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❇❡♥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 : F 2 n → F 2 ✳ P♦❧②♥♦♠✐❛❧ ❢♦r♠ f ❤❛s ❛ ✉♥✐q✉❡ tr❛❝❡ ❡①♣❛♥s✐♦♥ ♦❢ t❤❡ ❢♦r♠✿ � � a j x j � Tr o ( j ) + ǫ (1 + x 2 n − 1 ) , f ( x ) = a j ∈ F 2 o ( j ) , 1 j ∈ Γ n ✇❤❡r❡ Γ n ✐s t❤❡ s❡t ♦❢ ✐♥t❡❣❡rs ♦❜t❛✐♥❡❞ ❜② ❝❤♦♦s✐♥❣ ♦♥❡ ❡❧❡♠❡♥t ✐♥ ❡❛❝❤ ❝②❝❧♦t♦♠✐❝ ❝❧❛ss ♠♦❞✉❧♦ 2 n − 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 2 n − 1 − 2 n/ 2 − 1 ✱ ✐✳❡✳ ✐s ❛s ❢❛r ❛s ♣♦ss✐❜❧❡ ♦❢ ❛❧❧ ❛✣♥❡ ❢✉♥❝t✐♦♥s✳ ✸ ✴ ✸✾

❲❛❧s❤✲❍❛❞❛♠❛r❞ tr❛♥s❢♦r♠ ❲❛❧s❤✲❍❛❞❛♠❛r❞ tr❛♥s❢♦r♠ ❋♦r ω ∈ F 2 n ✱ t❤❡ ❲❛❧s❤✲❍❛❞❛♠❛r❞ tr❛♥s❢♦r♠ ♦❢ f ❛t ω ✐s � 1 ( ωx ) . ( − 1) f ( x )+Tr n χ f ( ω ) = � x ∈ F 2 n ✭❍②♣❡r✮✲❜❡♥t♥❡ss ❝❛♥ ❜❡ ❝❤❛r❛❝t❡r✐③❡❞ ✉s✐♥❣ t❤❡ ❲❛❧s❤✲❍❛❞❛♠❛r❞ tr❛♥s❢♦r♠✳ ❇❡♥t♥❡ss✿ ❆ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ f : F 2 n → F 2 ✐s s❛✐❞ t♦ ❜❡ ❜❡♥t ✐❢ n 2 ✱ ❢♦r ❛❧❧ ω ∈ F 2 n ✳ � χ f ( ω ) = ± 2 ❍②♣❡r✲❇❡♥t♥❡ss✿ ❆ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ f : F 2 n → F 2 ✐s s❛✐❞ t♦ ❜❡ ❤②♣❡r✲❜❡♥t ✐❢ t❤❡ ❢✉♥❝t✐♦♥ x �→ f ( x i ) ✐s ❜❡♥t✱ ❢♦r ❡✈❡r② ✐♥t❡❣❡r i ❝♦✲♣r✐♠❡ ✇✐t❤ 2 n − 1 ✳ ✹ ✴ ✸✾

❈♦♠♣✉t✐♥❣ t❤❡ ❲❛❧s❤✲❍❛❞❛♠❛r❞ tr❛♥s❢♦r♠ ❚❤❡ ❲❛❧s❤✲❍❛❞❛♠❛r❞ tr❛♥s❢♦r♠ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ q✉✐t❡ ❡❛s✐❧② ❛♥❞ ❡✣❝✐❡♥t❧②✿ ❛❧❣♦r✐t❤♠ ✐♥ O (2 m m 2 ) ❜✐t ♦♣❡r❛t✐♦♥s ❛♥❞ O (2 m m ) ♠❡♠♦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 ♦♥ F 2 m ❛r❡ � 1 ( ax + 1 ( − 1) Tr m x ) , a ∈ F 2 m . K m ( a ) = x ∈ F 2 m ❘❡♠❛r❦✿ ❚❤❡ ❢✉♥❝t✐♦♥ a �→ K m ( a ) ✐s t❤❡ ❲❛❧s❤✲❍❛❞❛♠❛r❞ tr❛♥s❢♦r♠ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ Tr m 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 ♦❢ K m ( a ) ❣✐✈❡ r✐s❡ t♦ ❜❡♥t ❢✉♥❝t✐♦♥s✳ Pr♦♣♦s✐t✐♦♥ ✭▼♦♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s❬❉✐❧✼✹✱ ▲❲✾✵✱ ▲❡❛✵✻✱ ❈●✵✽❪✮ ▲❡t f : F 2 n → F 2 ❜❡ ❞❡✜♥❡❞ ❛s � ax r (2 m − 1) � , gcd( r, 2 m + 1) = 1 . f ( x ) = Tr n 1 ❚❤❡♥ f ✐s ❤②♣❡r✲❜❡♥t ✐✛ K m ( 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 : F 2 n → F 2 ❜❡ ❞❡✜♥❡❞ ❛s � ax r (2 m − 1) � � � 2 n − 1 , gcd( r, 2 m + 1) = 1 . f ( x ) = Tr n + Tr 2 bx 3 1 1 ■❢ m ✐s ♦❞❞✱ t❤❡♥ f ✐s ❤②♣❡r❜❡♥t ✐✛ K m ( 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 { K m ( a ) , a ∈ F 2 m } ✐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 ∈ F 2 m ✳ ❚❤❡♥ K m ( a ) ≡ 0 (mod 8) ✐❢ ❛♥❞ ♦♥❧② ✐❢ Tr m 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 ∗ 2 m ✳ ❙✉♣♣♦s❡ t❤❛t t❤❡r❡ ❡①✐sts 2 m s✉❝❤ t❤❛t a = b 4 + b 3 ✳ t ∈ F ∗ ■❢ m ✐s ♦❞❞✱ t❤❡♥ K m ( a ) ≡ 1 (mod 3) ✳ ■❢ m ✐s ❡✈❡♥✱ t❤❡♥ K m ( a ) ≡ 0 (mod 3) ✐❢ Tr m 1 ( b ) = 0 ❛♥❞ K m ( a ) ≡ − 1 (mod 3) ✐❢ Tr m 1 ( b ) = 1 ✳ Pr♦♣♦s✐t✐♦♥ ✭❬❈❍❩✵✾❪✮ ▲❡t a ∈ F ∗ 2 m ✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿ � a 1 / 3 � ■❢ m ✐s ♦❞❞✱ t❤❡♥ K m ( a ) ≡ 1 (mod 3) ✐❢ ❛♥❞ ♦♥❧② ✐❢ Tr m = 0 ✳ 1 (1+ b ) 4 ❢♦r s♦♠❡ b ∈ F ∗ b ❚❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ a = 2 m ✳ ■❢ m ✐s ❡✈❡♥✱ t❤❡♥ K m ( a ) ≡ 1 (mod 3) ✐❢ ❛♥❞ ♦♥❧② ✐❢ a = b 3 ❢♦r s♦♠❡ b s✉❝❤ t❤❛t Tr m 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 : y 2 + xy = x 3 + bx 2 + a , ✇✐t❤ a � = 0 ❛♥❞ j ( E ) = 1 /a ✳ ▼♦r❡♦✈❡r ✐ts ✜rst ❞✐✈✐s✐♦♥ ♣♦❧②♥♦♠✐❛❧s ❛r❡ ❬❑♦❜✾✵✱ ❇❙❙✵✵❪ f 1 ( x ) = 1 , f 2 ( x ) = x, f 3 ( x ) = x 4 + x 3 + a, f 4 ( x ) = x 6 + ax 2 . ✶✶ ✴ ✸✾