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❈♦✉♥t✐♥❣ r❡❞✉❝✐❜❧❡✱ ♣♦✇❡r❢✉❧✱ ❛♥❞ r❡❧❛t✐✈❡❧② ✐rr❡❞✉❝✐❜❧❡ ♠✉❧t✐✈❛r✐❛t❡ ♣♦❧②♥♦♠✐❛❧s ♦✈❡r ✜♥✐t❡ ✜❡❧❞s

❆❧❢r❡❞♦ ❱✐♦❧❛

❯♥✐✈❡rs✐❞❛❞ ❞❡ ❧❛ ❘❡♣ú❜❧✐❝❛✱ ❯r✉❣✉❛②

❈❛♥❛❉❆▼ ✷✵✶✸

❏♦✐♥t ✇♦r❦ ✇✐t❤ ❏♦❛❝❤✐♠ ✈♦♥ ③✉r ●❛t❤❡♥ ❛♥❞ ❑♦♥st❛♥t✐♥ ❩✐❡❣❧❡r

❚❤❡ ♣r♦❜❧❡♠✳

▼♦st ✉♥✐✈❛r✐❛t❡ ♣♦❧②♥♦♠✐❛❧s ♦✈❡r ❛ ✜♥✐t❡ ✜❡❧❞ ❛r❡ r❡❞✉❝✐❜❧❡✳ ❘❛♥❞♦♠ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ✉♣ t♦ ♥ ❛r❡ ✐rr❡❞✉❝✐❜❧❡ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ❛❜♦✉t ✶❂♥✳ ■♥ t✇♦ ♦r ♠♦r❡ ✈❛r✐❛❜❧❡s✱ ♠♦st ♣♦❧②♥♦♠✐❛❧s ❛r❡ ✐rr❡❞✉❝✐❜❧❡✳ ❬✈♦♥ ③✉r ●❛t❤❡♥ ✷✵✵✽❪✱ ❣✐✈❡s ♣r❡❝✐s❡ ❛♣♣r♦①✐♠❛t✐♦♥s ✇✐t❤ ❡①♣♦♥❡♥t✐❛❧❧② ❞❡❝r❡❛s✐♥❣ ❡rr♦r t❡r♠ ✭❜✐✈❛r✐❛t❡✮✳ ❬❇♦❞✐♥ ✷✵✵✽❪✱ ❬❇♦❞✐♥ ✷✵✶✵❪ st✉❞✐❡s ❜✐✈❛r✐❛t❡ ❛♥❞ ❣❡♥❡r❛❧ ❝❛s❡✳ ■♥ ❙❡❝t✐♦♥ ✸✳✻ ♦❢ ❬▼✉❧❧❡♥ ❛♥❞ P❛♥❛r✐♦ ✷✵✶✸❪✱ t❤❡ ♥❡✇ ❍❛♥❞❜♦♦❦ ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s✱ t❤❡r❡ ✐s ❛ s✉r✈❡② ♦❢ t❤❡s❡ r❡s✉❧ts✳

▼❛✐♥ ❝♦♥tr✐❜✉t✐♦♥s✳

❊①❛❝t ❢♦r♠✉❧❛s ❢♦r t❤❡ ♥✉♠❜❡rs ♦❢ r❡❞✉❝✐❜❧❡✱ s✲♣♦✇❡r❢✉❧✱ ❛♥❞ r❡❧❛t✐✈❡❧② ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s✳ ❚❤❡ ❧❛tt❡r ❛❧s♦ ②✐❡❧❞s t❤❡ ♥✉♠❜❡r ♦❢ ❛❜s♦❧✉t❡❧② ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s✳ ❖❢ t❤❡s❡ ❘✳❱✳✱ ♦♥❧② r❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s ❤❛❞ ❜❡❡♥ tr❡❛t❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✱ ✉s✉❛❧❧② ✇✐t❤ ♠✉❝❤ ❧❛r❣❡r ❡rr♦r t❡r♠s✳ ❚❤❡ ❢♦r♠✉❧❛s ②✐❡❧❞ s✐♠♣❧❡✱ ②❡t ♣r❡❝✐s❡✱ ❛♣♣r♦①✐♠❛t✐♦♥s t♦ t❤❡s❡ ♥✉♠❜❡rs✱ ✇✐t❤ r❛♣✐❞❧② ❞❡❝❛②✐♥❣ r❡❧❛t✐✈❡ ❡rr♦rs✳

▼❡t❤♦❞♦❧♦❣✐❡s ✉s❡❞✳

• ❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s✳

❉✐✈❡r❣❡♥t ♣♦✇❡r s❡r✐❡s ✭❡①❝❡♣t ❛t ✵✮✳ P♦✇❡r s❡r✐❡s ✇✐t❤ s②♠❜♦❧✐❝ ❝♦❡✣❝✐❡♥ts✱ ♥❛♠❡❧② r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s ✐♥ ❛ ✈❛r✐❛❜❧❡ r❡♣r❡s❡♥t✐♥❣ t❤❡ ✜❡❧❞ s✐③❡✳ ❨✐❡❧❞s ✐♥ ❛ str❛✐❣❤t❢♦r✇❛r❞ ♠❛♥♥❡r ❡①❛❝t ❢♦r♠✉❧❛s✳ ❚❤❡♥✱ ❝♦❡✣❝✐❡♥t ❝♦♠♣❛r✐s♦♥s ②✐❡❧❞ t✐❣❤t ❛♣♣r♦①✐♠❛t✐♦♥s ✇✐t❤ ❡①♣♦♥❡♥t✐❛❧❧② ❞❡❝r❡❛s✐♥❣ r❡❧❛t✐✈❡ ❡rr♦r ✐♥ ❜✐t s✐③❡ ♦❢ t❤❡ ❞❛t❛✳

