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❙❛❣❡ ❞❛②s ✶✵✱ ◆❛♥❝②✱ ❋r❛♥❝❡

■♠♣❧❡♠❡♥t✐♥❣ t❤❡ ❲❡✐❧✱ ❚❛t❡ ❛♥❞ ❆t❡ ♣❛✐r✐♥❣s ✉s✐♥❣ ❙❛❣❡ s♦❢t✇❛r❡

◆❛❞✐❛ ❊▲ ▼❘❆❇❊❚ ▲■❘▼▼✱ ■✸▼✱ ❯♥✐✈❡rs✐té ▼♦♥t♣❡❧❧✐❡r ✷ ❙❛t✉r❞❛② ✶✶t❤ ❖❝t♦❜❡r ✷✵✵✽

❖✉t❧✐♥❡ ♦❢ t❤❡ ♣r❡s❡♥t❛t✐♦♥

✶✳ ❉❡✜♥✐t✐♦♥ ♦❢ ❛ ♣❛✐r✐♥❣ ✷✳ ❈♦♥str✉❝t✐♦♥ ♦❢ ❛ ♣❛✐r✐♥❣ ✸✳ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❛ ♣❛✐r✐♥❣

❲❤❛t ✐s ❛ ♣❛✐r✐♥❣ ❄

Pr♦♣❡rt✐❡s

▲❡t ●✶✱ ●✷ ❛♥❞ ●✸ ❜❡ t❤r❡❡ ❣r♦✉♣s ✇✐t❤ t❤❡ s❛♠❡ ♦r❞❡r r✳ ❆ ♣❛✐r✐♥❣ ✐s ❛ ♠❛♣ ✿ ❡ : ●✶ × ●✷ → ●✸ ✇❤✐❝❤ ✈❡r✐✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s ✿

• ◆♦♥ ❞❡❣❡♥❡r❛t❡ ❀

⋄ ∀P ∈ ●✶ {✵}∃◗ ∈ ●✷/❡(P, ◗) = ✶ ⋄ ∀◗ ∈ ●✷ {✵}∃P ∈ ●✶/❡(P, ◗) = ✶

• ❇✐❧✐♥❡❛r✐t② ✿ ∀P, P′ ∈ ●✶, ∀◗, ◗′ ∈ ●✷

⋄ ❡(P + P′, ◗) = ❡(P, ◗).❡(P′, ◗) ⋄ ❡(P, ◗ + ◗′) = ❡(P, ◗).❡(P, ◗′)

❲❤❛t ✐s ❛ ♣❛✐r✐♥❣ ❄

Pr♦♣❡rt✐❡s

▲❡t ●✶✱ ●✷ ❛♥❞ ●✸ ❜❡ t❤r❡❡ ❣r♦✉♣s ✇✐t❤ t❤❡ s❛♠❡ ♦r❞❡r r✳ ❆ ♣❛✐r✐♥❣ ✐s ❛ ♠❛♣ ✿ ❡ : ●✶ × ●✷ → ●✸ ✇❤✐❝❤ ✈❡r✐✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s ✿

• ◆♦♥ ❞❡❣❡♥❡r❛t❡ ❀
• ❇✐❧✐♥❡❛r✐t② ❀

❈♦♥s❡q✉❡♥❝❡

∀❥ ∈ N, ❡([❥]P, ◗) = ❡(P, ◗)❥ = ❡(P, [❥]◗)

❊❧❧✐♣t✐❝ ❈✉r✈❡ ❈r②♣t♦❣r❛♣❤② ❛♥❞ ♣❛✐r✐♥❣s

P❛rt ✶ ✲ ❈r②♣t❛♥❛❧②s❡

❚❤❡ ▼❖❱✴❋r❡② ❘ü❝❦ ❛tt❛❝❦ ❛❣❛✐♥st t❤❡ ❉▲P ♦♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡s ✐♥ ✶✾✾✸✱ ✶✾✾✹ ✿ ✉s✐♥❣ ♣❛✐r✐♥❣s✱ t❤❡ ❉▲P ♦♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡s ❜❡❝♦♠❡s ❛ ❉▲P ♦♥ ✜♥✐t❡ ✜❡❧❞✳

