t ts str ss - - PowerPoint PPT Presentation

❇❡♥t ❢✉♥❝t✐♦♥s✱ ❑❧♦♦st❡r♠❛♥ s✉♠s ❛♥❞ ♣♦✐♥t ❝♦✉♥t✐♥❣

❏❡❛♥✲P✐❡rr❡ ❋❧♦r✐✱ ❙✐❤❡♠ ▼❡s♥❛❣❡r ❛♥❞ ●ér❛r❞ ❈♦❤❡♥

❆◆❙❙■✱ ❯♥✐✈❡rs✐t② ♦❢ P❛r✐s ✽ ❛♥❞ ❚é❧é❝♦♠ P❛r✐s❚❡❝❤

◆♦✈❡♠❜❡r ✹✱ ✷✵✶✶

✶ ✴ ✸✾

❖✉t❧✐♥❡

✶

❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ❛♥❞ ❜❡♥t ❢✉♥❝t✐♦♥s

✷

❑❧♦♦st❡r♠❛♥ s✉♠s ❛♥❞ ❞✐✈✐s✐❜✐❧✐t② ♣r♦♣❡rt✐❡s

✸

❊❧❧✐♣t✐❝ ❝✉r✈❡s ✐♥ ❡✈❡♥ ❝❤❛r❛❝t❡r✐st✐❝

✹

❑❧♦♦st❡r♠❛♥ s✉♠s ✇✐t❤ ✈❛❧✉❡ 0

✺

❑❧♦♦st❡r♠❛♥ s✉♠s ✇✐t❤ ✈❛❧✉❡ 4

✻

❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts

✼

❋✉rt❤❡r ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ✐♥✈♦❧✈✐♥❣ ❤②♣❡r❡❧❧✐♣t✐❝ ❝✉r✈❡s

✷ ✴ ✸✾

❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s

❆ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ✐s ❛ ❢✉♥❝t✐♦♥ f : F2n → F2✳ P♦❧②♥♦♠✐❛❧ ❢♦r♠ f ❤❛s ❛ ✉♥✐q✉❡ tr❛❝❡ ❡①♣❛♥s✐♦♥ ♦❢ t❤❡ ❢♦r♠✿ f(x) =

• j∈Γn

Tro(j)

1

• ajxj

+ ǫ(1 + x2n−1), aj ∈ F2o(j) , ✇❤❡r❡ Γn ✐s t❤❡ s❡t ♦❢ ✐♥t❡❣❡rs ♦❜t❛✐♥❡❞ ❜② ❝❤♦♦s✐♥❣ ♦♥❡ ❡❧❡♠❡♥t ✐♥ ❡❛❝❤ ❝②❝❧♦t♦♠✐❝ ❝❧❛ss ♠♦❞✉❧♦ 2n − 1 ✱o(j) t❤❡ s✐③❡ ♦❢ t❤❡ ❝♦s❡t ❛♥❞ ǫ = wt(f) (mod 2)✳ ❇❡♥t♥❡ss ❆ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ f ✐s s❛✐❞ t♦ ❜❡ ❜❡♥t ✐❢ ✐t ❤❛s ♠❛①✐♠✉♠ ♥♦♥✲❧✐♥❡❛r✐t② 2n−1 − 2n/2−1✱ ✐✳❡✳ ✐s ❛s ❢❛r ❛s ♣♦ss✐❜❧❡ ♦❢ ❛❧❧ ❛✣♥❡ ❢✉♥❝t✐♦♥s✳

✸ ✴ ✸✾

❲❛❧s❤✲❍❛❞❛♠❛r❞ tr❛♥s❢♦r♠

❲❛❧s❤✲❍❛❞❛♠❛r❞ tr❛♥s❢♦r♠ ❋♦r ω ∈ F2n✱ t❤❡ ❲❛❧s❤✲❍❛❞❛♠❛r❞ tr❛♥s❢♦r♠ ♦❢ f ❛t ω ✐s

• χf(ω) =
• x∈F2n

(−1)f(x)+Trn

1 (ωx) .

✭❍②♣❡r✮✲❜❡♥t♥❡ss ❝❛♥ ❜❡ ❝❤❛r❛❝t❡r✐③❡❞ ✉s✐♥❣ t❤❡ ❲❛❧s❤✲❍❛❞❛♠❛r❞ tr❛♥s❢♦r♠✳ ❇❡♥t♥❡ss✿ ❆ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ f : F2n → F2 ✐s s❛✐❞ t♦ ❜❡ ❜❡♥t ✐❢

• χf(ω) = ±2

n 2 ✱ ❢♦r ❛❧❧ ω ∈ F2n✳

❍②♣❡r✲❇❡♥t♥❡ss✿ ❆ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ f : F2n → F2 ✐s s❛✐❞ t♦ ❜❡ ❤②♣❡r✲❜❡♥t ✐❢ t❤❡ ❢✉♥❝t✐♦♥ x → f(xi) ✐s ❜❡♥t✱ ❢♦r ❡✈❡r② ✐♥t❡❣❡r i ❝♦✲♣r✐♠❡ ✇✐t❤ 2n − 1✳

✹ ✴ ✸✾

❈♦♠♣✉t✐♥❣ t❤❡ ❲❛❧s❤✲❍❛❞❛♠❛r❞ tr❛♥s❢♦r♠

❚❤❡ ❲❛❧s❤✲❍❛❞❛♠❛r❞ tr❛♥s❢♦r♠ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ q✉✐t❡ ❡❛s✐❧② ❛♥❞ ❡✣❝✐❡♥t❧②✿ ❛❧❣♦r✐t❤♠ ✐♥ O(2mm2) ❜✐t ♦♣❡r❛t✐♦♥s ❛♥❞ O(2mm) ♠❡♠♦r②✱ ❝❛❝❤❡ ❡✣❝✐❡♥t✱ r✐❞✐❝✉❧♦✉s❧② s♠❛❧❧ ❝♦♥st❛♥t ❬❆r♥✶✵❪✳ ❆❧r❡❛❞② ✐♠♣❧❡♠❡♥t❡❞ ✐♥ ❙❛❣❡ ❬❙+✶✶❪ ✭✉s✐♥❣ ❈②t❤♦♥ ❬❇❈❙✶✵❪✮✳ ❍♦✇❡✈❡r t❤❡r❡ ❛r❡ s♦♠❡ ❞r❛✇❜❛❝❦s ✇✐t❤ t❤❡ ❝✉rr❡♥t ✐♠♣❧❡♠❡♥t❛t✐♦♥✿

✶

r❡t✉r♥s t❤❡ ♦♣♣♦s✐t❡ ♦❢ t❤❡ tr❛♥s❢♦r♠❀

✷

❧✐♠✐t❡❞ t♦ 32 ❜✐ts❀

✸

r❡t✉r♥s ❛ P②t❤♦♥ ❛rr❛②✳ ❙♦♠❡ ✐♠♣r♦✈❡♠❡♥ts ♣r♦✈✐❞❡❞ ✐♥ ❚r❛❝ t✐❝❦❡t ★✶✶✹✺✵✳

✺ ✴ ✸✾

❇✐♥❛r② ❑❧♦♦st❡r♠❛♥ ❙✉♠s

❚❤❡ ❜✐♥❛r② ❑❧♦♦st❡r♠❛♥ s✉♠s ♦♥ F2m ❛r❡ Km(a) =

• x∈F2m

(−1)Trm

1 (ax+ 1 x),

a ∈ F2m . ❘❡♠❛r❦✿ ❚❤❡ ❢✉♥❝t✐♦♥ a → Km(a) ✐s t❤❡ ❲❛❧s❤✲❍❛❞❛♠❛r❞ tr❛♥s❢♦r♠ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ Trm

1 (1/x)✳

❚❤❡r❡❢♦r❡✱ ❛❧❧ ✈❛❧✉❡s ♦❢ ❑❧♦♦st❡r♠❛♥ s✉♠s ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛t ♦♥❝❡ ✉s✐♥❣ ❛ ❢❛st ❲❛❧s❤✲❍❛❞❛♠❛r❞ tr❛♥s❢♦r♠✳