❈♦♠❜✐♥❛t♦r✐❛❧ ❝♦✉♥t✐♥❣✳

❨✐❡❧❞s ✏s❡❝♦♥❞ ♦r❞❡r✑ ❛♣♣r♦①✐♠❛t✐♦♥s ✇✐t❤ ❡①♣❧✐❝✐t ❝♦♥st❛♥ts ✐♥ t❤❡ ❡rr♦r t❡r♠

◆♦t❛t✐♦♥✳

▲❡t t❤❡ r✐♥❣ ❋❬①✶❀ ✿ ✿ ✿ ❀ ①r❪ ✐♥ r ✕ ✶ ✈❛r✐❛❜❧❡s ♦✈❡r ❛ ✜❡❧❞ ❋✳ P ❛❧❧

r❀♥✭❋✮ ❂ ❢❢ ✷ ❋❬①✶❀ ✿ ✿ ✿ ❀ ①r❪✿ ❞❡❣ ❢ ❂ ♥❣✳

P♦❧②✳ ♦❢ ❞❡❣✳ ❛t ♠♦st ♥ ❢♦r♠ ❛♥ ❋✲✈❡❝t♦r s♣❛❝❡ ♦❢ ❞✐♠❡♥s✐♦♥

❜r❀♥ ❂ ✒r ✰ ♥ r ✓ ❂ ✭r ✰ ♥✮r r✦ ❀ ✭r✰①✮r ❂ ✭r✰①✮✁✭r✶✰①✮ ✁ ✁ ✁ ✭✶✰①✮❀

❖✈❡r ❛ ✜♥✐t❡ ✜❡❧❞ Fq ✇✐t❤ q ❡❧❡♠❡♥ts✱ ✇❡ ❤❛✈❡ ★P ❛❧❧

r❀♥✭Fq✮ ❂ q❜r❀♥ q❜r❀♥✶ ❂ q❜r❀♥✭✶ q❜r✶❀♥✮✿

❚❤❡ ♠♦♥✐❝ ♣♦❧②♥♦♠✐❛❧s ❛r❡ t❤♦s❡ ✇✐t❤ ❧❡❛❞✐♥❣ ❝♦❡✣❝✐❡♥t ✶✳ ▲❡t Pr❀♥✭❋✮ ❂ ❢❢ ✷ P❛❧❧

r❀♥✭❋✮✿ ❢ ✐s ♠♦♥✐❝❣✿

❚❤❡♥ ★Pr❀♥✭Fq✮ ❂ ★P ❛❧❧

r❀♥✭Fq✮

q ✶ ❂ q❜r❀♥✶ ✶ q❜r✶❀♥ ✶ q✶ ✿

• ❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s✳

▲❡t ❈♥ ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts ♦❢ s✐③❡ ♥ ✐♥ ❛ ❝❧❛ss ❈✳ ❚❤❡♥ ❈✭③✮ ❂

❳

♥✕✵

❈♥③♥ ✷ Z✕✵ ③ ✿ ❆ ♣♦✇❡r s❡r✐❡s ✐s ♦r✐❣✐♥❛❧ ✐❢ ✐ts ❝♦♥st❛♥t t❡r♠ ✈❛♥✐s❤❡s✳ ❧♦❣✭✶ ③✮ ❂

❳

♥✕✶

③♥ ♥ ✷ Q ③ ✐s ♦r✐❣✐♥❛❧ ❛♥❞ s✉❜st✐t✉t✐♥❣ ❛ ♣♦✇❡r s❡r✐❡s ❢ ✐♥ ❛♥♦t❤❡r ♣♦✇❡r s❡r✐❡s ❣ ✐s ✇❡❧❧✲❞❡✜♥❡❞ ✐❢ ❢ ✐s ♦r✐❣✐♥❛❧✳ ❚✇♦ ❝♦♠❜✐♥❛t♦r✐❛❧ ❝❧❛ss❡s ❆ ❛♥❞ ❇ ❛r❡ ✐s♦♠♦r♣❤✐❝ ✐❢ t❤❡r❡ ✐s ❛ s✐③❡✲♣r❡s❡r✈✐♥❣ ❜✐❥❡❝t✐♦♥ ❆ ✦ ❇ ♦r ❡q✉✐✈❛❧❡♥t❧② ✐❢ ❆ ❂ ❇✳

❈♦♠❜✐♥❛t♦r✐❛❧ ❝❧❛ss❡s✳

▲❡t ❆✱ ❇ ❜❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ❝❧❛ss❡s✳ ❚❤❡ ❞✐s❥♦✐♥t ✉♥✐♦♥ ✐s ❆ ❴ ❬ ❇ ❂ ❢❢✵❣ ✂ ❆❣ ❬ ❢❢✶❣ ✂ ❇❣✿ ❚❤❡ s✐③❡ ♦❢ ❛♥ ❡❧❡♠❡♥t ✭✵❀ ❛✮ ♦r ✭✶❀ ❜✮ ✐s ❞❡✜♥❡❞ ❛s t❤❡ s✐③❡ ♦❢ ❛ ♦r ❜✱ r❡s♣❡❝t✐✈❡❧②✳ ❙❊◗✭❆✮ ❂ ❢✭☛✶❀ ✿ ✿ ✿ ❀ ☛❵✮✿ ❵ ✕ ✵❀ ☛✐ ✷ ❆❣ ✐s t❤❡ s❡q✉❡♥❝❡ ❝❧❛ss✱ ✇❤❡r❡ ❥✭☛✶❀ ✿ ✿ ✿ ❀ ☛❵✮❥ ❂ P

✐❥☛✐❥✳

■t ✐s ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ ❝❧❛ss✱ ✐❢ ❆ ❝♦♥t❛✐♥s ♥♦ ❡❧❡♠❡♥t ♦❢ s✐③❡ ✵✳ ❚❤❡ ♠✉❧t✐s❡t ❝❧❛ss ✐s ▼❙❊❚ ✭❆✮ ❂ ❙❊◗✭❆✮❂✘❀ ✇❤❡r❡ ✭☛✶❀ ✿ ✿ ✿ ❀ ☛❵✮ ✘ ✭☞✶❀ ✿ ✿ ✿ ❀ ☞❵✮ ✐❢ t❤❡r❡ ✐s ❛ ♣❡r♠✉t❛t✐♦♥ ✛ ♦❢ ❢✶❀ ✿ ✿ ✿ ❀ ❵❣ s✉❝❤ t❤❛t ☛✐ ❂ ☞✛✭✐✮ ❢♦r ❛❧❧ ✐✳ ❚❤✐s ❝❧❛ss ❝♦♥t❛✐♥s ❛❧❧ ✜♥✐t❡ s❡q✉❡♥❝❡s ♦❢ ❡❧❡♠❡♥ts ❢r♦♠ ❆ ✇❤❡r❡ r❡♣❡t✐t✐♦♥ ✐s ❛❧❧♦✇❡❞✱ ❜✉t ♦r❞❡r✐♥❣ ✐❣♥♦r❡❞✳