• ●✐✈❡♥ P ❛♥❞ ◗ = αP ∈ ❊(Fq)✱

t❤❡ ❉▲P ♦♥ ❊(Fq) ❝♦♥s✐sts ✐♥ ✜♥❞✐♥❣ α✳

• ▲❡t ❙ ∈ ❊(Fq) ❜❡ ❛ ♣♦✐♥t s✉❝❤ t❤❛t ❡(P, ❙) = ✶✱

❧❡t ❡(P, ❙) = ❣ ❛♥❞ ❡(◗, ❙) = ❤ ∈ ❊(Fq)✱ t❤❡♥

• t❤❡ ❉▲P ❜❡❝♦♠❡s ✜♥❞✐♥❣ α s✉❝❤ t❤❛t ❤ = ❣α ✐♥ ❛ ✜♥✐t❡ ✜❡❧❞✳

❊❧❧✐♣t✐❝ ❈✉r✈❡ ❈r②♣t♦❣r❛♣❤② ❛♥❞ ♣❛✐r✐♥❣s

P❛rt ✷ ✲ ❈r②♣t♦❣r❛♣❤②

P❛✐r✐♥❣s ❛❧❧♦✇ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ♥♦✈❡❧ ♣r♦t♦❝♦❧s ❛♥❞ s✐♠♣❧✐✜❝❛t✐♦♥ ♦❢ ❡①✐st✐♥❣ ♣r♦t♦❝♦❧s✳

• ❚❤❡ tr✐ ♣❛rt✐t❡ ❉✐✣❡ ❍❡❧❧♠❛♥ ❦❡② ❡①❝❤❛♥❣❡ ♣r♦t♦❝♦❧ ✭❏♦✉①

✷✵✵✶✮

• ❚❤❡ ■❞❡♥t✐t② ❇❛s❡❞ ❊♥❝r②♣t✐♦♥ ✭❇♦♥❡❤ ❛♥❞ ❋r❛♥❦❧✐♥ ✷✵✵✶✮
• ❙❤♦rt s✐❣♥❛t✉r❡ s❝❤❡♠❡ ✭❇♦♥❡❤✱ ▲②♥♥✱ ❙❝❤❛❝❦❛♠♠ ✷✵✵✶✮
• ●r♦✉♣ s✐❣♥❛t✉r❡s s❝❤❡♠❡s ✭❇♦♥❡❤✱ ❙❝❤❛❝❦❛♠♠✱ ✷✵✵✹✮

❊❧❧✐♣t✐❝ ❈✉r✈❡ ❈r②♣t♦❣r❛♣❤② ❛♥❞ ♣❛✐r✐♥❣s

P❛✐r✐♥❣s ✉s❡❞

❋♦✉r ♣❛✐r✐♥❣s ❛r❡ ♣r✐♥❝✐♣❛❧❧② ✉s❡❞ ✐♥ ❝r②♣t♦❣r❛♣❤② ✿

• t❤❡ ❲❡✐❧ ♣❛✐r✐♥❣✱
• t❤❡ ❚❛t❡ ♣❛✐r✐♥❣✱
• t❤❡ η❚ ♣❛✐r✐♥❣✱
• t❤❡ ❆t❡ ♣❛✐r✐♥❣✳

■ ❢♦❝✉s❡❞ ♦♥❧② ♦♥ t❤❡ ♣❛✐r✐♥❣s ❝♦♥str✉❝t❡❞ ❜② t❤❡ s❛♠❡ ✇❛②✳ ❚❤❡ ▼✐❧❧❡r ❛❧❣♦r✐t❤♠ ❝♦♥str✉❝t✐♥❣ t❤❡ ❢✉♥❝t✐♦♥ ❢r,P ✐s ❛ ❝❡♥tr❛❧ st❡♣ ❢♦r t❤❡ ❲❡✐❧✱ ❚❛t❡ ❛♥❞ ❆t❡ ♣❛✐r✐♥❣s✳

❈♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ♣❛✐r✐♥❣s

❉❛t❛

❚♦ ❝♦♠♣✉t❡ ❛ ♣❛✐r✐♥❣✱ ✇❡ ♥❡❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡❧❡♠❡♥ts ✿