✻ ✴ ✸✾

❈❤❛r❛❝t❡r✐③❛t✐♦♥ ✉s✐♥❣ t❤❡ ❱❛❧✉❡ 0

✭❍②♣❡r✮✲❜❡♥t♥❡ss ❝❛♥ ❜❡ ❝❤❛r❛❝t❡r✐③❡❞ ✉s✐♥❣ s✉❝❤ s✉♠s✳ ■t ✐s ❦♥♦✇♥ s✐♥❝❡ ✶✾✼✹ t❤❛t t❤❡ ③❡r♦s ♦❢ Km(a) ❣✐✈❡ r✐s❡ t♦ ❜❡♥t ❢✉♥❝t✐♦♥s✳ Pr♦♣♦s✐t✐♦♥ ✭▼♦♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s❬❉✐❧✼✹✱ ▲❲✾✵✱ ▲❡❛✵✻✱ ❈●✵✽❪✮ ▲❡t f : F2n → F2 ❜❡ ❞❡✜♥❡❞ ❛s f(x) = Trn

1

• axr(2m−1)

, gcd(r, 2m + 1) = 1 . ❚❤❡♥ f ✐s ❤②♣❡r✲❜❡♥t ✐✛ Km(a) = 0✳ ❙❡✈❡r❛❧ ♦t❤❡r ❢❛♠✐❧✐❡s ❛❞♠✐t ❛ s✐♠✐❧❛r ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ❬▼❡s❛r❪✳

✼ ✴ ✸✾

❈❤❛r❛❝t❡r✐③❛t✐♦♥ ✉s✐♥❣ t❤❡ ❱❛❧✉❡ 4

■t ✐s ♦♥❧② ✐♥ ✷✵✵✾ t❤❛t ▼❡s♥❛❣❡r ❤❛s s❤♦✇♥ t❤❛t t❤❡ ✈❛❧✉❡ 4 ❧❡❛❞s t♦ s✐♠✐❧❛r ❝♦♥tr✉❝t✐♦♥s ❬▼❡s✶✶❪✳ Pr♦♣♦s✐t✐♦♥ ✭❬▼❡s✶✶❪✮ ▲❡t f : F2n → F2 ❜❡ ❞❡✜♥❡❞ ❛s f(x) = Trn

1

• axr(2m−1)

+ Tr2

1

• bx

2n−1 3

• , gcd(r, 2m + 1) = 1 .

■❢ m ✐s ♦❞❞✱ t❤❡♥ f ✐s ❤②♣❡r❜❡♥t ✐✛ Km(a) = 4✳ ■❢ m ✐s ❡✈❡♥✱ t❤✐s ✐s ❛ ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥✳ ▼♦r❡ ❢❛♠✐❧✐❡s ❛r❡ ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ s❛♠❡ ♣❛♣❡r ❬▼❡s✶✶❪✳

✽ ✴ ✸✾

❈❧❛ss✐❝❛❧ ❞✐✈✐s✐❜✐❧✐t② r❡s✉❧ts

❉✐✈✐s✐❜✐❧✐t② ♦❢ ❑❧♦♦st❡r♠❛♥ s✉♠s ❤❛s ❜❡❡♥ st✉❞✐❡❞ ❢♦r ❛ ❧♦♥❣ t✐♠❡✳ Pr♦♣♦s✐t✐♦♥ ✭❬▲❲✾✵❪✮ ▲❡t m ≥ 3 ❜❡ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✳ ❚❤❡ s❡t {Km(a), a ∈ F2m} ✐s t❤❡ s❡t ♦❢ ❛❧❧ t❤❡ ✐♥t❡❣❡r ♠✉❧t✐♣❧❡s ♦❢ 4 ✐♥ t❤❡ r❛♥❣❡ [−2(m+2)/2 + 1, 2(m+2)/2 + 1]✳ ▼♦st ❝❧❛ss✐❝❛❧ r❡s✉❧ts ❛r✐s❡ ❢r♦♠ t❤❡ st✉❞② ♦❢ t❤❡ ❧✐♥❦ ❜❡t✇❡❡♥ ❡①♣♦♥❡♥t✐❛❧ s✉♠s ❛♥❞ ❝♦s❡t ✇❡✐❣❤t ❞✐str✐❜✉t✐♦♥ ❬❍❩✾✾✱ ❈❍❩✵✾❪✳ Pr♦♣♦s✐t✐♦♥ ✭❬❍❩✾✾❪✮ ▲❡t m ≥ 3 ❜❡ ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❛♥❞ a ∈ F2m✳ ❚❤❡♥ Km(a) ≡ 0 (mod 8) ✐❢ ❛♥❞ ♦♥❧② ✐❢ Trm

1 (a) = 0✳

❚❤❡s❡ ❝♦♥❞✐t✐♦♥s ❝❛♥ ❜❡ ✉s❡❞ t♦ ✜❧t❡r ♦✉t t❤❡ a✬s t♦ t❡st ✇❤✐❧❡ ♣❡r❢♦r♠✐♥❣ ❛ r❛♥❞♦♠ s❡❛r❝❤✳

✾ ✴ ✸✾

❋✉rt❤❡r ❞✐✈✐s✐❜✐❧✐t② ♣r♦♣❡rt✐❡s ♠♦❞ 3✳ Pr♦♣♦s✐t✐♦♥ ✭❬❍❩✾✾❪✮ ▲❡t m ≥ 3 ❜❡ ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ❛♥❞ a ∈ F∗

2m✳ ❙✉♣♣♦s❡ t❤❛t t❤❡r❡ ❡①✐sts

t ∈ F∗

2m s✉❝❤ t❤❛t a = b4 + b3✳

■❢ m ✐s ♦❞❞✱ t❤❡♥ Km(a) ≡ 1 (mod 3)✳ ■❢ m ✐s ❡✈❡♥✱ t❤❡♥ Km(a) ≡ 0 (mod 3) ✐❢ Trm

1 (b) = 0 ❛♥❞

Km(a) ≡ −1 (mod 3) ✐❢ Trm

1 (b) = 1✳

Pr♦♣♦s✐t✐♦♥ ✭❬❈❍❩✵✾❪✮ ▲❡t a ∈ F∗

2m✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿

■❢ m ✐s ♦❞❞✱ t❤❡♥ Km(a) ≡ 1 (mod 3) ✐❢ ❛♥❞ ♦♥❧② ✐❢ Trm

1

• a1/3

= 0✳ ❚❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ a =

b (1+b)4 ❢♦r s♦♠❡ b ∈ F∗ 2m✳

■❢ m ✐s ❡✈❡♥✱ t❤❡♥ Km(a) ≡ 1 (mod 3) ✐❢ ❛♥❞ ♦♥❧② ✐❢ a = b3 ❢♦r s♦♠❡ b s✉❝❤ t❤❛t Trm

2 (b) = 0✳

✶✵ ✴ ✸✾

❊q✉❛t✐♦♥s

❍❡r❡ ❛r❡ s♦♠❡ s♣❡❝✐✜❝ r❡s✉❧ts t♦ ❡❧❧✐♣t✐❝ ❝✉r✈❡s ✐♥ ❡✈❡♥ ❝❤❛r❛❝t❡r✐st✐❝✳ E ✐s ♦r❞✐♥❛r② ✐✛ j(E) = 0✳ ■t ❝❛♥ t❤❡♥ ❜❡ ❞❡s❝r✐❜❡❞ ❛s E : y2 + xy = x3 + bx2 + a , ✇✐t❤ a = 0 ❛♥❞ j(E) = 1/a✳ ▼♦r❡♦✈❡r ✐ts ✜rst ❞✐✈✐s✐♦♥ ♣♦❧②♥♦♠✐❛❧s ❛r❡ ❬❑♦❜✾✵✱ ❇❙❙✵✵❪ f1(x) = 1, f2(x) = x, f3(x) = x4 + x3 + a, f4(x) = x6 + ax2 .