❈♦♠❜✐♥❛t♦r✐❛❧ ❝❧❛ss❡s ❛♥❞ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s✳

❋❛❝t

▲❡t ❆✱ ❇✱ ❛♥❞ ❈ ❜❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ❝❧❛ss❡s✳

✶ ■❢ ❆ ❂ ❇ ❴

❬ ❈✱ t❤❡♥ ❆ ❂ ❇ ✰ ❈✳

✷ ■❢ ❆ ❂ ▼❙❊❚ ✭❇✮ ❛♥❞ ❇✵ ❂ ✵✱ t❤❡♥

❇ ❂

❳

❦✕✶

✖✭❦✮ ❦ ❧♦❣✭❆✭③❦✮✮❀ ✇❤❡r❡ ✖ ✐s t❤❡ ♥✉♠❜❡r✲t❤❡♦r❡t✐❝ ▼ö❜✐✉s✲❢✉♥❝t✐♦♥✱ ❞❡✜♥❡❞ ❛s ✖✭❦✮ ❂

✽ ❃ ❁ ❃ ✿

✶ ✐❢ ❦ ❂ ✶❀ ✭✶✮❵ ✐❢ ❦ ✐s t❤❡ ♣r♦❞✉❝t ♦❢ ❵ ❞✐st✐♥❝t ♣r✐♠❡s❀ ✵ ♦t❤❡r✇✐s❡✿

❘❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s✳

▲❡t ■r❀♥✭❋✮ ❂ ❢❢ ✷ Pr❀♥✭❋✮✿ ❢ ✐rr❡❞✉❝✐❜❧❡❣✱ ❘r❀♥✭❋✮ ❂ Pr❀♥✭❋✮ ♥ ■r❀♥✭❋✮✿ ❚❤❡ ♣♦❧②♥♦♠✐❛❧ ✶ ✐s ♥❡✐t❤❡r r❡❞✉❝✐❜❧❡ ♥♦r ✐rr❡❞✉❝✐❜❧❡✳ ■t ✐s ♥❛t✉r❛❧ t♦ ❤❛✈❡ ❘r❀✵✭❋✮ ❂ ❢✶❣ ❛♥❞ ■r❀✵✭❋✮ ❂ ∅✳ P ❂ ❙

♥✕✵ Pr❀♥✭Fq✮✱ ■ ❂ ❙ ♥✕✵ ■r❀♥✭Fq✮✱ ❛♥❞ ❘ ❂ P ♥ ■✱ ❛r❡

❝♦♠❜✐♥❛t♦r✐❛❧ ❝❧❛ss❡s ✇✐t❤ t❤❡ t♦t❛❧ ❞❡❣r❡❡ ❛s s✐③❡ ❢✉♥❝t✐♦♥s✳ ❚❤❡♥✱ P♥ ❂ Pr❀♥✭Fq✮ ❂ ★Pr❀♥✭Fq✮ ❂ q❜r❀♥✶ ✶ q❜r✶❀♥ ✶ q✶ ✿ ❇② ✉♥✐q✉❡ ❢❛❝t♦r✐③❛t✐♦♥✱ ❡✈❡r② ❡❧❡♠❡♥t ✐♥ P ❝♦rr❡s♣♦♥❞s t♦ ❛♥ ✉♥♦r❞❡r❡❞ ✜♥✐t❡ s❡q✉❡♥❝❡ ♦❢ ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s✱ ✇❤❡r❡ r❡♣❡t✐t✐♦♥ ✐s ❛❧❧♦✇❡❞✳ ❍❡♥❝❡ P ✐s ✐s♦♠♦r♣❤✐❝ t♦ ▼❙❊❚ ✭■✮✳ ❚❤❡♥ ■ ❂

❳

❦✕✶

✖✭❦✮ ❦ ❧♦❣ P✭③❦✮✿

❘❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s✳

❙✐♠♣❧❡ ▼❛♣❧❡ ♣r♦❣r❛♠ t♦ ❝♦♠♣✉t❡ ❡①❛❝t ✈❛❧✉❡s✳ ❙♠❛❧❧ ✈❛❧✉❡s ♦❢ ♥ ❛♥❞ r✳

❘❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s✳

❉✐✈❡r❣❡♥t s❡r✐❡s✳

❆♥❛❧②t✐❝ ❛♣♣r♦❛❝❤ ❛s ✐♥ ❬❋❧❛❥♦❧❡t ❛♥❞ ❙❡❞❣❡✇✐❝❦ ✷✵✵✾❪ ❝❛♥♥♦t ❜❡ ✉s❡❞✳ ▼♦✈❡ ❢r♦♠ ♣♦✇❡r s❡r✐❡s ✐♥ Q ③ t♦ ♣♦✇❡r s❡r✐❡s ✐♥ Q✭q✮ ③✱ ✇❤❡r❡ q ✐s ❛ s②♠❜♦❧✐❝ ✈❛r✐❛❜❧❡ r❡♣r❡s❡♥t✐♥❣ t❤❡ ✜❡❧❞ s✐③❡✳ ❋♦r r ✕ ✷ ❛♥❞ ♥ ✕ ✵ ✇❡ ❧❡t P♥✭q✮ ❂ Pr❀♥✭q✮ ❂ q❜r❀♥✶ ✶ q❜r✶❀♥ ✶ q✶ ✷ Z❬q❪❀ ❲❡ ❞❡✜♥❡ t❤❡ ♣♦✇❡r s❡r✐❡s P❀ I❀ R ✷ Q✭q✮ ③ ❜② P✭q❀ ③✮ ❂