• ❊ ❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ♦✈❡r Fq ✿

❊ : ②✷ = ①✸ + ❛① + ❜✱ ✇❤❡r❡ ❛✱ ❜ ∈ Fq✳

• r ❛ ♣r✐♠❡ ❞✐✈✐❞✐♥❣ ❝❛r❞(❊(Fq))✱

❝♦♥s✐❞❡r ❊[r] ✿ ❊[r] = {P ∈ ❊(Fq), [r]P = P∞}✳

• ❚❤❡ ❡♠❜❡❞❞✐♥❣ ❞❡❣r❡❡ ❦ ✿ ♠✐♥✐♠❛❧ ✐♥t❡❣❡r s✉❝❤ t❤❛t

r|(q❦ − ✶) ✿ ■❢ ❣❝❞(r, q) = ✶✱ t❤❡♥ ❊[r] ∼ = Z/rZ × Z/rZ✱ ■❢ ❦ > ✶ t❤❡♥ ❊[r] = ❊(Fq❦)[r]✳

• ❆ ❢✉♥❝t✐♦♥ ❢r,P ❞❡s❝r✐❜❡❞ ❧❛t❡❧②✳

❈♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ♣❛✐r✐♥❣s

❚❤❡ ❲❡✐❧ ♣❛✐r✐♥❣

▲❡t P ∈ ❊[r] ❛♥❞ ◗ ∈ ❊[r] ❚❤❡ ❲❡✐❧ ♣❛✐r✐♥❣ ✐s t❤❡ ❜✐❧✐♥❡❛r ♠❛♣ ✿ ❡❲ : ❊[r] × ❊[r] → F∗

q❦

(P, ◗) → ❢r,P(◗) ❢r,◗(P)

❈♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ♣❛✐r✐♥❣s

❚❤❡ ❚❛t❡ ♣❛✐r✐♥❣

▲❡t P ∈ ❊(Fq)[r]✱ ◗ ∈ ❊(Fq❦)/r❊(Fq❦) ❛♥❞ ❦ ❜❡ t❤❡ ❡♠❜❡❞❞✐♥❣ ❞❡❣r❡❡ ♦❢ t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡✳ ❚❤❡ ❚❛t❡ ♣❛✐r✐♥❣ ✐s t❤❡ ❜✐❧✐♥❡❛r ♠❛♣ ✿ ❡❚ : ❊(Fq)[r] × ❊(Fq❦)/r❊(Fq❦) → F∗

q❦

(P, ◗) → ❢r,P(◗)

q❦ −✶ r

❈♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ♣❛✐r✐♥❣s

❚❤❡ ❆t❡ ♣❛✐r✐♥❣

❚❤❡ ❆t❡ ♣❛✐r✐♥❣ ✐s t❤❡ ❧❛t❡st ♦♣t✐♠✐s❛t✐♦♥ ♦❢ t❤❡ ❚❛t❡ ♣❛✐r✐♥❣✳ ■t ✐s ❝♦♥str✉❝t❡❞ ❜② t❤❡ s❛♠❡ ✇❛②✳ ❚❤❡ ❆t❡ ♣❛✐r✐♥❣ ❡❛ts t❤❡ ❚ ✐♥ ❚❛t❡✱ ❛♥❞ ✉s❡s ✐t ✐♥ ♦r❞❡r t♦ ❜❡ ❝♦♠♣✉t❡❞ ✇✐t❤ ❧❡ss ✐t❡r❛t✐♦♥s✳ ▲❡t πq ❜❡ t❤❡ ❋r♦❜❡♥✐✉s ♠❛♣ ♦✈❡r t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ✿ πq([①, ②]) = [①q, ②q] t ❞❡♥♦t❡s t❤❡ tr❛❝❡ ♦❢ t❤❡ ❋r♦❜❡♥✐✉s ♦✈❡r ❊(Fq) ❛♥❞ ❚ = t − ✶✳

