Criptografia e curvas elípticas

❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❈❛♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦

❈r✐♣t♦❣r❛✜❛ ❡ ❈✉r✈❛s ❊❧í♣t✐❝❛s

❱❛♥✐❛ ❇❛t✐st❛ ❙❝❤✉♥❝❦ ❋❧♦s❡

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❯♥✐✈❡rs✐tár✐❛ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜✲ t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡

❖r✐❡♥t❛❞♦r

Pr♦❢✳ ❉r✳ ❍❡♥r✐q✉❡ ▲❛③❛r✐

✶✶✶ ❳✶✶✶①

❋❧♦s❡✱ ❱❛♥✐❛ ❇✳ ❙✳

❈r✐♣t♦❣r❛✜❛ ❡ ❈✉r✈❛s ❊❧í♣t✐❝❛s✴ ❱❛♥✐❛ ❇❛t✐st❛ ❙❝❤✉♥❝❦ ❋❧♦s❡✲ ❘✐♦ ❈❧❛r♦✿ ❬s✳♥✳❪✱ ✷✵✶✶✳

✺✺ ❢✳ ✿ ✜❣✳

❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛✱ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s✳

❖r✐❡♥t❛❞♦r✿ ❍❡♥r✐q✉❡ ▲❛③❛r✐

✶✳ ❈r✐♣t♦❣r❛✜❛✳ ✷✳ ❈♦r♣♦ ✜♥✐t♦✳ ✸✳ Pr♦❜❧❡♠❛ ❞♦ ▲♦❣❛r✐t♠♦ ❉✐s❝r❡t♦✳ ✹✳ ❚❡♦r❡♠❛ ❞❡ ❍❛ss❡✳ ■✳ ❚ít✉❧♦

❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖

❱❛♥✐❛ ❇❛t✐st❛ ❙❝❤✉♥❝❦ ❋❧♦s❡

❈r✐♣t♦❣r❛❢✐❛ ❡ ❈✉r✈❛s ❊❧í♣t✐❝❛s

❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❯♥✐✈❡rs✐tár✐❛ ❞♦ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑✱ ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛✲ ❞♦r❛✿

Pr♦❢✳ ❉r✳ ❍❡♥r✐q✉❡ ▲❛③❛r✐ ❖r✐❡♥t❛❞♦r

Pr♦❢✳ ❉r✳ ❏❛✐♠❡ ❊❞♠✉♥❞♦ ❆♣❛③❛ ❘♦❞r✐❣✉❡③ ❋❊ ✲ ❯◆❊❙P ✲ ■❧❤❛ ❙♦❧t❡✐r❛

Pr♦❢❛✳ ❉r❛✳❈❛r✐♥❛ ❆❧✈❡s ■●❈❊ ✲ ❯◆❊❙P ✲ ❘✐♦ ❈❧❛r♦

❆❣r❛❞❡❝✐♠❡♥t♦s

Pr✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s q✉❡ é ♦♥✐♣r❡s❡♥t❡✱ ♦♥✐s❝✐❡♥t❡ ❡ ♦♥✐♣♦t❡♥t❡✳

➚ ♠✐♥❤❛ ❢❛♠í❧✐❛ ♣❡❧♦ ❝❛r✐♥❤♦✱ ❝♦♠♣r❡❡♥sã♦ ❡ ❡stí♠✉❧♦ ❡♠ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛ ♠ã❡✱ q✉❡ ♠❡s♠♦ s❡♠ s❛❜❡r ♠✉✐t♦ s♦❜r❡ ♠❛t❡♠át✐❝❛✱ s❡ ❡s❢♦rç♦✉ ♣❛r❛ ❡♥s✐♥❛r✲♠❡ ❛s ♣r✐♠❡✐✲ r❛s ✏❝♦♥t✐♥❤❛s✑✳

