❘❡❝♦♥str✉çã♦ ❞❛ ❈❤❛✈❡ ❙❡❝r❡t❛
❞♦ ❘❙❆ ▼✉❧t✐✲♣r✐♠♦
❘❡②♥❛❧❞♦ ❈á❝❡r❡s ❱✐❧❧❡♥❛
❉✐ss❡rt❛çã♦ ❆♣r❡s❡♥t❛❞❛
❛♦
■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛
❞❛
❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦
♣❛r❛
♦❜t❡♥çã♦ ❞♦ tít✉❧♦
❞❡
▼❡str❡ ❡♠ ❈✐ê♥❝✐❛s
Pr♦❣r❛♠❛✿ ❈✐ê♥❝✐❛s ❞❛ ❈♦♠♣✉t❛çã♦
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❘♦✉t♦ ❚❡r❛❞❛
❉✉r❛♥t❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦ ♦ ❛✉t♦r r❡❝❡❜❡✉ ❛✉①í❧✐♦ ✜♥❛♥❝❡✐r♦ ❞❛ ❈❆P❊❙
❘❡❝♦♥str✉çã♦ ❞❛ ❈❤❛✈❡ ❙❡❝r❡t❛
❞♦ ❘❙❆ ▼✉❧t✐✲♣r✐♠♦
❊st❛ ✈❡rsã♦ ❞❛ ❞✐ss❡rt❛çã♦ ❝♦♥té♠ ❛s ❝♦rr❡çõ❡s ❡ ❛❧t❡r❛çõ❡s s✉❣❡r✐❞❛s ♣❡❧❛ ❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛ ❞✉r❛♥t❡ ❛ ❞❡❢❡s❛ ❞❛ ✈❡rsã♦ ♦r✐❣✐♥❛❧ ❞♦ tr❛❜❛❧❤♦✱ r❡❛❧✐③❛❞❛ ❡♠ ✷✸✴✵✾✴✷✵✶✸✳ ❯♠❛ ❝ó♣✐❛ ❞❛ ✈❡rsã♦ ♦r✐❣✐♥❛❧ ❡stá ❞✐s♣♦♥í✈❡❧ ♥♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✳
❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛✿
❆❣r❛❞❡❝✐♠❡♥t♦s
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢✳ ❉r✳ ❘♦✉t♦ ❚❡r❛❞❛ q✉❡ ♠❡ ✐♥tr♦❞✉③✐✉ à ❢❛s❝✐♥❛♥t❡ ❝✐ê♥❝✐❛ ❞❛ ❝r✐♣t♦❧♦❣✐❛ ❡ ♠❡ ❛✉①í❧✐♦ ♥❛ ❡❧❛❜♦r❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❆♦s ♠❡✉s ❛♠✐❣♦s ❞♦ ❧❛❜♦r❛tór✐♦ ❞❡ s❡❣✉r❛♥ç❛ ❞❡ ❞❛❞♦s ❞♦ ■▼❊✲❯❙P✳ ❉❡♥tr❡ ❡❧❡s ❛❣r❛❞❡ç♦ ❡s♣❡❝✐❛❧♠❡♥t❡ ❛ ❉❡♥✐s❡ ❍✳ ●♦②❛ ❡ ❛ ❘❛❢❛❡❧ ❲✐❧❧ ▼✳ ❞❡ ❆r❛✉❥♦ ♣❡❧❛s ♦❜s❡r✈❛çõ❡s ✐♠♣♦rt❛♥t❡s ♥♦ ❞❡❝♦rr❡r ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❛ ❞✐ss❡rt❛çã♦✳
❆ t♦❞❛ ♠✐♥❤❛ ❢❛♠í❧✐❛❀ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛ ♠❡✉s ♣❛✐s ●✉✐❧❧❡r♠♦ ❈✳ ◆✐❡t♦ ❡ P❡tr♦♥✐❧❛ ❱✳ ❍✐♥♦❥♦s❛ ♣❡❧❛ ❛❥✉❞❛ ✜♥❛♥❝❡✐r❛✱ ✐♥t❡❧❡❝t✉❛❧ ❡ ❡♠♦❝✐♦♥❛❧✳
➚ ❈❆P❊❙ ♣❡❧♦ s✉♣♦rt❡ ✜♥❛♥❝❡✐r♦✳
❘❡s✉♠♦
❊♠ ✷✵✵✾✱ ◆✳ ❍❡♥✐♥❣❡r ❡ ❍✳ ❙❤❛❝❤❛♠ ❛♣r❡s❡♥t❛r❛♠ ✉♠ ❛❧❣♦r✐t♠♦ ❞❡ r❡❝♦♥str✉çã♦ q✉❡ ♣❡r♠✐t❡ r❡❝✉♣❡r❛r ❛ ❝❤❛✈❡ s❡❝r❡t❛ sk ❞♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❜ás✐❝♦ ❡♠ t❡♠♣♦ ♣♦❧✐♥♦♠✐❛❧ t❡♥❞♦ ❡♠ ❢♦r♠❛
❛❧❡❛tór✐❛ ✷✼ ✪ ❞♦s s❡✉s ❜✐ts✳ ❙❛❜❡♠♦s q✉❡ ♣♦❞❡♠♦s ♦❜t❡r ✉♠❛ ✈❡rsã♦ ❝♦♠ ❡rr♦s ✭❜✐ts ♠♦❞✐✜❝❛❞♦s✮ ❞❛ ❝❤❛✈❡ s❡❝r❡t❛ ❘❙❆ ❣r❛ç❛s ❛♦s ❛t❛q✉❡s ❝♦❧❞ ❜♦♦t✳ ❖ ❛❧❣♦r✐t♠♦ ❛♣r❡s❡♥t❛❞♦ ♣♦r ❍❡♥✐♥❣❡r✲❙❤❛❝❤❛♠ ❝♦rr✐❣❡ ❡ss❡s ❡rr♦s ❢❛③❡♥❞♦ ✉s♦ ❞❛s r❡❧❛çõ❡s ♠❛t❡♠át✐❝❛s q✉❡ ❡①✐st❡ ❡♥tr❡ ❛s ❝❤❛✈❡s ♣ú❜❧✐❝❛ ❡ s❡❝r❡t❛ ❞♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❜ás✐❝♦✳ ❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❡st✉❞❛r ❡ss❡ ❛❧❣♦r✐t♠♦ ♣❛r❛ ✐♠♣❧❡♠❡♥t❛r ❡ ❛♥❛❧✐s❛r s❡✉ ❛♥á❧♦❣♦ ♣❛r❛ ♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ♠✉❧t✐✲♣r✐♠♦✳ ❖s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♠♦str❛♠ q✉❡ ♣❛r❛ r❡❝♦♥str✉✐r ❛ ❝❤❛✈❡ s❡❝r❡t❛sk❞♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆u✲♣r✐♠♦s é ♣r❡❝✐s♦ t❡r ✉♠❛ ❢r❛çã♦ ❞❡ ❜✐ts
❝♦rr❡t♦s ♠❛✐♦r ❛ 2−22uu+2+1✱ ♠♦str❛♥❞♦ ❛ss✐♠ q✉❡ ❛ s❡❣✉r❛♥ç❛ ♦❢❡r❡❝✐❞❛ ♣❡❧♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ♠✉❧t✐✲♣r✐♠♦ ✭u≥3✮ é ♠❛✐♦r ❝♦♠ r❡❧❛çã♦ ❛♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❜ás✐❝♦ ✭u= 2✮✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❘❡❝♦♥str✉çã♦ ❞❛ ❝❤❛✈❡ s❡❝r❡t❛✱ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ♠✉❧t✐✲♣r✐♠♦✱ ❆t❛q✉❡s ❈♦❧❞✲ ❇♦♦t✳
❆❜str❛❝t
■♥ ✷✵✵✾✱ ◆✳ ❍❡♥✐♥❣❡r ❛♥❞ ❍✳ ❙❤❛❝❤❛♠ ♣r❡s❡♥t❡❞ ❛♥ ❛❧❣♦r✐t♠ ❢♦r r❡❝♦♥str✉❝t✐♥❣ t❤❡ s❡❝r❡t ❦❡②sk
♦❢ t❤❡ ❜❛s✐❝ ❘❙❆ ❝r②♣t♦s②st❡♠ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✇✐t❤ ❛ ❢r❛❝t✐♦♥ ♦❢ r❛♥❞♦♠ ❜✐ts ❣r❡❛t❡r ♦r ❡q✉❛❧ t♦ ✵✳✷✼ ♦❢ ✐ts ❜✐ts✳ ❲❡ ❦♥♦✇ t❤❛t s❡❝r❡t ❦❡② ✇✐t❤ ❡rr♦rssk˜ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ ❉❘❆▼ ✉s✐♥❣ ❝♦❧❞✲❜♦♦t
❛tt❛❝❦s✳ ❚❤❡ ❍❡♥✐♥❣❡r ❛♥❞ ❙❤❛❝❤❛♠✬s ❛❧❣♦r✐t❤♠ ✜①❡s t❤❡s❡ ❡rr♦rs ✉s✐♥❣ t❤❡ r❡❞✉♥❞❛♥❝② ♦❢ s❡❝r❡t ❛♥❞ ♣✉❜❧✐❝ ❦❡② ♦❢ ❜❛s✐❝ ❘❙❆ ❝r②♣t♦s②st❡♠✳ ■♥ t❤✐s ✇♦r❦✱ t❤❡ t♦♣✐❝ ✐s t♦ st✉❞② t❤✐s ❛❧❣♦r✐t♠ t♦ ✐♠♣❧❡♠❡♥t ❛♥❞ ❛♥❛❧②③❡ ✐ts ❛♥❛❧♦❣♦✉s ❢♦r t❤❡ ♠✉❧t✐✲♣r✐♠❡ ❘❙❆ ❝r②♣t♦s②st❡♠✳ ❖✉r ♦❜t❛✐♥❡❞ r❡s✉❧ts s❤♦✇ t❤❡ s❡❝r❡t ❦❡② sk ♦❢ ♠✉❧t✐✲♣r✐♠❡ ❘❙❆ ❝r②♣t♦s②st❡♠ ❝❛♥ ❜❡ r❡❝♦♥str✉❝t❡❞ ❤❛✈✐♥❣ ❛ ❢r❛❝t✐♦♥ ❡q✉❛❧ ♦r
❣r❡❛t❡r t❤❛♥ 2−22uu+2+1 ♦❢ r❛♥❞♦♠ ❜✐ts✳ t❤❡r❡❢♦r❡ t❤❡ s❡❝✉r✐t② ♦❢ ♠✉❧t✐✲♣r✐♠❡ ❘❙❆ ❝r②♣t♦s②st❡♠ ✭u≥3✮ ✐s ❣r❡❛t❡r t❤❛♥ ❜❛s✐❝ ❘❙❆ ❝r②♣t♦s②st❡♠ ✭u= 2✮✳
❑❡②✇♦r❞s✿ ❙❡❝r❡t ❦❡② r❡❝♦♥str✉❝t✐♥❣✱ ▼✉❧t✐✲♣r✐♠❡ ❘❙❆ ❝r②♣t♦s②st❡♠✱ ❈♦❧❞ ❜♦♦t ❛tt❛❝❦s✳
❙✉♠ár✐♦
▲✐st❛ ❞❡ ❆❜r❡✈✐❛t✉r❛s ①✐
◆♦t❛çã♦ ❡ ▲✐st❛ ❞❡ ❙í♠❜♦❧♦s ①✐✐✐
✶ ■♥tr♦❞✉çã♦ ✶
✶✳✶ ❖❜❥❡t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ ❘❡s✉❧t❛❞♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✸ ❖r❣❛♥✐③❛çã♦ ❞♦ ❚r❛❜❛❧❤♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷
✷ ❈♦♥❝❡✐t♦s ✸