✶✶ ✴ ✸✾

◗✉❛❞r❛t✐❝ t✇✐st

■❢ E ✐s ♦r❞✐♥❛r②✱ t❤❡♥ t❤❡ q✉❛❞r❛t✐❝ t✇✐st E ✐s ❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ✇✐t❤ t❤❡ s❛♠❡ j✲✐♥✈❛r✐❛♥t ❛s E✱ ❜✉t ♥♦♥✲✐s♦♠♦r♣❤✐❝ t♦ ✐t ♦✈❡r Fq ✭✐t ❜❡❝♦♠❡s s♦ ♦✈❡r Fq2✮✳ ■t ❝❛♥ ❜❡ ❣✐✈❡♥ ❜② t❤❡ ❲❡✐❡rstr❛ss ❡q✉❛t✐♦♥

• E : y2 + xy = x3 +

bx2 + a , ✇❤❡r❡ b ✐s ❛♥② ❡❧❡♠❡♥t ♦❢ Fq s✉❝❤ t❤❛t Trm

1

• b
• = 1 − Trm

1 (b) ❬❊♥❣✾✾❪✳

❚❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ♦❢ ❛ ❝✉r✈❡ ❛♥❞ ✐ts q✉❛❞r❛t✐❝ t✇✐st ❛r❡ ❝❧♦s❡❧② r❡❧❛t❡❞ ❬❊♥❣✾✾✱ ❇❙❙✵✵❪✿ #E + # E = 2q + 2 .

✶✷ ✴ ✸✾

❈✉r✈❡s ✇✐t❤ ❛ ❣✐✈❡♥ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts

❚❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ❝✉r✈❡ ✐s ❣✐✈❡♥ ❜② t❤❡ tr❛❝❡ ♦❢ ✐ts ❋r♦❜❡♥✐✉s✿ #E = q + 1 − t . ■❢ E ✐s ♦r❞✐♥❛r②✱ t❤❡♥ 2 ∤ t ❛♥❞ t❤❡ ❡♥❞♦♠♦r♣❤✐s♠ r✐♥❣ ♦❢ E ✐s ❛♥ ♦r❞❡r ✐♥ K = Q[α] ❝♦♥t❛✐♥✐♥❣ t❤❡ ♦r❞❡r Z[α] ♦❢ ❞✐s❝r✐♠✐♥❛♥t ∆ ✇❤❡r❡ α = t+

√ ∆ 2

❛♥❞ ∆ = t2 − 4q✳ ❚❤✐s ✐♠♣❧✐❡s t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ s✉❝❤ ❝✉r✈❡s ✐s ❣✐✈❡♥ ❜② t❤❡ ❑r♦♥❡❝❦❡r ❝❧❛ss ♥✉♠❜❡r ❬❙❝❤✽✼✱ ❈♦①✽✾❪ H(∆) =

• Z[α]⊂O⊂K

h(O) . ■t ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✉s✐♥❣ ♠♦r❡ ❝❧❛ss✐❝❛❧ q✉❛♥t✐t✐❡s ❛s H(∆) = h(OK)

• d|f

d [O∗

K : O]

• p|d
• 1 −

∆K p 1 p

• .

✶✸ ✴ ✸✾

❊❧❧✐♣t✐❝ ❝✉r✈❡s ❛♥❞ ❑❧♦♦st❡r♠❛♥ s✉♠s

❚❤❡ ✜rst r❡s✉❧t ❛❜♦✈❡ ✐s ✐♥ ❢❛❝t ♣r♦✈❡❞ ✉s✐♥❣ ❡❧❧✐♣t✐❝ ❝✉r✈❡s✦ ❚❤❡♦r❡♠ ✭❬▲❲✽✼✱ ❑▲✽✾❪✮ ▲❡t m ≥ 3 ❜❡ ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✱ a ∈ F∗

2m ❛♥❞ Em(a) t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡

❞❡✜♥❡❞ ♦✈❡r F2m ❜② t❤❡ ❡q✉❛t✐♦♥ Em(a) : y2 + xy = x3 + a . ❚❤❡♥ #Em(a) = 2m + Km(a) . ❚❤❡ t❤❡♦r② ♦❢ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❝❛♥ ❜❡ ✉s❡❞ ♠✉❝❤ ❢✉rt❤❡r✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❢❛❝t t❤❛t t❤❡ ❑❧♦♦st❡r♠❛♥ s✉♠s ❛r❡ ❞✐✈✐s✐❜❧❡ ❜② 4 ✐s ♥♦t❤✐♥❣ ❜✉t t❤❡ ❢❛❝t t❤❛t ❡✈❡r② s✉❝❤ ❡❧❧✐♣t✐❝ ❝✉r✈❡s ❤❛s ❛ 4✲t♦rs✐♦♥ ♣♦✐♥t✳

✶✹ ✴ ✸✾

❘❡✜♥✐♥❣ ❍❩ ❘❡s✉❧t

Pr♦♣♦s✐t✐♦♥ ▲❡t a ∈ F∗

2m✳

■❢ m ✐s ♦❞❞✱ t❤❡♥ Km(a) ≡ 1 (mod 3) ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts t ∈ F2m s✉❝❤ t❤❛t a = t4 + t3✳ ■❢ m ✐s ❡✈❡♥✱ t❤❡♥✿

Km(a) ≡ 0 (mod 3) ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts t ∈ F2m s✉❝❤ t❤❛t a = t4 + t3 ❛♥❞ Trm

1 (t) = 0❀

Km(a) ≡ −1 (mod 3) ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts t ∈ F2m s✉❝❤ t❤❛t a = t4 + t3 ❛♥❞ Trm

1 (t) = 1✳

■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢✿

✶

❖♥❡ ✇❛② ✐s ❣✐✈❡♥ ❜② ❬❍❩✾✾❪✳

✷

❋♦r t❤❡ ♦t❤❡r ✇❛②✱ ❧♦♦❦ ❛t t❤❡ 3✲❞✐✈✐s✐♦♥ ♣♦❧②♥♦♠✐❛❧ ♦❢ E ♦r E✳

✶✺ ✴ ✸✾

❇❛s✐❝ s❡❛r❝❤ ❛❧❣♦r✐t❤♠

❚❤❡ ❛❜♦✈❡ ❞✐s❝✉ss✐♦♥ ❛❧r❡❛❞② ❣✐✈❡s ❛♥ ❡✣❝✐❡♥t ♠❡t❤♦❞ t♦ ✜♥❞ s♣❡❝✐✜❝ ✈❛❧✉❡s ♦❢ ❑❧♦♦st❡r♠❛♥ s✉♠s✳

✶

P✐❝❦ ❛ r❛♥❞♦♠ a ∈ F2m✳

✷

❚r❛♥s❢♦r♠ ✐t t♦ ❤❛✈❡ ❛ ❣✐✈❡♥ s❤❛♣❡✳

✸

❈❤❡❝❦ ❢♦r ❛❞❞✐t✐♦♥❛❧ ❞✐✈✐s✐❜✐❧✐t② ♣r♦♣❡rt✐❡s✳

✹

❈♦♠♣✉t❡ t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ Em(a)✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧✐t② ✐s ✐♥❞❡❡❞ q✉❛❞r❛t✐❝ ✐♥ m ❬❍❛r✵✷✱ ❱❡r✵✸❪✿ O(m2 log2 m log log m) .

✶✻ ✴ ✸✾

❋✐♥❞✐♥❣ ③❡r♦s

❚❤❡ ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡ ▲❛❝❤❛✉❞✲❲♦❧❢♠❛♥♥ t❤❡♦r❡♠ ✐s #Em(a) = 2m . ❚❤❡♥✱ ❛s ❛ ❣r♦✉♣ Em(a) ≃ Z/2mZ , ❛♥❞ ❤❛❧❢ ✐ts ♣♦✐♥ts ❤❛✈❡ ❡①❛❝t ♦r❞❡r 2m✳ ❋r♦♠ t❤❡s❡ ❢❛❝ts✱ ▲✐s♦♥➙❦ ❬▲✐s✵✽❪ ❞❡❞✉❝❡❞ t❤❛t t♦ ❝❤❡❝❦ t❤❛t Em(a) ✐♥❞❡❡❞ ❤❛s ❛ s✉❝❤ str✉❝t✉r❡ ✐t ✐s ❡♥♦✉❣❤ t♦ t❛❦❡ ❛ r❛♥❞♦♠ ♣♦✐♥t ❛♥❞ ❝❤❡❝❦ ✐t ❤❛s ♦r❞❡r ❡①❛❝t❧② 2m✳ ■❢ ❛ s✉❝❤ ♣♦✐♥t ✐s ❢♦✉♥❞✱ t❤❡♥ t❤❡ ❍❛ss❡✲❲❡✐❧ t❤❡♦r❡♠ ❡♥s✉r❡s t❤❛t Em(a) ✐s ✐♥❞❡❡❞ ♦❢ ❝❛r❞✐♥❛❧✐t② 2m✳ ❚❤✐s ❣✐✈❡s ❛♥ ❡✣❝✐❡♥t ♣r♦❜❛❜✐❧✐st✐❝ ❛❧❣♦r✐t❤♠ t♦ ✜♥❞ ③❡r♦s ♦❢ ❑❧♦♦st❡r♠❛♥ s✉♠s ❛♥❞ ❤❡ ❝♦✉❧❞ ✜♥❞ ③❡r♦s ♦❢ ❑❧♦♦st❡r♠❛♥ s✉♠s ❢♦r m ✉♣ t♦ 64✳

