Pr rtt - PowerPoint PPT Presentation

❚❤❡ ❈♦♥❥✉❣❛❝② Pr♦❜❧❡♠✿ ❈r②♣t♦❛♥❛❧②t✐❝ ❛♣♣r♦❛❝❤❡s t♦ ❉❡❤♥✬s Pr♦❜❧❡♠ ❇♦❛③ ❚s❛❜❛♥ P❛rt✐❛❧❧② ❥♦✐♥t ✇✐t❤ ❉❛✈✐❞ ●❛r❜❡r✱ ❆r❦❛❞✐✉s ❑❛❧❦❛✱ ▼✐♥❛ ❚❡✐❝❤❡r✱ ●❛r② ❱✐♥♦❦✉r ❇❛r✲■❧❛♥ ❯♥✐✈❡rs✐t② ●❆●❚❆✲✻✱ ❏✉❧②✴❆✉❣✉st ✷✵✶✷ ❈❊

P❛rt ■ ❑❡② ❊①❝❤❛♥❣❡ Pr♦t♦❝♦❧s ❛♥❞ ❘❡♣r❡s❡♥t❛t✐♦♥ ❛tt❛❝❦s

❑❡② ❊①❝❤❛♥❣❡ Pr♦t♦❝♦❧s ✭❑❊Ps✮ ❆❧✐❝❡ ❛♥❞ ❇♦❜ ✇✐s❤ t♦ ❝♦♠♠✉♥✐❝❛t❡ ♦✈❡r ❛♥ ✐♥s❡❝✉r❡ ❝❤❛♥♥❡❧✳ ∃ ❊✣❝✐❡♥t ✫ s❡❝✉r❡ ♠❡t❤♦❞s ✐❢ t❤❡② s❤❛r❡ ❛ s❡❝r❡t ✭✏❦❡②✑✮✿ ❙②♠♠❡tr✐❝ ❡♥❝r②♣t✐♦♥ ✭❆❊❙✱✳ ✳ ✳ ✮✳ ❍♦✇ t♦ ❞❡❝✐❞❡ ❛ s❤❛r❡❞ s❡❝r❡t ❦❡② ♦✈❡r ❛♥ ✐♥s❡❝✉r❡ ❝❤❛♥♥❡❧❄ ❉✐✣❡✕❍❡❧❧♠❛♥ ✶✾✼✻✳ ❑❡② ❊①❝❤❛♥❣❡ Pr♦t♦❝♦❧✳ ❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❜r❡❛❦t❤r♦✉❣❤ ✐♥ ❝r②♣t♦❣r❛♣❤②✳ ■♥ t❤✐s ♠✐♥✐❝♦✉rs❡✿ ❖♥❧② ♣❛ss✐✈❡ ❛❞✈❡rs❛r✐❡s✳ ❚❤❡ ❦❡r♥❡❧ ♦♥ ✇❤✐❝❤ ♠♦r❡ ✐♥✈♦❧✈❡❞ P❑❈ ✐s ❜✉✐❧t✳

❑❡② ❊①❝❤❛♥❣❡ Pr♦t♦❝♦❧ ✲ t❤❡ ❝♦♥❝❡♣t ❈♦✉rt❡s② ♦❢ ❲✐❦✐♣❡❞✐❛

� � ❚❤❡ ❉✐✣❡✕❍❡❧❧♠❛♥ ❑❊P ❆❧✐❝❡ P✉❜❧✐❝ ❇♦❜ a ∈ { ✵ , ✶ , . . . , p − ✶ } G = � g � , | G | = p b ∈ { ✵ , ✶ , . . . , p − ✶ } g a g b K = g a b = g ab K = g b a = g ab ❊①♣♦♥❡♥t✐❛t✐♦♥✳ x �→ g x ✈✐❛ sq✉❛r❡ ❛♥❞ ♠✉❧t✐♣❧②✱ O ( ❧♦❣ ✷ p ) ✳