❳

♥✕✵

P♥✭q✮③♥❀ I✭q❀ ③✮ ❂

❳

❦✕✶

✖✭❦✮ ❦ ❧♦❣ P✭q❀ ③❦✮❀ R✭q❀ ③✮ ❂ P✭q❀ ③✮ I✭q❀ ③✮✿

❉✐✈❡r❣❡♥t s❡r✐❡s✳

❚❤❡♥ ✶ P✭q❀ ③❦✮ ✐s ❛♥ ♦r✐❣✐♥❛❧ ♣♦✇❡r s❡r✐❡s✱ ❛♥❞ ❧♦❣ P✭q❀ ③❦✮ ❛♥❞ I ❛r❡ ✇❡❧❧✲❞❡✜♥❡❞✱ ✇✐t❤ I✭q❀ ✵✮ ❂ ✵✳ ❋♦r q ✷ Q✱ t❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s ✐♥ Q✭q✮ ✇✐t❤♦✉t ♣♦❧❡ ❛t q ✥ q ❢♦r♠ ❛ r✐♥❣✳ ❚❤❡ ❡✈❛❧✉❛t✐♦♥ ♠❛♣ ✇❤✐❝❤ s✉❜st✐t✉t❡s ❛♥ ✐♥t❡❣❡r q ❢♦r q ✐s ❛ r✐♥❣ ❤♦♠♦♠♦r♣❤✐s♠✳ ❙✐♥❝❡ P♥ ✐s ❛❝t✉❛❧❧② ❛ ♣♦❧②♥♦♠✐❛❧ ✐♥ q✱ ❡✈❛❧✉❛t✐♥❣ q ✥ q ♠❛♣s P✭q❀ ③✮ t♦ P✭③✮ ❝♦❡✣❝✐❡♥t✇✐s❡✳ ❋✉rt❤❡r♠♦r❡✱ ❬③♥❪I✭q❀ ③✮ ❂ ■♥❀ ❬③♥❪R✭q❀ ③✮ ❂ ❘♥✿

❆s②♠♣t♦t✐❝ r❡s✉❧ts✳

❖✭q♠✮ ✇✐t❤ ♠ ❃ ✵ ♠❡❛♥s t❤❡ ❡①✐st❡♥❝❡ ♦❢ s♦♠❡ ❢ ✇✐t❤ ❞❡❣r❡❡ ❛t ♠♦st ♠ t❤❛t ♠❛❦❡s t❤❡ ❡q✉❛t✐♦♥ ✈❛❧✐❞✳ ❘❡s✉❧ts ✈❛❧✐❞ ❢♦r ❛♥② ✏✜①❡❞✑ r ❛♥❞ ♥✳ ■❢ ❛ t❡r♠ ❖✭q♠✮ ❛♣♣❡❛rs✱ t❤❡♥ ✇❡ ♠❛② ❝♦♥❝❧✉❞❡ ❛ ♥✉♠❡r✐❝❛❧ ❛s②♠♣t♦t✐❝ r❡s✉❧t ❢♦r ❣r♦✇✐♥❣ ♣r✐♠❡ ♣♦✇❡rs q✳

❚❤❡♦r❡♠

▲❡t r ✕ ✷ ❛♥❞ ✚r❀♥✭q✮ ❂ q✭

r✰♥✶ r

✮✰r✶

✶qr ✭✶q✶✮✷ ✷ Q✭q✮✿

❚❤❡♥ R✵ ❂ ✶✱ R✶ ❂ ✵✱ R✷ ❂ ✚r❀✷✭q✮

✷

✁ ✭✶ qr✶✮✱ R✸ ❂ ✚r❀✸✭q✮ ✏ ✶ q✭

r✰✶ ✷ ✮ ✰ q✭ r ✷✮ ✶ ✷qr ✰ ✷q✷r✶ q✷r✷

✸✭✶ q✶✮ ✑ ❀ R✹ ❂ ✚r❀✹✭q✮ ✁ ✏ ✶ ✰ q✭

r✰✶ ✸ ✮ ✁ ✶ ✰ ❖✭qr✭r✶✮❂✷✮

✷✭✶ qr✮ ✑ ❀ ❛♥❞ ❢♦r ♥ ✕ ✺ R♥ ❂ ✚r❀♥✭q✮ ✁ ✏ ✶ ✰ q✭

r✰♥✷ r✶ ✮✰r✭r✰✶✮❂✷ ✁ ✶✰❖✭qr✭r✶✮❂✷✮

✶qr

✑ ✿

❈♦♠❜✐♥❛t♦r✐❛❧ ❆♣♣r♦❛❝❤✿ Pr❡❝✐s❡ ❡rr♦r ❜♦✉♥❞s✳

❊①♣♦♥❡♥t✐❛❧❧② ❞❡❝❛②✐♥❣ ✐♥ r✱ ♥✱ ❛♥❞ ❧♦❣✷ q✱ ✇❤❡♥ t❤❡ ♦t❤❡r t✇♦ ❛r❡ ✜①❡❞✳

❚❤❡♦r❡♠

▲❡t q❀ r ✕ ✷ ❛♥❞ ✚r❀♥ ❛s ❜❡❢♦r❡✳ ❲❡ ❤❛✈❡ ★❘r❀✵✭Fq✮ ❂ ✶✱ ★❘r❀✶✭Fq✮ ❂ ✵✱ ★❘r❀✷✭Fq✮ ❂ ✚r❀✷✭q✮

