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■♠♣r♦✈✐♥❣ ◆❋❙ ❢♦r t❤❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ♣r♦❜❧❡♠ ✐♥ ♥♦♥✲♣r✐♠❡ ✜♥✐t❡ ✜❡❧❞s P♦❧②♥♦♠✐❛❧ s❡❧❡❝t✐♦♥ ❛♥❞ ✐♥❞✐✈✐❞✉❛❧ ❧♦❣❛r✐t❤♠

❘❛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❛✐♥❡

❆●❈❚ ✷✵✶✺✱ ▼❛② ✷✵t❤

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✶ ✴ ✶✼

❖✉r ❲♦r❦

Fp2✿ t❛r❣❡t ❣r♦✉♣ ♦❢ ♣❛✐r✐♥❣✲❜❛s❡❞ ❝r②♣t♦s②st❡♠s ❘❡❝♦r❞ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❛ ❉✐s❝r❡t❡ ▲♦❣❛r✐t❤♠ ✭❉▲✮ ✐♥ Fp2 ♦❢ ✻✵✵ ❜✐ts ✭log2 p = 300 ❜✐ts✮ ❉▲ ✐♥ Fp2 ✐s ✷✻✵ t✐♠❡s ❢❛st❡r t❤❛♥ ❉▲ ✐♥ Fp′ ♦❢ 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❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✷ ✴ ✶✼

❉✐s❝r❡t❡ ▲♦❣ ✐♥ Fpn ✇✐t❤ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡

❈♦♥t❡①t ✿ ❉✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ♣r♦❜❧❡♠ ✭❉▲P✮ ✐♥ F∗

pn

■♥ ❛ s✉❜❣r♦✉♣ g ♦❢ F∗

pn ♦❢ ♦r❞❡r ℓ✱

(g, x) → gx ✐s ❡❛s② ✭♣♦❧②♥♦♠✐❛❧ t✐♠❡✮ (g, gx) → x ✐s ✭✐♥ ✇❡❧❧✲❝❤♦s❡♥ s✉❜❣r♦✉♣✮ ❤❛r❞✿ ❉▲P✳ ■♥ ♦✉r ✇♦r❦✿ ❲❡ ❛tt❛❝❦ ❉▲ ✐♥ Fp2✱ st❛rt✐♥❣ ♣♦✐♥t ♦❢ Fp3✱ Fp4, . . . Fp12 p ✐s ❧❛r❣❡✿ q✉❛s✐ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦✳ ❞♦❡s ◆❖❚ ❛♣♣❧② ❉▲P ✐♥ t❤❡s❡ Fpn st✐❧❧ ❛s②♠♣t♦t✐❝❛❧❧② ❛s ❤❛r❞ ❛s ✐♥ t❤❡ ✾✵✬s ❝♦♥s❡q✉❡♥❝❡s ❢♦r ♣❛✐r✐♥❣✲❜❛s❡❞ ❝r②♣t♦✿ Fp2 t❛r❣❡t ❣r♦✉♣

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✸ ✴ ✶✼

❉✐s❝r❡t❡ ▲♦❣ ✐♥ Fpn ✇✐t❤ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡

Pr❛❝t✐❝❛❧ ✐♠♣r♦✈❡♠❡♥ts ❛♥❞ ♥❡✇ ❛s②♠♣t♦t✐❝ ❝♦♠♣❧❡①✐t✐❡s

L✲♥♦t❛t✐♦♥✿ Q = pn✱ LQ[1/3, c] = ❡(c+o(1))(log Q)1/3 (log log Q)2/3 ❢♦r c > 0✳ ❉▲ ✐♥ Fpn✱ s♠❛❧❧ n✱ ❧❛r❣❡ p✿ ❝♦♠♣❧❡①✐t② ✐♥ Lpn[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❦ ❉▲ ✐♥ Fpn✱ p = LQ(2/3, c′) 1.92 < c < 2.42 ✗ 1.74 ✓ ❉▲ ✐♥ Fpn✱ ♠❡❞✐✉♠ p 2.42 ✗ 2.20 ✓ ▼◆❋❙ ✈❛r✐❛♥ts✿ s❡❡ ❬P✐❡rr♦t✶✺❪✱ ❊✉r♦❝r②♣t ✷✵✶✺✳

