Open Criptografia RSA e a Teoria dos Números

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛

Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚

❈r✐♣t♦❣r❛✜❛ ❘❙❆ ❡ ❛ ❚❡♦r✐❛ ❞♦s

◆ú♠❡r♦s

♣♦r

❘♦❜❡r✈❛❧ ❞❛ ❈♦st❛ ▲✐♠❛

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦

❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆ ✲ ❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❆❣♦st♦✴✷✵✶✸ ❏♦ã♦ P❡ss♦❛ ✲ P❇

▲✼✸✷❝ ▲✐♠❛✱ ❘♦❜❡r✈❛❧ ❞❛ ❈♦st❛✳

❈r✐♣t♦❣r❛✜❛ ❘❙❆ ❡ ❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ✴ ❘♦❜❡r✈❛❧ ❞❛ ❈♦st❛ ▲✐♠❛✳✕ ❏♦ã♦ P❡ss♦❛✱ ✷✵✶✸✳

✼✻❢✳

❖r✐❡♥t❛❞♦r✿ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦ ❉✐ss❡rt❛çã♦ ✭▼❡str❛❞♦✮ ✲ ❯❋P❇✴❈❈❊◆

✶✳ ▼❛t❡♠át✐❝❛✳ ✷✳ ❈r✐♣t♦❣r❛✜❛ ❘❙❆✳ ✸✳ ❚❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✳ ✹✳ ❈♦♥❣r✉ê♥❝✐❛s✳ ✺✳ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✳

❆❣r❛❞❡❝✐♠❡♥t♦s

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢✳ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈✳ ❘✐❜❡✐r♦ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ♣❡❧❛ ❝♦♠✲ ♣r❡❡♥sã♦ ❡ ♣❡❧❛ ♦r✐❡♥t❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❆♦ Pr♦❢✳ ❆♥tô♥✐♦ ❞❡ ❆♥❞r❛❞❡ ❡ ❙✐❧✈❛ ♣❡❧♦ ✐♥❝❡♥t✐✈♦✱ ✐♥s✐st❡♥t❡ ❡ s✐st❡♠át✐❝♦✳ ■♥❢❡❧✐③♠❡♥t❡✱ ♥ã♦ ❢✉✐ s❡✉ ❛❧✉♥♦✱ ♠❛s✱ t❡♥❤♦ ♣♦r ❡❧❡ ✉♠❛ ❣r❛♥❞❡ ❛❞♠✐r❛çã♦✳ ●r❛♥❞❡ Pr♦❢✳ ❆♥❞r❛❞❡✳

❆♦s ♠❡✉s ♣r♦❢❡ss♦r❡s ❞❡ ❞✐✈❡rs❛s é♣♦❝❛s ♣♦r ♠♦str❛r❡♠ ❝♦♠ ♠❛❡str✐❛ ❛ ❜❡❧❡③❛ ❞❛ ♠❛t❡♠át✐❝❛✱ ❝✐t♦ ❛q✉✐ ♦ ♥♦♠❡ ❞❡ ❛❧❣✉♥s✿ ❏♦ã♦ ▼♦♥t❡♥❡❣r♦✱ ▲❡♥✐♠❛r✱ ▼❛r✐✈❛❧❞♦✱ ❆❜✲ ❞♦r❛❧ ✭✐♥ ♠❡♠♦r✐❛♠✮✱ ▼❛rs✐❝❛♥♦ ✭✐♥ ♠❡♠♦r✐❛♠✮✱ ❈❤✐❛♥❝❛ ✭✐♥ ♠❡♠♦r✐❛♠✮✱ ❆♥t♦♥✐♦ ❈❛r❧♦s ✭❯❋P❊✲✶✾✾✸✮✱ ❏♦r❣❡ ❍♦✉♥✐❡ ✭❯❋P❊✲✶✾✾✸✮ ❡ ❘♦❜❡rt♦ ❇❡❞r❡❣❛❧ ✭❯❋P❊✲✶✾✾✸✮✳

❆ ♠✐♥❤❛ ❡s♣♦s❛ ❏✉❝✐❧❡♥❡ ❡ ♠✐♥❤❛s ✜❧❤❛s ❏✉❝✐❛♥❡ ❡ ❏✉❝✐❡❧❡♥ ♣❡❧❛ ❝♦♠♣r❡❡♥sã♦ ❡ ❛♣♦✐♦ ♥❛ r❡❛❧✐③❛çã♦ ❞❡ss❡ ♠❡str❛❞♦✳

❉❡❞✐❝❛tór✐❛

❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝r✐♣t♦❣r❛✜❛✱ ❞✐❢❡r❡♥❝✐❛♠♦s ❛ ❝r✐♣t♦✲ ❣r❛✜❛ s✐♠étr✐❝❛ ❞❛ ❝r✐♣t♦❣r❛✜❛ ❛ss✐♠étr✐❝❛ ❡ ♠♦str❛♠♦s ❝♦♠♦ ❢✉♥❝✐♦♥❛ ❛ ❝r✐♣t♦❣r❛✜❛ ❘❙❆✳ ❆❧é♠ ❞✐ss♦✱ ❞❡st❛❝❛♠♦s ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ♠❛t❡♠át✐❝♦s q✉❡ ❥✉st✐✜❝❛♠ ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞❡ss❡ ❝r✐♣t♦ss✐st❡♠❛ ❡ s✉❛ s❡❣✉r❛♥ç❛✱ t❛✐s ❝♦♠♦✿ ❝♦♥❣r✉ê♥❝✐❛s✱ ❚❡♦✲ r❡♠❛ ❞❡ ❊✉❧❡r✱ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✱ ❚❡♦r❡♠❛ ❞❡ ❲✐❧s♦♥✱ ❈r✐tér✐♦ ❞❡ ❊✉❧❡r ♣❛r❛ r❡sí❞✉♦s q✉❛❞rát✐❝♦s✱ ▲❡✐ ❞❡ ❘❡❝✐♣r♦❝✐❞❛❞❡ ◗✉❛❞rát✐❝❛ ❡ t❡st❡s ❞❡ ♣r✐♠❛❧✐❞❛❞❡✳

P❛❧❛✈r❛s ❝❤❛✈❡s✿ ❈r✐♣t♦❣r❛✜❛ ❘❙❆✱ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✱ ❝♦♥❣r✉ê♥❝✐❛s✱ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✳

❆❜str❛❝t

■♥ t❤✐s ✇♦r❦ ✇❡ ♣r❡s❡♥t t❤❡ ❝♦♥❝❡♣t ♦❢ ❝r②♣t♦❣r❛♣❤②✱ ❤✐❣❤❧✐❣❤t✐♥❣ t❤❡ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ s②♠♠❡tr✐❝ ❡♥❝r②♣t✐♦♥ ❛♥❞ ❛s②♠♠❡tr✐❝ ❡♥❝r②♣t✐♦♥✳ ❲❡ ❛❧s♦ s❤♦✇ ❤♦✇ ❘❙❆ ❡♥❝r②♣t✐♦♥ ✇♦r❦s✳ ▼♦r❡♦✈❡r✱ ✇❡ st✉❞② t❤❡ ♠❛✐♥ ♠❛t❤❡♠❛t✐❝❛❧ r❡s✉❧ts t❤❛t ❥✉st✐❢② t❤❡ ♦♣❡r❛t✐♦♥ ♦❢ t❤✐s ❝r②♣t♦s②st❡♠ ❛♥❞ ✐ts s❡❝✉r✐t②✱ s✉❝❤ ❛s✿ ❝♦♥❣r✉❡♥❝❡s✱ ❊✉❧❡r✬s t❤❡♦r❡♠✱ ❋❡r♠❛t✬s ▲✐tt❧❡ ❚❤❡♦r❡♠✱ ❲✐❧s♦♥✬s ❚❤❡♦r❡♠✱ ❊✉❧❡r✬s ❝r✐t❡r✐♦♥ ❢♦r q✉❛❞r❛t✐❝ r❡s✐❞✉❡s✱ ▲❛✇ ♦❢ ◗✉❛❞r❛t✐❝ ❘❡❝✐♣r♦❝✐t② ❛♥❞ ♣r✐♠❛❧✐t② t❡sts✳

❑❡②✇♦r❞s✿ ❘❙❆ ❊♥❝r②♣t✐♦♥✱ ♥✉♠❜❡r t❤❡♦r②✱ ❝♦♥❣r✉❡♥❝❡✱ ❋❡r♠❛t✬s ▲✐tt❧❡ ❚❤❡♦✲ r❡♠✳

❙✉♠ár✐♦

✶ ❈♦♥❣r✉ê♥❝✐❛s ✶

✶✳✶ ❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❖s ❚❡♦r❡♠❛s ❞❡ ❊✉❧❡r ❡ ❲✐❧s♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸ ❘❡sí❞✉♦s ◗✉❛❞rát✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✷ ◆ú♠❡r♦s Pr✐♠♦s ✷✼

✷✳✶ ❈♦♠♦ ❊♥❝♦♥tr❛r ◆ú♠❡r♦s Pr✐♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✷ ❚❡st❡ ❞❡ Pr✐♠❛❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✸ ❈r✐♣t♦❣r❛✜❛ ❘❙❆ ✸✾

✸✳✶ ❈r✐♣t♦ss✐st❡♠❛s ❙✐♠étr✐❝♦s ❡ ❆ss✐♠étr✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✷ ●❡r❛çã♦ ❞❛s ❈❤❛✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✸✳✸ ❈♦❞✐✜❝❛çã♦ ❡ ❉❡❝♦❞✐✜❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✹ ❙❡❣✉r❛♥ç❛ ❞♦ ❘❙❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✺ ❆ss✐♥❛t✉r❛ ❉✐❣✐t❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✻ ❆♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

✹ ❈r✐♣t♦❣r❛✜❛ ❘❙❆ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ✺✹

✹✳✶ ▼❛t❡♠át✐❝❛ ❇ás✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✳✶✳✶ ❉✐✈✐sã♦ ❊✉❝❧✐❞✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✳✶✳✷ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

✹✳✶✳✸ ❈♦♥❣r✉ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✹✳✷ ❈r✐♣t♦❣r❛✜❛ ❘❙❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✻✹

■♥tr♦❞✉çã♦

❊st❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❡①♣❧♦r❛r ❛ ♠❛t❡♠át✐❝❛ ♥❡❝❡ssár✐❛ ♣❛r❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞♦ s✐st❡♠❛ ❝r✐♣t♦❣rá✜❝♦ ❘❙❆✳ ❆❧é♠ ❞✐ss♦✱ ❞❡✈❡ s❡r✈✐r ❝♦♠♦ ❢♦♥t❡ ✐♥s♣✐r❛❞♦r❛ ♣❛r❛ ❡st✐♠✉❧❛r ♣r♦❢❡ss♦r❡s ❡ ❛❧✉♥♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❛ ✈❡r q✉❡ ❛ ♠❛t❡♠át✐❝❛✱ ♠❡s♠♦ tã♦ ❛❜str❛t❛✱ ❝♦♠♦ é ♦ ❝❛s♦ ❞♦s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦s ❝❛♣ít✉❧♦ ✶ ❡ ✷✱ ♣♦ss✉✐ ✉♠❛ ❛♣❧✐❝❛çã♦ ❡①tr❡♠❛♠❡♥t❡ s✐♠♣❧❡s ❡ ✐♥✉s✐t❛❞❛ ❝♦♠♦ ❛ ❝r✐♣t♦❣r❛✜❛ ❘❙❆✳