✷✳✶ ❈r✐♣t♦❣r❛✜❛ ❡ ❈r✐♣t♦❣r❛✜❛ ❞❡ ❈❤❛✈❡ Pú❜❧✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✷✳✶✳✶ ❈r✐♣t♦❣r❛✜❛ ❞❡ ❈❤❛✈❡ Pú❜❧✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✷✳✶✳✷ ❈r✐♣t♦ss✐st❡♠❛ ❘❙❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✶✳✸ ▼ét♦❞♦ ❞❡ ◗✉✐sq✉❛t❡r✲❈♦✉✈r❡✉r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✶✳✹ ❈♦rr❡çã♦ ❞♦ ❘❙❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✶✳✺ ❙❡❣✉r❛♥ç❛ ❞♦ ❘❙❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✷ P✉❜❧✐❝ ❑❡② ❈r✐♣t♦❣r❛♣❤② ❙t❛♥❞❛r❞ ✲ P❑❈❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✷✳✶ P❑❈❙ ★✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✸ ❆t❛q✉❡s ❙✐❞❡✲❈❤❛♥♥❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✸✳✶ ❆t❛q✉❡s ❈♦❧❞✲❜♦♦t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✸✳✷ ■❞❡♥t✐✜❝❛çã♦ ❞❡ ❈❤❛✈❡s ❘❙❆ ♥❛ ▼❡♠ór✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✸ ❆❧❣♦r✐t♠♦ ❞❡ ❘❡❝♦♥str✉çã♦ ❞❛ ❈❤❛✈❡ ❙❡❝r❡t❛ ❞♦ ❈r✐♣t♦ss✐st❡♠❛ ❘❙❆ ✶✺ ✸✳✶ ❈á❧❝✉❧♦ ❞❡ ❱❛r✐á✈❡✐s ❆✉①✐❧✐❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✶✳✶ ❈❛s♦ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✸✳✶✳✷ ❈❛s♦ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✷ ❈♦rr❡çã♦ ❞❡ ❆❧❣✉♥s ❇✐ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✸ ▲❡♠❛ ❞❡ ❍❡♥s❡❧ ✲ ●❡r❛çã♦ ❞❛s ❡q✉❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✹ ❆❧❣♦r✐t♠♦ ❡ s❡✉ ❝♦♠♣♦rt❛♠❡♥t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✹✳✶ ❈♦♠♣❧❡①✐❞❛❞❡ ❞♦ ❆❧❣♦r✐t♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✹✳✷ ❆❧❣✉♥s ❘❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽
✹ ■♠♣❧❡♠❡♥t❛çã♦ ❡ ❘❡s✉❧t❛❞♦s ✺✶
✹✳✶ ■♠♣❧❡♠❡♥t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹✳✷ ❘❡s✉❧t❛❞♦s ♣❛r❛ ❈❤❛✈❡s ❞❡ ✷✵✹✽ ❇✐ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✳✷✳✶ ❘❡s✉❧t❛❞♦s ❞♦s ❊①♣❡r✐♠❡♥t♦s ❞♦ ❈r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❇ás✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
✈✐✐✐ ❙❯▼➪❘■❖
✹✳✷✳✷ ❘❡s✉❧t❛❞♦s ❞♦s ❊①♣❡r✐♠❡♥t♦s ❞♦ ❈r✐♣t♦ss✐st❡♠❛ ❘❙❆ ✸✲♣r✐♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✹✳✷✳✸ ❘❡s✉❧t❛❞♦s ❞♦s ❊①♣❡r✐♠❡♥t♦s ❞♦ ❈r✐♣t♦ss✐st❡♠❛ ❘❙❆ ✹✲♣r✐♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✹✳✷✳✹ ❈♦♠♣❛r❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✳✸ ❊①♣❡r✐♠❡♥t♦s ❊①tr❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✸✳✶ ❊①♣❡r✐♠❡♥t♦s ♣❛r❛ ❝❤❛✈❡s ❞❡ ✸✵✼✷ ❜✐ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✸✳✷ ❊①♣❡r✐♠❡♥t♦s ♣❛r❛ ❝❤❛✈❡s ❞❡ ✹✵✾✻ ❜✐ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
✺ ❈♦♥❝❧✉sõ❡s ✺✾
✺✳✶ ❈♦♥❝❧✉sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✺✳✷ ❘❡❝♦♠❡♥❞❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✺✳✸ ❚r❛❜❛❧❤♦s ❋✉t✉r♦s ❡ Pr♦❜❧❡♠❛s ❆❜❡rt♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶
❆ ❚❡♦r✐❛ ❞❡ ◆ú♠❡r♦s ✻✸
❆✳✶ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ❡ ❈♦♥❣r✉ê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ❆✳✶✳✶ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ❆✳✶✳✷ ❈♦♥❣r✉ê♥❝✐❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ❆✳✶✳✸ ❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s ❊st❡♥❞✐❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ❆✳✶✳✹ ■♥✈❡rs❛ ▼✉❧t✐♣❧✐❝❛t✐✈❛ ▼ó❞✉❧♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ❆✳✶✳✺ ●r✉♣♦ ❡ ❆♥❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ❆✳✶✳✻ ❆ ❋✉♥çã♦ ❊✉❧❡r ❡ ♦ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r✲❋❡r♠❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ❆✳✶✳✼ P♦❧✐♥ô♠✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ❆✳✷ ❊q✉❛çõ❡s ▼ó❞✉❧♦ m ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼
❆✳✷✳✶ ❊q✉❛çõ❡s ▲✐♥❡❛r❡s ▼ó❞✉❧♦m ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼
❆✳✷✳✷ ❈♦♥❣r✉ê♥❝✐❛s ❞❡ ●r❛✉ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ❆✳✸ ❈♦♥❝❡✐t♦s ❊①tr❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾
❇ Pr♦❜❛❜✐❧✐❞❛❞❡ ✼✶
❇✳✶ ❊①♣❡r✐♠❡♥t♦s ❆❧❡❛tór✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ❇✳✶✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ❇✳✶✳✷ ❊s♣❛ç♦ ❆♠♦str❛❧ ❡ ❊✈❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ❇✳✶✳✸ Pr♦❜❛❜✐❧✐❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷
❈ ❊st❛tíst✐❝❛ ✼✺
❙❯▼➪❘■❖ ✐①
▲✐st❛ ❞❡ ❆❜r❡✈✐❛t✉r❛s
❘❙❆ ❙✐st❡♠❛ ❈r✐♣t♦❣rá✜❝♦ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ❘✐✈❡st✱ ❙❤❛♠✐r ❡ ❆❞❡❧♠❛♥✳
▼■❚ ■♥st✐t✉t♦ ❚❡❝♥♦❧ó❣✐❝♦ ❞❡ ▼❛ss❛❝❤✉s❡tts ✭▼❛ss❛❝❤✉s❡tts ■♥st✐t✉t❡ ♦❢ ❚❡❝❤♥♦❧♦❣②✮✳ ◆P ❈❧❛ss❡ ❞❡ ❝♦♠♣❧❡①✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ❡♠ ♠♦❞❡❧♦ ♥ã♦ ❞❡t❡r♠✐♥íst✐❝♦
