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  1. P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥ ■❙❆ ✷✵✵✾✱ ❙é♦✉❧✱ ❙♦✉t❤ ❑♦r❡❛ ❲❤❛t ❆❜♦✉t ❱✉❧♥❡r❛❜✐❧✐t② t♦ ❛ ❋❛✉❧t ❆tt❛❝❦ ♦❢ t❤❡ ▼✐❧❧❡r✬s ❆❧❣♦r✐t❤♠ ❞✉r✐♥❣ ❛♥ ■❞❡♥t✐t② ❇❛s❡❞ Pr♦t♦❝♦❧ ❄ ◆❛❞✐❛ ❊▲ ▼❘❆❇❊❚ ▲■❘▼✱ ■✸▼✱ ❯♥✐✈❡rs✐té ▼♦♥t♣❡❧❧✐❡r ✷ ❚❤✉rs❞❛② ✷✺t❤ ❏✉♥❡ ✷✵✵✾

  2. P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥ ❖✉t❧✐♥❡ ♦❢ t❤❡ ♣r❡s❡♥t❛t✐♦♥ P❛✐r✐♥❣s Pr♦♣❡rt✐❡s ♦❢ ♣❛✐r✐♥❣s P❛✐r✐♥❣ ❜❛s❡❞ ❝r②♣t♦❣r❛♣❤② ■❞❡♥t✐t② ❇❛s❡❞ ❈r②♣t♦❣r❛♣❤② ❈♦♥str✉❝t✐♦♥ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ Pr♦♣❡rt✐❡s ❈♦♥str✉❝t✐♦♥ ❋❛✉❧t ❛tt❛❝❦ ❉❡✜♥✐t✐♦♥ ❆❣❛✐♥st t❤❡ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❈♦♥❝❧✉s✐♦♥

  3. P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥ ❖✉t❧✐♥❡ P❛✐r✐♥❣s Pr♦♣❡rt✐❡s ♦❢ ♣❛✐r✐♥❣s P❛✐r✐♥❣ ❜❛s❡❞ ❝r②♣t♦❣r❛♣❤② ■❞❡♥t✐t② ❇❛s❡❞ ❈r②♣t♦❣r❛♣❤② ❈♦♥str✉❝t✐♦♥ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ Pr♦♣❡rt✐❡s ❈♦♥str✉❝t✐♦♥ ❋❛✉❧t ❛tt❛❝❦ ❉❡✜♥✐t✐♦♥ ❆❣❛✐♥st t❤❡ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❈♦♥❝❧✉s✐♦♥

  4. ❈♦♥s❡q✉❡♥❝❡s ❡ P ◗ ❥ ❥ ❡ ❥ P ◗ ❡ P ❥ ◗ P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥ ❲❤❛t ✐s ❛ ♣❛✐r✐♥❣ ❄ ▲❡t G ✶ ✱ G ✷ ❛♥❞ G ✸ ❜❡ t❤r❡❡ ❣r♦✉♣s ✇✐t❤ t❤❡ s❛♠❡ ♦r❞❡r r ✳ ❆ ♣❛✐r✐♥❣ ✐s ❛ ♠❛♣ ✿ ❡ : G ✶ × G ✷ → G ✸ ✇❤✐❝❤ ✈❡r✐✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s ✿ • ◆♦♥ ❞❡❣❡♥❡r❛t❡ ❀ • ❇✐❧✐♥❡❛r✐t② ❀

  5. P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥ ❲❤❛t ✐s ❛ ♣❛✐r✐♥❣ ❄ ▲❡t G ✶ ✱ G ✷ ❛♥❞ G ✸ ❜❡ t❤r❡❡ ❣r♦✉♣s ✇✐t❤ t❤❡ s❛♠❡ ♦r❞❡r r ✳ ❆ ♣❛✐r✐♥❣ ✐s ❛ ♠❛♣ ✿ ❡ : G ✶ × G ✷ → G ✸ ✇❤✐❝❤ ✈❡r✐✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s ✿ • ◆♦♥ ❞❡❣❡♥❡r❛t❡ ❀ • ❇✐❧✐♥❡❛r✐t② ❀ ❈♦♥s❡q✉❡♥❝❡s ∀ ❥ ∈ N , ❡ ([ ❥ ] P , ◗ ) = ❡ ( P , ◗ ) ❥ = ❡ ( P , [ ❥ ] ◗ )