❆ ❝r✐♣t♦❣r❛✜❛ é ♦ ❡st✉❞♦ ❞❡ ♠ét♦❞♦s q✉❡ ♣❡r♠✐t❛♠ ❡s❝r❡✈❡r ♠❡♥s❛❣❡♥s ❡♠ ❝✐❢r❛s ♦✉ ❝ó❞✐❣♦s✱ ❞❡ ♠♦❞♦ q✉❡ ❛♣❡♥❛s ♦s ❧❡❣ít✐♠♦s ❞❡st✐♥❛tár✐♦s s❡❥❛♠ ❝❛♣❛③❡s ❞❡ ❞❡❝♦✲ ❞✐✜❝❛r ❡ ❧❡r ❛s ♠❡♥s❛❣❡♥s✳ ❍♦❥❡ ❡♠ ❞✐❛ ♠❛✐s ❡ ♠❛✐s ♣❡ss♦❛s ❡ ❡♠♣r❡s❛s ✉t✐❧✐③❛♠ ❛ ✐♥t❡r♥❡t ♣❛r❛ s❡ ❝♦♠✉♥✐❝❛r✱ t♦r♥❛♥❞♦ ♦ ✉s♦ ❞❛ ❝r✐♣t♦❣r❛✜❛ ♣❛r❛ ♠❛♥t❡r ♦ s✐❣✐❧♦ ❞❡ss❛ ❝♦♠✉♥✐❝❛çã♦ ❝❛❞❛ ✈❡③ ♠❛✐s ✐♠♣♦rt❛♥t❡✳ P♦r ❡①❡♠♣❧♦✱ ❜❛♥❝♦s ❝♦♠❡r❝✐❛✐s ♥❡❝❡ss✐t❛♠ ❣❛r❛♥t✐r q✉❡ ❛s tr❛♥s❛çõ❡s ❡♥tr❡ s❡✉s ❝❧✐❡♥t❡s ❡ ♦ ❜❛♥❝♦ t❡♥❤❛ ❛ ♠á①✐♠❛ s❡❣✉r❛♥ç❛ ♣♦ssí✈❡❧✳ ◆♦ ❝❛s♦ ❞♦s ❜❛♥❝♦s ❡ss❛ s❡❣✉r❛♥ç❛ é ❛✐♥❞❛ ♠❛✐s ✐♠♣♦rt❛♥t❡✱ ♣♦✐s✱ q✉❛s❡ t♦❞❛s ❛s s✉❛s tr❛♥s❛çõ❡s ♣❛ss❛♠ ♣❡❧❛ ✐♥t❡r♥❡t ❡ ❡♥✈♦❧✈❡♠ ❣r❛♥❞❡s ✈❛❧♦r❡s ♠♦♥❡tá✲ r✐♦s✱ ♠❡s♠♦ ❛q✉❡❧❛s r❡❛❧✐③❛❞❛s ♥❛s ❛❣ê♥❝✐❛s ❜❛♥❝ár✐❛s✳ ◆♦ss♦ tr❛❜❛❧❤♦ s❡rá ❜❛s❡❛❞♦ ♥❛ ❝r✐♣t♦❣r❛✜❛ ❘❙❆✳ ❊ss❡ s✐st❡♠❛ ❝r✐♣t♦❣rá✜❝♦ ❢✉♥❝✐♦♥❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ sã♦ ❝r✐❛❞❛s ❞✉❛s ❝❤❛✈❡s✱ ✉♠❛ ❝❤❛✈❡ ❞❡ ❝♦❞✐✜❝❛çã♦ q✉❡ s❡rá ♣ú❜❧✐❝❛ ❡ ✉♠❛ ❝❤❛✈❡ ❞❡ ❞❡❝♦❞✐✜❝❛çã♦ q✉❡ s❡rá ♣r✐✈❛❞❛✳ ❆ss✐♠✱ s❡ ✉♠ ✉s✉ár✐♦ ❆ ❞❡s❡❥❛ ❡♥✈✐❛r ✉♠❛ ♠❡♥s❛✲ ❣❡♠ ♣❛r❛ ✉♠ ✉s✉ár✐♦ ❇✱ ❡♥tã♦ ❆ ✉s❛ ❛ ❝❤❛✈❡ ❞❡ ❝♦❞✐✜❝❛çã♦ ❞❡ ❇ ♣❛r❛ ❝♦❞✐✜❝❛r ❛ ♠❡♥s❛❣❡♠ ❡ ❡♥✈✐❛ ❡ss❛ ♠❡♥s❛❣❡♠ ❝♦❞✐✜❝❛❞❛ ♣❛r❛ ❇✱ q✉❛♥❞♦ ❇ r❡❝❡❜❡ ❛ ♠❡♥s❛❣❡♠

❝♦❞✐✜❝❛❞❛ ✉s❛ s✉❛ ❝❤❛✈❡ ❞❡ ❞❡❝♦❞✐✜❝❛çã♦✱ q✉❡ ❛♣❡♥❛s ❡❧❡ ❝♦♥❤❡❝❡✱ ❡ ❞❡❝♦❞✐✜❝❛ ❛ ♠❡♥s❛❣❡♠ ❝♦❞✐✜❝❛❞❛✱ ♦❜t❡♥❞♦ ❛ss✐♠ ❛ ♠❡♥s❛❣❡♠ ♦r✐❣✐♥❛❧✳

❆♦ ❧♦♥❣♦ ❞♦s três ♣r✐♠❡✐r♦s ❝❛♣ít✉❧♦s ♣r♦❝✉r❛♠♦s ♠♦str❛r ❞❡ ❢♦r♠❛ s✐st❡♠át✐❝❛ ❡ ❢♦r♠❛❧ ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ♠❛t❡♠át✐❝♦s q✉❡ t♦r♥❛♠ ♣♦ssí✈❡❧ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞♦ s✐st❡♠❛ ❝r✐♣t♦❣rá✜❝♦ ❘❙❆✳ ❊✱ ♥♦ ú❧t✐♠♦ ❝❛♣ít✉❧♦ t❡♥t❛♠♦s ❡①♣❧♦r❛r ❡ss❡ ❝r✐♣t♦s✲ s✐st❡♠❛ ❝♦♠ ✉♠ ❡♥❢♦q✉❡ ♠❡♥♦s ❢♦r♠❛❧ ❞❡ ♠♦❞♦ q✉❡ s❡❥❛ ♣♦ssí✈❡❧ ❡①♣♦r ❡ss❡ ❛ss✉♥t♦ ♣❛r❛ ❛❧✉♥♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳

P❛r❛ ❞❡s❡♥✈♦❧✈❡r ♦s ❝♦♥t❡ú❞♦s ❛q✉✐ ❡st✉❞❛❞♦s ❛ss✉♠✐♠♦s ❝♦♠♦ ❝♦♥❤❡❝✐❞♦s ❛❧❣✉♥s tó♣✐❝♦s ❡❧❡♠❡♥t❛r❡s ❞❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✱ ❡♥tr❡ ♦✉tr♦s✱ ❝✐t❛♠♦s ♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦✱ ♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛ ❡ ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠✳ ◆♦ ❡♥t❛♥t♦✱ ♥♦ ú❧t✐♠♦ ❝❛♣ít✉❧♦ ❛❜♦r❞❛♠♦s ✐♥❢♦r♠❛❧♠❡♥t❡ ♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛ ❡ ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠✳

❆ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❛ ❝r✐♣t♦❣r❛✜❛ ❘❙❆ ❡stá t♦t❛❧♠❡♥t❡ ❜❛s❡❛❞❛ ♥❛ ❛r✐t♠ét✐❝❛ ❞❛s ❝♦♥❣r✉ê♥❝✐❛s✳ ❉❛í✱ ❝♦♠❡ç❛♠♦s ♥♦ss♦ tr❛❜❛❧❤♦ ❞❡s❡♥✈♦❧✈❡♥❞♦ ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛✲ ❞♦s s♦❜r❡ ❛s ❝♦♥❣r✉ê♥❝✐❛s✳ ◆❛ s❡çã♦ ✶ ❞♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❞❡s❝r❡✈❡♠♦s ♦s ❛s♣❡❝t♦s ❛r✐t♠ét✐❝♦s ❞❛s ❝♦♥❣r✉ê♥❝✐❛s✱ ♠♦str❛♥❞♦ ❝♦♠♦ sã♦ ❢❡✐t❛s ❛s ♠❛♥✐♣✉❧❛çõ❡s ❝♦♠ ❝♦♥✲ ❣r✉ê♥❝✐❛s ❡ ✐♥tr♦❞✉③✐♥❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✱ ♠✉✐t♦ út✐❧ ❡♠ ❛❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s✱ ❝♦♠♦ ❢♦✐ ♦ ❝❛s♦ ❞❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r ❡ ♦ t❡♦r❡♠❛ ❞❡ ❲✐❧s♦♥✳ ◆❛ s❡çã♦ ✷✱ ✐♥tr♦❞✉③✐♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❡q✉❛çã♦ ♠♦❞✉❧❛r ❡ ❛ ❢✉♥çã♦ ✜ ❞❡ ❊✉❧❡r✳ ◆❡st❛ s❡çã♦ ❛♣❡s❛r ❞♦ tít✉❧♦ ❞á ❛ ❡♥t❡♥❞❡r q✉❡ ♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ s❡r✐❛ ♦ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r ❡ ♦ ❚❡♦r❡♠❛ ❞❡ ❲✐❧s♦♥✱ ❛q✉✐ ❞❡♠♦♥str❛♠♦s ♦ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t q✉❡ s❡rá ✉s❛❞♦ ♣❛r❛ ❥✉st✐✜❝❛r ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞♦ s✐st❡♠❛ ❘❙❆✱ ❛♣r❡✲ s❡♥t❛♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ♦r❞❡♠ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ Zn✱ ❛❧❣✉♥s r❡s✉❧t❛❞♦s

s♦❜r❡ ❡ss❛ ♦r❞❡♠ ❡ ❛ ❞❡✜♥✐çã♦ ❞❡ r❛✐③ ♣r✐♠✐t✐✈❛✳ ◆❛ s❡çã♦ ✸✱ ❡st✉❞❛♠♦s ♦s r❡sí❞✉♦s q✉❛❞rát✐❝♦s✱ ✐♥tr♦❞✉③✐♠♦s ♦ ❙í♠❜♦❧♦ ❞❡ ▲❡❣❡♥❞r❡✱ ❞❡♠♦♥str❛♠♦s ♦ ❈r✐tér✐♦ ❞❡ ❊✉✲ ❧❡r ❡ ❛ ▲❡✐ ❞❡ ❘❡❝✐♣r♦❝✐❞❛❞❡ ◗✉❛❞rát✐❝❛✱ ❡ss❡s r❡s✉❧t❛❞♦s ♥ã♦ sã♦ ♥❡❝❡ssár✐♦s ♣❛r❛ ❛

✐♠♣❧❡♠❡♥t❛çã♦ ❞♦ s✐st❡♠❛ ❘❙❆✱ ♠❛s✱ sã♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ ❞❡♠♦♥str❛r ✉♠ ❞♦s t❡st❡s ❞❡ ♣r✐♠❛❧✐❞❛❞❡ ❡st✉❞❛❞♦s ♥♦ ❝❛♣ít✉❧♦ ✷✳

◆♦ ❝❛♣ít✉❧♦ ✷ ❡st✉❞❛♠♦s✱ ♥❛ s❡çã♦ ✶✱ ❝♦♠♦ ❡♥❝♦♥tr❛r ♥ú♠❡r♦s ♣r✐♠♦s ❝♦♠ ♠❛✐s ❞❡ ✶✵✵ ❛❧❣❛r✐s♠♦s✳ ❆q✉✐ ❛♣r❡s❡♥t❛♠♦s ❛ ❢✉♥çã♦π(x)✱ ♦ t❡♦r❡♠❛ ❞♦s ♥ú♠❡r♦s ♣r✐♠♦s

