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■♠♣r♦✈✐♥❣ ◆❋❙ ❢♦r t❤❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ♣r♦❜❧❡♠ ✐♥ ♥♦♥✲♣r✐♠❡ ✜♥✐t❡ ✜❡❧❞s ❘❛③✈❛♥ ❇❛r❜✉❧❡s❝✉✱ P✐❡rr✐❝❦ ●❛✉❞r②✱ ❆✉r♦r❡ ●✉✐❧❧❡✈✐❝ ✱ ❋r❛♥☛ ❝♦✐s ▼♦r❛✐♥ ■♥st✐t✉t ♥❛t✐♦♥❛❧ ❞❡ r❡❝❤❡r❝❤❡ ❡♥ ✐♥❢♦r♠❛t✐q✉❡ ❡t ❡♥ ❛✉t♦♠❛t✐q✉❡ ✭■◆❘■❆✮ ✁ ❊❝♦❧❡ P♦❧②t❡❝❤♥✐q✉❡✴▲■❳ ❈❡♥tr❡ ♥❛t✐♦♥❛❧ ❞❡ ❧❛ r❡❝❤❡r❝❤❡ s❝✐❡♥t✐✜q✉❡ ✭❈◆❘❙✮ ❯♥✐✈❡rs✐t✁ ❡ ❞❡ ▲♦rr❛✐♥❡ ❊✉r♦❝r②♣t ✷✵✶✺✱ ❆♣r✐❧ ✷✼t❤ ❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ F pn ❆♣r✐❧ ✷✼t❤✱ ✷✵✶✺ ✶ ✴ ✶✸

❖✉r ❲♦r❦ F p 2 ✿ t❛r❣❡t ❣r♦✉♣ ♦❢ ♣❛✐r✐♥❣✲❜❛s❡❞ ❝r②♣t♦s②st❡♠s ❘❡❝♦r❞ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ ❉✐s❝r❡t❡ ▲♦❣❛r✐t❤♠ ✭❉▲✮ ✐♥ F p 2 ♦❢ ✻✵✵ ❜✐ts ✭ log 2 p = 300 ❜✐ts✮ ❉▲ ✐♥ F p 2 ✐s ✷✻✵ t✐♠❡s ❢❛st❡r t❤❛♥ ❉▲ ✐♥ F p ′ ♦❢ s❛♠❡ s✐③❡ ➙ s❡r✐♦✉s ❝♦♥s❡q✉❡♥❝❡s ❢♦r ♣❛✐r✐♥❣✲❜❛s❡❞ ❝r②♣t♦ s♦✉r❝❡ ❝♦❞❡✿ ❤tt♣✿✴✴❝❛❞♦✲♥❢s✳❣❢♦r❣❡✳✐♥r✐❛✳❢r✴ ❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ F pn ❆♣r✐❧ ✷✼t❤✱ ✷✵✶✺ ✷ ✴ ✶✸

❉✐s❝r❡t❡ ▲♦❣ ✐♥ F pn ✇✐t❤ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ❈♦♥t❡①t ✿ ❉✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ♣r♦❜❧❡♠ ✭❉▲P✮ ✐♥ F ∗ p n ■♥ ❛ s✉❜❣r♦✉♣ � g � ♦❢ F ∗ p n ♦❢ ♦r❞❡r ℓ ✱ ( g , x ) �→ g x ✐s ❡❛s② ✭♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ ( g , g x ) �→ x ✐s ✭✐♥ ✇❡❧❧✲❝❤♦s❡♥ s✉❜❣r♦✉♣✮ ❤❛r❞✿ ❉▲P✳ ■♥ ♦✉r ✇♦r❦✿ ❲❡ ❛tt❛❝❦ ❉▲ ✐♥ F p 2 ✱ st❛rt✐♥❣ ♣♦✐♥t ♦❢ F p 3 ✱ F p 4 , . . . F p 12 p ✐s ❧❛r❣❡✿ q✉❛s✐ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦✳ ❞♦❡s ◆❖❚ ❛♣♣❧② ❉▲P ✐♥ t❤❡s❡ F p n st✐❧❧ ❛s②♠♣t♦t✐❝❛❧❧② ❛s ❤❛r❞ ❛s ✐♥ t❤❡ ✾✵✬s ❝♦♥s❡q✉❡♥❝❡s ❢♦r ♣❛✐r✐♥❣✲❜❛s❡❞ ❝r②♣t♦✿ F p 2 t❛r❣❡t ❣r♦✉♣ ❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ F pn ❆♣r✐❧ ✷✼t❤✱ ✷✵✶✺ ✸ ✴ ✶✸