❙❡❝✉r✐t② ♦❢ t❤❡ ❉✐✣❡✕❍❡❧❧♠❛♥ ❑❊P ❉✐✣❡✕❍❡❧❧♠❛♥ Pr♦❜❧❡♠✳ ( g a , g b ) �→ g ab ✳ ❉✐s❝r❡t❡ ▲♦❣❛r✐t❤♠ Pr♦❜❧❡♠✳ g x �→ x ✳ ❉▲P ≥ ❉❍P✳ ❇♦t❤ ❛r❡ ǫ ✲❤❛r❞✳ ❚s ✷✵✵✻✳ ◆♦♥❡ ❞❡♣❡♥❞s ♦♥ ❣❡♥❡r❛t♦r ❝❤♦✐❝❡✳

❚❤❡ ❉✐s❝r❡t❡ ▲♦❣❛r✐t❤♠ Pr♦❜❧❡♠ ❉✐s❝r❡t❡ ▲♦❣❛r✐t❤♠ Pr♦❜❧❡♠✳ g x �→ x ✳ ❉❡♣❡♥❞s ♦♥ t❤❡ ❣r♦✉♣✦ G = ( Z p , +) ✳ g = ✶✳ ✏ g x ✑ = x · g = x · ✶ = x ✳ G ≤ ( Z ∗ p , · ) ✳ ◗✉✐t❡✱ ❜✉t ♥♦t ❡♥♦✉❣❤✱ ❤❛r❞✿ ✷ ( ✶ . ✸✸ + o ( ✶ )) n ✶ / ✸ ( ❧♦❣ ✷ n ) ✷ / ✸ ✳ ◆❋❙✳ n := ❧♦❣ ✷ ( p ) ✿ n ◆❋❙ ❲♦r❦ Pr❡❞✐❝t✐♦♥ ❨❡❛r ❇r♦❦❡♥ ✷ ✹✼ ✺✷✺ ✷✵✵✷ ✷ ✹✾ ✺✼✽ ✷✵✵✸ ✷ ✺✷ ✻✻✹ ✷✵✵✺ ✷ ✺✺ ✼✻✽ ✷✵✵✾ ✷ ✻✷ ✶✵✷✹ ✷✵✶✻❄ ✶✵ , ✵✵✵ ❜✐ts ♣r✐♠❡ ❢♦r ✏❡t❡r♥❛❧✑ s❡❝✉r✐t②❄ ■♠♣r❛❝t✐❝❛❧✳

❚❤❡ ❢✉t✉r❡ ♦❢ ❝r②♣t♦❣r❛♣❤② G ≤ ❊❧❧✐♣t✐❝ ❈✉r✈❡✳ ◆♦t❤✐♥❣ ❜❡tt❡r t❤❛♥ ✷ n / ✷ ✳ ❨❡t✳ ❊❈❈✳ ❘✐❝❤ ♠❛t❤❡♠❛t✐❝s → · · · → ❛❧❣♦r✐t❤♠✐❝ ❜r❡❛❦t❤r♦✉❣❤s❄ ◗✉❛♥t✉♠ ❈♦♠♣✉t❡rs✳ ❇r❡❛❦ ❛❧❧ ❉✐✣❡✕❍❡❧❧♠❛♥ ❑❊Ps✳ ❚❤❡♦r❡t✐❝✳ ❇✉t ✇❤❛t ✐s ②♦✉r ❛❧t❡r♥❛t✐✈❡❄ ❘✐✈❡st✲❙❤❛♠✐r✲❆❞❧❡♠❛♥ ✭❘❙❆✱ ✶✾✼✽✮✳ ❆s ❡❛s② ❛s ❉▲P ✐♥ Z ∗ p ✳ ▲❛tt✐❝❡✲❜❛s❡❞❄ ▼❛②❜❡✳ ❍♦✇ ❛❜♦✉t ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❣r♦✉♣s❄ ❲■◆✴❲■◆✿ ◆❡✇ ❑❊P ✴ ❡✣❝✐❡♥t ❛❧❣♦r✐t❤♠s✳