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✹ ✴ ✶✼

❉✐s❝r❡t❡ ▲♦❣ ✐♥ Fpn ✇✐t❤ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡

◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ❛❧❣♦r✐t❤♠ ❢♦r ❉▲ ✐♥ Fpn

✶✳ P♦❧②♥♦♠✐❛❧ ❙❡❧❡❝t✐♦♥✿ ❝♦♠♣✉t❡

❞❡✜♥❡ ♥✉♠❜❡r ✜❡❧❞s ✳ ✷✳ ❘❡❧❛t✐♦♥ ❝♦❧❧❡❝t✐♦♥ ❜❡t✇❡❡♥ ✐❞❡❛❧s ♦❢ ❡❛❝❤ ♥✉♠❜❡r ✜❡❧❞✳ ✸✳ ▲✐♥❡❛r ❛❧❣❡❜r❛ ♠♦❞✉❧♦ ✳ ❤❡r❡ ✇❡ ❦♥♦✇ t❤❡ ❞✐s❝r❡t❡ ❧♦❣ ♦❢ ❛ s✉❜s❡t ♦❢ ✐❞❡❛❧s ♦❢ ✱ ✳ ✹✳ ■♥❞✐✈✐❞✉❛❧ ▲♦❣❛r✐t❤♠✳

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✺ ✴ ✶✼

❉✐s❝r❡t❡ ▲♦❣ ✐♥ Fpn ✇✐t❤ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡

◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ❛❧❣♦r✐t❤♠ ❢♦r ❉▲ ✐♥ Fpn

✶✳ P♦❧②♥♦♠✐❛❧ ❙❡❧❡❝t✐♦♥✿ ❝♦♠♣✉t❡ f (x), g(x) ➙❞❡✜♥❡ ♥✉♠❜❡r ✜❡❧❞s

Kf , Kg✳ ✷✳ ❘❡❧❛t✐♦♥ ❝♦❧❧❡❝t✐♦♥ ❜❡t✇❡❡♥ ✐❞❡❛❧s ♦❢ ❡❛❝❤ ♥✉♠❜❡r ✜❡❧❞✳ ✸✳ ▲✐♥❡❛r ❛❧❣❡❜r❛ ♠♦❞✉❧♦ ✳ ❤❡r❡ ✇❡ ❦♥♦✇ t❤❡ ❞✐s❝r❡t❡ ❧♦❣ ♦❢ ❛ s✉❜s❡t ♦❢ ✐❞❡❛❧s ♦❢ ✱ ✳ ✹✳ ■♥❞✐✈✐❞✉❛❧ ▲♦❣❛r✐t❤♠✳

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✺ ✴ ✶✼

❉✐s❝r❡t❡ ▲♦❣ ✐♥ Fpn ✇✐t❤ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡

◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ❛❧❣♦r✐t❤♠ ❢♦r ❉▲ ✐♥ Fpn

✶✳ P♦❧②♥♦♠✐❛❧ ❙❡❧❡❝t✐♦♥✿ ❝♦♠♣✉t❡ f (x), g(x) ➙❞❡✜♥❡ ♥✉♠❜❡r ✜❡❧❞s

Kf , Kg✳ ✷✳ ❘❡❧❛t✐♦♥ ❝♦❧❧❡❝t✐♦♥ ❜❡t✇❡❡♥ ✐❞❡❛❧s ♦❢ ❡❛❝❤ ♥✉♠❜❡r ✜❡❧❞✳ Q[x] Q[x]/(f (x)) Q[y]/(g(y)) ✸✳ ▲✐♥❡❛r ❛❧❣❡❜r❛ ♠♦❞✉❧♦ ✳ ❤❡r❡ ✇❡ ❦♥♦✇ t❤❡ ❞✐s❝r❡t❡ ❧♦❣ ♦❢ ❛ s✉❜s❡t ♦❢ ✐❞❡❛❧s ♦❢ ✱ ✳ ✹✳ ■♥❞✐✈✐❞✉❛❧ ▲♦❣❛r✐t❤♠✳

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✺ ✴ ✶✼

❉✐s❝r❡t❡ ▲♦❣ ✐♥ Fpn ✇✐t❤ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡

◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ❛❧❣♦r✐t❤♠ ❢♦r ❉▲ ✐♥ Fpn

✶✳ P♦❧②♥♦♠✐❛❧ ❙❡❧❡❝t✐♦♥✿ ❝♦♠♣✉t❡ f (x), g(x) ➙❞❡✜♥❡ ♥✉♠❜❡r ✜❡❧❞s

Kf , Kg✳ ✷✳ ❘❡❧❛t✐♦♥ ❝♦❧❧❡❝t✐♦♥ ❜❡t✇❡❡♥ ✐❞❡❛❧s ♦❢ ❡❛❝❤ ♥✉♠❜❡r ✜❡❧❞✳ Q[x] Q[x]/(f (x)) Q[y]/(g(y)) Fpn = Fp[z]/(ϕ(z)) ρf : x → z ρg : y → z ✸✳ ▲✐♥❡❛r ❛❧❣❡❜r❛ ♠♦❞✉❧♦ ✳ ❤❡r❡ ✇❡ ❦♥♦✇ t❤❡ ❞✐s❝r❡t❡ ❧♦❣ ♦❢ ❛ s✉❜s❡t ♦❢ ✐❞❡❛❧s ♦❢ ✱ ✳ ✹✳ ■♥❞✐✈✐❞✉❛❧ ▲♦❣❛r✐t❤♠✳