❆♦ ♣r♦❢❡ss♦r ❍❡♥r✐q✉❡ ▲❛③❛r✐ ♣❡❧❛ ❝♦♠♣r❡❡♥sã♦ ❡ ♣❛❝✐ê♥❝✐❛✳

❆♦s ♣r♦❢❡ss♦r❡s ❞❛ ❯◆❊❙P ❞❡ ■❧❤❛ ❙♦❧t❡✐r❛ q✉❡ ❢♦r❛♠ ♠❛✐s ❞♦ q✉❡ ♣r♦❢❡ss♦r❡s✳

❆♦s ♠❡✉s ❛♠✐❣♦s q✉❡ ❞❡ ❞✐✈❡rs❛s ❢♦r♠❛s ♠❡ ❛❥✉❞❛r❛♠ ♥❛ ♠✐♥❤❛ ❝❛♠✐♥❤❛❞❛ ✧♠❛✲ t❡♠át✐❝❛✧✿ ❆❞r✐❛♥❛✱ ❆♥❛ P❛✉❧❛✱ ❆♥❞ré✐❛✱ ❇r✉♥♦✱ ❉✐✈❛♥❡✱ ❉♦✉❣❧❛s✱ ❊❧❡♥✱ ▼❛r✐♥é✐❛✱ ❘❛❢❛❡❧✱ ❘❡❥❛♥❡✱ ❚❤❛❧✐t❛ ❡ ❚❤✐❛❣♦✳

❆♦s ♠❡✉s ❛♠✐❣♦s q✉❡ ♠❡ ❛❥✉❞❛r❛♠ ❡♠ ♦r❛çã♦ ♥♦s ♠♦♠❡♥t♦s ❞❡❝✐s✐✈♦s✿ ❉❡♥✐s❡✱ ❊❧✐s❛✱ ➱r✐❝❛✱ ❋❡r♥❛♥❞♦✱ ❍❡✐❞✐✱ ❏♦ã♦ ❡ ❙❡❧♠❛✳

❆♦s ♠❡✉s ❛♠✐❣♦s ❞❛ ■❣r❡❥❛ ❇❛t✐st❛ ❈❡♥tr❛❧✱ ❡♠ ❡s♣❡❝✐❛❧ ❛s ❢❛♠í❧✐❛s ❆♠❛r❛❧ ❡ ❇❛t✐st❛ s❡♠♣r❡ r❡❝❡♣t✐✈♦s ♣❛rt✐❧❤❛♥❞♦ ❞♦ ❛❝♦♥❝❤❡❣♦ ❞❡ s❡✉s ❧❛r❡s✱ ❡ ❛♦s ❛♠✐❣♦s ❞❛ Pr✐♠❡✐r❛ ❇❛t✐st❛ ❞❡ ❙ã♦ ❈❛❡t❛♥♦ ❞♦ ❙✉❧ ♣❡❧♦ ❛♣♦✐♦✳

❆♦s ♠❡✉s ❛❧✉♥♦s ❡ ❛♦s ❛♠✐❣♦s ❞❡ tr❛❜❛❧❤♦ ♥♦ ■♥st✐t✉t♦ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ P❛✉❧♦✱ ♣❡❧♦ ❡stí♠✉❧♦✳