✭◆♦♥✲❉❡t❡r♠✐♥✐st✐❝ P♦❧②♥♦♠✐❛❧ t✐♠❡✮✳
◆❋❙ ➱ ♦ ❛❧❣♦r✐t♠♦ ♠❛✐s ❡✜❝✐❡♥t❡ ♣❛r❛ ❢❛t♦r❛r ✐♥t❡✐r♦s ❞❡ ♠❛✐s ❞❡ ✶✵✵ ❞í❣✐t♦s ✭◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡✮✳
P❑❈❙ P❛❞rã♦ ❞❡ ❈r✐♣t♦❣r❛✜❛ ❞❡ ❈❤❛✈❡ Pú❜❧✐❝❛ ✭P✉❜❧✐❝ ❑❡② ❈r✐♣t♦❣r❛♣❤② ❙t❛♥❞❛r❞s✳✮ ❚❈❘ ❚❡♦r❡♠❛ ❈❤✐♥ês ❞♦ ❘❡st♦✳
❉❘❆▼ ▼❡♠ór✐❛ ❞❡ ❛❝❡ss♦ ❛❧❡❛tór✐♦ ❞✐♥á♠✐❝❛ ✭❉②♥❛♠✐❝ ❘❛♥❞♦♠✲❆❝❝❡ss ▼❡♠♦r②✮✳ ▼❙❇ ❇✐ts ♠❛✐s s✐❣♥✐✜❝❛t✐✈♦s ✭▼♦st ❙✐❣♥✐✜❝❛♥t ❇✐ts✮✳
▲❙❇ ❇✐ts ♠❡♥♦s s✐❣♥✐✜❝❛t✐✈♦s ✭▲❡❛st ❙✐❣♥✐✜❝❛♥t ❇✐ts✮✳
❊❈▼ ▼ét♦❞♦ ❞❡ ❝✉r✈❛s ❡❧í♣t✐❝❛s ✉s❛❞♦ ♣❛r❛ ❝❛❧❝✉❧❛r ❢❛t♦r❡s ♣r✐♠♦s ❞❡ ✉♠ ✐♥t❡✐r♦ ❝♦♠♣♦st♦✳ ✭❊❧❧✐♣t✐❝ ❝✉r✈❡ ♠❡t❤♦❞✮✳
▼❉❈ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠✳
◆♦t❛çã♦ ❡ ▲✐st❛ ❞❡ ❙í♠❜♦❧♦s
N ▼✉❧t✐♣❧✐❝❛çã♦ ❞♦s ♣r✐♠♦s ♣❛r❛ ♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆✳ n ◆ú♠❡r♦ ❞❡ ❜✐ts ❞❛ ✈❛r✐á✈❡❧ N✳
u ◆ú♠❡r♦ ❞❡ ❢❛t♦r❡s ♣r✐♠♦s ❞♦ ✐♥t❡✐r♦ N✳ sk ❈❤❛✈❡ s❡❝r❡t❛ ❞♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆✳
˜
sk sk ❝♦♠ ❛❧❣✉♥s ❡rr♦s ✭❜✐ts ♠♦❞✐✜❝❛❞♦s✮✳ δ ❋r❛çã♦ ❞❡ ❜✐ts ❝♦rr❡t♦s ❡♠ sk˜✳
≈ ❙í♠❜♦❧♦ ❞❡ ❛♣r♦①✐♠❛çã♦✳
pk ❈❤❛✈❡ ♣ú❜❧✐❝❛ ✭❞♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆✮✳ ri ■✲és✐♠♦ ❢❛t♦r ♣r✐♠♦ ❞♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆✳ ⌊b⌋ ❋✉♥çã♦ ♣✐s♦ ❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ ❜✳
p ❖✉tr❛ ❞❡✜♥✐çã♦ ❞❡ r1✳ q ❖✉tr❛ ❞❡✜♥✐çã♦ ❞❡ r2✳
φ(a) ❋✉♥çã♦ φ❞❡ ❊✉❧❡r ❛♣❧✐❝❛❞♦ ❛ ✉♠ ✐♥t❡✐r♦a✳ e ❊①♣♦❡♥t❡ ❞❡ ❡♥❝r✐♣t❛çã♦ ❞♦ ❘❙❆✳
d ❊①♣♦❡♥t❡ ❞❡ ❞❡❝r✐♣t❛çã♦ ❞♦ ❘❙❆✳
ZN ❈♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣♦r a✱ ♦♥❞❡ 0≤a≤N−1✳
ZN∗ ❈♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣♦r a✱ ♦♥❞❡ 0≤a≤N−1❡ aé ❝♦✲♣r✐♠♦ ❛N✳ M ❚❡①t♦ ❧❡❣í✈❡❧✳
C ❈r✐♣t♦❣r❛♠❛ ❞♦ t❡①t♦ ❧❡❣í✈❡❧M✳
di ■✲és✐♠♦ ❡①♣♦❡♥t❡ ❚❈❘✳ ❙❡✉ ✈❛❧♦r ❡stá ❞❛❞♦ ♣♦r d mod (ri−1)✳
dp ❖✉tr❛ ❞❡✜♥✐çã♦ ❞❡ d1✳
dq ❖✉tr❛ ❞❡✜♥✐çã♦ ❞❡ d2✳
r−21 ❈♦❡✜❝✐❡♥t❡ ❚❈❘✳ ❙❡✉ ✈❛❧♦r ❡stá ❞❛❞♦ ♣♦r r−21 mod r1✳
t−i 1 ■✲és✐♠♦ ❝♦❡✜❝✐❡♥t❡ ❚❈❘✳ ❙❡✉ ✈❛❧♦r ❡stá ❞❛❞♦ ♣♦r (Qi−1
j=1ri)−1 modri✳
Mi ■✲és✐♠♦ ♠❡♥s❛❣❡♠✳ s❡✉ ✈❛❧♦r ❡stá ❞❛❞♦ ♣♦r Mi =Cdi mod ri ≡ ❙í♠❜♦❧♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛
6≡ ❙í♠❜♦❧♦ ❞❡ ♥ã♦ ❝♦♥❣r✉ê♥❝✐❛✳
a|b ❖ ✐♥t❡✐r♦a ❞✐✈✐❞❡ ❛♦ ✐♥t❡✐r♦b✳ a6 |b ❖ ✐♥t❡✐r♦a ♥ã♦ ❞✐✈✐❞❡ ❛♦ ✐♥t❡✐r♦ ❜✳
known ■♥❞✐❝❛ ♦ ❡st❛❞♦ ❞❡ ✉♠ ❜✐t ❞❡ s❡r ❝♦♥❤❡❝✐❞♦✳ unknown ❊st❛❞♦ ❞❡ ✉♠ ❜✐t ❞❡ s❡r ❞❡s❝♦♥❤❡❝✐❞♦✳ mdc(a, b) ▼á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞♦s ✐♥t❡✐r♦s a❡b✳ min(a, b) ▼í♥✐♠♦ ✈❛❧♦r ❡♥tr❡ a❡b✳
max(a, b) ▼á①✐♠♦ ✈❛❧♦r ❡♥tr❡ a❡ b✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ ❈❤❛✈❡ s❡❝r❡t❛ ❘❙❆ ❡♠ ❆❙◆✳✶ ✭❋♦♥t❡✿P❑❈❙ ★✶✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✷ ❉❡s❛♣❛r❡❝✐♠❡♥t♦ ❣r❛❞✉❛❧ ❞❡ ❞❛❞♦s ❡♠ ✉♠ ❝❤✐♣ ❞❡ ♠❡♠ór✐❛ ❘❆▼ ✭❋♦♥t❡✿ ❈❡♥t❡r ❢♦r
■♥❢♦r♠❛t✐♦♥ ❚❡❝❤♥♦❧♦❣② ❛t Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② ✲ ✷✵✵✾✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✸ Pr♦♣r✐❡❞❛❞❡ ❞❡ r❡♠❛♥❡s❝ê♥❝✐❛ ❞❛ ♠❡♠ór✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✹ ❈♦❞✐✜❝❛çã♦ ❇❊❘ ❞❛ ❝❤❛✈❡ s❡❝r❡t❛ ❘❙❆✳ ❆ ♣❛rt❡ s♦♠❜r❡❛❞❛ ♠♦str❛ ❛ ❝❛❞❡✐❛ ✵✷✲✵✶✲
✵✵✲✵✷✳ ✭❋♦♥t❡✿ ❈❡♥t❡r ❢♦r ■♥❢♦r♠❛t✐♦♥ ❚❡❝❤♥♦❧♦❣② ❛t Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t②✮ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✸✳✶ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ❛❧❣♦r✐t♠♦ ❞❡ r❡❝♦♥str✉çã♦ ♦♥❞❡ ❛s ❝✐r❝✉♥❢❡rê♥✲
❝✐❛s✿ s✐♠♣❧❡s sã♦ ❛s r❛í③❡s ✐♥❝♦rr❡t❛s✱ ❛s ❞✉♣❧❛s sã♦ ❛s r❛í③❡s ❜♦❛s ❡ ♣♦r ú❧t✐♠♦ t❡♠♦s ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞✉♣❧❛ ❝✐♥③❛✱ ❛ q✉❛❧ é ❛ r❡♣r❡s❡♥t❛çã♦ ❞❛ r❛✐③ ❝♦rr❡t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✷ ❘❡♣r❡s❡♥t❛çã♦ ❞♦s ♠♦♠❡♥t♦s ❞❡ ♦r❞❡♠ ✶ ❡ ✷ ❞❛ ✈❛r✐á✈❡❧ ❞✐s❝r❡t❛ G✱ ♦♥❞❡ ♣♦❞❡♠♦s
♦❜s❡r✈❛r q✉❡ ❛♠❜♦s ♠♦♠❡♥t♦s sã♦ ✐❣✉❛✐s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✸ ❘❡♣r❡s❡♥t❛çã♦ ❞♦s ♠♦♠❡♥t♦s ❞❡ ♦r❞❡♠ ✶ ❡ ✷ ❞❛ ✈❛r✐á✈❡❧ ❞✐s❝r❡t❛ B✱ ♦♥❞❡ ♣♦❞❡♠♦s
♦❜s❡r✈❛r ❛♠❜♦s ♠♦♠❡♥t♦s✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✶ ❈♦♠♣❛r❛çã♦ ❞❛s ♠é❞✐❛s ❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ r❛í③❡s ❛♥❛❧✐s❛❞❛s ♣❛r❛ ❝❤❛✈❡s ❞❡ ✷✵✹✽ ❜✐ts✳ ✺✹ ✹✳✷ ❈♦♠♣❛r❛çã♦ ❞❛s ♠é❞✐❛s ❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ r❛í③❡s ❛♥❛❧✐s❛❞❛s ♣❛r❛ ❝❤❛✈❡s ❞❡ ✸✵✼✷ ❜✐ts✳ ✺✻ ✹✳✸ ❈♦♠♣❛r❛çã♦ ❞❛s ♠é❞✐❛s ❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ r❛í③❡s ❛♥❛❧✐s❛❞❛s ♣❛r❛ ❝❤❛✈❡s ❞❡ ✹✵✾✻ ❜✐ts✳ ✺✻
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✷✳✶ ▼á①✐♠♦ ♥ú♠❡r♦ ❞❡ ♣r✐♠♦s ♣❡r♠✐t✐❞♦s ❡♠ ✉♠ ♠ó❞✉❧♦ N ✭❋♦♥t❡✿ ❈♦♠♣❛q ❈♦♠♣✉t❡r