� � ❚❤❡ ❇r❛✐❞ ❉✐✣❡✕❍❡❧❧♠❛♥ ❑❊P ❉✐✣❡✕❍❡❧❧♠❛♥ ❑❊P ✶✾✼✻✳ ❆❧✐❝❡ P✉❜❧✐❝ ❇♦❜ a ∈ { ✵ , ✶ , . . . , p − ✶ } G = � g � , | G | = p b ∈ { ✵ , ✶ , . . . , p − ✶ } g a g b K = g b a K = g a b = g ab = g ab

� � ❚❤❡ ❇r❛✐❞ ❉✐✣❡✕❍❡❧❧♠❛♥ ❑❊P ❑♦✕▲❡❡✕❈❤❡♦♥✕❍❛♥✕❑❛♥❣✕P❛r❦ ✷✵✵✵✳ G ♥♦♥❝♦♠♠✉t❛t✐✈❡✳ g x := x − ✶ gx ✳ ❆❧✐❝❡ P✉❜❧✐❝ ❇♦❜ a ∈ A A , B ≤ G , g ∈ G , [ A , B ] = ✶ b ∈ B g a g b K = g b a K = g a b = g ab = g ba

❉❡❤♥✬s Pr♦❜❧❡♠s ✶✾✶✶ G = � X | R � ✳ ❲♦r❞ Pr♦❜❧❡♠✳ ❉❡❝✐❞❡ ✇❤❡t❤❡r g = ✶✳ ❈♦♥❥✉❣❛❝② Pr♦❜❧❡♠✳ ❉❡❝✐❞❡ ✇❤❡t❤❡r g , h ❛r❡ ❝♦♥❥✉❣❛t❡✳ ✭❆❑❆ ●❡♥❡r❛❧✐③❡❞ ❲♦r❞ Pr♦❜❧❡♠✳✮ ■s♦♠♦r♣❤✐s♠ Pr♦❜❧❡♠✳ ❉❡❝✐❞❡ ✇❤❡t❤❡r G , H ❛r❡ ✐s♦♠♦r♣❤✐❝✳ ❖r✐❣✐♥❛❧❧②✱ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠s✳ ❈r②♣t♦ ✉s❡s t❤❡ s❡❛r❝❤ ✈❡rs✐♦♥s✳ ❯♥❧✐❦❡ t❤❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠s✱ t❤❡ s❡❛r❝❤ ♣r♦❜❧❡♠s ❛r❡ ❞❡❝✐❞❛❜❧❡✱ ❜✉t ✇❡ ❛s❦ ❢♦r ❡✣❝✐❡♥t s♦❧✉t✐♦♥s✳ Pr♦♣♦s❡❞ ♣❧❛t❢♦r♠✳ ❆rt✐♥✬s ❜r❛✐❞ ❣r♦✉♣✳ ✭❚❇❉✮ ▼♦t✐✈❛t❡❞ ❛ ♥❡✇ ❧✐♥❡ ♦❢ r❡s❡❛r❝❤ ✐♥ ❝♦♠❜✐♥❛t♦r✐❛❧ ❣r♦✉♣ t❤❡♦r②✳