✏❆ ♠❛t❡♠át✐❝❛ é ❛ r❛✐♥❤❛ ❞❛s ❝✐ê♥❝✐❛s ❡ ❛ ❛r✐t♠ét✐❝❛ é ❛ r❛✐♥❤❛ ❞❛ ♠❛t❡♠át✐❝❛✑

❘❡s✉♠♦

❈♦♠ ♦ ❝r❡s❝✐♠❡♥t♦ ❞❛ ❝♦♠✉♥✐❝❛çã♦ ♥♦s ❞✐❛s ❛t✉❛✐s✱ ❛ s❡❣✉r❛♥ç❛ ♥❛ tr♦❝❛ ❞❡ ✐♥❢♦r♠❛✲ çõ❡s t❡♠ s❡ t♦r♥❛❞♦ ❝❛❞❛ ✈❡③ ♠❛✐s ✐♠♣♦rt❛♥t❡ ♦ q✉❡ t❡♠ ❞❛❞♦ ❞❡st❛q✉❡ ❛ ❈r✐♣t♦❣r❛✜❛✳ ❆ ❝r✐♣t♦❣r❛✜❛ ❝♦♥s✐st❡ ❞❡ té❝♥✐❝❛s ❜❛s❡❛❞❛s ❡♠ ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s q✉❡ t❡♠ ♣♦r ♦❜❥❡t✐✈♦ tr❛♥s♠✐t✐r ✐♥❢♦r♠❛çõ❡s s✐❣✐❧♦s❛s ❢♦r♠❛ s❡❣✉r❛ ❛tr❛✈és ❞❡ ❝❛♥❛✐s ♠♦♥✐t♦r❛❞♦s ♣♦r t❡r❝❡✐r♦s✳ ❯♠ r❛♠♦ ❞❛ ❈r✐♣t♦❣r❛✜❛ q✉❡ ✈❡♠ ❝r❡s❝❡♥❞♦ ❡stá ❧✐❣❛❞♦ ❛♦ ❡st✉❞♦ ❞❡ ❝✉r✈❛s ❡❧í♣t✐❝❛s✱ q✉❡ é ✉♠❛ ❞❛s ár❡❛s ♠❛✐s r✐❝❛s ❞❛ ♠❛t❡♠át✐❝❛✳ ❖ ♥♦♠❡ ✏❝✉r✈❛s ❡❧í♣✲ t✐❝❛s✑ é ❞❡ ❝❡rt❛ ❢♦r♠❛ ❡♥❣❛♥♦s♦✱ ♣♦✐s ❞✐❢❡r❡♥t❡ ❞♦ s❡♥t✐❞♦ ❧✐t❡r❛❧ ❞❛ ♣❛❧❛✈r❛✱ q✉❡ ❧❡✈❛ ❛ ♣❡♥s❛r ❡♠ ❡❧✐♣s❡s✱ s❡ tr❛t❛ ❞❡ ❡q✉❛çõ❡s r❡❧❛❝✐♦♥❛❞❛s ❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ t✐♣♦ ❞❡ ❝✉r✈❛ ❛❧❣é❜r✐❝❛✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ❛s ❝✉r✈❛s ❡❧í♣t✐❝❛s s❡rã♦ ❡st✉❞❛❞❛s ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛ á❧❣❡❜r❛ ❡ ❞❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❝♦♥❤❡❝❡r ❛ ❈r✐♣t♦❣r❛✜❛ ❞❡ ❈✉r✈❛s ❊❧í♣t✐❝❛s q✉❡ é ✉♠❛ ✈❛r✐❛çã♦ ❞♦ Pr♦❜❧❡♠❛ ❞♦ ▲♦❣❛r✐t♠♦ ❉✐s❝r❡t♦✳

❆❜str❛❝t

❲✐t❤ t❤❡ ❣r♦✇t❤ ♦❢ ❝♦♠♠✉♥✐❝❛t✐♦♥ t❤❡s❡ ❞❛②s✱ s❡❝✉r✐t② ✐♥ ❡①❝❤❛♥❣❡ ❢♦r ✐♥❢♦r♠❛✲ t✐♦♥ ❤❛s ❜❡❝♦♠❡ ✐♥❝r❡❛s✐♥❣❧② ✐♠♣♦rt❛♥t ✇❤❛t ❤❛s ❣✐✈❡♥ ♣r♦♠✐♥❡♥❝❡ t♦ ❈r②♣t♦❣r❛♣❤②✳ ❊♥❝r②♣t✐♦♥ t❡❝❤♥✐q✉❡s ✐s ❜❛s❡❞ ♦♥ ❝♦♥❝❡♣ts ♠❛t❤❡♠❛t✐❝❛❧ ❛✐♠s t♦ tr❛♥s♠✐t s❡♥s✐t✐✈❡ ✐♥❢♦r♠❛t✐♦♥ s❡❝✉r❡❧② t❤r♦✉❣❤ ❝❤❛♥♥❡❧s ♠♦♥✐t♦r❡❞ ❜② t❤✐r❞ ♣❛rt✐❡s✳ ❆ ❜r❛♥❝❤ ♦❢ ❝r②♣✲ t♦❣r❛♣❤② t❤❛t ❤❛s ❣r♦✇✐♥❣ ✉♣ ✐s ❝♦♥♥❡❝t❡❞ t♦ t❤❡ st✉❞② ♦❢ ❡❧❧✐♣t✐❝ ❝✉r✈❡s✱ ✇❤✐❝❤ ✐s ♦♥❡ ♦❢ t❤❡ ♠♦st r✐❝❤ ♠❛t❤❡♠❛t✐❝s✳ ❚❤❡ ♥❛♠❡ ✏ ❡❧❧✐♣t✐❝ ❝✉r✈❡s✑✐s s♦♠❡✇❤❛t ♠✐s❧❡❛❞✐♥❣✱ ❛s ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ❧✐t❡r❛❧ s❡♥s❡ ♦❢ t❤❡ ✇♦r❞✱ ✇❤✐❝❤ ♠❛❦❡s ♦♥❡ t❤✐♥❦ ♦❢ ❡❧❧✐♣s❡s ✐❢ ❡q✉❛t✐♦♥s ✐s r❡❧❛t❡❞ t♦ ❛ ❝❡rt❛✐♥ t②♣❡ ♦❢ ❛❧❣❡❜r❛✐❝ ❝✉r✈❡✳ ✐♥ t❤✐s ✇♦r❦✱ ❡❧❧✐♣t✐❝ ❝✉r✈❡s ❛r❡ st✉❞✐❡❞ ❢r♦♠ t❤❡ ✈✐❡✇♣♦✐♥t ♦❢ ❛❧❣❡❜r❛ ❛♥❞ ♦❢ ♥✉♠❜❡r t❤❡♦r② ✐♥ ♦r❞❡r t♦ ❦♥♦✇ t❤❡ ❈✉r✈❡ ❈r②♣t♦❣r❛♣❤② ❊❧❧✐♣t✐❝ ✐s ❛ ✈❛r✐❛t✐♦♥ ♦❢ t❤❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ♣r♦❜❧❡♠✳