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✺ ✴ ✶✼

❉✐s❝r❡t❡ ▲♦❣ ✐♥ Fpn ✇✐t❤ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡

◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ❛❧❣♦r✐t❤♠ ❢♦r ❉▲ ✐♥ Fpn

✶✳ P♦❧②♥♦♠✐❛❧ ❙❡❧❡❝t✐♦♥✿ ❝♦♠♣✉t❡ f (x), g(x) ➙❞❡✜♥❡ ♥✉♠❜❡r ✜❡❧❞s

Kf , Kg✳ ✷✳ ❘❡❧❛t✐♦♥ ❝♦❧❧❡❝t✐♦♥ ❜❡t✇❡❡♥ ✐❞❡❛❧s ♦❢ ❡❛❝❤ ♥✉♠❜❡r ✜❡❧❞✳ Q[x] Q[x]/(f (x)) Q[y]/(g(y)) Fpn = Fp[z]/(ϕ(z)) ρf : x → z ρg : y → z a − bx ∈ ∋ a − by ✸✳ ▲✐♥❡❛r ❛❧❣❡❜r❛ ♠♦❞✉❧♦ ✳ ❤❡r❡ ✇❡ ❦♥♦✇ t❤❡ ❞✐s❝r❡t❡ ❧♦❣ ♦❢ ❛ s✉❜s❡t ♦❢ ✐❞❡❛❧s ♦❢ ✱ ✳ ✹✳ ■♥❞✐✈✐❞✉❛❧ ▲♦❣❛r✐t❤♠✳

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✺ ✴ ✶✼

❉✐s❝r❡t❡ ▲♦❣ ✐♥ Fpn ✇✐t❤ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡

◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ❛❧❣♦r✐t❤♠ ❢♦r ❉▲ ✐♥ Fpn

✶✳ P♦❧②♥♦♠✐❛❧ ❙❡❧❡❝t✐♦♥✿ ❝♦♠♣✉t❡ f (x), g(x) ➙❞❡✜♥❡ ♥✉♠❜❡r ✜❡❧❞s

Kf , Kg✳ ✷✳ ❘❡❧❛t✐♦♥ ❝♦❧❧❡❝t✐♦♥ ❜❡t✇❡❡♥ ✐❞❡❛❧s ♦❢ ❡❛❝❤ ♥✉♠❜❡r ✜❡❧❞✳ Q[x] Q[x]/(f (x)) Q[y]/(g(y)) Fpn = Fp[z]/(ϕ(z)) a − bx ∈ ∋ a − by ρf : a − bx → a − bz ρg : a − by → a − bz ✸✳ ▲✐♥❡❛r ❛❧❣❡❜r❛ ♠♦❞✉❧♦ ✳ ❤❡r❡ ✇❡ ❦♥♦✇ t❤❡ ❞✐s❝r❡t❡ ❧♦❣ ♦❢ ❛ s✉❜s❡t ♦❢ ✐❞❡❛❧s ♦❢ ✱ ✳ ✹✳ ■♥❞✐✈✐❞✉❛❧ ▲♦❣❛r✐t❤♠✳

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✺ ✴ ✶✼

❉✐s❝r❡t❡ ▲♦❣ ✐♥ Fpn ✇✐t❤ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡

◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ❛❧❣♦r✐t❤♠ ❢♦r ❉▲ ✐♥ Fpn

✶✳ P♦❧②♥♦♠✐❛❧ ❙❡❧❡❝t✐♦♥✿ ❝♦♠♣✉t❡ f (x), g(x) ➙❞❡✜♥❡ ♥✉♠❜❡r ✜❡❧❞s

Kf , Kg✳ ✷✳ ❘❡❧❛t✐♦♥ ❝♦❧❧❡❝t✐♦♥ ❜❡t✇❡❡♥ ✐❞❡❛❧s ♦❢ ❡❛❝❤ ♥✉♠❜❡r ✜❡❧❞✳ Q[x] Q[x]/(f (x)) Q[y]/(g(y)) Fpn = Fp[z]/(ϕ(z)) a − bx ∈ ∋ a − by ρf : a − bx → a − bz ρg : a − by → a − bz ✸✳ ▲✐♥❡❛r ❛❧❣❡❜r❛ ♠♦❞✉❧♦ ℓ | pn − 1✳ ➙ ❤❡r❡ ✇❡ ❦♥♦✇ t❤❡ ❞✐s❝r❡t❡ ❧♦❣ ♦❢ ❛ s✉❜s❡t ♦❢ ✐❞❡❛❧s ♦❢ Kf ✱ Kg✳ ✹✳ ■♥❞✐✈✐❞✉❛❧ ▲♦❣❛r✐t❤♠✳