❡ ✜③❡♠♦s ✉♠❛ ❡st✐♠❛t✐✈❛ ❞♦ ♥ú♠❡r♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s ❝♦♠ ✶✵✵ ❛❧❣❛r✐s♠♦s q✉❡ ❞❡✈❡♠♦s t❡st❛r ♣❛r❛ ❡♥❝♦♥tr❛r ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ ❝♦♠ ✶✵✵ ❛❧❣❛r✐s♠♦s✳ ◆❛ s❡çã♦ ✷ ❞❡♠♦♥str❛♠♦s ❛❧❣✉♥s t❡st❡s ❞❡ ♣r✐♠❛❧✐❞❛❞❡ ❡ ❛♣r❡s❡♥t❛♠♦s✱ ♠❛s✱ ♥ã♦ ❞❡♠♦♥str❛♠♦s ♦ t❡st❡ ❆❑❙▲✳ ◆❛ ✈❡r❞❛❞❡ é ♥❡ss❛ s❡çã♦ ❡ ♥❛ s❡çã♦ ✸ ❞♦ ❝❛♣ít✉❧♦ ✶ q✉❡ r❡s✐❞❡ ❛ ♠❛t❡♠át✐❝❛ ♠❛✐s s♦✜st✐❝❛❞❛✳ ❊ss❛ ♠❛t❡♠át✐❝❛ ❡stá ✐♥t✐♠❛♠❡♥t❡ ❧✐❣❛❞❛ à s❡❣✉r❛♥ç❛ ❞♦ s✐st❡♠❛ ❘❙❆✱ ♣♦✐s✱ ♣❛r❛ ❞✐✜❝✉❧t❛r ❛ ❢❛t♦r❛çã♦ ❞❡ n=pq♥❡❝❡ss✐t❛♠♦s t❡r ❝❡rt❡③❛

q✉❡ ♦s ♥ú♠❡r♦s✱ p ❡ q✱ ❡s❝♦❧❤✐❞♦s s❡❥❛♠ r❡❛❧♠❡♥t❡ ♣r✐♠♦s✳

◆♦ ❝❛♣ít✉❧♦ ✸ ❛♣r❡s❡♥t❛♠♦s✱ ♥❛ s❡çã♦ ✶✱ ✉♠❛ ✈✐sã♦ ❣❡r❛❧ s♦❜r❡ ♦ q✉❡ é ✉♠ s✐st❡♠❛ ❝r✐♣t♦❣rá✜❝♦ ❞✐❢❡r❡♥❝✐❛♥❞♦ ♦s ❝r✐♣t♦ss✐st❡♠❛s s✐♠étr✐❝♦s ❡ ❛ss✐♠étr✐❝♦s✳ ◆❛s s❡çõ❡s ✷ ❡ ✸ ❞❡s❝r❡✈❡♠♦s✱ ♣❛ss♦ ❛ ♣❛ss♦✱ ❝♦♠♦ ❢✉♥❝✐♦♥❛ ♦ s✐st❡♠❛ ❘❙❆ ❡ ♥❛s s❡çõ❡s ✹ ❡ ✺ ❛❜♦r❞❛♠♦s ❛❧❣✉♥s ❛s♣❡❝t♦s ❧✐❣❛❞♦s ❛ s❡❣✉r❛♥ç❛ ❞♦ s✐st❡♠❛ ❘❙❆ ❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❛ss✐♥❛t✉r❛ ❞✐❣✐t❛❧✳ ❊✱ ♣♦r ✜♠✱ ♥❛ s❡çã♦ ✻ ✜③❡♠♦s ❞♦✐s ❡①❡♠♣❧♦s ♣❛r❛ q✉❡ ♦ ❧❡✐t♦r ✈❡❥❛ ❝♦♠♦ ❢✉♥❝✐♦♥❛ ❡ss❡ s✐st❡♠❛ ❝r✐♣t♦❣rá✜❝♦✳

❖ ❝❛♣ít✉❧♦ ✹ ❛♣r❡s❡♥t❛ ❛ ❝r✐♣t♦❣r❛✜❛ ❘❙❆ ❝♦♠ ✉♠❛ ❛❜♦r❞❛❣❡♠ ✐♥❢♦r♠❛❧✳ ❇✉s✲ ❝❛♥❞♦ ❛tr❛✈és ❞❡ ❡①❡♠♣❧♦s ♠♦str❛r t♦❞❛ ❛ ♠❛t❡♠át✐❝❛ ♥❡❝❡ssár✐❛ à ✐♠♣❧❡♠❡♥t❛çã♦ ❞❡ss❡ s✐st❡♠❛✱ s❡♠ ❛ ♣r❡♦❝✉♣❛çã♦ ❞❡ ❥✉st✐✜❝❛r ♣♦rq✉❡ ❡ss❡ s✐st❡♠❛ ❢✉♥❝✐♦♥❛ ❡ s❡♠ ♥♦s ♣r❡♦❝✉♣❛r♠♦s ❝♦♠ ❛ s❡❣✉r❛♥ç❛✳ ❆ ❛❜♦r❞❛❣❡♠ ❞❛❞❛ ❛♦s ❝♦♥t❡ú❞♦s ❛♣r❡s❡♥t❛❞♦s ♥❡ss❡ ❝❛♣ít✉❧♦ t❡♠ ❝♦♠♦ ♣r♦♣ós✐t♦ ❢❛❝✐❧✐t❛r ❛ ❝♦♠♣r❡❡♥sã♦ ❞♦ s✐st❡♠❛ ❘❙❆ ♣❛r❛ ♣r♦❢❡ss♦r❡s ❡ ❛❧✉♥♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡✈❡ ❛s s❡çõ❡s ✶ ❡ ✷ ❞♦ ❝❛♣ít✉❧♦ ✶ ❜❛s❡❛❞❛s ❡♠ [✸, ✺] ❡ ❛

s❡çã♦ ✸ ❢♦✐ ✐♥✢✉❡♥❝✐❛❞❛ ♣♦r [✻]✳ ❏á ♦ ❝❛♣ít✉❧♦ ✷ ❢♦✐ ✐♥s♣✐r❛❞♦ ❡♠ [✶, ✹]✳ ❆s s❡çõ❡s ✶

❡ ✷ ❞♦ ❝❛♣ít✉❧♦ ✸ t❡✈❡ ❢♦rt❡ ✐♥✢✉ê♥❝✐❛ ❞❡ [✼, ✽] ❡ ❛s ❞❡♠❛✐s s❡çõ❡s ❢♦r❛♠ ❜❛s❡❛❞❛s

❡♠ [✷]✳ ❖ ❝❛♣ít✉❧♦ ✹ é ✉♠❛ ❛❞❛♣t❛çã♦ ❢❡✐t❛ ♣❡❧♦ ❛✉t♦r ❞❡ss❡ tr❛❜❛❧❤♦ ❞♦ ❝♦♥t❡ú❞♦

❛♣r❡s❡♥t❛❞♦ ♥♦ ❝❛♣ít✉❧♦ ✸ ♣❛r❛ q✉❡ s❡❥❛ ♣♦ssí✈❡❧ ❛♣❧✐❝á✲❧♦ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✳

❈❛♣ít✉❧♦ ✶

❈♦♥❣r✉ê♥❝✐❛s

❆♣r❡s❡♥t❛r❡♠♦s ♥❡st❡ ❝❛♣ít✉❧♦ ❛s ❝♦♥❣r✉ê♥❝✐❛s✳ ❈♦♠♦ ♦ s✐st❡♠❛ ❝r✐♣t♦❣rá✜❝♦ ❘❙❆ ✉s❛ ❡ss❡♥❝✐❛❧♠❡♥t❡ ❛ ❛r✐t♠ét✐❝❛ ❞♦s r❡st♦s ❡♥tã♦ ❞❛r❡♠♦s ❛ ❡ss❡ ❝❛♣ít✉❧♦ ✉♠❛ ❛t❡♥çã♦ ♠❛✐♦r✱ ♠♦str❛♥❞♦ ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s s♦❜r❡ ❛s ❝♦♥❣r✉ê♥❝✐❛s✳ ❆❧é♠ ❞✐ss♦✱ ❛s ❝♦♥❣r✉ê♥❝✐❛s ❡stã♦ r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ t♦❞♦s ♦s r❡s✉❧t❛❞♦s ✉s❛❞♦s✱ t❛♥t♦ ♣❛r❛ ❥✉st✐✜❝❛r ♣♦rq✉❡ ♦ s✐st❡♠❛ ❘❙❆ ❢✉♥❝✐♦♥❛ q✉❛♥t♦ ♣❛r❛ ❞❡♠♦♥str❛r ♦s t❡st❡s ❞❡ ♣r✐♠❛❧✐❞❛❞❡✳

✶✳✶ ❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s

❆q✉✐ ♠♦str❛r❡♠♦s q✉❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ♥♦s ❞❛rá ✉♠❛ ❛r✐t♠ét✐❝❛ ♣❛r❛ ♦s r❡st♦s ❞❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛ ♣♦r ✉♠ ❞❛❞♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✳ ❱❡r❡♠♦s ♥♦ ❝❛♣ít✉❧♦ ✸ q✉❡ ❡ss❛ ❛r✐t♠ét✐❝❛ é ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❛ ❝r✐♣t♦❣r❛✜❛ ❘❙❆✳

❉❡✜♥✐çã♦ ✶✳✶✳✶ ❉❛❞♦s ❛✱ ❜ ❡ ♥ ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ❝♦♠ n > 1✳ ❉✐③❡♠♦s q✉❡ ❛ ❡ ❜

sã♦ ❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ ♥ s❡ ♥❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛ ♣♦r ♥ ❞❡✐①❛♠ ♦ ♠❡s♠♦ r❡st♦✳ ❊ ♥❡ss❡ ❝❛s♦ ❡s❝r❡✈❡♠♦s

a_≡b modn✳

❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❈❛♣ít✉❧♦ ✶

◆❛ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦ t❡r❡♠♦s ✉♠ ♠ét♦❞♦ ♣❛r❛ ❡st❛❜❡❧❡❝❡r s❡ ✉♠ ❞❛❞♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ é ♦✉ ♥ã♦ ❝♦♥❣r✉❡♥t❡ ❛ ♦✉tr♦ ♥ú♠❡r♦ ✐♥t❡✐r♦✳ ❆❧❣✉♥s ❛✉t♦r❡s ✉s❛♠ ❡ss❛ ♣r♦♣♦s✐çã♦ ❝♦♠♦ ❞❡✜♥✐çã♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✶ ❙❡❥❛ n _∈Z✱ ❝♦♠ n >1✳ ❙❡❥❛♠ a, b_∈Z✳ ❊♥tã♦

a_≡b modn _⇔ n_|(a₋b)

❉❡♠♦♥str❛çã♦✿ ❙❡ a_≡b mod n✱ ❡♥tã♦ ❡①✐st❡♠ q1, q2, r ∈Zt❛✐s q✉❡a=q1n+r ❡ b = q₂n+r✱ ❝♦♠ 0 _≤ r < n✳ ❆ss✐♠✱ a₋b = q₁n₋q₂n = (q_{1 −}q₂)n✳ P♦rt❛♥t♦✱

t❡♠♦s q✉❡ n_|(a₋b)✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡n_|(a₋b)✳ ❊♠ s❡❣✉✐❞❛✱ ❞✐✈✐❞❛ a