✷

✁ ✭✶ qr✶✮✱ ❥★❘r❀✸✭Fq✮ ✚r❀✸✭q✮❥ ✚r❀✸✭q✮ ❂ qr✭r✶✮❂✷ ✶ ✷qr ✰ ✷q✷r✶ q✷r✷ ✸✭✶ q✶✮ ✔ qr✭r✶✮❂✷❀ ❛♥❞ ❢♦r ♥ ✕ ✹ ❥★❘r❀♥✭Fq✮ ✚r❀♥✭q✮❥ ✚r❀♥✭q✮ ✔ q✭r✰♥✷

r✶ ✮✰r✭r✰✶✮❂✷

✭✶ q✶✮✭✶ qr✮ ✔ ✸q✭r✰♥✷

r✶ ✮✰r✭r✰✶✮❂✷✿

Pr♦♣♦rt✐♦♥ ♦❢ r❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s✳

❈♦r♦❧❧❛r②

❋♦r q❀ r ✕ ✷ ❛♥❞ ♥ ✕ ✺✱ ✇❡ ❤❛✈❡ ✶ ✹q✭r✰♥✶

r✶ ✮✰r ✔ ★❘r❀♥✭Fq✮

★Pr❀♥✭Fq✮ ✔ ✸q✭r✰♥✶

r✶ ✮✰r✿

❈♦r♦❧❧❛r②

▲❡t q❀ r ✕ ✷ ❛♥❞ ✚r❀♥ ❛s ❜❡❢♦r❡✳ ❲❡ ❤❛✈❡ ★■r❀✶✭Fq✮ ❂ ★Pr❀✶✭Fq✮✱ ★■r❀✷✭Fq✮ ❂ ★Pr❀✷✭Fq✮ ✚r❀✷✭q✮

✷

✁ ✭✶ qr✶✮✱ ❥★■r❀✸✭Fq✮ ✭★Pr❀✸✭Fq✮ ✚r❀✸✭q✮✮❥ ✔ ✚r❀✸✭q✮ ✁ q✭r✶✮r❂✷❀ ❛♥❞ ❢♦r ♥ ✕ ✹ ❥★■r❀♥✭Fq✮ ✭★Pr❀♥✭Fq✮ ✚r❀♥✭q✮✮❥ ✔ ✚r❀♥✭q✮ ✁ ✸q✭r✰♥✷

r✶ ✮✰✭r✰✶ ✷ ✮✿

P♦✇❡r❢✉❧ ♣♦❧②♥♦♠✐❛❧s✳

❋♦r ❛♥ ✐♥t❡❣❡r s ✕ ✷✱ ❛ ♣♦❧②♥♦♠✐❛❧ ✐s ❝❛❧❧❡❞ s✲♣♦✇❡r❢✉❧ ✐❢ ✐t ✐s ❞✐✈✐s✐❜❧❡ ❜② t❤❡ st❤ ♣♦✇❡r ♦❢ s♦♠❡ ♥♦♥❝♦♥st❛♥t ♣♦❧②♥♦♠✐❛❧✱ ❛♥❞ s✲♣♦✇❡r❢r❡❡ ♦t❤❡r✇✐s❡❀ ✐t ✐s sq✉❛r❡❢r❡❡ ✐❢ s ❂ ✷✳ ▲❡t ◗r❀♥❀s✭❋✮ ❂ ❢❢ ✷ Pr❀♥✭❋✮✿ ❢ ✐s s✲♣♦✇❡r❢✉❧❣✱ ❙r❀♥❀s✭❋✮ ❂ Pr❀♥✭❋✮ ♥ ◗r❀♥❀s✭❋✮✳ ▲❡t t❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ❝❧❛ss❡s ◗ ❂ ❙

♥✕✵ ◗r❀♥❀s ❛♥❞ ❙ ❂ P ♥ ◗✳

❆♥② ♠♦♥✐❝ ♣♦❧②♥♦♠✐❛❧ ❢ ❢❛❝t♦rs ✉♥✐q✉❡❧② ❛s ❢ ❂ ❣ ✁ ❤s ✇✐t❤ ❣ ♠♦♥✐❝ s✲❢r❡❡ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ❤ ❛r❜✐tr❛r② ♠♦♥✐❝✳ ❍❡♥❝❡ ❛s ●✳❋✳ P ❂ ❙ ✁ P✭③s✮❀ ◗ ❂ P ❙✿

• ✐✈❡s s✐♠♣❧❡ ▼❛♣❧❡ ♣r♦❣r❛♠s t♦ ❝♦♠♣✉t❡ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ ◗✳