❙✉♠ár✐♦

✶ Pré✲r❡q✉✐s✐t♦s ✾

✶✳✶ ➪❧❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✳✶ ●r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✳✷ ❆♥é✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✶✳✸ ❈♦r♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✶✳✹ ❆♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✷ ❊❧❡♠❡♥t♦s ❞❡ ❚❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✷✳✶ ❆❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✷✳✷ ❖s t❡♦r❡♠❛s ❞❡ ❊✉❧❡r✱ ❋❡r♠❛t ❡ ❲✐❧s♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷✳✸ ❊q✉❛çõ❡s ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❡ ♦ ❚❡♦r❡♠❛ ❝❤✐♥ês ❞♦ r❡st♦ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷✳✹ ❘❡sí❞✉♦s q✉❛❞rát✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✷✳✺ ❙í♠❜♦❧♦ ❞❡ ▲❡❣❡♥❞r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✷ ❈r✐♣t♦❣r❛✜❛ ✷✵

✷✳✶ ❯♠ ♣♦✉❝♦ ❞❡ ❍✐stór✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷ ❈♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❡ ❝r✐♣t♦❣r❛✜❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸ ❈r✐♣t♦❣r❛✜❛ ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛ ♦✉ ❛ss✐♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸✳✶ Pr♦❜❧❡♠❛ ❞♦ ❧♦❣❛r✐t♠♦ ❞✐s❝r❡t♦ ✭P▲❉✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸✳✷ ❆ tr♦❝❛ ❞❡ ❝❤❛✈❡s ❉✐✣❡✲❍❡❧❧♠❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸✳✸ ❖ ♠ét♦❞♦ ❞❡ ❊❧●❛♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✸✳✹ ❘❙❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✹ ◆♦çõ❡s ❞❡ t❡♦r✐❛ ❞❡ ❝♦♠♣❧❡①✐❞❛❞❡ ❝♦♠♣✉t❛❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✹✳✶ ❆❧❣♦r✐t♠♦s ❞❡ ♣r✐♠❛❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

✸ ❈✉r✈❛s ❊❧í♣t✐❝❛s ✷✾

✸✳✷✳✷ ❈♦❞✐✜❝❛♥❞♦ t❡①t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✷✳✸ Pr♦❜❧❡♠❛ ❞♦ ❧♦❣❛r✐t♠♦ ❞✐s❝r❡t♦ ♥♦ ✉s♦ ❞❡ ❝✉r✈❛s ❡❧í♣t✐❝❛s ✳ ✳ ✳ ✳ ✹✵ ✸✳✷✳✹ ❆ tr♦❝❛ ❞❡ ❝❤❛✈❡s ❞❡ ❉✐✣❡✲❍❡❧❧♠❛♥ ❝♦♠ ❝✉r✈❛s ❡❧í♣t✐❝❛s ✳ ✳ ✳ ✳ ✹✶ ✸✳✷✳✺ ❆ ❛♥❛❧♦❣✐❛ ❞❡ ▼❛ss❡②✲❖♠✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✷✳✻ ❆♥❛❧♦❣✐❛ ❞❡ ❊❧●❛♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✷✳✼ ❆ ❡s❝♦❧❤❛ ❞♦ ♣♦♥t♦ ♥❛ ❝✉r✈❛ ❡ s❡❧❡çã♦ ✧❛❧❡❛tór✐❛✧❞❡ (E, B) ✳ ✳ ✳ ✹✹