❡ b ♣♦r n✳ ❆ss✐♠✱ ❡①✐st❡♠ c₁, c₂, r₁, r_{2 ∈}Z t❛✐s q✉❡ a =c₁n+r1✱ ❝♦♠ 0 _≤r₁ < n❡ b =c2n+r2✱ ❝♦♠ 0 ≤r2 < n✳ ❉❛í✱ r❡s✉❧t❛ q✉❡ a−b = (c1−c2)n+r1−r2✱ ❛❧é♠ ❞✐ss♦✱ t❡♠♦s q✉❡ n_|(a₋b)✳ P♦rt❛♥t♦✱ a₋b _≡ r₁₋r₂ modn ❡ a₋b _≡ 0 mod n✳

▼❛s✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ t❡♠♦s q✉❡✿ ❝♦♠♦ a₋b é ❝♦♥❣r✉❡♥t❡ ❛ r1 −r2✱ t❡♠♦s q✉❡ ❡ss❡s ♥ú♠❡r♦s ❞❡✐①❛♠ ♦ ♠❡s♠♦ r❡st♦ ♥❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛ ♣♦r n✱ ♣♦r

♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ a₋b é ❝♦♥❣r✉❡♥t❡ ❛ 0✱ t❡♠♦s q✉❡ ❡❧❡s ❞❡✐①❛♠ ♦ ♠❡s♠♦ r❡st♦ ♥❛

❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛ ♣♦r n✱ ❛ss✐♠✱ t❡♠♦s q✉❡ r_{1 −}r₂ ❡ 0 ❞❡✐①❛♠ ♦ ♠❡s♠♦ r❡st♦ ♥❛

❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛ ♣♦r n✳ P♦rt❛♥t♦✱ ♦❜t❡♠♦s r1−r2 ≡0 mod n✱ ♠❛s✱ s❛❜❡♠♦s q✉❡

0 _≤ r₁ < n ❡ 0 _≤ r₂ < n✱ ❛ss✐♠✱ ♦❜t❡♠♦s r₁ = r2✳ ▲♦❣♦✱ t❡♠♦s q✉❡ a _≡ b mod n

❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳

❆ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r ♥♦s ❞✐③ q✉❡ ❛ ❝♦♥❣r✉ê♥❝✐❛ é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ s♦❜ ♦ ❝♦♥❥✉♥t♦ Z✳ ❆❧é♠ ❞❡ s❡r ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♣♦❞❡r♠♦s ❞❡s❡♥✲

✈♦❧✈❡r ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❛s ❝♦♥❣r✉ê♥❝✐❛s ❡❧❛ é ❡ss❡♥❝✐❛❧ ♣❛r❛ ❡st❛❜❡❧❡❝❡r♠♦s ♦ ❝♦♥❥✉♥t♦ ❞❛s ❝❧❛ss❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❡ ❞❡s❡♥✈♦❧✈❡r ❛ ❛r✐t♠ét✐❝❛ ❞❛s ❝❧❛ss❡s ❞❡ ❡q✉✐✲ ✈❛❧ê♥❝✐❛✳ ❆❧é♠ ❞✐ss♦✱ ❡❧❛ ❡st❛❜❡❧❡❝❡ ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❛s ❝♦♥❣r✉ê♥❝✐❛s ❡ ❛s ❝❧❛ss❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✱ q✉❡ s❡rá ♠✉✐t♦ út✐❧ ❡♠ ♠✉✐t❛s ❞❡♠♦♥str❛çõ❡s ❝♦♠♦ ✈❡r❡♠♦s ❛❞✐❛♥t❡✳

❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❈❛♣ít✉❧♦ ✶

Pr♦♣♦s✐çã♦ ✶✳✶✳✷ ❙❡❥❛ n _∈Z✱ ❝♦♠ n >1✳ P❛r❛ t♦❞♦s a, b, c_∈Z✱ t❡♠✲s❡ q✉❡✿

✭✶✮ a_≡a mod n✱

✭✷✮ ❙❡ a _≡b mod n✱ ❡♥tã♦ b_≡a mod n✱

✭✸✮ ❙❡ a _≡b mod n ❡ b_≡c modn✱ ❡♥tã♦ a_≡c mod n✳

❉❡♠♦♥str❛çã♦✿

✭✶✮ ❈♦♠♦ n_|0✱ t❡♠♦s q✉❡ n_|(a₋a)✳ ❉❛í✱ ♦❜t❡♠♦s q✉❡ a_≡a mod n✳

✭✷✮ ❙❡ a _≡ b mod n✱ ❡♥tã♦ n_|(a₋b)✱ ✐st♦ é✱ ❡①✐st❡ ✉♠ q _∈ Z t❛❧ q✉❡ a₋b = qn✳

❉❛í✱ t❡♠♦s q✉❡ b₋a = (₋q)n✱ ❝♦♠ (₋q) _∈ Z✱ ❛ss✐♠✱ t❡♠♦s q✉❡ n_|(b₋a)✳ ▲♦❣♦✱

b _≡a modn✳

✭✸✮ ❙❡ a_≡b modn ❡ b_≡c modn✱ ❡♥tã♦ t❡r❡♠♦s q✉❡n_|(a₋b) ❡ n_|(b₋c)✱ ♦ q✉❡

✐♠♣❧✐❝❛ n_|[(a₋b) + (b₋c)] = (a₋c)✳ ▲♦❣♦✱ t❡♠♦s q✉❡ a_≡c mod n✳

❆ ♣r♦♣♦s✐çã♦ s❡❣✉✐♥t❡ ♠♦str❛ q✉❡ ❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❛❞❛ ♣❡❧❛ ❝♦♥❣r✉ê♥❝✐❛ é ❝♦♠♣❛tí✈❡❧ ❝♦♠ ❛s ♦♣❡r❛çõ❡s ❞❡ s♦♠❛ ❡ ♣r♦❞✉t♦ ❡♠ Z✳ ❊st❛ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ é

❢✉♥❞❛♠❡♥t❛❧ ♥❛ ♠❛♥✐♣✉❧❛çã♦ ❛r✐t♠ét✐❝❛ ❞♦s r❡st♦s✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✸ ❙❡❥❛♠ a, b, c, d, n _∈ Z✱ ❝♦♠ n > 1✳ ❙❡ a _≡ b modn ❡ c_≡d modn✱ ❡♥tã♦

a+c_≡b+d modn e ac_≡bd modn✳

❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ a _≡ b modn ❡ c _≡ d mod n✱ t❡♠♦s q✉❡ n_|(a ₋b) ❡

n_|(c₋d)✱ ♦ q✉❡ ✐♠♣❧✐❝❛ n_|(a₋ b) + (c₋ d) = (a +c)₋(b +d)✳ ❉❛í✱ ♦❜t❡♠♦s

a+c_≡b+d mod n✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ n_|(a₋b)❡n_|(c₋d)✱ t❡♠♦s q✉❡ ❡①✐st❡♠

q₁ ❡q₂ ♥ú♠❡r♦s ✐♥t❡✐r♦s t❛✐s q✉❡a₋b =q₁n ❡c₋d=q₂n ♦ q✉❡ ✐♠♣❧✐❝❛ a=b+q₁n

❡ c = d+q2n✳ ❉❛í✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦ ❛s ❞✉❛s ✐❣✉❛❧❞❛❞❡ ♦❜t❡♠♦s

ac=bd+ (bq₂+dq₁+q₁q₂)n✱ ✐st♦ é✱ ac₋bd= (bq₂+dq₁)n ♦ q✉❡ ✐♠♣❧✐❝❛n_|(ac₋bd)✳

P♦rt❛♥t♦✱ t❡♠♦s q✉❡ ac_≡bd mod n✳

❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❈❛♣ít✉❧♦ ✶

❆s ❞✉❛s ♣r♦♣♦s✐çõ❡s q✉❡ s❡ s❡❣✉❡♠ ♠♦str❛♠ q✉❡ ✈❛❧❡ ❛ ❧❡✐ ❞♦ ❝❛♥❝❡❧❛♠❡♥t♦ ♥❛ ❛❞✐çã♦ ❡ ❡♠ ❝❡rt♦s ❝❛s♦s ❡s♣❡❝í✜❝♦s ✈❛❧❡✱ t❛♠❜é♠✱ ❛ ❧❡✐ ❞♦ ❝❛♥❝❡❧❛♠❡♥t♦ ♥❛ ♠✉❧t✐♣❧✐❝❛çã♦✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✹ ❙❡❥❛♠ a, b, c, n_∈Z✱ ❝♦♠ n >1✳ ❊♥tã♦

a+c_≡b+c mod n _⇔ a_≡b modn✳

❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ a+c _≡ b+c mod n t❡♠♦s✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✶✱ q✉❡ n_|[(a +c)₋(b+c)] = (a₋b)✳ P♦rt❛♥t♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✶ t❡♠♦s q✉❡ a _≡ b

mod n✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ a _≡b mod n✱ ❡♥tã♦ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✶ n_|(a₋b) = [(a+c)₋(b+c)]✳ ▲♦❣♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✶ t❡♠♦s q✉❡ a+c_≡b+c modn✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✺ ❙❡❥❛♠ a, b, c, n_∈Z✱ ❝♦♠ n >1 ❡ mdc(c, n) = 1✳ ❊♥tã♦

ac_≡bc mod n _⇔ a_≡b mod n✳

❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ ac_≡bc mod n✱ t❡♠♦s q✉❡ n_|(ac₋bc) = (a₋b)c✳ ❈♦♠♦ mdc(c, n) = 1 ❡ n_|(a ₋ b)c✱ t❡r❡♠♦s q✉❡ n_|(a ₋b)✳ ❙❡♥❞♦ ❛ss✐♠✱ ♦❜t❡♠♦s a _≡ b modn✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ a _≡ b mod n✱ ❡♥tã♦ n_|(a ₋ b)✳ ❉❛í✱ t❡♠♦s q✉❡

n_|(a₋b)c= (ac₋bc)✳ ▲♦❣♦✱ t❡♠♦s q✉❡ ac_≡bc mod n✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✻ ❙❡❥❛♠ a, b, c, n_∈Z✱ ❝♦♠ n >1 ❡ mdc(c, n) = d✳ ❊♥tã♦ ac_≡bc mod n _⇒ a_≡b mod n

d✳

❉❡♠♦♥str❛çã♦✿ ❙❡ ac _≡ bc modn✱ ❡♥tã♦ ❡①✐st❡ k _∈ Z t❛❧ q✉❡ ac ₋ bc = (a₋b)c = kn✳ ❉❛í✱ t❡♠♦s q✉❡ (a₋b)c

d = k n

d✳ ▼❛s✱ mdc( c d,

n

d) = 1 ✭❛q✉✐ ✉s❛♠♦s

✉♠ r❡s✉❧t❛❞♦ s♦❜r❡ ♠❞❝ q✉❡ ♥ã♦ ❞❡♠♦♥str❛♠♦s✳ ❖ ❧❡✐t♦r ✐♥t❡r❡ss❛❞♦ ♣♦❞❡rá ✈❡r ✉♠❛ ❞❡♠♦♥str❛çã♦ ❡♠ [✸]✮✱ ♦ q✉❡ ✐♠♣❧✐❝❛ n

d ❞✐✈✐❞❡ a−b✱ ✐st♦ é✱ a≡b mod n d ❝♦♠♦

q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳

❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❈❛♣ít✉❧♦ ✶

❉❡✜♥✐çã♦ ✶✳✶✳✷ ❯♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ ♥ é ✉♠ ❝♦♥❥✉♥t♦ ❙ ❞❡ ♥ ♥ú♠❡r♦s q✉❡ ❞❡✐①❛♠ t♦❞♦s ♦s r❡st♦s ♣♦ssí✈❡✐s ❞❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛ ❞❡ss❡s ♥ú♠❡r♦s ♣♦r ♥✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✼ ❙❡❥❛ S = _{r1, r2, . . . , rn} ⊆ Z ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s