❈♦r♣♦r❛t✐♦♥✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✸✳✶ ◆ú♠❡r♦ ❞❡ r❛í③❡s ❣❡r❛❞❛s ❛ ♣❛rt✐r ❞❡ ✉♠❛ r❛✐③ ❜♦❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✷ ◆ú♠❡r♦ ❞❡ r❛í③❡s ✐♥❝♦rr❡t❛s ❣❡r❛❞❛s ❛ ♣❛rt✐r ❞❡ ✉♠❛ r❛✐③ ❜♦❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✸ ❊st❛❞♦ ❞♦ ❜✐t ri[j]❝♦♠ r❡❧❛çã♦ ❛♦ ❡st❛❞♦ ❞♦ ❜✐t di[j+τ(ki)]♥❛ r❛✐③ ❜♦❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✹ ◆ú♠❡r♦ ❞❡ r❛í③❡s ✐♥❝♦rr❡t❛s ❣❡r❛❞❛s ❛ ♣❛rt✐r ❞❡ ✉♠❛ r❛✐③ ✐♥❝♦rr❡t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✺ ❊st❛❞♦ ❞♦ ❜✐t ri[j]❝♦♠ r❡❧❛çã♦ ❛♦ ❡st❛❞♦ ❞♦ ❜✐t di[j+τ(ki)]♥❛ r❛✐③ ✐♥❝♦rr❡t❛✳✳ ✳ ✳ ✳ ✸✽ ✹✳✶ ❘❡s✉❧t❛❞♦s ❞♦s ❡①♣❡r✐♠❡♥t♦s ❞❛ ❘❡❝♦♥str✉çã♦ ❞❛ ❈❤❛✈❡ ❙❡❝r❡t❛ ❘❙❆ ❇ás✐❝♦ ✭u= 2✮✳ ✺✷
✹✳✷ ❘❡s✉❧t❛❞♦s ❞♦s ❡①♣❡r✐♠❡♥t♦s ❞❛ ❘❡❝♦♥str✉çã♦ ❞❛ ❈❤❛✈❡ ❙❡❝r❡t❛ ❘❙❆ ✸✲♣r✐♠♦s✳ ✳ ✳ ✳ ✺✸ ✹✳✸ ❘❡s✉❧t❛❞♦s ❞♦s ❡①♣❡r✐♠❡♥t♦s ❞❛ ❘❡❝♦♥str✉çã♦ ❞❛ ❈❤❛✈❡ ❙❡❝r❡t❛ ❘❙❆ ✹✲♣r✐♠♦s✳ ✳ ✳ ✳ ✺✹ ✹✳✹ ❘❡s✉❧t❛❞♦s ❞♦s ❡①♣❡r✐♠❡♥t♦s ❞❛ ❘❡❝♦♥str✉çã♦ ❞❛ ❈❤❛✈❡ ❙❡❝r❡t❛ ❘❙❆ ♠ó❞✉❧♦ ✸✵✼✷✳ ✳ ✺✺ ✹✳✺ ❘❡s✉❧t❛❞♦s ❞♦s ❡①♣❡r✐♠❡♥t♦s ❞❛ ❘❡❝♦♥str✉çã♦ ❞❛ ❈❤❛✈❡ ❙❡❝r❡t❛ ❘❙❆ ♠ó❞✉❧♦ ✹✵✾✻✳ ✳ ✺✼ ✺✳✶ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ♦s ❝r✐♣t♦ss✐st❡♠❛s ❘❙❆ ❜ás✐❝♦✱ ✸✲♣r✐♠♦s ❡ ✹✲♣r✐♠♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✺✳✷ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ♦s ❝r✐♣t♦ss✐st❡♠❛s ❘❙❆ ♠✉❧t✐✲♣♦tê♥❝✐❛✱ ❜ás✐❝♦ ❡ ♠✉❧t✐✲♣r✐♠♦✳ ✳ ✳ ✳ ✻✵ ✺✳✸ ◆ú♠❡r♦ ❞❡ r❛í③❡s ✐♥❝♦rr❡t❛s ❣❡r❛❞❛s ❛ ♣❛rt✐r ❞❡ ✉♠❛ r❛✐③ ✐♥❝♦rr❡t❛ q✉❛♥❞♦ t❡♠♦s só
✉♠❛ ♣♦r❝❡♥t❛❣❡♠ ❞❡ ❜✐ts ❝♦♥❤❡❝✐❞♦s ❞❡d✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
❖ ❘❙❆ é ✉♠ ❞♦s ❝r✐♣t♦ss✐st❡♠❛s ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛ ♠❛✐s ✉s❛❞♦ ❡ ✐♠♣❧❡♠❡♥t❛❞♦ ❬P❛✐✵✸❪✱ s❡✉ ♥♦♠❡ é ❞❡r✐✈❛❞♦ ❞❛s ✐♥✐❝✐❛✐s ❞♦s s❡✉s ❝r✐❛❞♦r❡s✿ ❘♦♥ ✭❘✮✐✈❡st✱ ❆❞✐ ✭❙✮❤❛♠✐r ❡ ▲❡♥ ✭❆✮❞❧❡♠❛♥✳ ❊st❡ ❝r✐♣t♦ss✐st❡♠❛ ❢♦✐ ❝r✐❛❞♦ ❡♠ ❛❣♦st♦ ❞❡ ✶✾✼✼ ♥♦ ▼■❚✶ ❡ ♣✉❜❧✐❝❛❞♦ ❡♠ ❢❡✈❡r❡✐r♦ ❞❡ ✶✾✼✽ ❬❘❙❆✼✽❪✳
❉❡s❞❡ s✉❛ ♣✉❜❧✐❝❛çã♦✱ ♥❡♥❤✉♠ ❛t❛q✉❡ ❝♦♥s❡❣✉✐✉ q✉❡❜rá✲❧♦✱ ♣♦rt❛♥t♦ ♥ã♦ ❢♦✐ ♣r❡❝✐s♦ ♠✉❞❛r s✉❛ ❡str✉t✉r❛✳ ◆♦ ❡♥t❛♥t♦✱ ♥❛ ❧✐t❡r❛t✉r❛ ❝✐❡♥t✐✜❝❛ ❡①✐st❡♠ ♣❡sq✉✐s❛s s♦❜r❡ ❝❛s♦s ♦♥❞❡ ♦ ❘❙❆ é ✐♥s❡❣✉r♦✱ ♠❛s ✐ss♦ é ❞❡✈✐❞♦ ❛♦ ✉s♦ ✐♥❛❞❡q✉❛❞♦ ❞♦ ♠❡s♠♦✳
❆ s❡❣✉r❛♥ç❛ ❞♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❡stá ❜❛s❡❛❞❛ ♥♦ ♣r♦❜❧❡♠❛ ❞❡ ❢❛t♦r❛çã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ✭♦ q✉❛❧ é ✉♠ ♣r♦❜❧❡♠❛ ◆P✮ ❬❋❡♦❪✱ ♥♦ ❘❙❆ ♦ ✐♥t❡✐r♦ ❛ ❢❛t♦r❛r é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♠ó❞✉❧♦ N ❡ é ♦
r❡s✉❧t❛❞♦ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡u♣r✐♠♦s ❛❧❡❛tór✐♦s ❣r❛♥❞❡s ❞♦ ♠❡s♠♦ t❛♠❛♥❤♦✳ ◗✉❛♥❞♦ t❡♠♦s ♦ ✈❛❧♦r u = 2 é ❞❡♥♦♠✐♥❛❞♦ ❝♦♠♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❜ás✐❝♦ ❡ ♣❛r❛u ≥ 3 é ❝❤❛♠❛❞♦ ❞❡ ❝r✐♣t♦ss✐st❡♠❛
❘❙❆ ♠✉❧t✐✲♣r✐♠♦✳ ❖ ❛❧❣♦r✐t♠♦ ♠❛✐s rá♣✐❞♦ ❛té ❛ ❞❛t❛ ✭✷✵✶✸✮ ♣❛r❛ ❛ s♦❧✉çã♦ ❞❡ ❢❛t♦r❛çã♦ ❞❡ ✐♥t❡✐r♦s é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ◆❋❙ ✭◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡✮✱ ♦ q✉❛❧ ❡st❛❜❡❧❡❝❡✉ ✉♠ ♥♦✈♦ r❡❝♦r❞❡ ❡♠ ❢❛t♦r❛r ✉♠ ♠ó❞✉❧♦ N ❞❡ ✼✻✽ ❜✐ts ✭✷✸✷ ❞❡❝✐♠❛✐s✮ ❞♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❜ás✐❝♦ ❡♠ ❞❡③❡♠❜r♦ ❞♦ ✷✵✵✾ ❬❈♦♥❪✳
❖❜s❡r✈❛♥❞♦ ❛ ❞✐✜❝✉❧❞❛❞❡ ❞❡ ❢❛t♦r❛r ✉♠ ✐♥t❡✐r♦ s✉r❣✐✉ ❛ ✐❞❡✐❛ ❞❡ ❞❛r s♦❧✉çã♦ ❛ ❡ss❡ ♣r♦❜❧❡♠❛ ❢❛③❡♥❞♦ ✉s♦ ❞❡ ❛❧❣✉♠ t✐♣♦ ❞❡ ✐♥❢♦r♠❛çã♦ ❡①tr❛ ✭♥❡st❡ ❝❛s♦✱ ❜✐ts ❞♦s ❢❛t♦r❡s ♣r✐♠♦s ❞❡ N ♦✉ ♦✉tr♦s ❞❛❞♦s✮
q✉❡ s❡r✐❛ ❢♦r♥❡❝✐❞❛ ♣♦r ✉♠ ♦r❛❝✉❧♦✳
❊♠ ❝r✐♣t♦❣r❛✜❛✱ ✉♠ ♦rá❝✉❧♦ é ✉♠ ♣r♦❣r❛♠❛ q✉❡ r❡s♣♦♥❞❡ ♣❡r❣✉♥t❛s ❝♦♠ ✉♠❛ r❡s♣♦st❛ ❜♦♦❧❡❛♥❛ ✭✉♠ ❙■▼ ♦✉ ✉♠ ◆➹❖✮✳ P♦r ❡①❡♠♣❧♦✱ ♣♦❞❡♠♦s ♣❡r❣✉♥t❛r ✧q✉❛❧ é ♦ ✐✲és✐♠♦ ❜✐t ❞♦ ♠❡♥♦r ❢❛t♦r ❞❡
N❄✧✳ ◆♦ ❝❛s♦ tr✐✈✐❛❧ ♣❛r❛ ♦ ♠ó❞✉❧♦N ❞♦ ❝r✐♣t♦s✐st❡♠❛ ❘❙❆ ❇ás✐❝♦✱ só ♣r❡❝✐s❛♠♦s ❞❡n/2♣❡r❣✉♥t❛s
♣❛r❛ ❢❛t♦r❛r N✱ ♦♥❞❡ n = lgN é ♦ ♥ú♠❡r♦ ❞❡ ❜✐ts ❞❡N ❬▼❛✉✾✷❪✳ ❈♦♠ ❡ss❡ ♥♦✈♦ ❝♦♥❝❡✐t♦ s✉r❣✐✉
✉♠❛ ♥♦✈❛ ❧✐♥❤❛ ❞❡ ♣❡sq✉✐s❛ ❛ss✐st✐❞❛ ♣♦r ✉♠ ♦rá❝✉❧♦ q✉❡ ♣r♦❝✉r❛ ♠✐♥✐♠✐③❛r ♦ ♥ú♠❡r♦ ❞❡ ♣❡r❣✉♥t❛s ❢❡✐t❛s ❛♦ ♦rá❝✉❧♦✳ ❈♦rr❡s♣♦♥❞❡♥t❡ ❛ ❡ss❛ ❧✐♥❤❛ ❞❡ ♣❡sq✉✐s❛ t❡♠✲s❡ ✈ár✐♦s r❡s✉❧t❛❞♦s ❡✜❝✐❡♥t❡s ♦♥❞❡ ❛ ❢❛t♦r❛çã♦ ❞♦ ♠♦❞✉❧♦N ❞♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❜ás✐❝♦ é ♣♦ssí✈❡❧ ❡♠ t❡♠♣♦ ♣♦❧✐♥♦♠✐❛❧ t❡♥❞♦✿
• ♦s n/4 ❜✐ts ♠❡♥♦s s✐❣♥✐✜❝❛t✐✈♦s ✭▲❙❇✮ ♦✉ ♠❛✐s s✐❣♥✐✜❝❛t✐✈♦s ✭▼❙❇✮ ❞❡ ✉♠ ❞♦s s❡✉s ❢❛t♦r❡s
♣r✐♠♦sp ❬❈♦♣✾✼❪✳
• ♦sn/4 ▲❙❇ ❞♦ ❡①♣♦❡♥t❡ ❞❡ ❞❡❝r✐♣t❛çã♦d❬❇❉❋✾✽❪✳
• ♦sn/4 ▲❙❇ ❞❡ ❡①♣♦❡♥t❡ ❞❡ ❞❡❝r✐♣t❛çã♦dp ❬❇❉❋✾✽❪✳
• ✉♠ ♠á①✐♠♦ ❞❡log log(N) ❜❧♦❝♦s ❞❡s❝♦♥❤❡❝✐❞♦s ❡ ✉♠❛ ❢r❛çã♦ln(2) = 0.70 ❞❡ ❜✐ts ❝♦♥❤❡❝✐❞♦s
❞❡ p❬❍▼✵✽❪✳
❖ ❛❧❣♦r✐t♠♦ ❞❡ r❡❝♦♥str✉çã♦ ❞❛ ❝❤❛✈❡ s❡❝r❡t❛ sk ❞♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❜ás✐❝♦ ❞❡ ❍❡♥✐♥❣❡r