P♦✇❡r❢✉❧ ♣♦❧②♥♦♠✐❛❧s✳

❙✐♠♣❧❡ ▼❛♣❧❡ ♣r♦❣r❛♠ t♦ ❝♦♠♣✉t❡ ❡①❛❝t ✈❛❧✉❡s✳ ❙♠❛❧❧ ✈❛❧✉❡s ♦❢ ♥❀ s ❛♥❞ r✳

P♦✇❡r❢✉❧ ♣♦❧②♥♦♠✐❛❧s✳

P♦✇❡r❢✉❧ ♣♦❧②♥♦♠✐❛❧s✳

❚❤❡♦r❡♠

▲❡t r❀ s ✕ ✷✱ ♥ ✕ ✵✱ ❛♥❞ ✑r❀♥❀s✭q✮ ❂ q

• r✰♥s

r

✁

✰r✶ ✭✶ qr✮✭✶ q

• r✰♥s✶

r✶

✁

✮ ✭✶ q✶✮✷ ✷ Q✭q✮❀ ✍ ❂ r ✰ ♥ s r

✁

r ✰ ♥ ✷s r

✁

r✭r ✰ ✶✮ ✷ ✿

✶

■❢ ♥ ✕ ✷s✱ t❤❡♥ ✍ ✕ r✳

✷

Q♥ ❂

✽ ❃ ❃ ❃ ❃ ❃ ❁ ❃ ❃ ❃ ❃ ❃ ✿

✵ ❢♦r ♥ ❁ s✱ ✑r❀♥❀s✭q✮ ❢♦r s ✔ ♥ ❁ ✷s✱ ✑r❀♥❀s✭q✮

✏

✶ ✰ q✍ ✁ ✶q

✭♥✰r✷s✶ r✶

✮

✶q

✭♥✰rs✶ r✶

✮ ✁

✏

✶qr✭r✰✶✮❂✷ ✶qr

qr✭r✶✮❂✷ ✶qr

✶q✶

✑✑

❢♦r ✷s ✔ ♥ ❁ ✸s✳

P♦✇❡r❢✉❧ ♣♦❧②♥♦♠✐❛❧s✳

❚❤❡♦r❡♠ ✭❝♦♥t✳✮

✶

❋♦r ✭♥❀ s✮ ❂ ✭✻❀ ✷✮ ❛♥❞ r ✕ ✷✱ ✇❡ ❤❛✈❡ Q✻ ❂ ✑r❀✻❀✷✭q✮

✏

✶ ✰ q

• r✰✸

✹

✁

r✰✶ ✁

✏

q✶ ✭✶q✶✮

✶q✭r✰✷

✸ ✮✁

✭✶qr✮ ✶q✭r✰✸

✹ ✮✁

✰ q✭r✸✼r✰✻✮❂✻

✭✶qr✭r✰✶✮❂✷✮✷ ✭✶qr✮ ✶q✭r✰✸

✹ ✮✁

q✭r✸✰✸r✷✶✵r✰✻✮❂✻ ✭✶qr✮✭✶qr✭r✰✶✮❂✷✮

✭✶q✶✮ ✶q✭r✰✸

✹ ✮✁

✷q✭r✸✰✸r✷✰✹r✻✮❂✻ ✶qr✭r✰✶✮❂✷

✶q✭r✰✸

✹ ✮

✰ q✭r✸✰✻r✷✼r✰✻✮❂✻

✭✶qr✮✷ ✭✶q✶✮ ✶q✭r✰✸

✹ ✮✁

✑✑

❂ ✑r❀✻❀✷✭q✮ ✶ ✰ q✍✰✭r✷✮✭r✶✮✭r✰✸✮❂✻✭✶ ✰ ❖✭q✶✮✮✁ ✿

✷

❋♦r ♥ ✕ ✷s ❛♥❞ ✭♥❀ s✮ ✻❂ ✭✻❀ ✷✮✱ ✇❡ ❤❛✈❡ Q♥ ❂ ✑r❀♥❀s✭q✮ ✶ ✰ q✍✭✶ ✰ ❖✭q✶✮✮✁ ✿

▼♦r❡ ♣r❡❝✐s❡ ❛s②♠♣t♦t✐❝ ❜♦✉♥❞s✳

❚❤❡♦r❡♠

✶

❋♦r ✭♥❀ s✮ ❂ ✭✻❀ ✷✮✱ ✇❡ ❤❛✈❡ ✍ ❂ r✭r ✰ ✶✮✭r✷ ✰ ✾r ✰ ✷✮❂✷✹ ❛♥❞ ❥★◗r❀✻❀✷✭Fq✮ ✑r❀✻❀✷✭q✮❥ ✔ ✑r❀✻❀✷✭q✮ ✁ ✷q✍✰✭r✷✮✭r✶✮✭r✰✸✮❂✻✿

✷

❋♦r ♥ ✕ ✸s ❛♥❞ ✭♥❀ s✮ ✻❂ ✭✻❀ ✷✮✱ ✇❡ ❤❛✈❡ ❥★◗r❀♥❀s✭Fq✮ ✑r❀♥❀s✭q✮❥ ✔ ✑r❀♥❀s✭q✮ ✁ ✻q✍✿

❈♦r♦❧❧❛r②

❋♦r q❀ r❀ s ✕ ✷ ❛♥❞ ♥ ✕ s✱ ✇❡ ❤❛✈❡ ✶ ✷q

• r✰♥s

r

✁

✰r✶ ✔ ★◗r❀♥❀s✭Fq✮ ✔ ✶✵✷✽

✸ q

• r✰♥s

r

✁

✰r✶❀

✶ ✷q

• r✰♥

r

✁

✰

r✰♥s r

✁

✰r ✔ ★◗r❀♥❀s✭Fq✮

★Pr❀♥✭Fq✮ ✔ ✺q

• r✰♥

r

✁

✰

r✰♥s r

✁

✰r❀

✶ ✻q

• r✰♥✶

r

✁

✰

r✰♥s r

✁

✔ ★◗r❀♥❀s✭Fq✮ ★❘r❀♥✭Fq✮ ✔ ✶✾q

• r✰♥✶

r

✁

✰

r✰♥s r

✁

✿

s✲♣♦✇❡r❢r❡❡ ♣♦❧②♥♦♠✐❛❧s✳

❈♦r♦❧❧❛r②

▲❡t q❀ r❀ s ✕ ✷✱ ♥ ✕ ✵✱ ❛♥❞ ✑r❀♥❀s ❛♥❞ ✍ ❛s ❜❡❢♦r❡✳ ❲❡ ❤❛✈❡ ★Pr❀♥✭Fq✮ ✸✑r❀♥❀s✭q✮ ✔ ★❙r❀♥❀s✭Fq✮ ✔ ★Pr❀♥✭Fq✮❀ ❛♥❞ ♠♦r❡ ♣r❡❝✐s❡❧② ★❙r❀♥❀s✭Fq✮ ❂

✽ ❃ ❃ ❃ ❃ ❃ ❁ ❃ ❃ ❃ ❃ ❃ ✿

★Pr❀♥✭Fq✮ ❢♦r ♥ ❁ s✱ ★Pr❀♥✭Fq✮ ✑r❀♥❀s✭q✮ ❢♦r s ✔ ♥ ❁ ✷s✱ ★Pr❀♥✭Fq✮ ✑r❀♥❀s✭q✮

✏

✶ ✰ q✍ ✁ ✶q

✭♥✰r✷s✶ r✶

✮

✶q

✭♥✰rs✶ r✶

✮ ✁

✏

✶qr✭r✰✶✮❂✷ ✶qr

qr✭r✶✮❂✷ ✶qr

✶q✶

✑✑

❢♦r ✷s ✔ ♥ ❁ ✸s✱ ❥★❙r❀✻❀✷✭Fq✮ ✭★Pr❀♥✭Fq✮ ✑r❀✻❀✷✭q✮✮❥ ✔ ✑r❀✻❀✷✭q✮ ✁ ✷q✍✰✭r✷✮✭r✶✮✭r✰✸✮❂✻❀ ❛♥❞ ❢♦r ♥ ✕ ✸s ✇✐t❤ ✭♥❀ s✮ ✻❂ ✭✻❀ ✷✮ ❥★❙r❀♥❀s✭Fq✮ ✭★Pr❀♥✭Fq✮ ✑r❀♥❀s✭q✮✮❥ ✔ ✑r❀♥❀s✭q✮ ✁ ✻q✍✿