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✺ ✴ ✶✼

❉✐s❝r❡t❡ ▲♦❣ ✐♥ Fpn ✇✐t❤ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡

◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ❛❧❣♦r✐t❤♠ ❢♦r ❉▲ ✐♥ Fpn

✶✳ P♦❧②♥♦♠✐❛❧ ❙❡❧❡❝t✐♦♥✿ ❝♦♠♣✉t❡ f (x), g(x) ➙❞❡✜♥❡ ♥✉♠❜❡r ✜❡❧❞s

Kf , Kg✳ ✷✳ ❘❡❧❛t✐♦♥ ❝♦❧❧❡❝t✐♦♥ ❜❡t✇❡❡♥ ✐❞❡❛❧s ♦❢ ❡❛❝❤ ♥✉♠❜❡r ✜❡❧❞✳ Q[x] Q[x]/(f (x)) Q[y]/(g(y)) Fpn = Fp[z]/(ϕ(z)) a − bx ∈ ∋ a − by ρf : a − bx → a − bz ρg : a − by → a − bz ✸✳ ▲✐♥❡❛r ❛❧❣❡❜r❛ ♠♦❞✉❧♦ ℓ | pn − 1✳ ➙ ❤❡r❡ ✇❡ ❦♥♦✇ t❤❡ ❞✐s❝r❡t❡ ❧♦❣ ♦❢ ❛ s✉❜s❡t ♦❢ ✐❞❡❛❧s ♦❢ Kf ✱ Kg✳ ✹✳ ■♥❞✐✈✐❞✉❛❧ ▲♦❣❛r✐t❤♠✳

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✺ ✴ ✶✼

❉✐s❝r❡t❡ ▲♦❣ ✐♥ Fpn ✇✐t❤ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡

❘❡❧❛t✐♦♥ ❝♦❧❧❡❝t✐♦♥

❲❡ ♥❡❡❞ ❛ ❤✐❣❤ s♠♦♦t❤♥❡ss ♣r♦❜❛❜✐❧✐t② ♦❢ ✐❞❡❛❧s (a − bx) ∈ Kf ✱ (a − by) ∈ Kg✱ |a|, |b| < E ✐♥t❡❣❡rs NormKf /Q(a − bx) ❛♥❞ NormKg/Q(a − by) ✇❡ ❛♣♣r♦①✐♠❛t❡ |NormKf /Q(a − bx)| ≤ E deg f ||f ||∞ ✇✐t❤ ||f ||∞ = max1≤i≤deg f |fi| ✇❡ ✇❛♥t t♦ ♠✐♥✐♠✐③❡ t❤❡ ♣r♦❞✉❝t ♦❢ ♥♦r♠s✿ E deg f ||f ||∞E deg g||g||∞ ❲❡ ♥❡❡❞ f , g ♦❢ s♠❛❧❧ ❞❡❣r❡❡s f , g ♦❢ s♠❛❧❧ ❝♦❡✣❝✐❡♥ts ❲❡ ❝❛♥♥♦t ❤❛✈❡ ❜♦t❤✱ ✇❡ ♥❡❡❞ t♦ ❜❛❧❛♥❝❡ ❞❡❣r❡❡s ❛♥❞ ❝♦❡✣❝✐❡♥t s✐③❡s✳