❡ ❙❤❛❝❤❛♠ ❬❍❙✵✾❪ ❡stá ❜❛s❡❛❞♦ ❡♠ ❛t❛q✉❡s ❛ss✐st✐❞♦s ♣♦r ✉♠ ♦rá❝✉❧♦✱ ♠❛s ❛ ❞✐❢❡r❡♥ç❛ ❝♦♠ ♦s ❛t❛q✉❡s ❛♥t❡s ♠❡♥❝✐♦♥❛❞♦s é q✉❡ ♦ ❛t❛❝❛♥t❡ ♥ã♦ t❡♠ ❝♦♥tr♦❧❡ s♦❜r❡ ❛s ♣♦s✐çõ❡s ❞♦s ❜✐ts✱ ♦✉ s❡❥❛✱ ♦ ❛t❛❝❛♥t❡ só r❡❝❡❜❡ ❜✐ts ❛❧❡❛tór✐♦s✳ ❖s r❡s✉❧t❛❞♦s ❞❡ ❍❡♥✐♥❣❡r ❡ ❙❤❛❝❤❛♠ ♠♦str❛♠ q✉❡ é ♣♦ssí✈❡❧ ❛ r❡❝♦♥str✉çã♦ ❞❛ ❝❤❛✈❡ s❡❝r❡t❛ sk ❞♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❜ás✐❝♦ ❡♠ t❡♠♣♦ ♣♦❧✐♥♦♠✐❛❧ ❝♦♠ ✉♠❛
❣r❛♥❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ t❡♥❞♦ ✉♠❛ ❢r❛çã♦δ ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛0.27 ❞❡ ❜✐ts ❝♦rr❡t♦s ❡♠sk˜✷✳
✶■♥st✐t✉t♦ ❚❡❝♥♦❧ó❣✐❝♦ ❞❡ ▼❛ss❛❝❤✉s❡tts
✷❊str✉t✉r❛ ❞❡ ❞❛❞♦s q✉❡ ❝♦♥t❡♠ ❛❧❣✉♥s ❜✐ts ❝♦rr❡t♦s ❞❡sk
✷ ■◆❚❘❖❉❯➬➹❖ ✶✳✸
✶✳✶ ❖❜❥❡t✐✈♦s
❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡ss❡ ♣r♦❥❡t♦ é ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❡ ✉♠ ❛❧❣♦r✐t♠♦ ❞❡ r❡❝♦♥str✉çã♦ ❞❛ ❝❤❛✈❡ s❡❝r❡t❛ ♣❛r❛ ♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ♠✉❧t✐✲♣r✐♠♦(u≥3)✱ ❜❛s❡❛❞♦ ♥♦ ❛❧❣♦r✐t♠♦ ❞❡ r❡❝♦♥str✉çã♦ ♣❛r❛
♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❜ás✐❝♦ ✭u= 2✮ ♣r♦♣♦st♦ ♣♦r ❍❡♥✐♥❣❡r ❡ ❙❤❛❝❤❛♠✳
❈♦♠♦ ♦❜❥❡t✐✈♦s ❡s♣❡❝í✜❝♦s ❞❡st❡ ♣r♦❥❡t♦ ❡stã♦✿
✶✳ ❊st✉❞♦ ❡ ❛♥á❧✐s❡ ❞♦s ❝r✐♣t♦ss✐st❡♠❛s ❘❙❆ ❜ás✐❝♦ ❡ ❘❙❆ ♠✉❧t✐✲♣r✐♠♦✳
✷✳ ❊st✉❞♦ ❡ ■♠♣❧❡♠❡♥t❛çã♦ ❞♦ ❛❧❣♦r✐t♠♦ ❞❡ r❡❝♦♥str✉çã♦ ❞❛ ❝❤❛✈❡ s❡❝r❡t❛ ♣❛r❛ ♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❜ás✐❝♦✳
✸✳ ❊st✉❞♦ ❞❛s r❡❧❛çõ❡s q✉❡ ❡①✐st❡♠ ❡♥tr❡ ♦s ❜✐ts ❞❛s ✈❛r✐á✈❡✐s ❞❛ ❝❤❛✈❡ s❡❝r❡t❛sk ❞♦s ❝r✐♣t♦ss✐s✲
t❡♠❛s ❘❙❆ ❜ás✐❝♦ ❡ ❘❙❆ ♠✉❧t✐✲♣r✐♠♦✳
✹✳ ■♠♣❧❡♠❡♥t❛çã♦ ❡ ❛♥á❧✐s❡ ❞♦ ❛❧❣♦r✐t♠♦ ❞❡ r❡❝♦♥str✉çã♦ ❞❛ ❝❤❛✈❡ s❡❝r❡t❛ ♣❛r❛ ♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ♠✉❧t✐✲♣r✐♠♦✳
✶✳✷ ❘❡s✉❧t❛❞♦s
✶✳ P❛r❛ ♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❜ás✐❝♦ ♦✉ ♠✉❧t✐✲♣r✐♠♦ ✭♦♥❞❡ s❡✉ ♠ó❞✉❧♦ N ❡stá ❞❛❞♦ ♣♦r N =
Qu
i=1ri ♣❛r❛u≥2✮ ❢♦✐ ❞❡t❡r♠✐♥❛❞♦ q✉❡ é ♣r❡❝✐s♦ t❡r ✉♠❛ ❢r❛çã♦δ❞❡ ❜✐ts ❝♦rr❡t♦s ♠❛✐♦r q✉❡
2−22uu+2+1 ❡♠sk˜ ♣❛r❛ ♣♦❞❡r r❡❝♦♥str✉✐r ❛ ❝❤❛✈❡ s❡❝r❡t❛ sk ❡♠ t❡♠♣♦ ♣♦❧✐♥♦♠✐❛❧ O(n2) ❝♦♠ ✉♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♠❛✐♦r q✉❡1−n12✳ ❊ss❡ r❡s✉❧t❛❞♦ ♥♦s ♣❡r♠✐t❡ ❞✐③❡r q✉❡ ❛ r❡❝♦♥str✉çã♦ ❞❡ ✉♠❛ ❝❤❛✈❡ s❡❝r❡t❛ sk ♣❛r❛ ♦ ❝r✐♣t♦ss✐st❡♠❛✿
• ❘❙❆ ❜ás✐❝♦ é ♣r❡❝✐s♦ t❡r ✉♠❛ ❢r❛çã♦δ ≥2−245 ≈0.27 • ❘❙❆ ✸✲♣r✐♠♦s é ♣r❡❝✐s♦ t❡r ✉♠❛ ❢r❛çã♦δ ≥2−257 ≈0.37 • ❘❙❆ ✹✲♣r✐♠♦s é ♣r❡❝✐s♦ t❡r ✉♠❛ ❢r❛çã♦δ ≥2−269 ≈0.42 ❞❡ ❜✐ts ❝♦rr❡t♦s ❞❡ sk✳
✷✳ ❈♦♠ ♦ r❡s✉❧t❛❞♦ ❛❝✐♠❛ ❢♦✐ ❝♦♠♣r♦✈❛❞♦ q✉❡ ♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ♠✉❧t✐✲♣r✐♠♦ ♦❢❡r❡❝❡ ♠❛✐s s❡❣✉r❛♥ç❛ ❝♦♠ r❡❧❛çã♦ ❛♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❜ás✐❝♦✳
δ >2−22uu+2+1 >2−2 4
5 ♣❛r❛u≥3
✸✳ ❆ r❡❝♦♥str✉çã♦ ❞❡ ✉♠❛ ❝❤❛✈❡ s❡❝r❡t❛sk❞♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❜ás✐❝♦ ♦✉ ♠✉❧t✐✲♣r✐♠♦ s❡♠♣r❡
é ❢❡✐t❛ ❡♠ t❡♠♣♦ ♣♦❧✐♥♦♠✐❛❧ q✉❛♥❞♦ t❡♠ s❡ ✉♠❛ ❢r❛çã♦δ ❞❡ ❜✐ts ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ ✵✳✺✾✳ δ > lim
u→∞2−2
u+2
2u+1 ≈0.5858
✶✳✸ ❖r❣❛♥✐③❛çã♦ ❞♦ ❚r❛❜❛❧❤♦
◆♦ ❈❛♣ít✉❧♦ ✷✱ ❛♣r❡s❡♥t❛♠♦s ♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❡ ❝r✐♣t♦❣r❛✜❛✱ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆✱ ♣❛❞rã♦ ❞❡ ❝r✐♣t♦❣r❛✜❛ ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛ P❑❈❙ ✭P✉❜❧✐❝ ❦❡② ❝r②♣t♦❣r❛♣❤② st❛♥❞❛rt✮ ❞♦ ❘❙❆ ❡ ❛t❛q✉❡s s✐❞❡ ❝❤❛♥♥❡❧✳ ❖ ❈❛♣ít✉❧♦✸é ❞❡s❝r✐t♦ ♦ ❛❧❣♦r✐t♠♦ ❞❡ r❡❝♦♥str✉çã♦ ❞❛ ❝❤❛✈❡ s❡❝r❡t❛sk ❝♦♥t❡♥❞♦✿ ♦s ♣r❡✲
❝á❧❝✉❧♦s q✉❡ ❞❡✈❡♠ s❡r ❢❡✐t♦s✱ ♦ ❛❧❣♦r✐t♠♦ ❡ ✉♠❛ ❛♥á❧✐s❡ ❞❛ s✉❛ ❝♦♠♣❧❡①✐❞❛❞❡✳ ◆♦ ❈❛♣ít✉❧♦ ✹ sã♦ ♠♦str❛❞♦s ♦s r❡s✉❧t❛❞♦s ❞❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞♦ ❛❧❣♦r✐t♠♦✳ ❋✐♥❛❧♠❡♥t❡ ♥♦ ❈❛♣ít✉❧♦ ✺ ❡♥❝♦♥tr❛♠✲s❡ ❛s ❝♦♥❝❧✉sõ❡s ❞❡ss❡ tr❛❜❛❧❤♦✳
❈❛♣ít✉❧♦ ✷
❈♦♥❝❡✐t♦s
P❛r❛ ✐♥✐❝✐❛r ♥♦ss♦ ❡st✉❞♦ s♦❜r❡ ♦ ❛❧❣♦r✐t♠♦ ❞❡ r❡❝♦♥str✉çã♦ ❞❡ ❝❤❛✈❡s s❡❝r❡t❛s ❘❙❆ ❞❡ ❍❡♥✐♥❣❡r ❡ ❙❤❛❝❤❛♠ ♣r❡❝✐s❛♠♦s r❡❧❡♠❜r❛r ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ ❝♦♥❝❡✐t♦s ❞❡ ❝r✐♣t♦❣r❛✜❛✱ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❡ ❛t❛q✉❡s ❝♦❧❞✲❜♦♦t ❡ ❛ss✐♠ ❢♦r♠❛r ♦ ♥❡❝❡ssár✐♦ ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ss❡ ❛❧❣♦r✐t♠♦✳ P♦rt❛♥t♦ ❡st❡ ❝❛♣ít✉❧♦ s❡r✈❡ ❝♦♠♦ ❜❛s❡ ♣❛r❛ t♦❞♦s ♦s ❛❧❣♦r✐t♠♦s ❡ ✐❞❡✐❛s ❛♣r❡s❡♥t❛❞❛s ❛♦ ❧♦♥❣♦ ❞❡ss❡ ❞♦❝✉♠❡♥t♦ ❡ ❞❡✈❡ s❡r ❝♦♥s✉❧t❛❞♦ ❛ss✐♠ q✉❡ ❛❧❣✉♠❛ ❞ú✈✐❞❛ s✉r❣✐r✳
✷✳✶ ❈r✐♣t♦❣r❛✜❛ ❡ ❈r✐♣t♦❣r❛✜❛ ❞❡ ❈❤❛✈❡ Pú❜❧✐❝❛