✶✼ ✴ ✸✾

❙②❧♦✇ ❣r♦✉♣

❆❤♠❛❞✐ ❛♥❞ ●r❛♥❣❡r s✉❜s❡q✉❡♥t❧② ❜✉✐❧t ❛♥ ❡✣❝✐❡♥t ❞❡t❡r♠✐♥✐st✐❝ ❛❧❣♦r✐t❤♠ ❢r♦♠ t❤❡ ❛❜♦✈❡ ♦❜s❡r✈❛t✐♦♥s ❬❆●✶✶❪✳ ❘❛t❤❡r t❤❛♥ ❝♦♠♣✉t✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ♦❢ t❤❡ r❛♥❞♦♠❧② ❝❤♦s❡♥ ❝✉r✈❡s✱ ✐t ✐s ✐♥❞❡❡❞ ❡♥♦✉❣❤ t♦ ❝♦♠♣✉t❡ t❤❡ s✐③❡ ♦❢ t❤❡ 2✲❙②❧♦✇ s✉❜❣r♦✉♣ ♦❢ Em(a)✳ ❚❤✐s ❝❛♥ ❜❡ ❡✣❝✐❡♥t❧② ❞♦♥❡ ❜② ♣♦✐♥t ❤❛❧✈✐♥❣✳ ❚❤❡ ❛✈❡r❛❣❡ ❜✐t ❝♦♠♣❧❡①✐t② ❢♦r ♦♥❡ ❝✉r✈❡ ✐s O(m log m log log m) ✇❤❡r❡❛s ✐t ✐s O(m2 log2 m log log m) ❢♦r ♣♦✐♥t ❝♦✉♥t✐♥❣✳

✶✽ ✴ ✸✾

❊①t❡♥❞✐♥❣ t♦ t❤❡ ✈❛❧✉❡ 4

▲♦♦❦✐♥❣ ❢♦r t❤❡ ✈❛❧✉❡ 4✱ t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ t❤❡ ❝✉r✈❡ ❤❛s ❛ ✇❛② ❧❡ss s♣❡❝✐❛❧ ❢♦r♠✿ #Em(a) = 2m + 4 = 4(2m−2 + 1) , ❛♥❞ t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ t❤❡ t✇✐st❡❞ ❝✉r✈❡ ✐s ♥♦t ❜❡tt❡r # Em(a) = 2m − 2 = 2(2m−1 − 1) . ❲❡ ❝❛♥ ❤♦✇❡✈❡r ❞❡❞✉❝❡ ❢r♦♠ t❤❡s❡ ❡q✉❛❧✐t✐❡s s♦♠❡ ✜❧t❡r✐♥❣ ♣r♦♣❡rt✐❡s✳ Km(a) ≡ 4 (mod 8)✱ s♦ t❤❛t Trm

1 (a) = 1❀

Km(a) ≡ 1 (mod 3)✱ s♦ t❤❛t✿

✐❢ m ✐s ♦❞❞✱ t❤❡♥ a ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s t4 + t3❀ ✐❢ m ✐s ❡✈❡♥✱ t❤❡♥ a ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s t3 ✇✐t❤ Trm

2 (t) = 0✳

✶✾ ✴ ✸✾

❆❧❣♦r✐t❤♠ ❢♦r m ♦❞❞

■♥♣✉t✿ ❆ ♣♦s✐t✐✈❡ ♦❞❞ ✐♥t❡❣❡r m ≥ 3 ❖✉t♣✉t✿ ❆♥ ❡❧❡♠❡♥t a ∈ F2m s✉❝❤ t❤❛t Km(a) = 4

✶ a ←R F2m ✷ a ← a3(a + 1) ✸ ✐❢ Trm 1 (a) = 0 t❤❡♥ ✹

• ♦ t♦ st❡♣ ✶

✺ P ←R Em(a) ✻ ✐❢ [2m + 4]P = 0 t❤❡♥ ✼

• ♦ t♦ st❡♣ ✶

✽ ✐❢ #Em(a) = 2m + 4 t❤❡♥ ✾

• ♦ t♦ st❡♣ ✶

✶✵ r❡t✉r♥ ❛

✷✵ ✴ ✸✾

■♠♣❧❡♠❡♥t❛t✐♦♥ ❢♦r m ♦❞❞

❙♦♠❡ r❡❛s♦♥❛❜❧② ❡✣❝✐❡♥t ♣♦✐♥t ❝♦✉♥t✐♥❣ ♦♥ F2n ✐s ♥❡❡❞❡❞✳ ❊❛s② s♦❧✉t✐♦♥✿ ✉s❡ ▼❛❣♠❛✳ ▲❡ss ❡❛s② s♦❧✉t✐♦♥✿ ✉s❡ ❨❡♦❤✬s ●P s❝r✐♣t ❬❨❡♦❪✳ ❍❛r❞❡r s♦❧✉t✐♦♥✿ ✉s❡ ❙❛❣❡ ✇✐t❤ ❚r❛❝ t✐❝❦❡t ★✶✶✹✹✽ ♦r ★✶✶✺✹✽✳ ❍❛r❞❡st s♦❧✉t✐♦♥✿ ✐♠♣❧❡♠❡♥t ✐t ✐♥ ❛ ❈ ❧✐❜r❛r② ❛♥❞ ✐♥t❡r❢❛❝❡ ✐t ❢r♦♠ ❙❛❣❡✳ ❆s ❛ r❡s✉❧t ♦❢ ♦✉r ❡①♣❡r✐♠❡♥ts✱ ✇❡ ❢♦✉♥❞ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ✈❛❧✉❡ ♦❢ a ❢♦r m = 55 ❣✐✈❡s ❛ ✈❛❧✉❡ 4 ♦❢ ❜✐♥❛r② ❑❧♦♦st❡r♠❛♥ s✉♠✳

❚❤❡ ✜♥✐t❡ ✜❡❧❞ F255 ✐s r❡♣r❡s❡♥t❡❞ ❛s F2[x]/(x55 + x11 + x10 + x9 + x7 + x4 + 1)❀ a ✐s t❤❡♥ ❣✐✈❡♥ ❛s a = x53 + x52 + x51 + x50 + x47 + x43 + x41 + x38 + x37 + x35 + x33 + x32 + x30 + x29 + x28 + x27 + x26 + x25 + x24 + x22 + x20 + x19 + x17 + x16 + x15 + x13 + x12 + x5 .

❙♦♠❡ ❝❛❝❤✐♥❣ ♠❛♥❛❣❡♠❡♥t ♣r♦❜❧❡♠s ✐♥ ❙❛❣❡ ❛r❡ s♦♠❡❤♦✇ ❧✐♠✐t✐♥❣✳ ❙❡❡ ❚r❛❝ t✐❝❦❡ts ★✼✶✺ ❛♥❞ ★✶✶✺✷✶✳

✷✶ ✴ ✸✾

m ❊✈❡♥

■♥ t❤❡ ❝❛s❡ ✇❤❡r❡ m ✐s ❡✈❡♥✱ t❤❡ ❝♦♥❞✐t✐♦♥ ❣✐✈❡♥ ❜② ▼❡s♥❛❣❡r ❤❛s ♦♥❧② ❜❡❡♥ s❤♦✇♥ t♦ ❜❡ ♥❡❝❡ss❛r②✳ ■t ✐s ♦❢ ✐♥t❡r❡st t♦ ❝❤❡❝❦ ❝♦♠♣✉t❛t✐♦♥❛❧❧② ✇❤❡t❤❡r ❝♦✉♥t❡r❡①❛♠♣❧❡s ❝❛♥ ❜❡ ❢♦✉♥❞ ❢♦r s♠❛❧❧ ✈❛❧✉❡s ♦❢ m✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❝♦♠♣✉t✐♥❣ ❛❧❧ ❡❧❡♠❡♥ts ❣✐✈✐♥❣ ❛ s♣❡❝✐✜❝ ✈❛❧✉❡✱ r❛t❤❡r t❤❛♥ ❧♦♦❦✐♥❣ ❢♦r ♦♥❡✱ ♠✉st ❜❡ ❤❛♥❞❧❡❞ ❞✐✛❡r❡♥❧②✳ ❆ ❢❛st ❲❛❧s❤✲❍❛❞❛♠❛r❞ tr❛♥s❢♦r♠ s❤♦✉❧❞ ❜❡ ✉s❡❞✳ ▼♦r❡♦✈❡r✱ t♦ t❡st ❛❧❧ ❢✉♥❝t✐♦♥s ✐♥ t❤❡ ❢❛♠✐❧② ❞❡✜♥❡❞ ❜② ▼❡s♥❛❣❡r✿ fa,b(x) = Trn