♠ó❞✉❧♦ ♥ ❡ s❡❥❛♠ a, b_∈Z✱ ❝♦♠ mdc(a, n) = 1✱ ❡♥tã♦

S′ _={ar

1+b, ar2+b, . . . , arn+b}

t❛♠❜é♠ é ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ ♥✳

❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ {r₁, r₂, . . . , rn}é ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ❡ a, b∈

Z✱ t❡♠♦s ♣❛r❛ ❝❛❞❛ i = 1,2, . . . , n✱ q✉❡ ari +b ≡ rj mod n✱ ♣❛r❛ ❛❧❣✉♠ j =

1,2, . . . , n✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ ♦s ri′s ❢♦r♠❛♠ ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s✱

s❡ i ₆= j✱ t❡♠♦s q✉❡ ri 6≡ rj mod n✳ ❈♦♠♦ mdc(a, n) = 1 t❡♠♦s ♣❡❧❛ Pr♦♣♦s✐çã♦

✶✳✶✳✺ q✉❡ ari 6≡ arj modn ❡✱ ❞❛í✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✹✱ t❡♠♦s q✉❡ ari +b 6≡

arj +b mod n✳ P♦rt❛♥t♦✱ ♣r♦✈❛♠♦s q✉❡ ♦s ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦S′ sã♦ ❞♦✐s ❛ ❞♦✐s

✐♥❝♦♥❣r✉❡♥t❡s ❡ ❝♦♠♦ S′ _{♣♦ss✉✐ n ❡❧❡♠❡♥t♦s q✉❡ sã♦ ❝♦♥❣r✉❡♥t❡s ❛♦s ❡❧❡♠❡♥t♦s ❞❡}

S✱ ❝♦♥❝❧✉í♠♦s q✉❡ _{ar1+b, ar2+b, . . . , arn+b}é ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s✳

❈♦♠♦ ❞❛❞♦ a _∈ Z✱ ♣❡❧♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛✱ t❡♠♦s q✉❡ ❡①✐st❡♠

q, r _∈ Z t❛✐s q✉❡ a = qn +r✱ ❝♦♠ 0 _≤ r < n✱ ❡♥tã♦ ❛ Pr♦♣♦s✐çã♦ ❛♥t❡r✐♦r ♥♦s

❣❛r❛♥t❡ q✉❡ q✉❛❧q✉❡r s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ n t❡♠ s❡✉s ❡❧❡♠❡♥t♦s

❝♦♥❣r✉❡♥t❡s ❛♦s ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ {0,1,2, . . . , n₋1_}✱ ❡♠ ❛❧❣✉♠❛ ♦r❞❡♠✳

❆ ♣ró①✐♠❛ ❞❡✜♥✐çã♦ s❡r✈✐rá ♣❛r❛ ❡st❛❜❡❧❡❝❡r♠♦s ✉♠❛ ♥♦✈❛ ❢♦r♠❛ ❞❡ ❧✐❞❛r♠♦s ❝♦♠ ❛s ❝♦♥❣r✉ê♥❝✐❛s✱ ❡♠ ❛❧❣✉♥s ❝❛s♦s s✐♠♣❧✐✜❝❛♥❞♦ ♦s ❝á❧❝✉❧♦s ❡♠ ❛❧❣✉♠❛s ❞❡✲ ♠♦♥str❛çõ❡s✳

❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❈❛♣ít✉❧♦ ✶

❉❡✜♥✐çã♦ ✶✳✶✳✸ ❉❛❞♦ a_≡ b mod n✱ t❡♠♦s q✉❡ _∃k_∈Z t❛❧ q✉❡ a₋b = kn✳ ❈❤❛✲

♠❛♠♦s ❞❡ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ❛ ❡♠ r❡❧❛çã♦ ❛ ❝♦♥❣r✉ê♥❝✐❛ ♠ó❞✉❧♦ ♥ ♦ ❝♦♥❥✉♥t♦

a=_{a+kn:k _∈Z_}✳

❆ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦ ♥♦s ♠♦str❛ ❝♦♠♦ r❡❧❛❝✐♦♥❛r ❛s ❝♦♥❣r✉ê♥❝✐❛s ❝♦♠ ❛s ❝❧❛ss❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳ ❊❧❛ ♥♦s ♣❡r♠✐t❡ ♣❛ss❛r ❞❡ ✉♠❛ ❢♦r♠❛ ❞❡ ♥♦t❛çã♦ ♣❛r❛ ♦✉tr❛✳ ■st♦ é✱ ♠♦str❛ ❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝❛ ❡♥tr❡ ❛s ❞✉❛s ❢♦r♠❛s ❞❡ tr❛t❛r ❝♦♠ ♦s r❡st♦s ❞❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛ ♣♦r ✉♠ ❞❛❞♦ ♥ú♠❡r♦✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✽ ❙❡❥❛♠ a, b, n _∈ Z✱ n > 1✳ ❊♥tã♦✱ a _≡ b mod n s❡✱ ❡ s♦♠❡♥t❡

s❡✱ a=b✳

❉❡♠♦♥str❛çã♦✿ ❙❡x_∈a✱ ❡♥tã♦ ❡①✐st❡k _∈Zt❛❧ q✉❡x=a+kn✱ ✐st♦ é✱x₋a=kn✱

♦ q✉❡ ✐♠♣❧✐❝❛ x _≡ a modn✳ ❈♦♠♦ a _≡ b mod n✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✷ ✐t❡♠ ✭✸✮✱

t❡♠♦s q✉❡ x_≡ b mod n✳ P♦rt❛♥t♦✱ ❡①✐st❡ k′

∈Z t❛❧ q✉❡ x₋b=k′_{n✳ ▲♦❣♦✱ x}

∈ b✳

P♦r ♦✉tr♦ ❧❛❞♦✱ s❡x_∈b✱ ❡♥tã♦ ❡①✐st❡ ✉♠c_∈Zt❛❧ q✉❡x=b+cn✱ ✐st♦ é✱ x₋b =cn✳

P♦rt❛♥t♦✱x_≡b modn✳ ❈♦♠♦a_≡b mod n✱ ♥♦✈❛♠❡♥t❡ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✷ ✐t❡♠

✭✷✮ ❡ ✭✸✮✱ t❡♠♦s q✉❡x_≡a mod n✳ ❉❛í✱ t❡♠♦s q✉❡ ❡①✐st❡c′ _{∈Zt❛❧ q✉❡x−a=c}′_n✱ ✐st♦ é✱ x = a+c′_{n✳ ▲♦❣♦✱ x}

∈ a✳ P♦rt❛♥t♦✱ t❡♠♦s q✉❡ a = b✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ a=b✱ ❡♥tã♦a+ 0n =a_∈b✱ ✐st♦ é✱ ❡①✐st❡k _∈Zt❛❧ q✉❡ a=b+kn✳ ❉❛í✱ t❡♠♦s q✉❡ a₋b =kn✱ ✐st♦ é✱a_≡b modn✳

❉❡✜♥✐çã♦ ✶✳✶✳✹ ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❝❧❛ss❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ é ❝❤❛♠❛❞♦ ❞❡ ❝♦♥✲ ❥✉♥t♦ q✉♦❝✐❡♥t❡ ❞❡ Z ♣❡❧❛ r❡❧❛çã♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ♠ó❞✉❧♦ ♥✳ ❊ s❡rá ❞❡♥♦t❛❞♦ ♣♦r Zn✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✾ ❙❡ n_∈Z❡ n >1✱ ❡♥tã♦Zn ={0,1,2, . . . , n−1}é ✉♠ ❝♦♥❥✉♥t♦

❝♦♥t❡♥❞♦ ❡①❛t❛♠❡♥t❡ ♥ ❝❧❛ss❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛s✳

❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❈❛♣ít✉❧♦ ✶

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ a, b _∈ Z t❛✐s q✉❡ 0 _≤ a < b < n✳ ❊♥tã♦ b ₋a ₆= 0 ❡

n ∤ (b₋a)✳ P♦rt❛♥t♦✱ t❡r❡♠♦s q✉❡ a _6≡ b modn✳ ▲♦❣♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ❛♥t❡r✐♦r

t❡♠♦s q✉❡ a ₆=b✳ ❆ss✐♠✱ _{0,1,2, . . . , n₋1_{} ⊆} Zn é ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠ ❡①❛t❛♠❡♥t❡

n ❡❧❡♠❡♥t♦s✳ P❛r❛ ♣r♦✈❛r ❛ ✐❣✉❛❧❞❛❞❡ é s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡✿ ❞❛❞♦ a _∈ Z ❡♥tã♦

a _{∈ {}0,1,2, . . . , n₋1_}✳ P❡❧♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ t❡♠♦s q✉❡✱ _∃q, r_∈Z t❛✐s q✉❡

a =qn+r✱ ❝♦♠ 0_≤r < n✳ ❉❛í✱ t❡r❡♠♦s q✉❡a _≡r mod n✱ ❝♦♠ 0_≤r < n✱ ✐st♦ é✱ a=r _{∈ {}0,1,2, . . . , n₋1_} ❝♦♠♦ q✉❡rí❛♠♦s ♠♦str❛r✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✶✵ ❙❡❥❛ n _∈Z✳ ❙❡ a=b ❡ c=d✱ ❡♥tã♦

✭✶✮ a+c=b+d

✭✷✮ ac=bd

❉❡♠♦♥str❛çã♦✿ ❙❡❣✉❡ ✐♠❡❞✐❛t♦ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✸ ❡ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✽✳

❉❡✜♥✐çã♦ ✶✳✶✳✺ ❙❡❥❛ ♥ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✱ n > 1✳ ❉❡✜♥✐♠♦s ❛ s♦♠❛ ❡ ♦ ♣r♦❞✉t♦

❞❡ ❞✉❛s ❝❧❛ss❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ♣♦r

✭✶✮ a+b=a+b

✭✷✮ a_·b =ab

❖❜s❡r✈❡ q✉❡ ❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✶✵ ♥♦s ❣❛r❛♥t❡ q✉❡ ❡st❛s ♦♣❡r❛çõ❡s ❡stã♦ ❞❡✜♥✐❞❛s✳

❉❡✜♥✐çã♦ ✶✳✶✳✻ ❉❛❞♦ a _∈ Zn✳ ❉✐③❡♠♦s q✉❡ b é ♦ ✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ❞❡ a

q✉❛♥❞♦ a_·b= 1✳

❊ss❛ ❞❡✜♥✐çã♦ é ♠✉✐t♦ ✐♠♣♦rt❛♥t❡✱ ♣♦✐s✱ ❝♦♠♦ ✈❡r❡♠♦s ❛❞✐❛♥t❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s q✉❡ ♣♦ss✉❡♠ ✐♥✈❡rs♦ ❡♠Zné ❢❡❝❤❛❞♦ ❝♦♠ r❡❧❛çã♦ ❛ ♠✉❧t✐♣❧✐❝❛çã♦✳

❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❈❛♣ít✉❧♦ ✶

Pr♦♣♦s✐çã♦ ✶✳✶✳✶✶ ❆ ❝❧❛ss❡ a t❡♠ ✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ❡♠ Zn s❡✱ ❡ s♦♠❡♥t❡ s❡✱