❊♠ ❣r❡❣♦✱ ❦r②♣tós s✐❣♥✐✜❝❛ s❡❝r❡t♦ ♦✉ ♦❝✉❧t♦ ❡ ❣rá♣❤❡✐♥ s❡ r❡❢❡r❡ ❛ ❡s❝r✐t❛✱ ♣♦rt❛♥t♦ ♣♦❞❡♠♦s ❞❡✜♥✐r à ❝r✐♣t♦❣r❛✜❛ ❝♦♠♦ ♦ ❡st✉❞♦ ❞♦s ♣r✐♥❝í♣✐♦s ❡ té❝♥✐❝❛s ♣❡❧❛s q✉❛✐s ❛ ✐♥❢♦r♠❛çã♦ ♣♦❞❡ s❡r tr❛♥s❢♦r♠❛❞❛ ❞❛ s✉❛ ❢♦r♠❛ ♦r✐❣✐♥❛❧ ♣❛r❛ ♦✉tr❛ ✐❧❡❣í✈❡❧✱ ❞❡ ❢♦r♠❛ q✉❡ ♣♦ss❛ s❡r ❝♦♥❤❡❝✐❞❛ ❛♣❡♥❛s ♣♦r s❡✉ ❞❡st✐♥❛tár✐♦ ✭❞❡t❡♥t♦r ❞❛ ❝❤❛✈❡ s❡❝r❡t❛✮✱ ♦ q✉❡ ❛ t♦r♥❛ ❞✐❢í❝✐❧ ❞❡ s❡r ❧✐❞❛ ♣♦r ❛❧❣✉é♠ ♥ã♦ ❛✉t♦r✐③❛❞♦✳ P♦rt❛♥t♦✱ só ♦ r❡❝❡♣t♦r ❞❛ ♠❡♥s❛❣❡♠ ♣♦❞❡ ❧❡r ❛ ✐♥❢♦r♠❛çã♦ ❝♦♠ ❢❛❝✐❧✐❞❛❞❡ ❬❍♦♦✵✺❪✳ ❖ ♠ét♦❞♦ ♠❛✐s s✐♠♣❧❡s ❝♦♥s✐st❡ ❡♠ s✉❜st✐t✉✐r ✉♠❛ ❧❡tr❛ ♣❡❧❛ s❡❣✉✐♥t❡ ♥♦ ❛❧❢❛❜❡t♦✱ ✐st♦ é✱ tr❛♥s❧❛❞❛r ♦ ❛❧❢❛❜❡t♦ ✉♠❛ ❝❛s❛ ♣❛r❛ ❞✐❛♥t❡ ❞❡ ❢♦r♠❛ ❝✐r❝✉❧❛r✳ ❯♠❛ ❝♦❞✐✜❝❛çã♦ s❡♠❡❧❤❛♥t❡ ❛ ❡st❛ ❢♦✐ ✉t✐❧✐③❛❞❛ ♣♦r ❏ú❧✐♦ ❈és❛r ❛✜♠ ❞❡ ❡st❛❜❡❧❡❝❡r ✉♠❛ ❝♦♠✉♥✐❝❛çã♦ ❝♦♠ ❛❧❣✉♠❛ s❡❣✉r❛♥ç❛ ❝♦♠ ❛s ❧❡❣✐õ❡s ❡♠ ❝♦♠❜❛t❡ ♣❡❧❛ ❊✉r♦♣❛✱ ✜❝❛♥❞♦ ❛ss✐♠ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❝✐❢r❛✶❞❡ ❈és❛r✳ ❆❧é♠ ❞✐ss♦✱ é ✉♠ ❞♦s ♣r✐♠❡✐r♦s ♠ét♦❞♦s
❞❡ ❝♦❞✐✜❝❛çã♦ ❞❡ q✉❡ s❡ t❡♠ ♥♦tí❝✐❛✳
❆ ❝✐❢r❛ ❞❡ ❈és❛r ♣❡rt❡♥❝❡ ❛ ✉♠❛ ❝❧❛ss❡ ❞❡ ❛❧❣♦r✐t♠♦s ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ❛❧❣♦r✐t♠♦s ❞❡ ❝❤❛✈❡ s❡✲ ❝r❡t❛ ✭♦✉ s✐♠étr✐❝❛✮✳ ❆ ❝❤❛✈❡ s❡❝r❡t❛ ♥♦ ❝❛s♦ ❞❛ ❝✐❢r❛ ❞❡ ❈és❛r s❡r✐❛ ♦ ♥ú♠❡r♦ ❞❡ ♣♦s✐çõ❡s ❞❡s❧♦❝❛❞❛s ❡♠ r❡❧❛çã♦ ❛♦ ✐♥í❝✐♦ ❞♦ ❛❧❢❛❜❡t♦ ✭♥♦ ❝❛s♦ ❛❝✐♠❛ s♦♠❡♥t❡ ✉♠❛ ♣♦s✐çã♦✮✳ ❖❜s❡r✈❛♥❞♦ ❡st❡ ♠ét♦❞♦ ♥♦t❛♠♦s q✉❡ ❛ ❝❤❛✈❡ ❞❡✈❡ s❡r ♣r❡✈✐❛♠❡♥t❡ ❝♦♠❜✐♥❛❞❛ ❡♥tr❡ ♦ ❡♠✐ss♦r ❡ ♦ r❡❝❡♣t♦r ❞❛ ♠❡♥s❛❣❡♠ ❛tr❛✈és ❞❡ ✉♠ ♠❡✐♦ s✐❣✐❧♦s♦✱ ❛❧é♠ ❞✐ss♦✱ ❞❡✈❡ s❡r ♠❛♥t✐❞❛ ❡♠ s❡❣r❡❞♦ ♣❛r❛ ❡✈✐t❛r q✉❡ ✉♠❛ ♣❡ss♦❛ ♥ã♦ ❛✉t♦r✐③❛❞❛ ❝♦♥s✐❣❛ ❧❡r ❛ ♠❡♥s❛❣❡♠✳
✷✳✶✳✶ ❈r✐♣t♦❣r❛✜❛ ❞❡ ❈❤❛✈❡ Pú❜❧✐❝❛
❆ ✐❞❡✐❛ ❞❡ ❝r✐♣t♦ss✐st❡♠❛s ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛ ♦✉ ❛ss✐♠étr✐❝♦s ❢♦✐ ♣r♦♣♦st♦ ♣♦r ❉✐✣❡ ❡ ❍❡❧❧♠❛♥ ❡♠ ✶✾✼✻ ❬❉❍✼✻❪✳ ❊♠ ✉♠ ❝r✐♣t♦ss✐st❡♠❛ ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛ ❝❛❞❛ ❡♥t✐❞❛❞❡ A t❡♠ ✉♠❛ ❝❤❛✈❡ ♣ú❜❧✐❝❛ pk ✭❞✐str✐❜✉í❞❛ ❛ t♦❞♦s✮ ❡ ✉♠❛ ❝❤❛✈❡ s❡❝r❡t❛ ❝♦rr❡s♣♦♥❞❡♥t❡ sk ✭só A ❝♦♥❤❡❝❡ ❡st❡ ✈❛❧♦r✮✳ ❊♠
❝r✐♣t♦ss✐st❡♠❛s s❡❣✉r♦s✱ ❝❛❧❝✉❧❛r sk ❞❛❞♦ pk é ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡ ✐♥✈✐á✈❡❧✳ ❆ ❝❤❛✈❡ ♣ú❜❧✐❝❛ pk
❞❡✜♥❡ ✉♠ ♣r♦❝❡ss♦ ❞❡ ❡♥❝r✐♣t❛çã♦ Epk✱ ❡♥q✉❛♥t♦ ❛ ❝❤❛✈❡ s❡❝r❡t❛ ❡stá ❛ss♦❝✐❛❞❛ ❛ ✉♠ ♣r♦❝❡ss♦ ❞❡ ❞❡❝r✐♣t❛çã♦ Dsk✳ ❙❡ q✉❛❧q✉❡r ❡♥t✐❞❛❞❡ B ❞❡s❡❥❛ ❡♥✈✐❛r ✉♠❛ ♠❡♥s❛❣❡♠ M ♣❛r❛ A✱ B ❞❡✈❡ ♦❜t❡r ✉♠❛ ❝ó♣✐❛ ❞❛ ❝❤❛✈❡ ♣ú❜❧✐❝❛ pk❞❡ A ❡ ❛♣❧✐❝❛r ♦ ♣r♦❝❡ss♦ ❞❡ ❡♥❝r✐♣t❛çã♦ ♣❛r❛ ♦❜t❡r ♦ ❝r✐♣t♦❣r❛♠❛ C ❂Epk(M)✱ ❡ ❡♥✈✐á✲❧♦ ♣❛r❛ A✳ P❛r❛ ❞❡❝r✐♣t❛rC✱A ❛♣❧✐❝❛ ♦ ♣r♦❝❡ss♦ ❞❡ ❞❡❝r✐♣t❛çã♦ ✉s❛♥❞♦ s✉❛ ❝❤❛✈❡ s❡❝r❡t❛ sk ♣❛r❛ ♦❜t❡r ❛ ♠❡♥s❛❣❡♠ ♦r✐❣✐♥❛❧M ❂ Dsk(C)✳
❈♦♠♦ ❝❛❞❛ ❡♥t✐❞❛❞❡ A t❡♠ ✉♠❛ ❝❤❛✈❡ ♣ú❜❧✐❝❛ pkA ❡ ❡st❛ é ❞✐str✐❜✉í❞❛ ❞❡ ❢♦r♠❛ ♣ú❜❧✐❝❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❞✐str✐❜✉✐çã♦ ❞❡ ❝❤❛✈❡s ♥ã♦ ❡①✐st❡ ♥♦s ❝r✐♣t♦ss✐st❡♠❛s ❛ss✐♠étr✐❝♦s✳ ❆❧é♠ ❞❡ r❡s♦❧✈❡r
✶❈r✐♣t♦ss✐st❡♠❛s ❡❧❡♠❡♥t❛r❡s sã♦ ❝❤❛♠❛❞♦s ❞❡ ❝✐❢r❛s
✹ ❈❖◆❈❊■❚❖❙ ✷✳✶
❡ss❡ ♣r♦❜❧❡♠❛✱ ❛ ❝r✐♣t♦❣r❛✜❛ ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛ ❞á s♦❧✉çã♦ ❛ ♦✉tr♦s ♣r♦❜❧❡♠❛s ❝♦♠♦✿
✶✳ ❆✉t❡♥t✐❝❛çã♦ ❞❡ ❉❡st✐♥♦✳✲ ❊s❝♦♥❞❡r ✐♥❢♦r♠❛çõ❡s s✐❣✐❧♦s❛s ❞❛s ♣❡ss♦❛s q✉❡ ❝♦♥tr♦❧❛♠ ❛s ❧✐♥❤❛s ❞❡ ❝♦♠✉♥✐❝❛çã♦ ❡ ♦s ❝♦♠♣✉t❛❞♦r❡s ✐♥t❡r♠❡❞✐ár✐♦s ✭♣r♦✈❡❞♦r❡s✱ r♦t❡❛❞♦r❡s✮ ❣❛r❛♥t✐♥❞♦ q✉❡ ♦ ✈❡r❞❛❞❡✐r♦ ❞❡st✐♥❛tár✐♦ s❡❥❛ ❛ ú♥✐❝❛ ♣❡ss♦❛ q✉❡ ❝♦♥s✐❣❛ ❧❡r ❛ ✐♥❢♦r♠❛çã♦ ❡♥✈✐❛❞❛✳ ✷✳ ❆✉t❡♥t✐❝❛çã♦ ❞❡ ❖r✐❣❡♠✳✲ ❊✈✐t❛r q✉❡ ✉♠ t❡r❝❡✐r♦ ❢❛❧s✐✜q✉❡ ❛ ✐❞❡♥t✐❞❛❞❡ ❞♦ ❡♠✐ss♦r ❡♥✲
✈✐❛♥❞♦ ✐♥❢♦r♠❛çã♦ ♣❛r❛ ♦ ❞❡st✐♥❛tár✐♦✱ ❡♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♦ ❞❡st✐♥❛tár✐♦ ❞❡✈❡ t❡r ❝❡rt❡③❛ ❞❡ q✉❡ ❢♦✐ ♦ ✈❡r❞❛❞❡✐r♦ ❡♠✐ss♦r q✉❡ ❡♥✈✐♦ ❛ ✐♥❢♦r♠❛çã♦✳
✸✳ ■♥t❡❣r✐❞❛❞❡ ❞❛ ■♥❢♦r♠❛çã♦✳✲ ❊✈✐t❛r q✉❡ ✉♠ t❡r❝❡✐r♦ ❧❡✐❛ ❡ ❛❧t❡r❡ ❛ ✐♥❢♦r♠❛çã♦ s❡♠ s❡r ❞❡t❡❝t❛❞♦✱ ❡♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♦ ❞❡st✐♥❛tár✐♦ s❛❜❡ s❡ ❛ ♠❡♥s❛❣❡♠ ❢♦✐ ❛❧t❡r❛❞❛ ♥❛ ❧✐♥❤❛ ❞❡ ❝♦♠✉♥✐❝❛çã♦✳
✷✳✶✳✷ ❈r✐♣t♦ss✐st❡♠❛ ❘❙❆
❖ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❢♦✐ ✐♥✈❡♥t❛❞♦ ♣♦r ❘✳❘✐✈❡st✱ ❆✳ ❙❤❛♠✐r ❡ ▲✳❆❞❧❡♠❛♥ ❡♠ ✶✾✽✼ ❬❘❙❆✼✽❪ ❡ é ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ✉♥s ❞♦s ❝r✐♣t♦ss✐st❡♠❛s ♠❛✐s s❡❣✉r♦s ❥á q✉❡ ♥❡♥❤✉♠ ❛t❛q✉❡ ❝r✐♣t♦✲❛♥❛❧ít✐❝♦ ❝♦♥s❡❣✉✐✉ q✉❡❜rá✲❧♦ ❛té ❤♦❥❡ ✭✷✵✶✸✮✳
❖ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ três ❛❧❣♦r✐t♠♦s ♦s q✉❛✐s sã♦ ❡①♣❧✐❝❛❞♦s ❡♠❜❛✐①♦✳ ❉❡♥♦✲ t❛r❡♠♦s ♦ ❡♠✐ss♦r ♦✉ r❡♠❡t❡♥t❡ ❞❛ ♠❡♥s❛❣❡♠ ❝♦♠♦Beto❡ ♦ r❡❝❡♣t♦r ♦✉ ❞❡st✐♥❛tár✐♦ ❝♦♠♦ Alice✳
❊①✐st❡ t❛♠❜é♠ ✉♠ t❡r❝❡✐r♦ ❛t♦r ♠❛❧✲✐♥t❡♥❝✐♦♥❛❞♦ ❝❤❛♠❛❞♦ Carlos q✉❡ ❞❡s❡❥❛ ❧❡r ❛ ✐♥❢♦r♠❛çã♦
❡♥✈✐❛❞❛ ♥❛ ♠❡♥s❛❣❡♠✳
✶✳ ●❡r❛çã♦ ❞❡ ❈❤❛✈❡s✳✲ ❖ ❛❧❣♦r✐t♠♦ ❞❡ ❣❡r❛çã♦ ❞❡ ❝❤❛✈❡s t❡♠ ✉♠ ♣❛râ♠❡tr♦ ❞❡ ❡♥tr❛❞❛
n✱ ♦ q✉❛❧ ✐♥❞✐❝❛ ♦ ♥ú♠❡r♦ ❞❡ ❜✐ts ✭♣❛râ♠❡tr♦ ❞❡ s❡❣✉r❛♥ç❛✮ ❞❡ N✳ ❈❛❞❛ ✉s✉ár✐♦ Alice❞❡✈❡
♦❜t❡r ❛❧❡❛t♦r✐❛♠❡♥t❡u♥ú♠❡r♦s ♣r✐♠♦sr1✱r2✱✳✳✳✱ru−1 ❡ru ❞❡ t❛♠❛♥❤♦⌊nu⌋❜✐ts ❝❛❧❝✉❧❛❞♦s ♣♦r ❛❧❣✉♠ ❛❧❣♦r✐t♠♦ ❞❡t❡r♠✐♥íst✐❝♦ ♦✉ ♣r♦❜❛❜✐❧íst✐❝♦ ♣❛r❛ ❝❛❧❝✉❧❛r ♦ ✐♥t❡✐r♦N =Qu