1

• ax2m−1

+ Tr2

1

• bx

2n−1 3

• ,

✐t ✐s ❡♥♦✉❣❤ t♦ s❡t b = 1 ❛♥❞ t❡st ♦♥❡ a ✐♥ ❡❛❝❤ ❝②❝❧♦t♦♠✐❝ ❝❧❛ss✳

✷✷ ✴ ✸✾

❆❧❣♦r✐t❤♠ ❢♦r m ❡✈❡♥

❚❤❡ t❡st ❛❧❣♦r✐t❤♠ ✐s ❛s ❢♦❧❧♦✇s✿

✶

❈♦♠♣✉t❡ {| Km(a) | a ∈ F2m} ✇✐t❤ ❛ ❢❛st ❲❛❧s❤✲❍❛❞❛♠❛r❞ tr❛♥s❢♦r♠ ♦❢ Tr m1/x✳

✷

❙❡❧❡❝t ♦♥❡ a ✐♥ ❡❛❝❤ ❝②❝❧♦t♦♠✐❝ ❝❧❛ss s✉❝❤ t❤❛t Km(a) = 4✳

✸

❋♦r ❡❛❝❤ a ❝♦♠♣✉t❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥✳

✹

❋♦r ❡❛❝❤ ❢✉♥❝t✐♦♥ ❝❤❡❝❦ ✐ts ❜❡♥t♥❡ss ✉s✐♥❣ ❛ ❢❛st ❲❛❧s❤✲❍❛❞❛♠❛r❞ tr❛♥s❢♦r♠✳ ■♥ st❡♣ ✷ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❡✣❝✐❡♥t❧② t❡st ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ a ✐♥ ❡❛❝❤ ❝②❝❧♦t♦♠✐❝ ❝❧❛ss ✉s✐♥❣ ♥❡❝❦❧❛❝❡s ❬❉✉✈✽✽✱ ❘❙❲✾✷✱ ❘✉s✵✸❪✳ ❙t❡♣ ✸ ✐s t❤❡ ♠♦st t✐♠❡ ❝♦♥s✉♠✐♥❣ ♦♥❡✳ ❙t❡♣ ✹ ✐s t❤❡ ♠♦st ♠❡♠♦r② ❝♦♥s✉♠✐♥❣ ♦♥❡✳

✷✸ ✴ ✸✾

❊①♣❡r✐♠❡♥t❛❧ ❘❡s✉❧ts

❚❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✇❛s ♠❛❞❡ ✉s✐♥❣ ❙❛❣❡ ❬❙+✶✶❪ ❛♥❞ ❈②t❤♦♥ ❬❇❈❙✶✵❪✱ ♣❡r❢♦r♠✐♥❣ ❞✐r❡❝t ❝❛❧❧s t♦ ●✐✈❛r♦ ❬❉●●+✵✽❪✱ ◆❚▲ ❬❙❤♦✵✽❪ ❛♥❞ ❣❢✷① ❬❇●❚❩✵✽❪ ❧✐❜r❛r✐❡s ❢♦r ❡✣❝✐❡♥t ♠❛♥✐♣✉❧❛t✐♦♥ ♦❢ ✜♥✐t❡ ✜❡❧❞ ❡❧❡♠❡♥ts ❛♥❞ ❝♦♥str✉❝t✐♦♥ ♦❢ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s✳ m ◆❜✳ ♦❢ ❝②❝❧♦t♦♠✐❝ ❝❧❛ss❡s ❚✐♠❡ ❆❧❧ ❜❡♥t❄ ✹ ✶ ❁✶s ②❡s ✻ ✶ ❁✶s ②❡s ✽ ✷ ❁✶s ②❡s ✶✵ ✸ ✹s ②❡s ✶✷ ✻ ✶✸✵s ②❡s ✶✹ ✽ ✸✵✵✵s ②❡s ✶✻ ✶✹ ✽✷✵✵✵s ②❡s ✶✽ ✷✵ ✲ ✲

✷✹ ✴ ✸✾

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✳

✷✺ ✴ ✸✾

❈❤❛r♣✐♥✲●♦♥❣ ❝r✐t❡r✐♦♥

❈❤❛r♣✐♥ ❛♥❞ ●♦♥❣ ❬❈●✵✽❪ ❣❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❤②♣❡r❜❡♥t♥❡ss ❢♦r ❛ ❧❛r❣❡ ❝❧❛ss ♦❢ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s✳ ❚❤❡♦r❡♠ ✭❬❈●✵✽❪✮ ▲❡t far(x) =

• r∈R

Trn

1

• arxr(2m−1)

, ar ∈ F2m✱ ✇❤❡r❡ R ⊆ S✳ ▲❡t gar(x) =

r∈R Trm 1 (arDr(x))✳ ❚❤❡♥ far ✐s

❤②♣❡r❜❡♥t ✐✛

• x∈F∗

2m

χ

• Trm

1

• x−1

+ gar(x)

• = 2m − 2 wt(gar) − 1 .

✷✻ ✴ ✸✾

▼❡s♥❛❣❡r ❝r✐t❡r✐♦♥

▼❡s♥❛❣❡r ❬▼❡s✶✵❪ ❣❛✈❡ ❛ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❤②♣❡r❜❡♥t♥❡ss ❢♦r ❛♥♦t❤❡r ❧❛r❣❡ ❝❧❛ss ♦❢ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ❚❤❡♦r❡♠ ✭❬▼❡s✶✵❪✮ ▲❡t m ❜❡ ♦❞❞✱ b ❛ ♣r✐♠✐t✐✈❡ ❡❧❡♠❡♥t ♦❢ F∗

4 ❛♥❞

far,b(x) =

• r∈R

Trn

1

• arxr(2m−1)

+ Tr2

1

• bx

2n−1 3

• .

❚❤❡♥ far,b ✐s ❤②♣❡r❜❡♥t ✐✛

✶

• x∈F∗

2m,Trm 1 (x−1)=1

χ (gar(D3(x))) = −2❀

✷

• x∈F∗

2m

χ

• Trm

1

• x−1

+ gar(D3(x))

• = 2m − 2 wt(gar ◦ D3) + 3✳

✷✼ ✴ ✸✾

▲✐s♦♥➙❦✬s ✐❞❡❛

▲✐s♦♥➙❦ ❬▲✐s✵✽❪ ❡①t❡♥❞❡❞ t❤❡ ✐❞❡❛s ♦❢ ▲❛❝❤❛✉❞ ❛♥❞ ❲♦❧❢♠❛♥♥ t♦ r❡❢♦r♠✉❧❛t❡ t❤❡ ❈❤❛r♣✐♥✲●♦♥❣ ❝r✐t❡r✐♦♥ ✐♥ t❡r♠s ♦❢ ❤②♣❡r❡❧❧✐♣t✐❝ ❝✉r✈❡s✳ Pr♦♣♦s✐t✐♦♥ ▲❡t f : F2m → F2m ❜❡ ❛ ❢✉♥❝t✐♦♥ s✉❝❤ t❤❛t f(0) = 0✱ g = Trm

1 (f) ❛♥❞ Gf

❜❡ t❤❡ ✭❛✣♥❡✮ ❝✉r✈❡ ❞❡✜♥❡❞ ♦✈❡r F2m ❜② Gf : y2 + y = f(x) . ❚❤❡♥

• x∈F∗

2m

χ (g(x)) (= 2m − 1 − 2 wt(g)) = −2m − 1 + #Gf .