❆rt✐♥✬s ❜r❛✐❞ ❣r♦✉♣ ❇ ■❞❡♥t✐t② ❜r❛✐❞✿ • • •

❚❤❡ ♦r❞✐♥❛r② ❜r❛✐❞ ✶✿ • • •

❚❤❡ ♦r❞✐♥❛r② ❜r❛✐❞ σ − ✶ ✶ ✿ • • •

❚❤❡ ♦r❞✐♥❛r② ❜r❛✐❞ σ − ✶ ✶ σ ✷ ✿ • • •

❚❤❡ ♦r❞✐♥❛r② ❜r❛✐❞ σ − ✶ ✶ σ ✷ σ − ✶ ✶ ✿ • • •

❚❤❡ ♦r❞✐♥❛r② ❜r❛✐❞ σ − ✶ ✶ σ ✷ σ − ✶ ✶ σ ✷ ✿ • • •

❚❤❡ ♦r❞✐♥❛r② ❜r❛✐❞ σ − ✶ ✶ σ ✷ σ − ✶ ✶ σ ✷ σ − ✶ ✶ ✿ • • •

❚❤❡ ♦r❞✐♥❛r② ❜r❛✐❞ σ − ✶ ✶ σ ✷ σ − ✶ ✶ σ ✷ σ − ✶ ✶ σ ✷ ✿ • • •

❘❡❛❧ ❧✐❢❡ ❛♣♣❧✐❝❛t✐♦♥s ❆ ❈❤❛❧❧❛❤✳

❆rt✐♥✬s ❜r❛✐❞ ❣r♦✉♣ ❇ ❇ ✿ ❇r❛✐❞s ✴ ✐s♦t♦♣②✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ❈♦♥❝❛t❡♥❛t✐♦♥ ♦❢ ❜r❛✐❞s✳ ■♥✈❡rs✐♦♥✿ ▼✐rr♦r ❜r❛✐❞✳

●❡♥❡r❛t♦rs ♦❢ t❤❡ ❜r❛✐❞ ❣r♦✉♣ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ • • • σ ✶

●❡♥❡r❛t♦rs ♦❢ t❤❡ ❜r❛✐❞ ❣r♦✉♣ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ • • • σ ✷

●❡♥❡r❛t♦rs ♦❢ t❤❡ ❜r❛✐❞ ❣r♦✉♣ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ • • • σ ✸

●❡♥❡r❛t♦rs ♦❢ t❤❡ ❜r❛✐❞ ❣r♦✉♣ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ • • • σ ✹

●❡♥❡r❛t♦rs ♦❢ t❤❡ ❜r❛✐❞ ❣r♦✉♣ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ • • • σ ✺

●❡♥❡r❛t♦rs ♦❢ t❤❡ ❜r❛✐❞ ❣r♦✉♣ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ • • • σ ✻

●❡♥❡r❛t♦rs ♦❢ t❤❡ ❜r❛✐❞ ❣r♦✉♣ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ • • • σ ✼

●❡♥❡r❛t♦rs ♦❢ t❤❡ ❜r❛✐❞ ❣r♦✉♣ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ • • • σ ✽

●❡♥❡r❛t♦rs ♦❢ t❤❡ ❜r❛✐❞ ❣r♦✉♣ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ • • • σ ✾

❘❡❧❛t✐♦♥s ✐♥ t❤❡ ❜r❛✐❞ ❣r♦✉♣ ❋❛r ❈♦♠♠✉t❛t✐✈✐t②✿ σ i σ j = σ j σ i ❢♦r i + ✶ < j ✳ = ❚r✐♣❧❡ r❡❧❛t✐♦♥✿ σ i σ i + ✶ σ i = σ i + ✶ σ i σ i + ✶ ✳ =

◆♦r♠❛❧ ❢♦r♠s ❚❤✐♥❦ ❉❍ ❑❊P ✐♥ ( Z / p Z ) ∗ ✐♥st❡❛❞ ♦❢ Z ∗ p ✿ ✶✳ ▼❛② ♥♦t ❣❡t t❤❡ s❛♠❡ ❦❡② ✐❢ ❝❤♦✐❝❡ ♥♦t ❝❛♥♦♥✐❝❛❧✦ ✷✳ ❇r❡❛❦❛❜❧❡✦ ◆♦r♠❛❧ ❢♦r♠✿ n �→ ( n ♠♦❞ p ) ✿ ✶✳ ❊♥s✉r❡s s❛♠❡ ❦❡②✳ ✷✳ ❍✐❞❡s t❤❡ ❣❡♥❡r❛t✐♦♥ ✐♥❢♦✳ ❇r❛✐❞ ❉✐✣❡✕❍❡❧❧♠❛♥ ❑❊P ✉s❡s ❇ ❛s ♣❧❛t❢♦r♠ ❣r♦✉♣✳ ◆♦r♠❛❧ ❢♦r♠ ✐♥ ❇ ❄