i=1ri✳ ❆ s❡❣✉✐r✱ ❞❡t❡r♠✐♥❛♠♦s ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦e❝♦✲❝♦♣r✐♠♦ ❛φ(N) =Qu
i=1(ri−1)✳ ❊ss❡ ✈❛❧♦reé ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ❡①♣♦❡♥t❡ ❞❡ ❡♥❝r✐♣t❛çã♦ ✭❝♦♠✉♠❡♥t❡ ✜①❛❞♦ ❝♦♠ ♦ ✈❛❧♦r ❞❡e= 216+ 1 = 65537✮✳ ❆ ❝❤❛✈❡ ♣ú❜❧✐❝❛ ❞❡Aliceé ❞❛❞❛ ♣♦rhN, ei❡ ❛ ❝❤❛✈❡ s❡❝r❡t❛ é ❞❛❞❛ ♣♦rhN, di♦♥❞❡ ❛♠❜❛s ❝❤❛✈❡s
s❛t✐s❢❛③❡♠ ❛ ❝♦♥❣r✉ê♥❝✐❛ ed ≡1 mod φ(N)✳ ❖ ✈❛❧♦r d ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞♦ ❝♦♠ ♦ ❛❧❣♦r✐t♠♦
❞❡ ❊✉❝❧✐❞❡s✲❊st❡♥❞✐❞♦ ✭✈❡❥❛ ♦ ❆♣ê♥❞✐❝❡ ❆✳✶✳✸✮✳
✷✳ ❊♥❝r✐♣t❛çã♦✳✲ P❛r❛Beto❡♥✈✐❛r ✉♠❛ ♠❡♥s❛❣❡♠X ♣❛r❛Alice✱ ❡❧❡ ❞❡✈❡ ❢♦r♠❛t❛rX s❡❣✉♥❞♦
♦ ♣❛❞rã♦ P❑❈❙ ★✶✷ ❬❏❑✵✸❪ ❡ ♦❜t❡r ✉♠ ✐♥t❡✐r♦M ♣❡rt❡♥❝❡♥t❡ ❛♦ ❣r✉♣♦ZN✳ ❊♠ s❡❣✉✐❞❛Beto ♦❜té♠ ❛ ❝❤❛✈❡ ♣ú❜❧✐❝❛ ❞❡Alice pkA=hN, ei ❡ ❝❛❧❝✉❧❛ ♦ ❝r✐♣t♦❣r❛♠❛ C ❢❛③❡♥❞♦
C←Me mod N ✭✷✳✶✮
♣❛r❛ ❞❡♣♦✐sC s❡r ❡♥✈✐❛❞❛ ♣❛r❛Alice✳
✸✳ ❉❡❝r✐♣t❛çã♦✳✲ P❛r❛Alice♣♦❞❡r ❧❡r ❛ ♠❡♥s❛❣❡♠ ❡♥❝r✐♣t❛❞❛C❡♥✈✐❛❞❛ ♣♦rBeto✱Alice✉t✐❧✐③❛
❛ s✉❛ ❝❤❛✈❡ s❡❝r❡t❛skA=hN, di♣❛r❛ ❝❛❧❝✉❧❛r
M ←Cd modN. ✭✷✳✷✮
❆ s❡❣✉✐r✱ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦ ♣r♦❝❡ss♦ ✐♥✈❡rs♦ ❞❛ ❢♦r♠❛t❛çã♦ ♣❛r❛ ♦❜t❡r ♦ ✈❛❧♦r ❞❛ ♠❡♥s❛❣❡♠ ♦r✐❣✐♥❛❧ X ❛ ♣❛rt✐r ❞❡M✳
◗✉❛♥❞♦ t❡♠♦s u = 2 ❡♥tã♦ ♦ ❝r✐♣t♦ss✐st❡♠❛ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❘❙❆ ❜ás✐❝♦✳ P❛r❛ ✈❛❧♦r❡s u ≥3 é
❝❤❛♠❛❞♦ ❞❡ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ♠✉❧t✐✲♣r✐♠♦ ♦✉u✲♣r✐♠♦s✳
✷✳✶ ❈❘■P❚❖●❘❆❋■❆ ❊ ❈❘■P❚❖●❘❆❋■❆ ❉❊ ❈❍❆❱❊ PÚ❇▲■❈❆ ✺
❊①❡♠♣❧♦ ◆✉♠ér✐❝♦ ❞❡ ✉♠ ❈r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❇ás✐❝♦ ●❡r❛çã♦ ❞❡ ❈❤❛✈❡s✿
r1= 11✱r2 = 5
N =Q2
i=1ri = 55
φ(N) =Q2
i=1(ri−1) = 40
e= 3
d=e−1 mod φ(N) ❂ ✷✼ ❈❤❛✈❡ ♣ú❜❧✐❝❛ hN, ei ❂h55,3i
❈❤❛✈❡ s❡❝r❡t❛ hN, di ❂h55,27i
❊♥❝r✐♣t❛çã♦✿ P❛r❛ ✉♠❛ ♠❡♥s❛❣❡♠M ♦♥❞❡0≤M ≤N−1✱ ♥❡st❡ ❝❛s♦M = 4 t❡♠♦s C=Me mod N ❆❧❣♦r✐t♠♦ ✭✷✳✶✮ ❞❡ ❡♥❝r✐♣t❛çã♦ C= 43 mod 55
C= 9
❉❡❝r✐♣t❛çã♦✿
M =Cd mod N ❆❧❣♦r✐t♠♦ ✭✷✳✷✮ ❞❡ ❞❡❝r✐♣t❛çã♦ M = 927 mod 55
M = 4
❊①❡♠♣❧♦ ◆✉♠ér✐❝♦ ❞❡ ✉♠ ❈r✐♣t♦ss✐st❡♠❛ ❘❙❆ ✸✲♣r✐♠♦s ●❡r❛çã♦ ❞❡ ❈❤❛✈❡s✿
r1= 3✱r2 = 5✱r3= 7
N =Q3
i=1ri = 105
φ(N) =Q3
i=1(ri−1) = 48
e= 5
d=e−1 mod φ(N) = 29
❈❤❛✈❡ ♣ú❜❧✐❝❛ hN, ei ❂h105,5i
❈❤❛✈❡ s❡❝r❡t❛ hN, di ❂h105,29i
❊♥❝r✐♣t❛çã♦✿ P❛r❛ ✉♠❛ ♠❡♥s❛❣❡♠M ♦♥❞❡0≤M ≤N−1✱ ♥❡st❡ ❝❛s♦M = 4 t❡♠♦s C=Me mod N ❆❧❣♦r✐t♠♦ ✭✷✳✶✮ ❞❡ ❡♥❝r✐♣t❛çã♦ C= 45 mod 105
C= 79
❉❡❝r✐♣t❛çã♦✿
M =Cd mod N ❆❧❣♦r✐t♠♦ ✭✷✳✷✮ ❞❡ ❞❡❝r✐♣t❛çã♦ M = 7929 mod 105
M = 4
✷✳✶✳✸ ▼ét♦❞♦ ❞❡ ◗✉✐sq✉❛t❡r✲❈♦✉✈r❡✉r
❊♠ ✶✾✽✷ ❢♦✐ ♣r♦♣♦st❛ ✉♠❛ ♥♦✈❛ té❝♥✐❝❛ ❞❡ ❞❡❝r✐♣t❛çã♦ ♣❛r❛ ♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❜ás✐❝♦ ♣♦r ❏✲❏✳ ◗✉✐sq✉❛t❡r ❡ ❈✳ ❈♦✉✈r❡✉r ❬◗❈✽✷❪✳ ❊st❡ ♠ét♦❞♦ ❝♦♥s❡❣✉❡ ♦❜t❡r M ❞❡ C ❛ ♣❛rt✐r ❞❡ M1 =Cd
✻ ❈❖◆❈❊■❚❖❙ ✷✳✶
❛❣♦r❛ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ❛ ♠❡♥s❛❣❡♠M ❝♦♠ ❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦
M = ((M1−M2)r2−1 mod r1)r2+M2 ✭✷✳✸✮ ♦♥❞❡ d1 = d (modr1 −1)✱ d2 = d (modr2 −1)✱ M1 = Cd1 (modr1)✱ M2 = Cd2 (modr2) ❡
r2−1 é r−21 (modr1)✳ ❈♦♠♦ ♣♦❞❡♠♦s ♦❜s❡r✈❛r✱ ♦ ✈❛❧♦r d ❞❛ ❝❤❛✈❡ s❡❝r❡t❛ ❘❙❆ ♥ã♦ é ✉s❛❞❛ ♠❛✐s✱ ♣♦rt❛♥t♦✱ ❛ ❝❤❛✈❡ s❡❝r❡t❛ ♠✉❞❛ ❛ sk =hr1, r2, d1, d2, r2−1i✳ ❖ t❡♠♣♦ ❞❡ ❡①❡❝✉çã♦ ❞❛ ❋ór♠✉❧❛ ✭✷✳✸✮ é ❛té ✹ ✈❡③❡s ♠❛✐s rá♣✐❞♦ ❞♦ q✉❡ ❛ ❛♣❧✐❝❛çã♦ ❞❡M ←Cd mod N✱ ❛ ♣❛rt✐r ❞✐ss♦ ❡st❡ ♠ét♦❞♦ ✜❝♦✉
♠✉✐t♦ ♣♦♣✉❧❛r✳ ❊ss❡ ♠ét♦❞♦ ❛♣r❡s❡♥t❛❞♦ ♣♦r ❏✲❏✳ ◗✉✐sq✉❛t❡r ❡ ❈✳ ❈♦✉✈r❡✉r ❢♦✐ ❛❞❛♣t❛❞♦ ♣❛r❛ ♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❜ás✐❝♦ ❡ ♠✉❧t✐✲♣r✐♠♦ ❞❡✈✐❞♦ ❛♦ s❡✉ ♠❡♥♦r t❡♠♣♦ ♥♦ ♣r♦❝❡ss♦ ❞❡ ❞❡❝r✐♣t❛çã♦ ❝♦♠ r❡s♣❡✐t♦ ❛♦ ❆❧❣♦r✐t♠♦ ✭✷✳✷✮✳
P❛r❛ ♦ ✉s♦ ❞♦ ♠ét♦❞♦ ❞❡ ◗✉✐sq✉❛t❡r✲❈♦✉✈r❡✉r ♥♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❜ás✐❝♦ ❡ ♠✉❧t✐✲♣r✐♠♦ ❛ ❝❤❛✈❡ s❡❝r❡t❛ ❡stá ❞❡✜♥✐❞❛ ♣♦r
sk=hr1, r2, d1, d2, r2−1,hr3, d3, t3i, ..,hru, du, tuii
♦♥❞❡ ❛s ❝✐♥❝♦ ♣r✐♠❡✐r❛s ✈❛r✐á✈❡✐s s❡❣✉❡♠ t❡♥❞♦ ❛ ♠❡s♠❛ ❞❡✜♥✐çã♦ ❡ ❛s ✈❛r✐á✈❡✐s ❞❛s tr✐♣❧❛s ❡stã♦ ❞❡✜♥✐❞❛s ♣♦r
di≡d (mod ri−1)
tiRi≡1 (modri)
♦♥❞❡ Ri = Qij−=11 rj✱ ♣❛r❛ 3≤ i≤u✳ ❖ ♥♦✈♦ ♣r♦❝❡ss♦ ♣❛r❛ ❞❡❝r✐♣t❛r ♦ ❝r✐♣t♦❣r❛♠❛ C ❜❛s❡❛❞♦ ♥♦ ♠ét♦❞♦ ◗✉✐sq✉❛t❡r✲❈♦✉✈r❡✉r ❡stá ❞❡s❝r✐t♦ ♥♦ ❆❧❣♦r✐t♠♦✶ ❬❏❑✵✸❪✳
❆❧❣♦r✐t♠♦ ✶✿ ❉❡❝r✐♣t❛çã♦✲◗❈
❊♥tr❛❞❛✿ sk=hr1, r2, d1, d2, r−21,hr3, d3, t3i, ..,hru, du, tuii✱C ❙❛í❞❛✿M
M1 =Cd1 (modr1)❀ ✶
M2 =Cd2 (modr2)❀ ✷
♣❛r❛i= 3 ❛té u❢❛ç❛ ✸
Mi =Cdi mod ri;
✹
M = ((M1−M2)r2−1 modr1)r2+M2❀ ✺
R =r1❀ ✻
♣❛r❛i= 3 ❛té u❢❛ç❛ ✼
R =R∗ri−1❀ ✽
M = (Mi−M)ti mod ri)R+M❀
✾
✷✳✶ ❈❘■P❚❖●❘❆❋■❆ ❊ ❈❘■P❚❖●❘❆❋■❆ ❉❊ ❈❍❆❱❊ PÚ❇▲■❈❆ ✼
❊①❡♠♣❧♦ ◆✉♠ér✐❝♦ ❞♦ ▼ét♦❞♦ ◗✉✐sq✉❛t❡r✲❈♦✉✈r❡✉r ♣❛r❛ ♦ ❈r✐♣t♦s✐st❡♠❛ ❘❙❆ ❇ás✐❝♦ ●❡r❛çã♦ ❞❡ ❈❤❛✈❡s✿
r1= 11✱r2 = 5
N =Q2
i=1ri = 55
φ(N) =Q2
i=1(ri−1) = 40
e= 3
d=e−1 mod φ(N) = 27
d1 =d mod (r1−1) = 27 mod 10 = 7