❈♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ♣❛✐r✐♥❣s

❚❤❡ ❆t❡ ♣❛✐r✐♥❣

▲❡t P ∈ ❊[r] ∩ ❑❡r(πq − [✶]) ❛♥❞ ◗ ∈ ❊[r] ∩ ❑❡r(πq − [q])✱ ✐✳❡✳ ◗ ✈❡r✐❢②✐♥❣ πq(◗) = [q]◗✳ ❚❤❡ ❆t❡ ♣❛✐r✐♥❣ ✐s t❤❡ ❜✐❧✐♥❡❛r ♠❛♣ ✿ ❡❆ : ❊[r] ∩ ❑❡r(πq − [✶]) × ❊[r] ∩ ❑❡r(πq − [q]) → F∗

q❦

(P, ◗) → ❢❚,P(◗)

q❦ −✶ r

▼✐❧❧❡r ❛❧❣♦r✐t❤♠

❚❤❡ ❢✉♥❝t✐♦♥ ❢r,P

■♥ ♦r❞❡r t♦ ❝♦♠♣✉t❡ t❤❡ ♣❛✐r✐♥❣s✱ ✇❡ ♥❡❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ ❢✉♥❝t✐♦♥ ❢r,P✳ ❚❤❡ ♣r✐♥❝✐♣❛❧ ♣r♦♣❡rt② ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s t❤❛t ✿ ❉✐✈(❢r,P) = r❉✐✈(P) − r❉✐✈(P∞) ❱✐❝t♦r ▼✐❧❧❡r ❡st❛❜❧✐s❤❡❞ t❤❡ ▼✐❧❧❡r ❡q✉❛t✐♦♥ ✿ ❢✐+❥,P = ❢✐,P × ❢❥,P × ❧[✐]P,[❥]P ✈[✐+❥]P ✇❤❡r❡ ❧[✐]P+[❥]P ✐s t❤❡ ❧✐♥❡ ❥♦✐♥✐♥❣ t❤❡ ♣♦✐♥ts [✐]P ❛♥❞ [❥]P✱ ❛♥❞ ✈[✐+❥]P ✐s t❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ ♣♦✐♥t [✐ + ❥]P✳

▼✐❧❧❡r ❛❧❣♦r✐t❤♠

❊①❛♠♣❧❡

❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ ❢✼,P ✿

• ✼ ❂ ✻ ✰ ✶
• ❢✼,P = ❢✻,P × ❢✶,P ×

❧[✻]P,P ✈[✼]P

❢✶,P = ✶ ❢✼,P = ❢✻,P ×

❧[✻]P,P ✈[✼]P

• ❢✻,P = ❢✸,P × ❢✸,P ×

❧[✸]P,[✸]P ✈[✻]P

✇❤❡♥ ✐ = ❥✱ t❤❡ ❧✐♥❡ ❧ ✐s t❤❡ t❛♥❣❡♥t ❛t ♣♦✐♥t [✐]P

• ❢✻,P = ❢ ✷

✸,P × ❧[✸]P,[✸]P ✈[✻]P

❢✼,P = ❢ ✷

✸,P × ❧[✸]P,[✸]P ✈[✻]P

×

❧[✻]P,P ✈[✼]P

▼✐❧❧❡r ❛❧❣♦r✐t❤♠

❊①❛♠♣❧❡

❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ ❢✼,P ✿

• ❢✼,P = ❢ ✷

✸,P × ❧[✸]P,[✸]P ✈[✻]P

×

❧[✻]P,P ✈[✼]P

• ❢✸,P = ❢✷,P × ❢✶,P ×

❧[✷]P,P ✈[✸]P

❢✸,P = ❢✷,P ×

❧[✷]P,P ✈[✸]P

• ❢✷,P = ❢✶,P × ❢✶,P × ❧P,P

✈[✷]P

• ❢✼,P =

❧P,P

✈[✷]P × ❧[✷]P,P ✈[✸]P

✷ ×

❧[✸]P,[✸]P ✈[✻]P

×

❧[✻]P,P ✈[✼]P

❈♦♠♣✉t✐♥❣ ♣❛✐r✐♥❣s

▼✐❧❧❡r ❛❧❣♦r✐t❤♠ ✿ r❡t✉r♥ ❢r,P(◗)