✸✳✷✳✽ ❆ r❡❞✉çã♦ ●❧♦❜❛❧ ❞❡ (E, B) mod p ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

✸✳✷✳✾ ❖r❞❡♠ ❞♦ ♣♦♥t♦ B ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

✹ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✹✽

❘❡❢❡rê♥❝✐❛s ✹✾

❆ ❆❧❣✉♥s ❆❧❣♦r✐t♠♦s ✺✵

✶ Pré✲r❡q✉✐s✐t♦s

✶✳✶ ➪❧❣❡❜r❛

❱❛♠♦s ✈❡r ♥❡st❡ ❝❛♣ít✉❧♦ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s s❛ ➪❧❣❡❜r❛ ❆❜str❛t❛ ❡ ❞❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♥♦ss♦ ❡st✉❞♦

✶✳✶✳✶ ●r✉♣♦s

❯♠❛ ❞❛s ❡str✉t✉r❛s ♠❛✐s s✐♠♣❧❡s ❞❛ á❧❣❡❜r❛ é ♦ ❣r✉♣♦✱ q✉❡ ❝♦♥s✐st❡ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦✲✈❛③✐♦ ❡ ✉♠❛ ♦♣❡r❛çã♦ ✐♥✈❡rsí✈❡❧✳

❉❡✜♥✐çã♦ ✶✳✶✳ ❙❡❥❛ ∗ ✉♠❛ ♦♣❡r❛çã♦ ❞❡✜♥✐❞❛ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ G✳ ❉✐③❡♠♦s q✉❡ ♦ ♣❛r

(G,_∗) é ✉♠ ❣r✉♣♦ s❡ ❡ s♦♠❡♥t❡ s❡✿

• ❖ ❝♦♥❥✉♥t♦ G é ❢❡❝❤❛❞♦ s♦❜ ❛ ♦♣❡r❛çã♦ _∗✱ ✐st♦ é✱ _∀g, h_∈G✱ g_∗h_∈G✳

• ❆ ♦♣❡r❛çã♦ ∗ é ❛ss♦❝✐❛t✐✈❛✱ ✐st♦ é✱ ∀g, h, k_∈G✱ (g_∗h)_∗k=g_∗(h_∗k)

• ❊①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ ✐❞❡♥t✐❞❛❞❡ e _∈ G ♣❛r❛ _∗✱ ✐st♦ é✱ _∃e _∈ G✱ _∀g _∈ G✱ g _∗e =

e_∗g =g

• P❛r❛ t♦❞♦ ❡❧❡♠❡♥t♦ g _∈G ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ ✐♥✈❡rs♦ h_∈ G✱ _∀g _∈ G✱ _∃h_∈ G✱ g_∗h=h_∗g =e

❉❡✜♥✐çã♦ ✶✳✷✳ ❙❡❥❛ (G,_∗) ✉♠ ❣r✉♣♦✳ ❉✐③❡♠♦s q✉❡ ❡ss❡ ❣r✉♣♦ é ❛❜❡❧✐❛♥♦ s❡ _∗ ❢♦r ✉♠

♦♣❡r❛çã♦ ❝♦♠✉t❛t✐✈❛ ❡♠ G ✭✐st♦ é✱ _∀g, h_∈G, g_∗h=h_∗g✮✳

Pr♦♣♦s✐çã♦ ✶✳✶✳ ❙❡ (G,_∗) ✉♠ ❣r✉♣♦✱❡♥tã♦

✶✳ ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ é ú♥✐❝♦❀ ✷✳ ♦ ❡❧❡♠❡♥t♦ ✐♥✈❡rs♦ é ú♥✐❝♦❀

✸✳ s❡ a−1 _{❡ b}−1 _{sã♦ ♦s ✐♥✈❡rs♦s ❞❡ a ❡ b r❡s♣❡❝t✐✈❛♠♥t❡✱ ❡♥tã♦(a∗b)}−1 _=a−1_∗b−1_✳

➪❧❣❡❜r❛ ✶✵

❊①❡♠♣❧♦ ✶✳✶✳ (Z,+)✿ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❝♦♠ ❛ ♦♣❡r❛çã♦ ❞❡ ❛❞✐çã♦

✉s✉❛❧ é ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✱ ♣♦✐s ✈❛❧❡ ♦ ❛①✐♦♠❛ ❞❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡✱ ❝♦♠ ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ✵✱ ❡ ♦ ✐♥✈❡rs♦ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ a_∈ Z é ♦ ♦♣♦st♦ ❡♠Z✳ ❈♦♠♦ ❛ ♦♣❡r❛çã♦ ♥❡ss❡

