Fn

2 ✱ n ♦❞❞

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

2i + 1 ✇✐t❤ gcd(i, n) = 1✱

❬●♦❧❞ ✻✽❪✱❬◆②❜❡r❣ ✾✸❪

1 ≤ i ≤ (n − 1)/2

❑❛s❛♠✐

22i − 2i + 1 ✇✐t❤ gcd(i, n) = 1

❬❑❛s❛♠✐ ✼✶❪

2 ≤ i ≤ (n − 1)/2

❲❡❧❝❤

2

n−1 2

+ 3

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

2

n−1 2

+ 2

n−1 4

− 1 ✐❢ n ≡ 1 mod 4

❬❉♦❜❜❡rt✐♥ ✾✽❪

2

n−1 2

+ 2

3n−1 4

− 1 ✐❢ n ≡ 3 mod 4

❬❳✐❛♥❣✲❍♦❧❧♠❛♥♥ ✵✶❪ ◆♦♥✲♠♦♥♦♠✐❛❧ ♣❡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 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 ♦❢ 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❛♠✐ ✼✶❪

2

n 2 + 2 n 4 + 1

n ≡ 4 mod 8

❬❇r❛❝❦❡♥✲▲❡❛♥❞❡r ✶✵❪

2n − 2

❬▲❛❝❤❛✉❞✲❲♦❧❢♠❛♥♥ ✾✵❪ ❚❤❡ ❧❛st ♦♥❡ ✐s ❛✣♥❡❧② ❡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✐♦♥✦✦✦

❙♦♠❡ ✉s❡❢✉❧ ❧✐♥❦s✿

• ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ✭❛♥❞ r❡❧❛t❡❞ ❡♥tr✐❡s✮✱ ✐♥ ❊♥❝②❝❧♦♣❡❞✐❛ ♦❢ ❈r②♣✲

t♦❣r❛♣❤② ❛♥❞ ❙❡❝✉r✐t②✱ ❙♣r✐♥❣❡r✱ ✷✵✶✶✳

• ❍❛♥❞❜♦♦❦ ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s ✭●✳ ▼✉❧❧❡♥ ❛♥❞ ❉✳ P❛♥❛r✐♦✱ ❡❞s✳✮✱ ❈❘❈

Pr❡ss✱ ✷✵✶✸✳

✹✵