❛ ❡ ♥ sã♦ ❝♦♣r✐♠♦s✳

❉❡♠♦♥str❛çã♦✿ ❙❡ a t❡♠ ✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ❡♠ Zn✱ ❡♥tã♦ ❡①✐st❡ b ∈ Zn t❛❧

q✉❡ a_·b = 1✱ ✐st♦ é✱ ab = 1✳ P♦rt❛♥t♦✱ t❡♠♦s q✉❡ ab _≡ 1 mod n✳ ❉❛í✱ t❡♠♦s q✉❡ n_|(ab₋1)✱ ✐st♦ é✱ ❡①✐st❡ q _∈ Z t❛❧ q✉❡ ab₋1 = qn✳ ❆ss✐♠✱ ♦❜t❡♠♦s q✉❡ ❡①✐st❡♠ b, q _∈Zt❛✐s q✉❡ ba₋qn= 1✳ ▲♦❣♦✱ t❡♠♦s q✉❡a ❡n sã♦ ❝♦♣r✐♠♦s✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱

s❡ a ❡ n sã♦ ❝♦♣r✐♠♦s✱ ❡♥tã♦ ❡①✐st❡♠ x, y _∈ Z t❛✐s q✉❡ xa+yn= 1✳ ❆ss✐♠✱ t❡♠♦s

q✉❡ n_|(xa₋1)✱ ✐st♦ é✱xa_≡1 mod n✳ P♦rt❛♥t♦✱ t❡♠♦s q✉❡ xa=x_·a= 1✱ ♦✉ s❡❥❛✱

❡①✐st❡ ✉♠ x_∈Zn q✉❡ é ♦ ✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ❞❡ a✳

❉❡✜♥✐çã♦ ✶✳✶✳✼ ❖s ❡❧❡♠❡♥t♦s ✐♥✈❡rtí✈❡✐s ❞❡ Zn ❢♦r♠❛♠ ✉♠ ❝♦♥❥✉♥t♦ q✉❡ ❝❤❛♠❛✲

♠♦s ❞❡ ❝♦♥❥✉♥t♦ r❡❞✉③✐❞♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ ♥✳ ❊ s❡rá ❞❡♥♦t❛❞♦ ♣♦r Z∗

n✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✶✷ ❙❡ a, b_∈Z∗

n✱ ❡♥tã♦ ab∈Z∗n✳

❉❡♠♦♥str❛çã♦✿ ❈♦♠♦a, b_∈Z∗

nt❡♠♦s q✉❡∃x, y ∈Z∗nt❛✐s q✉❡a·x= 1❡b·y= 1✳

P♦rt❛♥t♦✱ t❡r❡♠♦s q✉❡ ab_·xy = abxy = axby = a_·x_·b_·y = 1_·1 = 1✳ P♦rt❛♥t♦

t❡♠♦s q✉❡ ab_∈Z∗

n✳

◆❛ Pr♦♣♦s✐çã♦ ❛♥t❡r✐♦r ♣r♦✈❛♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ Z∗

n é ❢❡❝❤❛❞♦ ❝♦♠ r❡❧❛çã♦ ❛♦

♣r♦❞✉t♦✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✶✸ ❙❡❥❛ b _∈Z∗

n✳ ❊♥tã♦✱ ∀a∈Z t❛❧ q✉❡ mdc(a, n) = 1 t❡♠✲s❡ q✉❡

ab_∈Z∗

n✳

❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ b _∈Z∗

n t❡♠♦s ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✶✶✱ q✉❡ mdc(b, n) = 1✱

❞❛í✱ t❡♠♦s q✉❡ ❡①✐st❡♠ x1, y1 ∈Z t❛✐s q✉❡ bx1+ny1 = 1✳ ▼❛s✱ mdc(a, n) = 1✱ ❞❛í✱ ❡①✐st❡♠ x₂, y_{2 ∈}Z t❛✐s q✉❡ax₂+ny₂ = 1✳ ❆ss✐♠✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦

❖s ❚❡♦r❡♠❛s ❞❡ ❊✉❧❡r ❡ ❲✐❧s♦♥ ❈❛♣ít✉❧♦ ✶

✐♠♣❧✐❝❛ mdc(ab, n) = 1✳ P♦rt❛♥t♦✱ ♥♦✈❛♠❡♥t❡ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✶✶✱ t❡♠♦s q✉❡

ab_∈Z∗

n ❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳

◆❡st❛ s❡çã♦ ❞❡s❝r❡✈❡♠♦s ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s s♦❜r❡ ❛s ❝♦♥❣r✉ê♥❝✐❛s ❡ ✜③❡♠♦s ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ❛s ❝♦♥❣r✉ê♥❝✐❛s ❡ ❛s ❝❧❛ss❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳ ❊st❡s r❡s✉❧t❛❞♦s s❡rã♦ ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ♣r♦✈❛r♠♦s ♦s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s ♥❛s ❞✉❛s ♣ró①✐♠❛s s❡çõ❡s✳

✶✳✷ ❖s ❚❡♦r❡♠❛s ❞❡ ❊✉❧❡r ❡ ❲✐❧s♦♥

◆❡st❛ s❡çã♦ ✐♥tr♦❞✉③✐♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❡q✉❛çã♦ ♠♦❞✉❧❛r ❡ ❛ ❢✉♥çã♦ ✜ ❞❡ ❊✉❧❡r✳ ❆❧é♠ ❞✐ss♦✱ ❞❡♠♦♥str❛♠♦s ♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r✱ ♦ t❡♦r❡♠❛ ❞❡ ❲✐❧s♦♥ ❡ ♦ ♣❡q✉❡♥♦ t❡♦r❡♠❛ ❞❡ ❋❡r♠❛t q✉❡ s❡rá ✉s❛❞♦ ♣❛r❛ ❥✉st✐✜❝❛r ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞♦ s✐st❡♠❛ ❘❙❆✳ ❚❛♠❜é♠ ❡st✉❞❛♠♦s ✉♠ ♣♦✉❝♦ s♦❜r❡ r❛✐③ ♣r✐♠✐t✐✈❛ ❡ s♦❜r❡ ❛ ♦r❞❡♠ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ Zn✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✶ ❙❡❥❛♠ a, n _∈ Z✱ ❝♦♠ n > 1✱ ❡♥tã♦ ❛ ❝♦♥❣r✉ê♥❝✐❛ aX _≡ 1 mod n ♣♦ss✉✐ ✉♠❛ s♦❧✉çã♦ x0 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ❡ ♥ sã♦ ❝♦♣r✐♠♦s✳ ❆❧é♠ ❞✐ss♦✱ ① é ♦✉tr❛ s♦❧✉çã♦ ❞❛ ❝♦♥❣r✉ê♥❝✐❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x_≡x₀ mod n✳

❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ x₀ s❡❥❛ s♦❧✉çã♦ ❞❛ ❝♦♥❣r✉ê♥❝✐❛ ❛❝✐♠❛✳ ■st♦ é✱ ax0 ≡ 1 mod n✳ ❆ss✐♠✱ n|(ax0 −1)✱ ✐st♦ é✱ ∃y ∈Z t❛❧ q✉❡ ax0 −1 = yn✳ ❆ss✐♠✱ t❡r❡♠♦s q✉❡x₀a₋yn= 1✳ P♦rt❛♥t♦✱ t❡♠♦s q✉❡a❡nsã♦ ❝♦♣r✐♠♦s✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱

s❡a❡nsã♦ ❝♦♣r✐♠♦s✱ ❡♥tã♦ ❡①✐st❡♠x0, y ∈Zt❛✐s q✉❡x0a+yn= 1✱ ❛ss✐♠✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ax_{0 −}1 = (₋y)n✱ ♦♥❞❡ (₋y)_∈ Z✳ ❉❛í✱ t❡♠♦s q✉❡ ax_{0 ≡} 1 mod n✳ ▲♦❣♦✱ x0 é s♦❧✉çã♦ ❞❛ ❝♦♥❣r✉ê♥❝✐❛ aX ≡1 mod n✳

❆❧é♠ ❞✐ss♦✱ s❡ x é ♦✉tr❛ s♦❧✉çã♦ ❞❛ ❝♦♥❣r✉ê♥❝✐❛ aX _≡ 1 mod n✱ ❡♥tã♦ ax _≡1 mod n✳ P♦r ♦✉tr♦ ❧❛❞♦✱x0✱ t❛♠❜é♠ é s♦❧✉çã♦✱ ✐st♦ é✱ ax_{0 ≡}1 mod n✳ ❙❡♥❞♦ ❛ss✐♠✱

❖s ❚❡♦r❡♠❛s ❞❡ ❊✉❧❡r ❡ ❲✐❧s♦♥ ❈❛♣ít✉❧♦ ✶

t❡♠♦s q✉❡ ax_≡ax0 mod n✳ ❈♦♠♦ mdc(a, n) = 1t❡♠♦s ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✺✱ q✉❡ ♣♦❞❡♠♦s ❝❛♥❝❡❧❛r ♦ a ❡ ♦❜t❡rx_≡x₀ mod n✱ ❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳

❉❡✜♥✐çã♦ ✶✳✷✳✶ ❙❡❥❛ ♥ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ ❝♦♠ n > 1✳ ❈❤❛♠❛♠♦s ❞❡ ❢✉♥çã♦ ✜ ❞❡

❊✉❧❡r✱ ❞❡♥♦t❛❞❛ ♣♦r φ(n)✱ ❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣❡❧♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ✐♥✈❡rtí✈❡✐s

❡♠ Zn✳

❖❜s❡r✈❡ q✉❡ φ(n) _≤ n₋1✳ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s q✉❡ φ(n) = n₋1 s❡✱ ❡ s♦♠❡♥t❡

s❡✱ n é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✳

▼❛✐s ❛❞✐❛♥t❡ ♠♦str❛r❡♠♦s ❝♦♠♦ ❝❛❧❝✉❧❛r φ(n) ❡♠ ❣❡r❛❧✳ ■r❡♠♦s ❛❣♦r❛ ✈❡r✐✜❝❛r

❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❢✉♥çã♦ ✜ ❞❡ ❊✉❧❡r✳

❚❡♦r❡♠❛ ✶✳✷✳✶ ❙❡❥❛♠ m, n_∈N✱ ❝♦♠ mdc(m, n) = 1✳ ❊♥tã♦

φ(mn) =φ(m)φ(n)✳

❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ❞✐s♣♦r ♦s ♥ú♠❡r♦s ❞❡ ✶ ❛té ♠♥ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

1 m+ 1 2m+ 1 _{· · ·} (n₋1)m+ 1 2 m+ 2 2m+ 2 _{· · ·} (n₋1)m+ 2 3 m+ 3 2m+ 3 _{· · ·} (n₋1)m+ 3