  6. ●✐✈❡♥ P ❛♥❞ ◗ P ❊ q ✱ t❤❡ ❉▲P ♦♥ ❊ q ❝♦♥s✐sts ✐♥ ✜♥❞✐♥❣ ✳ ▲❡t ❙ ❊ q ❜❡ ❛ ♣♦✐♥t s✉❝❤ t❤❛t ❡ P ❙ ✶✱ ❧❡t ❡ P ❙ ❣ ❛♥❞ ❡ ◗ ❙ ❤ ❊ q ✱ t❤❡♥ t❤❡ ❉▲P ❜❡❝♦♠❡s ✜♥❞✐♥❣ s✉❝❤ t❤❛t ❤ ❣ ✐♥ ❛ ✜♥✐t❡ ✜❡❧❞✳ P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥ ❊❧❧✐♣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❡ ✜❡❧❞✳

  7. ▲❡t ❙ ❊ q ❜❡ ❛ ♣♦✐♥t s✉❝❤ t❤❛t ❡ P ❙ ✶✱ ❧❡t ❡ P ❙ ❣ ❛♥❞ ❡ ◗ ❙ ❤ ❊ q ✱ t❤❡♥ t❤❡ ❉▲P ❜❡❝♦♠❡s ✜♥❞✐♥❣ s✉❝❤ t❤❛t ❤ ❣ ✐♥ ❛ ✜♥✐t❡ ✜❡❧❞✳ P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥ ❊❧❧✐♣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 ❛♥❞ ◗ = α P ∈ ❊ ( F q ) ✱ t❤❡ ❉▲P ♦♥ ❊ ( F q ) ❝♦♥s✐sts ✐♥ ✜♥❞✐♥❣ α ✳

  8. P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥ ❊❧❧✐♣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 ❛♥❞ ◗ = α P ∈ ❊ ( F q ) ✱ t❤❡ ❉▲P ♦♥ ❊ ( F q ) ❝♦♥s✐sts ✐♥ ✜♥❞✐♥❣ α ✳ • ▲❡t ❙ ∈ ❊ ( F q ) ❜❡ ❛ ♣♦✐♥t s✉❝❤ t❤❛t ❡ ( P , ❙ ) � = ✶✱ ❧❡t ❡ ( P , ❙ ) = ❣ ❛♥❞ ❡ ( ◗ , ❙ ) = ❤ ∈ ❊ ( F q ) ✱ t❤❡♥ • t❤❡ ❉▲P ❜❡❝♦♠❡s ✜♥❞✐♥❣ α s✉❝❤ t❤❛t ❤ = ❣ α ✐♥ ❛ ✜♥✐t❡ ✜❡❧❞✳

  9. P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥ ❊❧❧✐♣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 ✭❇♦♥❡❤✱ ❙❝❤❛❝❦❛♠♠✱ ✷✵✵✹✮

  10. P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥ ❊❧❧✐♣t✐❝ ❈✉r✈❡ ❈r②♣t♦❣r❛♣❤② ❛♥❞ ♣❛✐r✐♥❣s ■❞❡♥t✐t② ❇❛s❡❞ ❈r②♣t♦❣r❛♣❤② ✭■❇❈✮ ❆♥ ■❞❡♥t✐t② ❇❛s❡❞ ♣r♦t♦❝♦❧ ✐s ❛♥ ❛s②♠❡tr✐❝ s②st❡♠ ✇❤❡r❡ ✿ • ❚❤❡ ♣✉❜❧✐❝ ❦❡② ♦❢ ❆❧✐❝❡ ✐ts ❤❡r ✐❞❡♥t✐t②✱ • ❤❡r ♣r✐✈❛t❡ ❑❡② ✐s ❝♦♥str✉❝t❡❞ ❜② ❛ ❚r✉st❡❞ ❛✉t❤♦r✐t②✳ ❊①❛♠♣❧❡ ❲❡ ✐❧❧✉str❛t❡ t❤❡ ♥♦t✐♦♥ ♦❢ ■❞❡♥t✐t② ❇❛s❡❞ ❈r②♣t♦❣r❛♣❤② ✇✐t❤ t❤❡ ❡①❛♠♣❧❡ ♦❢ ❛ ❦❡② ❡①❝❤❛♥❣❡ ❜❡t✇❡❡♥ ❆❧✐❝❡ ❛♥❞ ❇♦❜✳

  11. P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥ ■❇❈ ❙❡❝✉r❡ ❦❡② ❡①❝❤❛♥❣❡ ❜❡t✇❡❡♥ ❆❧✐❝❡ ❛♥❞ ❇♦❜

  12. P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥ ■❇❈ ❙❡❝✉r❡ ❦❡② ❡①❝❤❛♥❣❡ ❜❡t✇❡❡♥ ❆❧✐❝❡ ❛♥❞ ❇♦❜

  13. P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥ ■❇❈ ❙❡❝✉r❡ ❦❡② ❡①❝❤❛♥❣❡ ❜❡t✇❡❡♥ ❆❧✐❝❡ ❛♥❞ ❇♦❜

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