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✻ ✴ ✶✼

❉✐s❝r❡t❡ ▲♦❣ ✐♥ Fpn ✇✐t❤ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡

❘❡❧❛t✐♦♥ ❝♦❧❧❡❝t✐♦♥

❲❡ ♥❡❡❞ ❛ ❤✐❣❤ s♠♦♦t❤♥❡ss ♣r♦❜❛❜✐❧✐t② ♦❢ ✐❞❡❛❧s (a − bx) ∈ Kf ✱ (a − by) ∈ Kg✱ |a|, |b| < E ✐♥t❡❣❡rs NormKf /Q(a − bx) ❛♥❞ NormKg/Q(a − by) ✇❡ ❛♣♣r♦①✐♠❛t❡ |NormKf /Q(a − bx)| ≤ E deg f ||f ||∞ ✇✐t❤ ||f ||∞ = max1≤i≤deg f |fi| ✇❡ ✇❛♥t t♦ ♠✐♥✐♠✐③❡ t❤❡ ♣r♦❞✉❝t ♦❢ ♥♦r♠s✿ E deg f ||f ||∞E deg g||g||∞ ❲❡ ♥❡❡❞ f , g ♦❢ s♠❛❧❧ ❞❡❣r❡❡s f , g ♦❢ s♠❛❧❧ ❝♦❡✣❝✐❡♥ts ❲❡ ❝❛♥♥♦t ❤❛✈❡ ❜♦t❤✱ ✇❡ ♥❡❡❞ t♦ ❜❛❧❛♥❝❡ ❞❡❣r❡❡s ❛♥❞ ❝♦❡✣❝✐❡♥t s✐③❡s✳

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✻ ✴ ✶✼

❖✉r ◆❡✇ P♦❧②♥♦♠✐❛❧ ❙❡❧❡❝t✐♦♥ ❢♦r Fpn ❆✳ ❚❤❡ ❣❡♥❡r❛❧✐③❡❞ ❏♦✉①✲▲❡r❝✐❡r ♠❡t❤♦❞

❆✳ ●❡♥❡r❛❧✐③❡❞ ❏♦✉①✲▲❡r❝✐❡r ♠❡t❤♦❞

❙✐♠♣❧✐✜❡❞ ✈❡rs✐♦♥✿ deg f = n + 1✱ deg g = n ✶✳ ❝❤♦♦s❡ f ✱ deg f = n + 1✱ s✳t✳ ✷✳ f ≡ ˜ f ϕ mod p✱ ϕ ❛ ♠♦♥✐❝ ✐rr❡❞✉❝✐❜❧❡ ❢❛❝t♦r ♦❢ ❞❡❣r❡❡ n ♠♦❞✉❧♦ p ϕ(x) = ϕ0 + ϕ1x + · · · + xn ✸✳ ❘❡❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛tr✐① ✉s✐♥❣ ▲▲▲

M =      p ✳✳✳ p ϕ0 ϕ1 · · · 1           deg ϕ = n r♦✇s

• 1 r♦✇

→ LLL(M) =         g0 g1 · · · gn ∗        

✹✳ g = g0 + g1x + · · · + gnxn✱ ||g||∞ = O(pn/(n+1)) E deg f +deg g||f ||∞||g||∞ = E 2n+1O(pn/(n+1))

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✼ ✴ ✶✼

❖✉r ◆❡✇ P♦❧②♥♦♠✐❛❧ ❙❡❧❡❝t✐♦♥ ❢♦r Fpn ❆✳ ❚❤❡ ❣❡♥❡r❛❧✐③❡❞ ❏♦✉①✲▲❡r❝✐❡r ♠❡t❤♦❞

❆✳ ●❡♥❡r❛❧✐③❡❞ ❏♦✉①✲▲❡r❝✐❡r ♠❡t❤♦❞✿ ❡①❛♠♣❧❡

p = 10000000019 ❛♥❞ n = 2 f = x3 + x + 1 ϕ = x2 + 3402015304x + 6660167027 M =   p p ϕ0 ϕ1 1   LLL → g = 746193x2 + 914408x + 4935648 ||f ||∞ = O(1)✱ ||g||∞ = O(p2/3) ❍✐st♦r✐❝❛❧ r❡♠❛r❦✿ t❤✐s ❝♦♥str✉❝t✐♦♥ ❛♣♣❡❛rs ✐♥ ❇❛r❜✉❧❡s❝✉ P❤❉ t❤❡s✐s ✭✷✵✶✸✮ ■♥ ❏❛♥✉❛r② ✇❡ ✇❡r❡ t♦❧❞ ❛❜♦✉t ▼❛t②✉❦❤✐♥✬s ✇♦r❦