d2 =d mod (r2−1) = 27 mod 4 = 3
r−21 =r2−1 mod r1 = 5−1 mod 11 = 9 ❈❤❛✈❡ ♣ú❜❧✐❝❛ hN, ei=h55,3i
❈❤❛✈❡ s❡❝r❡t❛ hr1, r2, d1, d2, r2−1i=h11,5,7,3,9i
❊♥❝r✐♣t❛çã♦✿ P❛r❛ ✉♠❛ ♠❡♥s❛❣❡♠M ♦♥❞❡0≤M ≤N−1✱ ♥❡st❡ ❝❛s♦M = 4 t❡♠♦s C=Me mod N ❆❧❣♦r✐t♠♦ ✭✷✳✶✮ ❞❡ ❡♥❝r✐♣t❛çã♦ C= 43 mod 55
C= 9
❉❡❝r✐♣t❛çã♦ ✭✉s❛♥❞♦ ♦ ❆❧❣♦r✐t♠♦ ✶✮✿
M1=Cd1 mod r1
M1= 97 mod 11
M1= 4
M2=Cd2 mod r2
M2= 273 mod 5
M2= 4
M = ((M1−M2)r2−1 mod r1)r2+M2
✽ ❈❖◆❈❊■❚❖❙ ✷✳✶
❊①❡♠♣❧♦ ◆✉♠ér✐❝♦ ❞♦ ▼ét♦❞♦ ◗✉✐sq✉❛t❡r✲❈♦✉✈r❡✉r ♣❛r❛ ♦ ❈r✐♣t♦s✐st❡♠❛ ❘❙❆ ✸✲ ♣r✐♠♦s
●❡r❛çã♦ ❞❡ ❈❤❛✈❡s✿
r1= 3✱r2 = 5✱r3= 7
N =Q3
i=1ri = 105
φ(N) =Q2
i=1(ri−1) = 48
e= 5
d=e−1 mod φ(N) = 29
d1 =d mod (r1−1) = 29 mod 2 = 1
d2 =d mod (r2−1) = 29 mod 4 = 1
d3 =d mod (r3−1) = 29 mod 6 = 5
r−21 =r2−1 mod r1 = 5−1 mod 3 = 2
R3=r1r2= 3∗5 = 15
t3 = (R3)−1 mod r3= 15−1 mod 7 = 1 ❈❤❛✈❡ ♣ú❜❧✐❝❛ hN, ei=h105,5i
❈❤❛✈❡ s❡❝r❡t❛ hp, q, dp, dq, q−1,(r3, d3, t3)i=h3,5,1,1,2,h7,5,1ii
❊♥❝r✐♣t❛çã♦✿ P❛r❛ ✉♠❛ ♠❡♥s❛❣❡♠M ♦♥❞❡ 0≤M ≤N−1✱ ♥❡st❡ ❝❛s♦M = 4 t❡♠♦s C=Me mod N ❆❧❣♦r✐t♠♦ ✭✷✳✶✮ ❞❡ ❡♥❝r✐♣t❛çã♦ C= 45 mod 105
C= 79
❉❡❝r✐♣t❛çã♦ ✭✉s❛♥❞♦ ♦ ❆❧❣♦r✐t♠♦ ✶✮✿
M1 =Cd1 mod r1
M1 = 791 mod 3
M1 = 1
M2 =Cd2 mod r2
M2 = 791 mod 5
M2 = 4
M3 =Cd3 mod r3
M3 = 795 mod 7
M3 = 4
M = ((M1−M2)r2−1 modr1)r2+M2
M = ((1−4)2 mod 3)∗5 + 4 M = 4
R= 3
♣❛r❛u= 3
R=r2∗R= 3∗5 = 15
M = ((M3−M)t3−1 mod r3)R+M
✷✳✶ ❈❘■P❚❖●❘❆❋■❆ ❊ ❈❘■P❚❖●❘❆❋■❆ ❉❊ ❈❍❆❱❊ PÚ❇▲■❈❆ ✾
✷✳✶✳✹ ❈♦rr❡çã♦ ❞♦ ❘❙❆
❆ s❡❣✉✐r ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ✈❛♠♦s ♣r♦✈❛r q✉❡(Me)d=M
mod N✱ t❡♥❞♦ ed= 1 modφ(N)✱ ♦ q✉❛❧ ♣♦❞❡ s❡r ❡①♣r❡ss❛❞♦ ❝♦♠♦ ed= 1 +kφ(N) ♦♥❞❡ ké ✉♠
✐♥t❡✐r♦ ♣♦s✐t✐✈♦✳ ❆ ♣r♦✈❛ ♣❛r❛ M ∈Z∗
N é ❞❡s❝r✐t❛ ❛ s❡❣✉✐r✿
(Me)d=M1+kφ(N) modN
❱❛♠♦s ❧❡♠❜r❛r ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ed= 1 mod φ(N) é ✐❣✉❛❧ ❛ed= 1 +kφ(N)✳ ❈♦♥t✐✲
♥✉❛♥❞♦
(Me)d= (Mφ(N))k∗M mod N
=M mod N ♣❡❧♦ ❈♦r♦❧ár✐♦✶
Pr♦✈❛♠♦s q✉❡ ♦ ❘❙❆ ❢✉♥❝✐♦♥❛ ♣❛r❛ t♦❞♦M ∈Z∗
N✱ ♠❛s ❛❣♦r❛ ❞❡✈❡♠♦s ♣r♦✈❛r q✉❡ ♦ ❘❙❆ ❢✉♥❝✐♦♥❛ ♣❛r❛ t♦❞♦ M ∈ZN✳ P❛r❛ ✐ss♦✱ ❞❡✈❡♠♦s ♣r♦✈❛r q✉❡ (Me)d =M mod ri ♣❛r❛ t♦❞♦ 1≤ i≤ u ✭❥á q✉❡N =Qu
i=1ri ♦♥❞❡ r1, r2, ..., ru sã♦ ♣r✐♠♦s ❞✐st✐♥t♦s✮✱ ❡ ❛ s❡❣✉✐r r❡s♦❧✈❡r ♦ s✐st❡♠❛ ❝♦♠ ♦ ❚❈❘✳ ❋❛r❡♠♦s ❛ ♣r♦✈❛ ❛♣❡♥❛s ♣❛r❛r1 ❥á q✉❡ ♦ ❝❛❧❝✉❧♦ é ❛♥á❧♦❣♦ ❛ ri ♣❛r❛2≤i≤u✳
❚❡♠♦s q✉❡ed= 1 modφ(N)❡ ♣♦rt❛♥t♦ed= 1 +k(r1−1)Qui=2(ri−1)✭✈❡❥❛ ❛ ❉❡✜♥✐çã♦ ✶✵✮✱ ♣♦rt❛♥t♦ t❡♠♦s q✉❡✿
(Me)d=M1+k(r1−1)Qui=2(ri−1) modr 1
=M(M(r1−1))kQui=2(ri−1) mod r
1
❈♦♥s✐❞❡r❛♥❞♦ q✉❡ ♦ mdc(M, r1) = 1✱ t❡♠♦s ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r ✭✈❡❥❛ ♦ ❚❡♦r❡♠❛ ✷✮ q✉❡
Mφ(r1)≡1 mod r
1 ✭❡ ❝♦♠♦ r1 é ♣r✐♠♦ t❡♠♦s q✉❡φ(r1) =r1−1 ❛ss✐♠ ♣♦❞❡♠♦s ♠♦str❛r q✉❡✿
(Me)d = M(1)kQui=2(ri−1) mod r
1
= M mod r1
❉❛ ♠❡s♠❛ ❢♦r♠❛ ♣♦❞❡♠♦s ♣r♦✈❛r q✉❡ (Me)d = M mod ri ♣❛r❛ 2 ≤ i ≤ u✳ ❊♠ ♦✉tr❛ ♣❛❧❛✈r❛s✱ s❛❜❡♠♦s q✉❡ (Me)d =M mod r
i ♣❛r❛1 ≤i≤u ❡ ♣❡❧❛ ❡q✉❛çã♦ ❚❈❘ ✭✈❡❥❛ ❆✳✻✮ ❝♦♥❝❧✉í♠♦s q✉❡
(Me)d=M mod N ♣❛r❛ t♦❞♦M ∈Z N✳ ✷✳✶✳✺ ❙❡❣✉r❛♥ç❛ ❞♦ ❘❙❆
❖ ❘❙❆ ❝♦♠♦ s✐st❡♠❛ ❝r✐♣t♦❣rá✜❝♦ ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛ ❞❡✈❡ ❝✉♠♣r✐r q✉❡ ❛ r❡❝✉♣❡r❛çã♦ ❞❛ ❝❤❛✈❡ s❡❝r❡t❛ ❛ ♣❛rt✐r ❞❛ ❝❤❛✈❡ ♣ú❜❧✐❝❛ t❡♠ q✉❡ s❡r ✉♠ ♣r♦❜❧❡♠❛ ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡ ✐♥✈✐á✈❡❧✱ ♦✉ s❡❥❛✱ ♦ ❝á❧❝✉❧♦ ❞❡ d❞❡✈❡ s❡r ❞✐❢í❝✐❧ ❛♣❡♥❛s t❡♥❞♦ ♦s ✈❛❧♦r❡s ❞❡e❡ N✳ ❆ ú♥✐❝❛ ♠❛♥❡✐r❛ ♣❛r❛ ❝❛❧❝✉❧❛r dé
❛♣❧✐❝❛♥❞♦ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s✲❊st❡♥❞✐❞♦ ❛♦s ✈❛❧♦r❡se❡φ(N)✱ ♠❛s ♣❛r❛ ♦❜t❡r ♦ ✈❛❧♦r ❞❡φ(N)
♣r❡❝✐s❛♠♦s s❛❜❡r ♦s ✈❛❧♦r❡s ❞♦s ♣r✐♠♦s ri ✭♣❛r❛ 1 ≤i≤u✮ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ❢❛t♦r❛r N✳ P♦rt❛♥t♦✱ ❛ s❡❣✉r❛♥ç❛ ❞♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ ❡stá ❜❛s❡❛❞❛ ♥♦ ♣r♦❜❧❡♠❛ ❞❛ ❢❛t♦r❛çã♦ ❞❡ ✐♥t❡✐r♦s ♦ q✉❛❧ é ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡ ✐♥✈✐á✈❡❧ ♣❛r❛ ♣r✐♠♦s ❣r❛♥❞❡s✱ ♠❛s ♣♦❞❡♠♦s ♣r♦♣♦r ♦s s❡❣✉✐♥t❡s ❛t❛q✉❡s✿
✶✳ ❈á❧❝✉❧♦ ❞❡φ(N) s❡♠ ❢❛t♦r❛r N
✷✳ ❉❡t❡r♠✐♥❛çã♦ ❞❡ds❡♠ ❢❛t♦r❛r N ❡ s❡♠ ❝❛❧❝✉❧❛r φ(N)
✸✳ ❈á❧❝✉❧♦ ❞❡ ✉♠d′ ❡q✉✐✈❛❧❡♥t❡ ❛d
❈♦♠♦ é ♠♦str❛❞♦ ❡♠ ❬❚❡r✵✵❪✱ r❡❛❧✐③❛r ❡ss❡s ❛t❛q✉❡s é ♣❡❧♦ ♠❡♥♦s tã♦ ❝❛r♦ ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡ q✉❛♥t♦ ♦ ♠❡❧❤♦r ❛❧❣♦r✐t♠♦ ♣❛r❛ ❢❛t♦r❛r ✐♥t❡✐r♦s ❝♦♥❤❡❝✐❞♦ ❛té ❛ ❞❛t❛ ❡♠ q✉❡ ❡st❡ tr❛❜❛❧❤♦ ❢♦✐ ❡s❝r✐t♦ ✭✷✵✶✸✮✱ ♣♦rt❛♥t♦ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ♦ ❝r✐♣t♦ss✐st❡♠❛ ❘❙❆ é s❡❣✉r♦✳
❙❛❜❡♠♦s q✉❡ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❢❛t♦r❛çã♦ ❞❡ ✐♥t❡✐r♦s ♠❛✐s rá♣✐❞♦ ❡ ❣❡♥ér✐❝♦ é ♦ ◆❋❙ ❬▲❡♥✵✶❪✱ ♦♥❞❡ s❡✉ t❡♠♣♦ ❡s♣❡r❛❞♦ ❤❡✉ríst✐❝♦ ♣❛r❛ ❝❛❧❝✉❧❛r ✉♠ ❢❛t♦r ♥ã♦✲tr✐✈✐❛❧ ❞♦ ✐♥t❡✐r♦ N ❡stá ❞❛❞♦ ♣♦r
O(e1.923 log 1 3nlog
2 3logn