❉✐s❝r❡t❡ ▲♦❣ ✐♥ F pn ✇✐t❤ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ Pr❛❝t✐❝❛❧ ✐♠♣r♦✈❡♠❡♥ts ❛♥❞ ♥❡✇ ❛s②♠♣t♦t✐❝ ❝♦♠♣❧❡①✐t✐❡s L ✲♥♦t❛t✐♦♥✿ Q = p n ✱ L Q [1 / 3 , c ] = ❡ ( c + o (1))(log Q ) 1 / 3 (log log Q ) 2 / 3 ❢♦r c > 0 ✳ ❉▲ ✐♥ F p n ✱ s♠❛❧❧ n ✱ ❧❛r❣❡ p ✿ ❝♦♠♣❧❡①✐t② ✐♥ L p n [1 / 3 , 1 . 92] ✭❛s ❢♦r ❘❙❆ ♠♦❞✉❧✉s ❢❛❝t♦r✐③❛t✐♦♥✮ s✐♥❝❡ t❤❡ ✾✵✬s n ≥ 2 ✿ t✇♦ ♥❡✇ ♣♦❧②♥♦♠✐❛❧ s❡❧❡❝t✐♦♥ ♠❡t❤♦❞s ❣r❡❛t ✐♠♣r♦✈❡♠❡♥ts ✐♥ ♣r❛❝t✐❝❡ r❡❝♦r❞ ♦❢ ✻✵✵ ❜✐ts ❇♦♥✉s✿ ❛s②♠♣t♦t✐❝ ❝♦♠♣❧❡①✐t② ✐♠♣r♦✈❡♠❡♥ts ✐♥ ♠❡❞✐✉♠ ❝❛r❛❝t❡r✐st✐❝ ❝❛s❡ α = 1 / 3 c ✱ ♣r❡✈✐♦✉s ✇♦r❦ c ✱ ♦✉r ✇♦r❦ ❉▲ ✐♥ F p n ✱ p = L Q (2 / 3 , c ′ ) 1 . 92 < c < 2 . 42 ✗ 1 . 74 ✓ ❉▲ ✐♥ F p n ✱ ♠❡❞✐✉♠ p 2 . 42 ✗ 2 . 20 ✓ ▼◆❋❙ ✈❛r✐❛♥ts✿ s❡❡ ❬P✐❡rr♦t✶✺❪✱ ❊✉r♦❝r②♣t ✷✵✶✺✳ ❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ F pn ❆♣r✐❧ ✷✼t❤✱ ✷✵✶✺ ✹ ✴ ✶✸

❞❡✜♥❡ ♥✉♠❜❡r ✜❡❧❞s ✶✳ P♦❧②♥♦♠✐❛❧ ❙❡❧❡❝t✐♦♥✿ ❝♦♠♣✉t❡ ✳ ✷✳ ❘❡❧❛t✐♦♥ ❝♦❧❧❡❝t✐♦♥ ❜❡t✇❡❡♥ ✐❞❡❛❧s ♦❢ ❡❛❝❤ ♥✉♠❜❡r ✜❡❧❞✳ ✸✳ ▲✐♥❡❛r ❛❧❣❡❜r❛ ♠♦❞✉❧♦ ✳ ❤❡r❡ ✇❡ ❦♥♦✇ t❤❡ ❞✐s❝r❡t❡ ❧♦❣ ♦❢ ❛ s✉❜s❡t ♦❢ ✐❞❡❛❧s ♦❢ ✱ ✳ ✹✳ ■♥❞✐✈✐❞✉❛❧ ▲♦❣❛r✐t❤♠✳ ❉✐s❝r❡t❡ ▲♦❣ ✐♥ F pn ✇✐t❤ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ❛❧❣♦r✐t❤♠ ❢♦r ❉▲ ✐♥ F p n ❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ F pn ❆♣r✐❧ ✷✼t❤✱ ✷✵✶✺ ✺ ✴ ✶✸

✷✳ ❘❡❧❛t✐♦♥ ❝♦❧❧❡❝t✐♦♥ ❜❡t✇❡❡♥ ✐❞❡❛❧s ♦❢ ❡❛❝❤ ♥✉♠❜❡r ✜❡❧❞✳ ✸✳ ▲✐♥❡❛r ❛❧❣❡❜r❛ ♠♦❞✉❧♦ ✳ ❤❡r❡ ✇❡ ❦♥♦✇ t❤❡ ❞✐s❝r❡t❡ ❧♦❣ ♦❢ ❛ s✉❜s❡t ♦❢ ✐❞❡❛❧s ♦❢ ✱ ✳ ✹✳ ■♥❞✐✈✐❞✉❛❧ ▲♦❣❛r✐t❤♠✳ ❉✐s❝r❡t❡ ▲♦❣ ✐♥ F pn ✇✐t❤ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ❛❧❣♦r✐t❤♠ ❢♦r ❉▲ ✐♥ F p n ✶✳ P♦❧②♥♦♠✐❛❧ ❙❡❧❡❝t✐♦♥✿ ❝♦♠♣✉t❡ f ( x ) , g ( x ) ➙ ❞❡✜♥❡ ♥✉♠❜❡r ✜❡❧❞s K f , K g ✳ ❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ F pn ❆♣r✐❧ ✷✼t❤✱ ✷✵✶✺ ✺ ✴ ✶✸