❬❒➚ÒÞÕ➮❮ ✷✵✵✻❪✿ ÝÔÔ➴✃Ò➮➶❮Û➱ ➶➚Ð➮➚❮Ò ❒➴Ò❰➘➚ Ð➴Ø➴Ò➚ ×➮Ñ❐❰➶❰➹❰ Ï❰❐ß ➘❐ß ➘➮Ñ✃Ð➴Ò❮❰➹❰ ❐❰➹➚Ð➮Ô❒➮Ð❰➶➚❮➮ß ➶ Ï❰❐➴ ●❋(pk)✳

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✽ ✴ ✶✼

❖✉r ◆❡✇ P♦❧②♥♦♠✐❛❧ ❙❡❧❡❝t✐♦♥ ❢♦r Fpn ❇✳ ❚❤❡ ❈♦♥❥✉❣❛t✐♦♥ ▼❡t❤♦❞

❇✳ ❚❤❡ ❈♦♥❥✉❣❛t✐♦♥ ▼❡t❤♦❞ ❢♦r Fp2✿ ❡①❛♠♣❧❡

✶✳ p = 7 mod 8 ✷✳ f = x4 + 1 ✐rr❡❞✉❝✐❜❧❡ ♦✈❡r Z✱ s♠❛❧❧ ✸✳ f = (x2+ √ 2x + 1)(x2− √ 2x + 1) ♦✈❡r Q( √ 2) ✹✳ x2 − 2 ❤❛s t✇♦ r♦♦ts ±r mod p ✺✳ ϕ = x2 + rx + 1 ✐s ✐rr❡❞✉❝✐❜❧❡ ♦✈❡r Fp s✐♥❝❡ p ≡ 7 mod 8✱ ❛♥❞ ♦✈❡r Z ✻✳ ❝♦♠♣✉t❡ (u, v) s✳t✳ u/v ≡ r mod p✱ ✇✐t❤ |u|, |v| ∼ p1/2 ✇✐t❤ t❤❡ r❛t✐♦♥❛❧ r❡❝♦♥str✉❝t✐♦♥ ♠❡t❤♦❞ ✼✳ g = vx2 + ux + v ≡ v · ϕ mod p

• ❡♥❡r❛❧✐③❡ t♦ ❤✐❣❤❡r n✿

deg f = 2n✱ deg g = n✱ ||f ||∞ = O(1)✱ ||g||∞ = O(p1/2) E deg f +deg g||f ||∞||g||∞ = E 3nO(p1/2)

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✾ ✴ ✶✼

■♥❞✐✈✐❞✉❛❧ ▲♦❣❛r✐t❤♠

■♥❞✐✈✐❞✉❛❧ ❧♦❣❛r✐t❤♠

♣♦❧②♥♦♠✐❛❧ s❡❧❡❝t✐♦♥ ϕ(x)✱ Fpn = Fp[x]/(ϕ(x))✱ f ✱ ♥✉♠❜❡r ✜❡❧❞ K = Q[¯ x]/(f (¯ x)) = Q[α]✱ ♠❛♣ ρ : α → x ∈ Fpn ❦♥♦✇♥ ❧♦❣s ♦❢ {pi}✱ NormK/Q(pi) ≤ B r❛♥❞♦♠ t❛r❣❡t s = n−1

i=0 sixi ∈ Fpn

♣r❡✐♠❛❣❡ ¯ s = n−1

i=0 ¯

siαi ∈ K✱ ✇✐t❤ ρ(¯ si) = si✱ ♥❡❡❞s B✲s♠♦♦t❤ ¯ s✱ ✐✳❡✳ B✲s♠♦♦t❤ NormK/Q(¯ s) ❞❡❞✉❝❡ ✐♥❞✐✈✐❞✉❛❧ ❧♦❣❛r✐t❤♠ log ¯ s✱ t❤❡♥ log s ❇♦tt❧❡♥❡❝❦✿ ✜♥❞ B✲s♠♦♦t❤ ¯ s✳ ▲♦♦♣ ♦✈❡r ge · s✱ g ❣❡♥❡r❛t♦r✱ ✐♥ t✐♠❡ LQ[1/3, c]✳

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✶✵ ✴ ✶✼

■♥❞✐✈✐❞✉❛❧ ▲♦❣❛r✐t❤♠

◆♦r♠ ❜♦✉♥❞

NormK/Q(¯ s) ≤ ||f ||∞

deg ¯ s ||¯

s||∞

deg f

||¯ s||∞ = O(p) deg ¯ s = n − 1 ❏▲❙❱1✿ ||f ||∞ = O(p1/2) ✗

❣❏▲✱ ❈♦♥❥✿ ||f ||∞ = O(1) ✓ deg f = n ✭❏▲❙❱1✮✱ n + 1 ✭❣❏▲✮✱ 2n ✗✭❈♦♥❥✮ ➙❘❡❞✉❝❡ ||¯ s||∞ ➙❘❡❞✉❝❡ deg ¯ s ❉♦ ❜♦t❤✱ ✉s❡ s✉❜✜❡❧❞ str✉❝t✉r❡✳