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

m 2m 3m _{· · ·} nm

❙❡ ♥❛ ❧✐♥❤❛ r✱ ♦♥❞❡ ❡stã♦ ♦s t❡r♠♦s r, m+r,2m+r, ...,(n₋1)m+r✱ t✐✈❡r♠♦s mdc(m, r) =d > 1✱ ❡♥tã♦ ♥❡♥❤✉♠ t❡r♠♦ ❞❡ss❛ ❧✐♥❤❛ s❡rá ♣r✐♠♦ ❝♦♠ mn✱ ✉♠❛ ✈❡③

q✉❡ ❡st❡s t❡r♠♦s✱ s❡♥❞♦ ❞❛ ❢♦r♠❛ km+r✱ ❝♦♠ 0_≤ k _≤ n₋1✱ sã♦ t♦❞♦s ❞✐✈✐sí✈❡✐s

♣♦r d✳ ▲♦❣♦✱ ♣❛r❛ ❡♥❝♦♥tr❛r♠♦s ♦s ♥❛t✉r❛✐s ❞❡st❛ t❛❜❡❧❛ q✉❡ sã♦ ♣r✐♠♦s ❝♦♠ mn✱

❞❡✈❡♠♦s ♦❧❤❛r ♥❛ ❧✐♥❤❛ r s♦♠❡♥t❡ s❡ mdc(m, r) = 1✳ P♦rt❛♥t♦✱ t❡♠♦s φ(m) ❧✐♥❤❛s

❡♠ q✉❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s sã♦ ♣r✐♠♦s ❝♦♠ m✳

❖s ❚❡♦r❡♠❛s ❞❡ ❊✉❧❡r ❡ ❲✐❧s♦♥ ❈❛♣ít✉❧♦ ✶

❆❣♦r❛ ❞❡✈❡♠♦s ♣r♦❝✉r❛r ❡♠ ❝❛❞❛ ✉♠❛ ❞❡ss❛s φ(m) ❧✐♥❤❛s✱ q✉❛♥t♦s ❡❧❡♠❡♥t♦s

sã♦ ♣r✐♠♦s ❝♦♠ n✱ ✉♠❛ ✈❡③ q✉❡ t♦❞♦s sã♦ ♣r✐♠♦s ❝♦♠ m✳ ❈♦♠♦ mdc(m, n) = 1 ♦s

❡❧❡♠❡♥t♦sr, m+r,2m+r, ...,(n₋1)m+r❢♦r♠❛♠ ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s

♠ó❞✉❧♦ n✳ ▲♦❣♦✱ ❝❛❞❛ ✉♠❛ ❞❡st❛s ❧✐♥❤❛s ♣♦ss✉✐ φ(n) ❡❧❡♠❡♥t♦s ♣r✐♠♦s ❝♦♠ n ❡✱

♣♦rt❛♥t♦✱ ❝♦♠♦ ❡❧❡s sã♦ ♣r✐♠♦s ❝♦♠ m✱ ❡❧❡s sã♦ ♣r✐♠♦s ❝♦♠ mn✳ ■st♦ ♥♦s ❣❛r❛♥t❡

q✉❡ φ(mn) =φ(m)φ(n)✳

▲❡♠❛ ✶✳✷✳✶ ❙❡ ♣ é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ ❡ r_∈N✱ ❡♥tã♦ t❡♠♦s

φ(pr_{) = p}r

−pr−1 _=pr₍₁

− 1p)✳

❉❡♠♦♥str❛çã♦✿ ❉❡ ✶ ❛tépr_{✱ t❡♠♦sp}r_{♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❚❡♠♦s q✉❡ ❡①❝❧✉✐r ❞❡ss❡s}

♦s ♥ú♠❡r♦s q✉❡ ♥ã♦ sã♦ ♣r✐♠♦s ❝♦♠ pr_{✱ ♦✉ s❡❥❛✱ t♦❞♦s ♦s ♠ú❧t✐♣❧♦s ❞❡ p✱ q✉❡ sã♦}

p,2p, ..., pr−1_{p✱ ❝✉❥♦ ♥ú♠❡r♦ ép}r−1_{✳ P♦rt❛♥t♦✱ φ(p}r_{) = p}r

−pr−1_✳ ❚❡♦r❡♠❛ ✶✳✷✳✷ P❛r❛ n = pα1

1 pα22pα33· · ·prαr✱ ❝♦♠ pi ♣r✐♠♦ ❡ αi∈N✱ i = 1, . . . , r✱

t❡♠♦s

φ(n) =n(1₋ 1

p1)(1−

1

p2)· · ·(1−

1

pr)✳

❉❡♠♦♥str❛çã♦✿ ❖ ❚❡♦r❡♠❛ ✶✳✷✳✶ ♥♦s ❣❛r❛♥t❡ q✉❡ φ(n) = φ(pα1

1 pα22pα33· · ·pαrr) =

φ(pα1

1 )φ(p

α2

2 )· · ·φ(pαrr)✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♦ ▲❡♠❛ ✶✳✷✳✶✱ ♣❡r♠✐t❡ ❡s❝r❡✈❡r ❡st❡ ú❧t✐♠♦

r❡s✉❧t❛❞♦ ❛ss✐♠✿ φ(n) = pα1

1 (1− p11)p

α2

2 (1− p12)· · ·p

αr

r (1− p1r) = p

α1

1 pα22· · ·pαrr(1−

1

p1)(1−

1

p2)· · ·(1−

1

pr)✳ P♦rt❛♥t♦✱ t❡r❡♠♦s q✉❡ φ(n) = n(1−

1

p1)(1−

1

p2)· · ·(1−

1

pr)

❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳

❚❡♦r❡♠❛ ✶✳✷✳✸ ✭❊✉❧❡r✮ ❙❡❥❛♠ n, a_∈Z✱ ❝♦♠ mdc(a, n) = 1✳ ❊♥tã♦✱

❖s ❚❡♦r❡♠❛s ❞❡ ❊✉❧❡r ❡ ❲✐❧s♦♥ ❈❛♣ít✉❧♦ ✶

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ Z∗

n = {r1, r2, . . . , rφ(n)} ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s ✐♥✈❡r✲ tí✈❡✐s ❞❡ Zn✳ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s q✉❡ aφ(n) · r1 ·r2· · ·rφ(n) = ar1 · ar2· · ·arφ(n)✳

❈♦♠♦ mdc(a, n) = 1 t❡♠♦s ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✶✸✱ q✉❡ ari ∈ Z∗n✱ ♣❛r❛ ❝❛❞❛

i _{∈ {}1,2, . . . , φ(n)_}✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ i ₆= j✱ t❡♠✲s❡ q✉❡ r_{i 6}= r_j✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ari 6=arj✱ ♣♦✐s✱mdc(a, n) = 1✳ ❆ss✐♠✱ t❡r❡♠♦s q✉❡ar1·ar2· · ·arφ(n) =r1·r2· · ·rφ(n)✳

P♦rt❛♥t♦✱ aφ(n)_·r

1·r2· · ·rφ(n) =r1·r2· · ·rφ(n)✳ ▲♦❣♦✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❝❛❞❛ ♠❡♠❜r♦ ❞❛

ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ♣❡❧♦ r❡s♣❡❝t✐✈♦ ✐♥✈❡rs♦ ❞♦s r′

is✱ ♦❜t❡♠♦saφ(n) = 1✱ ✐st♦ é✱aφ(n)≡1

mod n ❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳

❈♦r♦❧ár✐♦ ✶✳✷✳✶ ✭P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✮ ❙❡❥❛♠ a, p_∈Z✱ ❝♦♠ ♣ ♣r✐♠♦

❡ mdc(a, p) = 1✳ ❊♥tã♦

ap−1 _{≡1 mod p✳}

❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ φ(p) =p₋1 ❡ mdc(a, p) = 1✱ ♦ r❡s✉❧t❛❞♦ s❡❣✉❡ ❞✐r❡t♦ ❞♦

❚❡♦r❡♠❛ ✶✳✷✳✸✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✷ ❙❡❥❛♠ a, n_∈Z✱ ❝♦♠ a >1 ❡ n >1✳ ❊♥tã♦

∃r _∈Z, r >0✱ t❛❧ q✉❡ ar_{≡1 mod n ⇔ mdc(a, n) = 1}

❉❡♠♦♥str❛çã♦✿ ❙❡ mdc(a, n) = 1✱ ❡♥tã♦ ♦ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r ♥♦s ❣❛r❛♥t❡ q✉❡

❡①✐st❡ φ(n) = r _∈ Z t❛❧ q✉❡ ar

≡ 1 mod n✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡ mdc(a, n)>1✳ ❊♥tã♦ ❛ ❡q✉❛çã♦ aX₋nY = 1 ♥ã♦ ♣♦ss✉✐ s♦❧✉çã♦ ❡✱ ❞❛í✱ ❛ ❡q✉❛çã♦

♠♦❞✉❧❛r aX _≡ 1 mod n ♥ã♦ ♣♦ss✉✐ s♦❧✉çã♦✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♥ã♦ ♣♦❞❡ ❡①✐st✐r r _∈Z✱r >0 t❛❧ q✉❡ ar _{≡1 mod n✳ ▲♦❣♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ mdc(a, n) = 1✳}

❖ ♣ró①✐♠♦ ❧❡♠❛ s❡r✈✐rá ♣❛r❛ ♣r♦✈❛r♠♦s ♦ t❡♦r❡♠❛ ❞❡ ❲✐❧s♦♥✳

▲❡♠❛ ✶✳✷✳✷ ❙❡❥❛ a_∈Z✱ ❝♦♠ a >4 ✉♠ ♥ú♠❡r♦ ❝♦♠♣♦st♦✱ ❡♥tã♦ a_|(a₋1)!✳

❖s ❚❡♦r❡♠❛s ❞❡ ❊✉❧❡r ❡ ❲✐❧s♦♥ ❈❛♣ít✉❧♦ ✶

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ a _∈ Z ✉♠ ♥ú♠❡r♦ ❝♦♠♣♦st♦ ❡ a > 4✳ ❊♥tã♦ ❡①✐st❡♠

a₁, a_{2 ∈} Z t❛✐s q✉❡ a = a₁a2✱ ❝♦♠ 1 < a_{1 ≤} a₂ < a✳ ❙❡♥❞♦ ❛ss✐♠✱ t❡r❡♠♦s q✉❡

(a₋1)! = 1_{· · ·}a1· · ·a2· · ·(a−1)✳ P♦rt❛♥t♦✱ t❡♠♦s q✉❡ a|(a−1)! ❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳

❚❡♦r❡♠❛ ✶✳✷✳✹ ✭❲✐❧s♦♥✮ ❙❡❥❛ p_∈Z✱ p >1✳ ❊♥tã♦

♣ é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ ⇔ (p₋1)! _≡p₋1 mod p✳

❉❡♠♦♥str❛çã♦✿ ❙❡ p é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ ❡♥tã♦ 1,2, . . . , p₋1 sã♦ ♦s ❡❧❡♠❡♥t♦s

✐♥✈❡rtí✈❡✐s ❞❡ Zp✳ ❆❧é♠ ❞✐ss♦✱ s❡x∈Zp é t❛❧ q✉❡ x2 = 1✱ ❡♥tã♦ x2 ≡1 mod p✱ ✐st♦

é✱ p_|(x2_{−1) = (x−1)(x+ 1)✳ ❉❛í✱ t❡♠♦s p|(x−1) ♦✉ p|(x+ 1)✳ ▼❛s✱ 0≤ x < p} ❡ p é ♣r✐♠♦✳ ❆ss✐♠✱ t❡r❡♠♦s q✉❡ x = 1 ♦✉ x = p₋1✳ ▲♦❣♦✱ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s