✷✽ ✴ ✸✾

❘❡❢♦r♠✉❧❛t✐♦♥ ♦❢ ❈● ❝r✐t❡r✐♦♥

❆♣♣❧✐❡❞ t♦ ❈● ❝r✐t❡r✐♦♥ ✇❡ ❣❡t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❤❛r❛❝t❡r✐③❛t✐♦♥✳ ❚❤❡♦r❡♠ ✭❬▲✐s✶✶❪✮ ▲❡t Har ❛♥❞ Gar ❜❡ t❤❡ ✭❛✣♥❡✮ ❝✉r✈❡s ❞❡✜♥❡❞ ♦✈❡r F2m ❜② Har : y2 + xy = x + x2

r∈R

arDr(x) , Gar : y2 + y =

• r∈R

arDr(x) . ❚❤❡♥ far ✐s ❤②♣❡r❜❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ #Har − #Gar = −1 .

✷✾ ✴ ✸✾

❈♦♠♣❧❡①✐t②

❚❤❡ s♠♦♦t❤ ♣r♦❥❡❝t✐✈❡ ♠♦❞❡❧s ♦❢ t❤❡ ❝✉r✈❡s Har ❛♥❞ Gar ❛r❡ ❤②♣❡r❡❧❧✐♣t✐❝✳ ❚❤❡ ♣♦❧②♥♦♠✐❛❧ ❞❡✜♥✐♥❣ Har ✭r❡s♣❡❝t✐✈❡❧② Gar✮ ✐s ♦❢ ❞❡❣r❡❡ rmax + 2 ✭r❡s♣❡❝t✐✈❡❧② rmax✮✱ s♦ t❤❡ ❝✉r✈❡ ✐s ♦❢ ❣❡♥✉s (rmax + 1)/2 ✭r❡s♣❡❝t✐✈❡❧② (rmax − 1)/2✮✳ ❚❤❡ ❝♦♠♣❧❡①✐t② ❢♦r t❡st✐♥❣ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ✐♥ t❤✐s ❢❛♠✐❧② ✐s t❤❡♥ ❞♦♠✐♥❛t❡❞ ❜② t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ ❛ ❝✉r✈❡ ♦❢ ❣❡♥✉s (rmax + 1)/2✱ ✇❤✐❝❤ ✐s ♣♦❧②♥♦♠✐❛❧ ✐♥ m ❢♦r ❛ ✜①❡❞ rmax ✭❛♥❞ s♦ ✜①❡❞ ❣❡♥❡r❛ ❢♦r t❤❡ ❝✉r✈❡s Har ❛♥❞ Gar✮✳ ❚❤❡♦r❡♠ ▲❡t H ❜❡ ❛♥ ❤②♣❡r❡❧❧✐♣t✐❝ ❝✉r✈❡ ♦❢ ❣❡♥✉s g ❞❡✜♥❡❞ ♦✈❡r F2m✳ ❚❤❡r❡ ❡①✐st ❛♥ ❛❧❣♦r✐t❤♠ t♦ ❝♦♠♣✉t❡ t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ H ✐♥ O(g3m3(g2 + log2 m log log m) log gm log log gm) ❜✐t ♦♣❡r❛t✐♦♥s ❛♥❞ O(g4m3) ♠❡♠♦r②✳

✸✵ ✴ ✸✾

❘❡❢♦r♠✉❧❛t✐♦♥ ♦❢ ▼❡s♥❛❣❡r ❝r✐t❡r✐♦♥

❚❤❡♦r❡♠ ▲❡t H3

ar ❛♥❞ G3 ar ❜❡ t❤❡ ✭❛✣♥❡✮ ❝✉r✈❡s ❞❡✜♥❡❞ ♦✈❡r F2m ❜②

H3

ar : y2 + xy = x + x2 r∈R

arDr(D3(x)) , G3

ar : y2 + y =

• r∈R

arDr(D3(x)) . ■❢ b ✐s ❛ ♣r✐♠✐t✐✈❡ ❡❧❡♠❡♥t ♦❢ F4✱ t❤❡♥ far,b ✐s ❤②♣❡r❜❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ #H3

ar − #G3 ar = 3 .

❲❡ ❤❛✈❡ t♦ ❝♦♠♣✉t❡ t❤❡ ❝❛r❞✐♥❛❧✐t✐❡s ♦❢ t✇♦ ❝✉r✈❡s ♦❢ ❣❡♥❡r❛ (3rmax + 1)/2 ❛♥❞ (3rmax − 1)/2✳

✸✶ ✴ ✸✾

▲✐tt❧❡ tr✐❝❦

❯s✐♥❣ t❤❡ ❢❛❝t t❤❛t x → D3(x) = x3 + x ✐s ❛ ♣❡r♠✉t❛t✐♦♥ ✇❤❡♥ m ✐s ♦❞❞✳ ❚❤❡♦r❡♠ ■❢ b ✐s ❛ ♣r✐♠✐t✐✈❡ ❡❧❡♠❡♥t ♦❢ F4✱ t❤❡♥ far,b ✐s ❤②♣❡r❜❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ #G3

ar − 1

2 (#Gar + #Har) = −3 2 . ❚❤✐s ✐s s❧✐❣❤t❧② ♠♦r❡ ❡✣❝✐❡♥t✳

✸✷ ✴ ✸✾

❘❡❢❡r❡♥❝❡s ■

❖♠r❛♥ ❆❤♠❛❞✐ ❛♥❞ ❘♦❜❡rt ●r❛♥❣❡r✳ ❆♥ ❡✣❝✐❡♥t ❞❡t❡r♠✐♥✐st✐❝ t❡st ❢♦r ❑❧♦♦st❡r♠❛♥ s✉♠ ③❡r♦s✳ ❈♦❘❘✱ ❛❜s✴✶✶✵✹✳✸✽✽✷✱ ✷✵✶✶✳ ❏✳ ❆r♥❞t✳ ▼❛tt❡rs ❈♦♠♣✉t❛t✐♦♥❛❧✿ ■❞❡❛s✱ ❆❧❣♦r✐t❤♠s✱ ❙♦✉r❝❡ ❈♦❞❡✳ ❙♣r✐♥❣❡r✱ ✷✵✶✵✳ ❘✳ ❇r❛❞s❤❛✇✱ ❈✳ ❈✐tr♦✱ ❛♥❞ ❉✳❙✳ ❙❡❧❥❡❜♦t♥✳ ❈②t❤♦♥✿ t❤❡ ❜❡st ♦❢ ❜♦t❤ ✇♦r❧❞s✳ ❈✐❙❊ ✷✵✶✶ ❙♣❡❝✐❛❧ P②t❤♦♥ ■ss✉❡✱ ♣❛❣❡ ✷✺✱ ✷✵✶✵✳ ❘✐❝❤❛r❞ P✳ ❇r❡♥t✱ P✐❡rr✐❝❦ ●❛✉❞r②✱ ❊♠♠❛♥✉❡❧ ❚❤♦♠é✱ ❛♥❞ P❛✉❧ ❩✐♠♠❡r♠❛♥♥✳ ❋❛st❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐♥ ●❋✭✷✮❬①❪✳ ■♥ ❆❧❢r❡❞ ❏✳ ✈❛♥ ❞❡r P♦♦rt❡♥ ❛♥❞ ❆♥❞r❡❛s ❙t❡✐♥✱ ❡❞✐t♦rs✱ ❆◆❚❙✱ ✈♦❧✉♠❡ ✺✵✶✶ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✶✺✸✕✶✻✻✳ ❙♣r✐♥❣❡r✱ ✷✵✵✽✳ ■✳ ❋✳ ❇❧❛❦❡✱ ●✳ ❙❡r♦✉ss✐✱ ❛♥❞ ◆✳ P✳ ❙♠❛rt✳ ❊❧❧✐♣t✐❝ ❝✉r✈❡s ✐♥ ❝r②♣t♦❣r❛♣❤②✱ ✈♦❧✉♠❡ ✷✻✺ ♦❢ ▲♦♥❞♦♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t② ▲❡❝t✉r❡ ◆♦t❡ ❙❡r✐❡s✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❈❛♠❜r✐❞❣❡✱ ✷✵✵✵✳ ❘❡♣r✐♥t ♦❢ t❤❡ ✶✾✾✾ ♦r✐❣✐♥❛❧✳