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✶✶ ✴ ✶✼

■♥❞✐✈✐❞✉❛❧ ▲♦❣❛r✐t❤♠

Fp2✱ ❈♦♥❥✉❣❛t✐♦♥

f = x4 + 1 s = s0 + s1x Fp2 s✉❜✜❡❧❞✿ Fp s ←(1/s1) ·s s♦ s1 = 1✳ ❉❡✜♥❡ ❧❛tt✐❝❡ L =     p s0 1 ϕ0 ϕ1 1 ϕ0 ϕ1 1     ▲▲▲ ♦✉t♣✉ts ¯ r ∈ Z[x]✱ ✇✐t❤ ||¯ r||∞ ≤ CLLL det(L)1/dim = CLLL p1/4✱ ♠❛♣ ¯ r ✐♥t♦ K ❤❡♥❝❡ Norm(¯ r) = O(p) ✐♥st❡❛❞ ♦❢ Norm(¯ s) = O(p2)

➙❉♦ ✇❡ ❤❛✈❡ log ρ(¯ r) = log s ❄

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✶✷ ✴ ✶✼

■♥❞✐✈✐❞✉❛❧ ▲♦❣❛r✐t❤♠

❲❡ ♥❡❡❞ log ρ(¯ r) = log s✿

▲▲▲ → ¯ r ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ L r♦✇s✳ ρ(¯ r) = ρ(a1p + a2¯ s + a3ϕ + a4xϕ) ≡ u · s mod (p, ϕ) ✇✐t❤ u ∈ Fp log u = 0 mod ℓ ✇✐t❤ ℓ | p + 1 s✐♥❝❡ u ∈ Fp ❤❡♥❝❡ log ρ(¯ r) ≡ log s mod ℓ✳ ❘✉♥♥✐♥❣✲t✐♠❡ ❢♦r ✜♥❞✐♥❣ ❛ B✲s♠♦♦t❤ ❞❡❝♦♠♣♦s✐t✐♦♥✱ Fp2 ✇✐t❤ ❈♦♥❥ ♠❡t❤♦❞ ✿ LQ[1/3, 1.14] ✐♥st❡❛❞ ♦❢ LQ[1/3, 1.44]

• ❡♥❡r❛❧✐③❛t✐♦♥✿

❏▲❙❱1✱ ❣❏▲✱ ❈♦♥❥ Fp2m✱ ✇✐t❤ u ∈ Fp2

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✶✸ ✴ ✶✼

❉✐s❝r❡t❡ ▲♦❣❛r✐t❤♠ ❘❡❝♦r❞ ✐♥ ✻✵✵✲❜✐t Fp2

❖✉r ❘❡❝♦r❞✿ ❉✐s❝r❡t❡ ▲♦❣❛r✐t❤♠ ✐♥ Fp2 ♦❢ ✻✵✵ ❜✐ts

p = 314159265358979323846264338327950288419716939\ (300 ❜✐ts) 937510582097494459230781640628620899877709223 p + 1 = 8 · ℓ ℓ = 392699081698724154807830422909937860524646174\ (295 ❜✐ts) 92188822762186807403847705078577612484713653 p − 1 = 6 · h0 ✇✐t❤ h0 ❛ ✷✾✺ ❜✐t ♣r✐♠❡ ❈r②♣t♦❣r❛♣❤✐❝ s✉❜❣r♦✉♣✿ G ♦❢ ♦r❞❡r ℓ ❋♦r ♦✉r r❡❝♦r❞✿ Q = p2✱ log2 Q = 600✱ ♦♣t✐♠❛❧ ✈❛❧✉❡ ♦❢ E ❛r♦✉♥❞ log2 E = 27 ❜✐ts✳

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✶✹ ✴ ✶✼

❉✐s❝r❡t❡ ▲♦❣❛r✐t❤♠ ❘❡❝♦r❞ ✐♥ ✻✵✵✲❜✐t Fp2

❖✉r ❘❡❝♦r❞✿ ❉✐s❝r❡t❡ ▲♦❣❛r✐t❤♠ ✐♥ Fp2 ♦❢ ✻✵✵ ❜✐ts

P♦❧②♥♦♠✐❛❧ s❡❧❡❝t✐♦♥✿

• ❡♥❡r❛❧✐③❡❞ ❏♦✉① ▲❡r❝✐❡r✿ f = x3 + x + 1✱ ||g||∞ = O(p2/3)✱ ◆♦r♠s

❜♦✉♥❞❡❞ ❜② E 5p2/3 ♦❢ 339 ❜✐ts ✗ ❈♦♥❥✉❣❛t✐♦♥✿ f = x4 + 1✱ ||g||∞ = O(p1/2)✱ ◆♦r♠s ❜♦✉♥❞❡❞ ❜② E 6p1/2 ♦❢ 317 ❜✐ts ➙✷✷ ❜✐ts ❧❡ss ✓ f = x4 + 1 g = 448225077249286433565160965828828303618362474 x2 − 296061099084763680469275137306557962657824623 x + 448225077249286433565160965828828303618362474 . ||g||∞ = 150 ❜✐ts ϕ = x2 + yx + 1, log2 y = log2 p ❚❛r❣❡t✿ s = ⌊(π(2298)/8)⌋x + ⌊(γ · 2298)⌋ ∈ Fp2 = Fp[x]/(ϕ(x)) gen = x + 2