❉❛t❛✿ r = (r♥ . . . ❧✵)✷✱ P ∈ ❊(Fq) ❛♥❞ ◗ ∈ ❊(Fq❦) ❀ ❘❡s✉❧t✿ ❢r,P(◗) ∈ F∗

q❦ ❀

✶ ✿ ❚ ← P ✱ ❢✶ ← ✶✱ ❢✷ ← ✶ ❀ ❢♦r ✐ = ♥ − ✶ t♦ ✵ ❞♦ ✷ ✿ ❚ ← [✷]❚ ❀ ✸ ✿ ❢✶ ← − ❢✶

✷ × ❧✶(◗) ❀

✹ ✿ ❢✷ ← − ❢✷

✷ × ✈✷(◗) ❀

✐❢ r✐ = ✶ t❤❡♥ ✺ ✿ ❚ ← ❚ + P ❀ ❀ ❀ ❡♥❞ ❡♥❞ r❡t✉r♥

❈♦♠♣✉t✐♥❣ ♣❛✐r✐♥❣s

▼✐❧❧❡r ❛❧❣♦r✐t❤♠ ✿ r❡t✉r♥ ❢r,P(◗)

❉❛t❛✿ r = (r♥ . . . ❧✵)✷✱ P ∈ ❊(Fq) ❛♥❞ ◗ ∈ ❊(Fq❦) ❀ ❘❡s✉❧t✿ ❢r,P(◗) ∈ F∗

q❦ ❀

✶ ✿ ❚ ← P ✱ ❢✶ ← ✶✱ ❢✷ ← ✶ ❀ ❢♦r ✐ = ♥ − ✶ t♦ ✵ ❞♦ ✷ ✿ ❚ ← [✷]❚✱❀ ✸ ✿ ❢✶ ← − ❢✶

✷ × ❧❞(◗) ❀

✹ ✿ ❢✷ ← − ❢✷

✷ × ✈❞(◗) ❀

✐❢ r✐ = ✶ t❤❡♥ ✺ ✿ ❚ ← ❚ + P ❀ ✻ ✿ ❢✶ ← − ❢✶ × ❧❛(◗) ❀ ✼ ✿ ❢✷ ← − ❢✷ × ✈❛(◗)❀ ❡♥❞ ❡♥❞ r❡t✉r♥

❢✶ ❢✷

■♠♣❧❡♠❡♥t❛t✐♦♥ ✉s✐♥❣ ❙❛❣❡

• ♦♦❞ ♣♦✐♥ts ♦❢ ❙❛❣❡
• ❡❛s② t♦ ✇r✐t❡ ♦♣❡r❛t✐♦♥ ♦♥ t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡ P + ◗✱ ❛♥❞ ✷ ∗ P

❢♦r ❛❞❞✐♥❣ ❛♥❞ ♠✉❧t✐♣❧②✐♥❣ ♣♦✐♥t✳

• t❤❡ tr❛❝❡ ♦❢ t❤❡ ❋r♦❜❡♥✐✉s ✐s ✐♠♣❧❡♠❡♥t❡❞
• r❛♥❞♦♠ ♣♦✐♥t ♦♥ t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡
• t❤❡ ✇♦r❦s❤❡❡t ✐s ✈❡r② ♥✐❝❡ t♦ ✉s❡
• ♣②t❤♦♥ q✉✐t❡ ❡❛s② t♦ ❧❡❛r♥

❈♦♥❝❧✉s✐♦♥

❚♦ ❝♦♠♣✉t❡ ♣❛✐r✐♥❣s✱ ✇❡ ❤❛✈❡ ✿

• ❛r✐t❤♠❡t✐❝ ♦❢ ✜♥✐t❡ ✜❡❧❞
• ♦♣❡r❛t✐♦♥ ♦♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡s

■t ✐s ✈❡r② ❡❛s② t♦ ✐♠♣❧❡♠❡♥t ✇✐t❤ ❙❛❣❡✳ ❆ ✧♥❛✐✈❡✧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❣✐✈❡s ❣♦♦❞ r❡s✉❧t ❝♦♠♣❛r❡ t♦ ▼❛❣♠❛✳ ■ ❤❛✈❡ t♦ ✐♠♣r♦✈❡ ♠② ✐♠♣❧❡♠❡♥t❛t✐♦♥✱ ✐♥ ♦r❞❡r t♦ ❤❛✈❡ ❜❡tt❡r ♣❡r❢♦r♠❛♥❝❡s✳