❞❡Z∗

p✱ ❝♦♠ ❡①❝❡ssã♦ ❞❡1❡p−1✱ ♣♦ss✉❡♠ ✐♥✈❡rs♦ ❞✐❢❡r❡♥t❡ ❞❡❧❡ ♣ró♣r✐♦✳ P♦rt❛♥t♦✱

t❡♠♦s q✉❡✿ 2_{· · ·}(p₋2) = 1✳ ❆ss✐♠✱ ♦❜t❡♠♦s 2_{· · ·}(p₋2)_≡1 mod p✳ ▲♦❣♦✱ t❡♠♦s

q✉❡ (p₋1)! = 1_·2_{· · ·}(p₋2)(p₋1) _≡ p₋1 mod p✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛

q✉❡ (p₋ 1)! _≡ p₋1 mod p ❡ q✉❡ p > 4 ♥ã♦ é ♣r✐♠♦✳ ❊♥tã♦ ♣❡❧♦ ▲❡♠❛ ✶✳✷✳✷

♣♦❞❡♠♦s ❣❛r❛♥t✐r q✉❡ p_|(p₋ 1)!✳ ❈♦♠♦ p > 4 p ∤ (p₋1)✳ ❙❡♥❞♦ ❛ss✐♠✱ t❡♠♦s

q✉❡ p ∤ [(p₋1)!₋(p₋1)]✳ ▲♦❣♦✱ (p₋1)! _6≡ p₋1 mod p ♦ q✉❡ ❝♦♥tr❛❞✐③ ♥♦ss❛

❤✐♣ót❡s❡✳ ❆❣♦r❛ ♦❜s❡r✈❡ q✉❡ 3! = 6_6≡3 mod 4,2! = 2_≡2 mod 3❡ 1!_≡1 mod 2✳

P♦rt❛♥t♦✱ s❡ (p₋1)!_≡p₋1 mod p ❡♥tã♦ p é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✳

❖❜s❡r✈❡ q✉❡ ♦ ❚❡♦r❡♠❛ ❞❡ ❲✐❧s♦♥ ❝❛r❛❝t❡r✐③❛ t♦❞♦s ♦s ♥ú♠❡r♦s ♣r✐♠♦s✱ ✐st♦ é✱ t♦❞♦ ♥ú♠❡r♦ q✉❡ ♦❜❡❞❡❝❡ ♦ t❡♦r❡♠❛ é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ ❡✱ r❡❝✐♣r♦❝❛♠❡♥t❡✱ t♦❞♦ ♥ú♠❡r♦ ♣r✐♠♦ ♦❜❡❞❡❝❡ ❡ss❡ t❡♦r❡♠❛✳

❖s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s ❛té ♦ ✜♥❛❧ ❞❡ss❛ s❡çã♦ s❡rã♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ ♣r♦✈❛r♠♦s ❛❧❣✉♥s t❡st❡s ❞❡ ♣r✐♠❛❧✐❞❛❞❡ ❛♣r❡s❡♥t❛❞♦s ♥♦ ❝❛♣ít✉❧♦ ✷✳ ❆ s❡❣✉✐r ❞❡s❝r❡✈❡r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s s♦❜r❡ ❛ ♦r❞❡♠ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ Zn ❡ s♦❜r❡ ❛ r❛✐③ ♣r✐♠✐t✐✈❛

♠ó❞✉❧♦ n✳

❖s ❚❡♦r❡♠❛s ❞❡ ❊✉❧❡r ❡ ❲✐❧s♦♥ ❈❛♣ít✉❧♦ ✶

P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r t❡♠♦s q✉❡ s❡a, n_∈Z✱ ❝♦♠ mdc(a, n) = 1 ❡n >1✱ ❡♥tã♦

aφ(n) _{≡ 1 mod n✳ P♦rt❛♥t♦✱ ❡①✐st❡ ✉♠ ♠❡♥♦r ❡①♣♦❡♥t❡ k t❛❧ q✉❡ a}k

≡ 1 mod n✳

❊st❡ ♠❡♥♦r ✈❛❧♦r ❞❡ k ♣♦❞❡rá s❡r ♠❡♥♦r ❞♦ q✉❡φ(n)✳ P♦r ❡①❡♠♣❧♦✱ mdc(3,8) = 1✱

❛ss✐♠✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r✱ t❡♠♦s q✉❡ 3φ(8) _{≡ 1 mod 8✱ ♠❛s✱ 3}2 _{≡ 1 mod 8 ❡}

2<4 =φ(8)✳

❉❡✜♥✐çã♦ ✶✳✷✳✷ ❖ ♠❡♥♦r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❦ ♣❛r❛ ♦ q✉❛❧ ak _{≡ 1 mod n✱ ♦♥❞❡}

mdc(a, n) = 1✱ é ❝❤❛♠❛❞♦ ❞❡ ♦r❞❡♠ ❞❡ ❛ ♠ó❞✉❧♦ ♥ ❡ ❞❡♥♦t❛❞♦ ♣♦r ordna✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✸ ❙❡❥❛ k =ordna✳ ❊♥tã♦

ah _{≡1 mod n⇔k|h✳}

❉❡♠♦♥str❛çã♦✿ ❉❛❞♦s h, k _∈ Z✱ ❝♦♠ k ₆= 0✱ ❡♥tã♦ ♣❡❧♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦

t❡♠♦s q✉❡ ❡①✐st❡♠ q, r _∈ Z t❛✐s q✉❡ h = qk+r✱ ❝♦♠ 0 _≤ r < k✳ ❙❡♥❞♦ ❛ss✐♠✱

t❡♠♦s q✉❡ ah _=aqk+r _{= (a}k₎q_ar_{✳ P♦r ♦✉tr♦ ❧❛❞♦✱ k =ord}

na✱ ✐st♦ é✱ ak ≡1 mod n✳

P♦rt❛♥t♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ ah _{= a}qk+r _{= (a}k₎q_ar _{≡ 1}q_ar _{≡ a}r _{modn✳ ❆❧é♠ ❞✐ss♦✱}

ah

≡ 1 mod n✱ ♣♦r ❤✐♣ót❡s❡✳ ▲♦❣♦✱ t❡♠♦s ar

≡ 1 mod n✳ ▼❛s✱ ❝♦♠♦0 _≤r < k ❡ k ♣♦r ❞❡✜♥✐çã♦ é ♦ ♠❡♥♦r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ t❛❧ q✉❡ ak _{≡1 mod n✱ t❡♠♦s q✉❡ r= 0✳}

P♦rt❛♥t♦✱ h =qk✱ ✐st♦ é✱ k_|h✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ k _| h✱ ❡♥tã♦ ❡①✐st❡ q _∈Z t❛❧ q✉❡

h=qk✱ ❞❛í✱ ah _=aqk _{= (a}k₎q _≡1q_{= 1 mod n✳}

❈♦r♦❧ár✐♦ ✶✳✷✳✷ ordna|φ(n)✳

❉❡♠♦♥str❛çã♦✿ P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r✱ t❡♠♦s q✉❡aφ(n) _{≡1 mod n✱ s❡mdc(a, n) =}

1✳ ❉❛í✱ ♦ ❚❡♦r❡♠❛ q✉❡ ❛❝❛❜❛♠♦s ❞❡ ❞❡♠♦♥str❛r ♥♦s ❣❛r❛♥t❡ q✉❡ ord_na_|φ(n)✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✹ ❙❡❥❛ k=ord_na✳ ❊♥tã♦at

≡ah _{mod n✱ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ t}

≡h

mod k✳

❖s ❚❡♦r❡♠❛s ❞❡ ❊✉❧❡r ❡ ❲✐❧s♦♥ ❈❛♣ít✉❧♦ ✶

❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r♦ ✈❛♠♦s s✉♣♦r q✉❡ at _{≡ a}h _{modn ❡ t ≥ h✳ ❆ss✐♠}

♣♦❞❡♠♦s ❡s❝r❡✈❡r✿ at _{= a}h_at−h _{❡ ❝♦♠♦ a}h

≡ at _{mod n✱ t❡♠♦s q✉❡ a}h

≡ ah_at−h

mod n✳ P♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s mdc(a, n) = 1 ♦ q✉❡ ✐♠♣❧✐❝❛ mdc(ah_{, n) = 1✳ P♦r✲}

t❛♥t♦✱ ♣♦❞❡♠♦s ❝❛♥❝❡❧❛r ah_{✱ ♥❡st❛ ú❧t✐♠❛ ❝♦♥❣r✉ê♥❝✐❛✱ ♦❜t❡♥❞♦ 1}

≡ at−h _{mod n✳}

❉❛í✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✷✳✸✱ t❡♠♦s q✉❡ k_|(t₋h)✱ ✐st♦ é✱ t _≡ h mod k✳ ❘❡❝✐♣r♦❝❛✲

♠❡♥t❡✱ s❡ t _≡ h modk✱ ❡♥tã♦✱ k_|(t ₋h) ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ ❡①✐st❡ m _∈ Z t❛❧ q✉❡

t = h+mk✳ P♦rt❛♥t♦✱ t❡♠♦s q✉❡ at ₌. _ah+mk_{✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ k = ord} na

t❡♠♦s q✉❡ ak

≡ 1 mod n✳ ❆ss✐♠✱ ♦❜t❡♠♦s at _{= a}h+mk _{= a}h_(ak₎m

≡ ah₁m _{= a}h

mod n ❝♦♠♦ q✉❡rí❛♠♦s ♣r♦✈❛r✳

❈♦r♦❧ár✐♦ ✶✳✷✳✸ ❙❡ k = ordna✱ ❡♥tã♦ ♦s ♥ú♠❡r♦s 1, a, a2, . . . , ak−1 sã♦ ✐♥❝♦♥❣r✉✲

❡♥t❡s ♠ó❞✉❧♦ ♥✳

❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ ❞♦✐s ❞❡st❡s ♥ú♠❡r♦s s❡❥❛♠ ❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ n✱

✐st♦ é✱ at

≡ah _{mod n✱ ♦♥❞❡t, h}

∈ {0,1,2, . . . , k₋1_}✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✷✳✹✱ t❡♠♦s

q✉❡ t_≡h mod k✳ P♦rt❛♥t♦✱ t❡♠♦s q✉❡k_|(t₋h)✱ ♠❛s✱ ❝♦♠♦ t, h_{∈ {}0,1,2, . . . , k₋

1_}✱ t❡r❡♠♦s q✉❡ t ₋ h = 0✱ ✐st♦ é✱ t = h✳ ❆ss✐♠✱ ❝♦♥❝❧✉í♠♦s q✉❡ ♦s ♥ú♠❡r♦s

1, a, a2, . . . , ak−1 _{sã♦ t♦❞♦s ✐♥❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦n✳}

❉❡✜♥✐çã♦ ✶✳✷✳✸ ◗✉❛♥❞♦ ordna =φ(n)❞✐③❡♠♦s q✉❡ ❛ é ✉♠❛ r❛✐③ ♣r✐♠✐t✐✈❛ ♠ó❞✉❧♦

♥✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✺ ❙❡ ❛ é ✉♠❛ r❛✐③ ♣r✐♠✐t✐✈❛✱ ❡♥tã♦ ♦s ♥ú♠❡r♦s a, a2, . . . , aφ(n) _❢♦r✲ ♠❛♠ ✉♠ s✐st❡♠❛ r❡❞✉③✐❞♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ ♥✳

❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ a é ✉♠❛ r❛✐③ ♣r✐♠✐t✐✈❛ ♠ó❞✉❧♦ n t❡♠♦s q✉❡ ordna =

φ(n)✳ P❡❧♦ ❈♦r♦❧ár✐♦ ✶✳✷✳✸✱ s❛❜❡♠♦s q✉❡1, a, a2, . . . , aφ(n)−1 _{sã♦ t♦❞♦s ✐♥❝♦♥❣r✉❡♥t❡s} ♠ó❞✉❧♦ n✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ mdc(a, n) = 1 t❡♠♦s✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✶✶✱ q✉❡

♦s ♥ú♠❡r♦s1, a, a2, . . . , aφ(n)−1 _{sã♦ t♦❞♦s ✐♥✈❡rtí✈❡✐s ❡♠Z}

n✳ ❉❛í ❢♦r♠❛♠ ✉♠ s✐st❡♠❛