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✶✺ ✴ ✶✼

❉✐s❝r❡t❡ ▲♦❣❛r✐t❤♠ ❘❡❝♦r❞ ✐♥ ✻✵✵✲❜✐t Fp2

❙♣❡❡❞✲✉♣ ♦❢ ❘❡❧❛t✐♦♥ ❈♦❧❧❡❝t✐♦♥ ❛♥❞ ▲✐♥❡❛r ❆❧❣❡❜r❛

• ❛❧♦✐s ❛✉t♦♠♦r♣❤✐s♠✿ x → 1/x ❜♦t❤ ❢♦r f = x4 + 1 ❛♥❞

g = vx2 + ux + v a − bx → −b + ax✿ ❛ s❡❝♦♥❞ r❡❧❛t✐♦♥ ❢♦r ❢r❡❡

➙ s♣❡❡❞✲✉♣ ❜② ❛ ❢❛❝t♦r ✷ ❢♦r r❡❧❛t✐♦♥ ❝♦❧❧❡❝t✐♦♥ ➙ s♣❡❡❞✲✉♣ ❜② ❛ ❢❛❝t♦r ✹ ❢♦r ❧✐♥❡❛r ❛❧❣❡❜r❛

♦t❤❡rs ✐♠♣♦rt❛♥t ❛❧❣❡❜r❛✐❝ s✐♠♣❧✐✜❝❛t✐♦♥ ❛♥❞ s♣❡❡❞✲✉♣ ❋✐♥❛❧❧②✱ loggen s ≡ 276214243617912804300337349268306605403758173\ 81941441861019832278568318885392430499058012 mod ℓ.

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✶✻ ✴ ✶✼

❉✐s❝r❡t❡ ▲♦❣❛r✐t❤♠ ❘❡❝♦r❞ ✐♥ ✻✵✵✲❜✐t Fp2

❘❡❝♦r❞ r✉♥♥✐♥❣✲t✐♠❡ ❝♦♠♣❛r✐s♦♥ ✐♥ ②❡❛rs ❢♦r ✻✵✵✲❜✐t ✐♥♣✉ts

r❡❧❛t✐♦♥ ❧✐♥❡❛r ❆❧❣♦r✐t❤♠ ❝♦❧❧❡❝t✐♦♥ ❛❧❣❡❜r❛ t♦t❛❧ ◆❋❙ ■♥t❡❣❡r ❋❛❝t♦r✐③❛t✐♦♥ ✺② ✵✳✺② ✺✳✺② ×11 ◆❋❙ ❉▲ ✐♥ Fp ✺✵② ✽✵② ✶✸✵② ×260 ❚❤✐s ✇♦r❦✿ ◆❋❙ ❉▲ ✐♥ Fp2 ✵✳✹② ✵✳✵✺② ✭●P❯✮ ✵✳✺② ×1 ❉▲ ✐♥ Fp2 < ■♥t❡❣❡r ❋❛❝t♦r✐③❛t✐♦♥ < ❉▲ ✐♥ Fp P❛♣❡r✿ ❤tt♣s✿✴✴❤❛❧✳✐♥r✐❛✳❢r✴❤❛❧✲✵✶✶✶✷✽✼✾ ❆❧❣❡❜r❛✐❝ s❡❝r❡ts✿ ❤tt♣s✿✴✴❤❛❧✳✐♥r✐❛✳❢r✴❤❛❧✲✵✶✵✺✷✹✹✾ ❙♦✉r❝❡ ❝♦❞❡✿ ❤tt♣✿✴✴❝❛❞♦✲♥❢s✳❣❢♦r❣❡✳✐♥r✐❛✳❢r✴

➙ ❉♦✇♥❧♦❛❞ ✐t ❛♥❞ s♦❧✈❡ ②♦✉r ♦✇♥ ❉▲ ✐♥ Fp2

❙t❛② t✉♥❡❞ ❢♦r ♠♦r❡ r❡❝♦r❞s ❞✉r✐♥❣ s✉♠♠❡r✳

❇❛r❜✉❧❡s❝✉✱ ●❛✉❞r②✱ ●✉✐❧❧❡✈✐❝✱ ▼♦r❛✐♥ ◆❋❙✲❉▲ ✐♥ Fpn ▼❛② ✷✵t❤✱ ✷✵✶✺ ✶✼ ✴ ✶✼