✸✸ ✴ ✸✾

❘❡❢❡r❡♥❝❡s ■■

P❛s❝❛❧❡ ❈❤❛r♣✐♥ ❛♥❞ ●✉❛♥❣ ●♦♥❣✳ ❍②♣❡r❜❡♥t ❢✉♥❝t✐♦♥s✱ ❑❧♦♦st❡r♠❛♥ s✉♠s✱ ❛♥❞ ❉✐❝❦s♦♥ ♣♦❧②♥♦♠✐❛❧s✳ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ■♥❢♦r♠❛t✐♦♥ ❚❤❡♦r②✱ ✺✹✭✾✮✿✹✷✸✵✕✹✷✸✽✱ ✷✵✵✽✳ P❛s❝❛❧❡ ❈❤❛r♣✐♥✱ ❚♦r ❍❡❧❧❡s❡t❤✱ ❛♥❞ ❱✐❝t♦r ❩✐♥♦✈✐❡✈✳ ❉✐✈✐s✐❜✐❧✐t② ♣r♦♣❡rt✐❡s ♦❢ ❝❧❛ss✐❝❛❧ ❜✐♥❛r② ❑❧♦♦st❡r♠❛♥ s✉♠s✳ ❉✐s❝r❡t❡ ▼❛t❤❡♠❛t✐❝s✱ ✸✵✾✭✶✷✮✿✸✾✼✺✕✸✾✽✹✱ ✷✵✵✾✳ ❉❛✈✐❞ ❆✳ ❈♦①✳ Pr✐♠❡s ♦❢ t❤❡ ❢♦r♠ x2 + ny2✳ ❆ ❲✐❧❡②✲■♥t❡rs❝✐❡♥❝❡ P✉❜❧✐❝❛t✐♦♥✳ ❏♦❤♥ ❲✐❧❡② ✫ ❙♦♥s ■♥❝✳✱ ◆❡✇ ❨♦r❦✱ ✶✾✽✾✳ ❋❡r♠❛t✱ ❝❧❛ss ✜❡❧❞ t❤❡♦r② ❛♥❞ ❝♦♠♣❧❡① ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ❏❡❛♥✲●✉✐❧❧❛✉♠❡ ❉✉♠❛s✱ ❚❤✐❡rr② ●❛✉t✐❡r✱ P❛s❝❛❧ ●✐♦r❣✐✱ ❏❡❛♥✲▲♦✉✐s ❘♦❝❤✱ ❛♥❞ ●✐❧❧❡s ❱✐❧❧❛r❞✳

• ✐✈❛r♦✲✸✳✷✳✶✸r❝✶✿ ❈✰✰ ❧✐❜r❛r② ❢♦r ❛r✐t❤♠❡t✐❝ ❛♥❞ ❛❧❣❡❜r❛✐❝ ❝♦♠♣✉t❛t✐♦♥s✱ ❙❡♣t❡♠❜❡r ✷✵✵✽✳

❤tt♣✿✴✴❧❥❦✳✐♠❛❣✳❢r✴❈❆❙❨❙✴▲❖●■❈■❊▲❙✴❣✐✈❛r♦✴✳ ❏♦❤♥ ❋r❛♥❝✐s ❉✐❧❧♦♥✳ ❊❧❡♠❡♥t❛r② ❍❛❞❛♠❛r❞ ❉✐✛❡r❡♥❝❡ ❙❡ts✳ Pr♦◗✉❡st ▲▲❈✱ ❆♥♥ ❆r❜♦r✱ ▼■✱ ✶✾✼✹✳ ❚❤❡s✐s ✭P❤✳❉✳✮✕❯♥✐✈❡rs✐t② ♦❢ ▼❛r②❧❛♥❞✱ ❈♦❧❧❡❣❡ P❛r❦✳

✸✹ ✴ ✸✾

❘❡❢❡r❡♥❝❡s ■■■

❏❡❛♥✲P✐❡rr❡ ❉✉✈❛❧✳

• é♥ér❛t✐♦♥ ❞✬✉♥❡ s❡❝t✐♦♥ ❞❡s ❝❧❛ss❡s ❞❡ ❝♦♥❥✉❣❛✐s♦♥ ❡t ❛r❜r❡ ❞❡s ♠♦ts ❞❡ ▲②♥❞♦♥ ❞❡

❧♦♥❣✉❡✉r ❜♦r♥é❡✳ ❚❤❡♦r✳ ❈♦♠♣✉t✳ ❙❝✐✳✱ ✻✵✿✷✺✺✕✷✽✸✱ ✶✾✽✽✳ ❆♥❞r❡❛s ❊♥❣❡✳ ❊❧❧✐♣t✐❝ ❈✉r✈❡s ❛♥❞ ❚❤❡✐r ❆♣♣❧✐❝❛t✐♦♥s t♦ ❈r②♣t♦❣r❛♣❤②✿ ❆♥ ■♥tr♦❞✉❝t✐♦♥✳ ❙♣r✐♥❣❡r✱ ✶st ❡❞✐t✐♦♥✱ ❆✉❣✉st ✶✾✾✾✳ ❘♦❜❡rt ❍❛r❧❡②✳ ❆s②♠♣t♦t✐❝❛❧❧② ♦♣t✐♠❛❧ ♣✲❛❞✐❝ ♣♦✐♥t✲❝♦✉♥t✐♥❣✳ ❊♠❛✐❧ t♦ ◆▼❇❘❚❍❘❨ ❧✐st✱ ❉❡❝❡♠❜❡r ✷✵✵✷✳ ❤tt♣✿✴✴❧✐sts❡r✈✳♥♦❞❛❦✳❡❞✉✴❝❣✐✲❜✐♥✴✇❛✳❡①❡❄❆✷❂✐♥❞✵✷✶✷✫▲❂♥♠❜rt❤r②✫❚❂✵✫P❂✶✸✹✸✳ ❚♦r ❍❡❧❧❡s❡t❤ ❛♥❞ ❱✐❝t♦r ❩✐♥♦✈✐❡✈✳ ❖♥ ❧✐♥❡❛r ●♦❡t❤❛❧s ❝♦❞❡s ❛♥❞ ❑❧♦♦st❡r♠❛♥ s✉♠s✳ ❉❡s✳ ❈♦❞❡s ❈r②♣t♦❣r❛♣❤②✱ ✶✼✭✶✲✸✮✿✷✻✾✕✷✽✽✱ ✶✾✾✾✳ ◆✐❝❤♦❧❛s ❑❛t③ ❛♥❞ ❘♦♥ ▲✐✈♥é✳ ❙♦♠♠❡s ❞❡ ❑❧♦♦st❡r♠❛♥ ❡t ❝♦✉r❜❡s ❡❧❧✐♣t✐q✉❡s ✉♥✐✈❡rs❡❧❧❡s ❡♥ ❝❛r❛❝tér✐st✐q✉❡s 2 ❡t 3✳ ❈✳ ❘✳ ❆❝❛❞✳ ❙❝✐✳ P❛r✐s ❙ér✳ ■ ▼❛t❤✳✱ ✸✵✾✭✶✶✮✿✼✷✸✕✼✷✻✱ ✶✾✽✾✳