❘❡❧❛t✐✈❡❧② ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s✳

❆ ♣♦❧②♥♦♠✐❛❧ ♦✈❡r ❋ ✐s ❛❜s♦❧✉t❡❧② ✐rr❡❞✉❝✐❜❧❡ ✐❢ ✐t ✐s ✐rr❡❞✉❝✐❜❧❡ ♦✈❡r ❛♥ ❛❧❣❡❜r❛✐❝ ❝❧♦s✉r❡ ♦❢ ❋✳ ■t ✐s r❡❧❛t✐✈❡❧② ✐rr❡❞✉❝✐❜❧❡ ✐❢ ✐t ✐s ✐rr❡❞✉❝✐❜❧❡ ♦✈❡r ❋ ❜✉t ❢❛❝t♦rs ♦✈❡r s♦♠❡ ❡①t❡♥s✐♦♥ ✜❡❧❞ ♦❢ ❋✳ ❆r❀♥✭❋✮ ❂ ❢❢ ✷ Pr❀♥✭❋✮✿ ❢ ✐s ❛❜s♦❧✉t❡❧② ✐rr❡❞✉❝✐❜❧❡❣ ✒ ■r❀♥✭❋✮❀ ❊r❀♥✭❋✮ ❂ ■r❀♥✭❋✮ ♥ ❆r❀♥✭❋✮✿ ❋♦r ❛ ✜❡❧❞ ❡①t❡♥s✐♦♥ Fq❦ ♦✈❡r Fq✱ ❧❡t ●❦ ❂ ●❛❧✭Fq❦ ✿ Fq✮✳ ■t ❛❝ts ♦♥ Fq❦❬①❪ ❝♦❡✣❝✐❡♥t✇✐s❡ ✇✐t❤ ❦ ❥ ♥ ❛♥❞ t❤❡ ✏♥♦r♠✑ ♠❛♣ ✬r❀♥❀❦ ✿ Pr❀♥❂❦✭Fq❦✮ ✦ Pr❀♥✭Fq✮❀ ❣ ✼✦

❨

✛✷●❦

❣✛❀ ❙✐♥❝❡ ✭✬r❀♥❀❦✭❣✮✮✜ ❂ ✬r❀♥❀❦✭❣✮ ❢♦r ❛♥② ✜ ✷ ●❦ ❛♥❞ t❤❡r❡❢♦r❡ ✬r❀♥❀❦✭❣✮ ✷ Pr❀♥✭Fq✮✱ t❤✐s ♠❛♣ ✐s ✇❡❧❧✲❞❡✜♥❡❞✳

❘❡❧❛t✐✈❡❧② ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s✳

❘❡❧❛t✐✈❡❧② ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s ✐♥ Pr❀♥✭Fq✮ ❛r❡ t❤❡ ♣r♦❞✉❝t ♦❢ ❛❧❧ ❝♦♥❥✉❣❛t❡s ♦❢ ❛♥ ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧ ❣ ❞❡✜♥❡❞ ♦✈❡r s♦♠❡ ❡①t❡♥s✐♦♥ ✜❡❧❞ Fq❦✳ ■❢ ❣ ✐ts❡❧❢ ✐s r❡❧❛t✐✈❡❧② ✐rr❡❞✉❝✐❜❧❡ ♦✈❡r Fq❦✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ❛♣♣r♦♣r✐❛t❡ ♠✉❧t✐♣❧❡ ❥ ♦❢ ❦ ❛♥❞ ❤ ✷ Pr❀♥❂❥✭Fq❥✮ ✇✐t❤ t❤❡ s❛♠❡ ✐♠❛❣❡ ✬r❀♥❀❦✭❣✮ ❂ ✬r❀♥❀❥✭❤✮ ✐♥ Pr❀♥✭Fq✮ ❛♥❞ t❤❡ ♣r♦♣❡rt② t❤❛t ❤ ✐s ❛❜s♦❧✉t❡❧② ✐rr❡❞✉❝✐❜❧❡✳ ❙♦✱ ❡✈❡r② r❡❧❛t✐✈❡❧② ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ✬r❀♥❀❦✭❆r❀♥❂❦✭Fq❦✮✮ ❢♦r ❛ ✉♥✐q✉❡ ❦ ❃ ✶ ❞✐✈✐❞✐♥❣ ♥✳ ❚❤❡♥ ❆r❀♥✭Fq✮ ❂ ✬r❀♥❀✶✭❆r❀♥✭Fq✮✮❀ ❊r❀♥✭Fq✮ ✒

❬

✶❁❦ ❥ ♥

✬r❀♥❀❦✭❆r❀♥❂❦✭Fq❦✮✮✿

❘❡❧❛t✐✈❡❧② ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s✳

▲❡t ❆✰

r❀♥❂❦✭Fq❦✮ ❂ ❆r❀♥❂❦✭Fq❦✮ ♥ ❙ s ❥ ❦❀ s✻❂❦ ❆r❀♥❂❦✭Fqs✮ ❜❡ t❤❡

s❡t ♦❢ ❛❜s♦❧✉t❡❧② ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s ♦✈❡r Fq❦ t❤❛t ❛r❡ ♥♦t ❞❡✜♥❡❞ ♦✈❡r ❛ ♣r♦♣❡r s✉❜✜❡❧❞ ❝♦♥t❛✐♥✐♥❣ Fq✱ ❛♥❞ ■r❀♥❀❦✭Fq✮ ❂ ✬r❀♥❀❦✭❆✰

r❀♥❂❦✭Fq❦✮✮✿

▲❡♠♠❛

✶

❲❡ ❤❛✈❡ t❤❡ ❞✐s❥♦✐♥t ✉♥✐♦♥ ■r❀♥✭Fq✮ ❂ ❴

❬

❦ ❥ ♥

■r❀♥❀❦✭Fq✮ ❛♥❞ ♠♦r❡ ♣r❡❝✐s❡❧② ❆r❀♥✭Fq✮ ❂ ■r❀♥❀✶✭Fq✮❀ ❊r❀♥✭Fq✮ ❂ ❴

❬

✶❁❦ ❥ ♥

■r❀♥❀❦✭Fq✮✿

✷

★■r❀♥❀❦✭Fq✮ ❂ ✶

❦ ★❆✰ r❀♥❂❦✭Fq❦✮✳

❘❡❧❛t✐✈❡❧② ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s✳

❲✐t❤ I✭q❀ ③✮ ❛s ❜❡❢♦r❡✱ ✇❡ ❣❡t t❤❡ ♣♦✇❡r s❡r✐❡s A❀ E ✷ Q✭q✮ ③ A✵✭q✮ ❂ I✵✭q✮ ❂ ✵❀ A♥✭q✮ ❂