✷✳ ❘❡❧❛t✐♦♥ ❝♦❧❧❡❝t✐♦♥ ❜❡t✇❡❡♥ ✐❞❡❛❧s ♦❢ ❡❛❝❤ ♥✉♠❜❡r ✜❡❧❞✳ ✸✳ ▲✐♥❡❛r ❛❧❣❡❜r❛ ♠♦❞✉❧♦ ✳ ❤❡r❡ ✇❡ ❦♥♦✇ t❤❡ ❞✐s❝r❡t❡ ❧♦❣ ♦❢ ❛ s✉❜s❡t ♦❢ ✐❞❡❛❧s ♦❢ ✱ ✳ ✹✳ ■♥❞✐✈✐❞✉❛❧ ▲♦❣❛r✐t❤♠✳ ❉✐s❝r❡t❡ ▲♦❣ ✐♥ F pn ✇✐t❤ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ❛❧❣♦r✐t❤♠ ❢♦r ❉▲ ✐♥ F p n ✶✳ P♦❧②♥♦♠✐❛❧ ❙❡❧❡❝t✐♦♥✿ ❝♦♠♣✉t❡ f ( x ) , g ( x ) ➙ ❞❡✜♥❡ ♥✉♠❜❡r ✜❡❧❞s K f , K g ✳ Q [ x ] Q [ x ] / ( f ( x )) Q [ y ] / ( g ( y )) ❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ F pn ❆♣r✐❧ ✷✼t❤✱ ✷✵✶✺ ✺ ✴ ✶✸

✷✳ ❘❡❧❛t✐♦♥ ❝♦❧❧❡❝t✐♦♥ ❜❡t✇❡❡♥ ✐❞❡❛❧s ♦❢ ❡❛❝❤ ♥✉♠❜❡r ✜❡❧❞✳ ✸✳ ▲✐♥❡❛r ❛❧❣❡❜r❛ ♠♦❞✉❧♦ ✳ ❤❡r❡ ✇❡ ❦♥♦✇ t❤❡ ❞✐s❝r❡t❡ ❧♦❣ ♦❢ ❛ s✉❜s❡t ♦❢ ✐❞❡❛❧s ♦❢ ✱ ✳ ✹✳ ■♥❞✐✈✐❞✉❛❧ ▲♦❣❛r✐t❤♠✳ ❉✐s❝r❡t❡ ▲♦❣ ✐♥ F pn ✇✐t❤ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ❛❧❣♦r✐t❤♠ ❢♦r ❉▲ ✐♥ F p n ✶✳ P♦❧②♥♦♠✐❛❧ ❙❡❧❡❝t✐♦♥✿ ❝♦♠♣✉t❡ f ( x ) , g ( x ) ➙ ❞❡✜♥❡ ♥✉♠❜❡r ✜❡❧❞s K f , K g ✳ Q [ x ] Q [ x ] / ( f ( x )) Q [ y ] / ( g ( y )) ρ f : x �→ z ρ g : y �→ z F p n = F p [ z ] / ( ϕ ( z )) ❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ F pn ❆♣r✐❧ ✷✼t❤✱ ✷✵✶✺ ✺ ✴ ✶✸

✸✳ ▲✐♥❡❛r ❛❧❣❡❜r❛ ♠♦❞✉❧♦ ✳ ❤❡r❡ ✇❡ ❦♥♦✇ t❤❡ ❞✐s❝r❡t❡ ❧♦❣ ♦❢ ❛ s✉❜s❡t ♦❢ ✐❞❡❛❧s ♦❢ ✱ ✳ ✹✳ ■♥❞✐✈✐❞✉❛❧ ▲♦❣❛r✐t❤♠✳ ❉✐s❝r❡t❡ ▲♦❣ ✐♥ F pn ✇✐t❤ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ❛❧❣♦r✐t❤♠ ❢♦r ❉▲ ✐♥ F p n ✶✳ P♦❧②♥♦♠✐❛❧ ❙❡❧❡❝t✐♦♥✿ ❝♦♠♣✉t❡ f ( x ) , g ( x ) ➙ ❞❡✜♥❡ ♥✉♠❜❡r ✜❡❧❞s K f , K g ✳ ✷✳ ❘❡❧❛t✐♦♥ ❝♦❧❧❡❝t✐♦♥ ❜❡t✇❡❡♥ ✐❞❡❛❧s ♦❢ ❡❛❝❤ ♥✉♠❜❡r ✜❡❧❞✳ Q [ x ] Q [ x ] / ( f ( x )) Q [ y ] / ( g ( y )) ∋ a − by a − bx ∈ ρ f : x �→ z ρ g : y �→ z F p n = F p [ z ] / ( ϕ ( z )) ❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ F pn ❆♣r✐❧ ✷✼t❤✱ ✷✵✶✺ ✺ ✴ ✶✸