✸✺ ✴ ✸✾

❘❡❢❡r❡♥❝❡s ■❱

◆❡❛❧ ❑♦❜❧✐t③✳ ❈♦♥str✉❝t✐♥❣ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❝r②♣t♦s②st❡♠s ✐♥ ❝❤❛r❛❝t❡r✐st✐❝ ✷✳ ■♥ ❆❧❢r❡❞ ▼❡♥❡③❡s ❛♥❞ ❙❝♦tt ❆✳ ❱❛♥st♦♥❡✱ ❡❞✐t♦rs✱ ❈❘❨P❚❖✱ ✈♦❧✉♠❡ ✺✸✼ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✶✺✻✕✶✻✼✳ ❙♣r✐♥❣❡r✱ ✶✾✾✵✳ ◆✳ ●✳ ▲❡❛♥❞❡r✳ ▼♦♥♦♠✐❛❧ ❜❡♥t ❢✉♥❝t✐♦♥s✳ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ■♥❢♦r♠❛t✐♦♥ ❚❤❡♦r②✱ ✺✷✭✷✮✿✼✸✽✕✼✹✸✱ ✷✵✵✻✳ P❡tr ▲✐s♦♥❡❦✳ ❖♥ t❤❡ ❝♦♥♥❡❝t✐♦♥ ❜❡t✇❡❡♥ ❑❧♦♦st❡r♠❛♥ s✉♠s ❛♥❞ ❡❧❧✐♣t✐❝ ❝✉r✈❡s✳ ■♥ ❙♦❧♦♠♦♥ ❲✳ ●♦❧♦♠❜✱ ▼❛tt❤❡✇ ●✳ P❛r❦❡r✱ ❆❧❡①❛♥❞❡r P♦tt✱ ❛♥❞ ❆r♥❡ ❲✐♥t❡r❤♦❢✱ ❡❞✐t♦rs✱ ❙❊❚❆✱ ✈♦❧✉♠❡ ✺✷✵✸ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✶✽✷✕✶✽✼✳ ❙♣r✐♥❣❡r✱ ✷✵✵✽✳ P❡tr ▲✐s♦♥➙❦✳ ❍②♣❡r❜❡♥t ❢✉♥❝t✐♦♥s ❛♥❞ ❤②♣❡r❡❧❧✐♣t✐❝ ❝✉r✈❡s✳ ❚❛❧❦ ❣✐✈❡♥ ❛t ❆r✐t❤♠❡t✐❝✱ ●❡♦♠❡tr②✱ ❈r②♣t♦❣r❛♣❤② ❛♥❞ ❈♦❞✐♥❣ ❚❤❡♦r② ✭❆●❈❚✲✶✸✮✱ ▼❛r❝❤ ✷✵✶✶✳

• ✐❧❧❡s ▲❛❝❤❛✉❞ ❛♥❞ ❏❛❝q✉❡s ❲♦❧❢♠❛♥♥✳

❙♦♠♠❡s ❞❡ ❑❧♦♦st❡r♠❛♥✱ ❝♦✉r❜❡s ❡❧❧✐♣t✐q✉❡s ❡t ❝♦❞❡s ❝②❝❧✐q✉❡s ❡♥ ❝❛r❛❝tér✐st✐q✉❡ 2✳ ❈✳ ❘✳ ❆❝❛❞✳ ❙❝✐✳ P❛r✐s ❙ér✳ ■ ▼❛t❤✳✱ ✸✵✺✭✷✵✮✿✽✽✶✕✽✽✸✱ ✶✾✽✼✳

✸✻ ✴ ✸✾

❘❡❢❡r❡♥❝❡s ❱

• ✐❧❧❡s ▲❛❝❤❛✉❞ ❛♥❞ ❏❛❝q✉❡s ❲♦❧❢♠❛♥♥✳

❚❤❡ ✇❡✐❣❤ts ♦❢ t❤❡ ♦rt❤♦❣♦♥❛❧s ♦❢ t❤❡ ❡①t❡♥❞❡❞ q✉❛❞r❛t✐❝ ❜✐♥❛r② ●♦♣♣❛ ❝♦❞❡s✳ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ■♥❢♦r♠❛t✐♦♥ ❚❤❡♦r②✱ ✸✻✭✸✮✿✻✽✻Ð✻✾✷✱ ✶✾✾✵✳ ❙✐❤❡♠ ▼❡s♥❛❣❡r✳ ❍②♣❡r✲❜❡♥t ❜♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ✇✐t❤ ♠✉❧t✐♣❧❡ tr❛❝❡ t❡r♠s✳ ■♥ ▼✳ ❆♥✇❛r ❍❛s❛♥ ❛♥❞ ❚♦r ❍❡❧❧❡s❡t❤✱ ❡❞✐t♦rs✱ ❲❆■❋■✱ ✈♦❧✉♠❡ ✻✵✽✼ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✾✼✕✶✶✸✳ ❙♣r✐♥❣❡r✱ ✷✵✶✵✳ ❙✐❤❡♠ ▼❡s♥❛❣❡r✳ ❆ ♥❡✇ ❝❧❛ss ♦❢ ❜❡♥t ❛♥❞ ❤②♣❡r✲❜❡♥t ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ✐♥ ♣♦❧②♥♦♠✐❛❧ ❢♦r♠s✳ ❉❡s✳ ❈♦❞❡s ❈r②♣t♦❣r❛♣❤②✱ ✺✾✭✶✲✸✮✿✷✻✺✕✷✼✾✱ ✷✵✶✶✳ ❙✐❤❡♠ ▼❡s♥❛❣❡r✳ ❙❡♠✐✲❜❡♥t ❢✉♥❝t✐♦♥s ❢r♦♠ ❉✐❧❧♦♥ ❛♥❞ ◆✐❤♦ ❡①♣♦♥❡♥ts✱ ❑❧♦♦st❡r♠❛♥ s✉♠s ❛♥❞ ❉✐❝❦s♦♥ ♣♦❧②♥♦♠✐❛❧s✳ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ■♥❢♦r♠❛t✐♦♥ ❚❤❡♦r②✱ ❚♦ ❛♣♣❡❛r✳ ❋r❛♥❦ ❘✉s❦❡②✱ ❈❛r❧❛ ❉✳ ❙❛✈❛❣❡✱ ❛♥❞ ❚❡rr② ▼✐♥ ❨✐❤ ❲❛♥❣✳

• ❡♥❡r❛t✐♥❣ ♥❡❝❦❧❛❝❡s✳

❏✳ ❆❧❣♦r✐t❤♠s✱ ✶✸✭✸✮✿✹✶✹✕✹✸✵✱ ✶✾✾✷✳

✸✼ ✴ ✸✾

❘❡❢❡r❡♥❝❡s ❱■

❋r❛♥❦ ❘✉s❦❡②✳ ❈♦♠❜✐♥❛t♦r✐❛❧ ●❡♥❡r❛t✐♦♥✳ ❯♥♣✉❜❧✐s❤❡❞ ♠❛♥✉s❝r✐♣t✱ ✷✵✵✸✳ ❲♦r❦✐♥❣ ❱❡rs✐♦♥ ✭✶❥✲❈❙❈ ✹✷✺✴✺✷✵✮✳ ❲✳ ❆✳ ❙t❡✐♥ ❡t ❛❧✳ ❙❛❣❡ ▼❛t❤❡♠❛t✐❝s ❙♦❢t✇❛r❡ ✭❱❡rs✐♦♥ ✹✳✼✮✳ ❚❤❡ ❙❛❣❡ ❉❡✈❡❧♦♣♠❡♥t ❚❡❛♠✱ ✷✵✶✶✳ ❤tt♣✿✴✴✇✇✇✳s❛❣❡♠❛t❤✳♦r❣✳ ❘❡♥é ❙❝❤♦♦❢✳ ◆♦♥s✐♥❣✉❧❛r ♣❧❛♥❡ ❝✉❜✐❝ ❝✉r✈❡s ♦✈❡r ✜♥✐t❡ ✜❡❧❞s✳ ❏✳ ❈♦♠❜✳ ❚❤❡♦r②✱ ❙❡r✳ ❆✱ ✹✻✭✷✮✿✶✽✸✕✷✶✶✱ ✶✾✽✼✳ ❱✐❝t♦r ❙❤♦✉♣✳ ◆❚▲ ✺✳✹✳✷✿ ❆ ❧✐❜r❛r② ❢♦r ❞♦✐♥❣ ♥✉♠❜❡r t❤❡♦r②✱ ▼❛r❝❤ ✷✵✵✽✳ ✇✇✇✳s❤♦✉♣✳♥❡t✴♥t❧✳ ❋r❡❞❡r✐❦ ❱❡r❝❛✉t❡r❡♥✳ ❈♦♠♣✉t✐♥❣ ③❡t❛ ❢✉♥❝t✐♦♥s ♦❢ ❝✉r✈❡s ♦✈❡r ✜♥✐t❡ ✜❡❧❞s✳ P❤❉ t❤❡s✐s✱ ❑❛t❤♦❧✐❡❦❡ ❯♥✐✈❡rs✐t❡✐t ▲❡✉✈❡♥✱ ✷✵✵✸✳

✸✽ ✴ ✸✾

❘❡❢❡r❡♥❝❡s ❱■■

❨❡♦❤✳

• P✴P❛r✐ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ♣♦✐♥t ❝♦✉♥t✐♥❣ ✐♥ ❝❤❛r❛❝t❡r✐st✐❝ ✷✳

❤tt♣✿✴✴♣❛❣❡s✳❝s✳✇✐s❝✳❡❞✉✴⑦②❡♦❤✴♥t✴s❛t♦❤✲❢❣❤✳❣♣✳

✸✾ ✴ ✸✾