❳

❦ ❥ ♥

✶ ❦

❳

s ❥ ❦

✖✭s✮I♥❂❦✭qs✮ ✷ Z❬q❪ ❢♦r ♥ ❃ ✵❀ A✭q❀ ③✮ ❂

❳

♥✕✵

A♥✭q✮③♥ ✷ Z❬q❪ ③ ❀ E✭q❀ ③✮ ❂ I✭q❀ ③✮ A✭q❀ ③✮ ❂

❳

✶❁❦ ❥ ♥

✶ ❦

❳

s ❥ ❦

✖✭s✮I♥❂❦✭qs✮ ✷ Z❬q❪ ③ ✿ ❚❤❡♥ A♥✭q✮ ❂ ★❆r❀♥✭Fq✮❀ E♥✭q✮ ❂ ★❊r❀♥✭Fq✮✿

❘❡❧❛t✐✈❡❧② ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s✳

❘❡❧❛t✐✈❡❧② ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s✳

❘❡❧❛t✐✈❡❧② ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s✳

❚❤❡ ❛♣♣r♦❛❝❤ ❜② ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ❣✐✈❡s

❚❤❡♦r❡♠

▲❡t r❀ ♥ ✕ ✷✱ ❧❡t ❵ ❜❡ t❤❡ s♠❛❧❧❡st ♣r✐♠❡ ❞✐✈✐s♦r ♦❢ ♥✱ ❛♥❞ ✎r❀♥✭q✮ ❂ q

❵✭

r✰♥❂❵ r

✁

✶✮

❵✭✶ q❵✮ ✷ Q✭q✮❀ ✔ ❂ ✭❵ ✶✮✭r ✶ ✰ ♥❂❵ r ✶

✁

r✮ ✰ ✶✿ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞✳

✶

E✶✭q✮ ❂ ✵✳

✷

■❢ ♥ ✐s ♣r✐♠❡✱ t❤❡♥ E♥✭q✮ ❂ ✎r❀♥✭q✮✭✶ q♥r✮

✏

✶ qr✭♥✶✮ ✭✶ qr✮✭✶ q♥✮ ✭✶ q✶✮✭✶ q♥r✮

✑

✿

✸

■❢ ♥ ✐s ❝♦♠♣♦s✐t❡✱ t❤❡♥ ✔ ✕ ✷ ❛♥❞ E♥✭q✮ ❂ ✎r❀♥✭q✮✭✶ ✰ ❖✭q✔✮✮✿

❘❡❧❛t✐✈❡❧② ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s✳

❚❤❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ❛♣♣r♦❛❝❤ ②✐❡❧❞s t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t✳

❚❤❡♦r❡♠

▲❡t r❀ q ✕ ✷✱ ❛♥❞ ✎r❀♥ ❛♥❞ ✔ ❛s ❜❡❢♦r❡✳

✶

★❊r❀✶✭Fq✮ ❂ ✵✳

✷

■❢ ♥ ✐s ♣r✐♠❡✱ t❤❡♥ ★❊r❀♥✭Fq✮ ❂ ✎r❀♥✭q✮✭✶ q♥r✮ ✏ ✶ qr✭♥✶✮ ✭✶ qr✮✭✶ q♥✮ ✭✶ q✶✮✭✶ q♥r✮ ✑ ❀ ✵ ✔ ✎r❀♥✭q✮ ★❊r❀♥✭Fq✮ ✔ ✸qr✭♥✶✮✿

✸

■❢ ♥ ✐s ❝♦♠♣♦s✐t❡✱ t❤❡♥ ❥★❊r❀♥✭Fq✮ ✎r❀♥✭q✮❥ ✔ ✎r❀♥✭q✮ ✁ ✸q✔✿

❘❡❧❛t✐✈❡❧② ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧s✳

❈♦r♦❧❧❛r②

▲❡t q❀ r❀ ♥ ✕ ✷✱ ❛♥❞ ❵ ❜❡ t❤❡ s♠❛❧❧❡st ♣r✐♠❡ ❞✐✈✐s♦r ♦❢ ♥✱ t❤❡♥ ✶ ✹❵q❵✭r✰♥❂❵

r ✮❵ ✔ ★❊r❀♥✭Fq✮ ✔ ✷

❵ q❵✭❧✰♥❂❵

r ✮❵❀

✶ ✽❵q✭r✰♥

r ✮✰❵✭r✰♥❂❵ r ✮❵✰✶ ✔ ★❊r❀♥✭Fq✮

★Pr❀♥✭Fq✮ ✔ ✷ ❵ q✭r✰♥

r ✮✰❵✭r✰♥❂❵ r ✮❵✰✶❀

✶ ✽❵q✭r✰♥

r ✮✰❵✭r✰♥❂❵ r ✮❵✰✶ ✔ ★❊r❀♥✭Fq✮

★■r❀♥✭Fq✮ ✔ ✷ ❵ q✭r✰♥

r ✮✰❵✭r✰♥❂❵ r ✮❵✰✶✿

❈♦r♦❧❧❛r②

▲❡t r❀ q❀ ♥ ✕ ✷ ❛♥❞ ✚r❀♥✭q✮ ❛s ✐♥ ✭✶✮✳ ❚❤❡♥ ★Pr❀♥✭Fq✮ ✹✚r❀♥✭q✮ ✔ ★❆r❀♥✭Fq✮ ✔ ★■r❀♥✭Fq✮ ✔ ★Pr❀♥✭Fq✮❀ ✇❤❡r❡ t❤❡ ✹ ❝❛♥ ❜❡ r❡♣❧❛❝❡❞ ❜② ✸ ❢♦r ♥ ✕ ✸✳