❣r✉♣♦ é ❛ ❛❞✐çã♦✱ ♦ ❝❤❛♠❛♠♦s ❞❡ ❣r✉♣♦ ❛❞✐t✐✈♦✳ ❉❡✜♥✐çã♦ ✶✳✸✳ ❙❡❥❛ n ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✳ ❉❡✜♥✐♠♦s

Z∗_n=_{a_∈Zn:mdc(a, n) = 1}

❊①❡♠♣❧♦ ✶✳✷✳ ❈♦♥s✐❞❡r❡♠♦s Z∗14✳ ❖s ❡❧❡♠❡♥t♦s ✐♥✈❡rtí✈❡✐s ♣❛r❛ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♠ó✲

❞✉❧♦14✱ q✉❡ ❞❡♥♦t❛♠♦s ♣♦r(_⊗)❡♠ Z∗14✱sã♦ ♦s ❡❧❡♠❡♥t♦s r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s ❝♦♠ ✶✹✱

❛ s❛❜❡r✿ ✶✱ ✸✱ ✺✱ ✾✱ ✶✶ ❡ ✶✸✳ ❆ss✐♠✱

Z∗₁₄ =_{1,3,5,9,11,13_}

❆ t❛❜❡❧❛ ❞❡ ⊗ ♣❛r❛ Z∗

14 é ❛ s❡❣✉✐♥t❡✿

❖s ✐♥✈❡rs♦s ❞♦s ❡❧❡♠❡♥t♦s ♥❡st❡ Z∗₁₄ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ♥❛ t❛❜❡❧❛ ❛❝✐♠❛✱ ❛ss✐♠✿

1−1 _{= 1 3}−1 _{= 5 5}−1 _{= 3}

9−1 _{= 11 11}−1 _{= 9 13}−1 _{= 13}

Pr♦♣♦s✐çã♦ ✶✳✷✳ ❙❡❥❛ n ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✳ ❊♥tã♦✱ (Z∗

n,⊗) é ✉♠ ❣r✉♣♦✳

❉❡✜♥✐çã♦ ✶✳✹✳ ❆ ♦r❞❡♠ ❞❡ ✉♠ ❣r✉♣♦ é ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ G✳ ❉❡✲

♥♦t❛♠♦s ♣♦r |G_|✳

❊①❡♠♣❧♦ ✶✳✸✳ ❙❡❥❛ ♦ ❣r✉♣♦ ❛❞✐t✐✈♦(Zn,⊕)✱ ♦♥❞❡Zn ={0,1,2, ..., n−1}❡⊕é ❛ s♦♠❛

s❡❣✉✐❞❛ ❞♦ ❝á❧❝✉❧♦ ❞♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ♣♦r n✳ ❉❡ss❛ ❢♦r♠❛ Zn t❡♠ n ❡❧❡♠❡♥t♦s✱ s❡♥❞♦

➪❧❣❡❜r❛ ✶✶

G ♣♦❞❡ s❡r ✉♠ ❣r✉♣♦ ❞❡ ♦r❞❡♠ ✐♥✜♥✐t❛✱ ♣♦r ❡①❡♠♣❧♦ (Z,+)✳ ◆❡st❡ ❝❛s♦ ❞✐③❡♠♦s

q✉❡ ❛ ♦r❞❡♠ ❞❡ Gé ✐♥✜♥✐t❛✳

❉❡✜♥✐çã♦ ✶✳✺✳ ❙❡❥❛ (G,_∗) ✉♠ ❣r✉♣♦✱ ❡ s❡❥❛ H _⊆G✳ ❙❡(H,_∗) ❢♦r t❛♠❜é♠ ✉♠ ❣r✉♣♦

❝♦♠ ❛ r❡str✐çã♦ à H ❞❛ ♦♣❡r❛çã♦ ❞❡ G✱ ♦ ❝❤❛♠❛♠♦s ❞❡ s✉❜❣r✉♣♦ ❞❡ (G,_∗)✳

❊①❡♠♣❧♦ ✶✳✹✳ H =_{0,2,4,6,8_} é ✉♠ s✉❜❣r✉♣♦ ❞❡ (Z₁₀,_⊕)✳

◆♦t❡ q✉❡ H ❝♦♥té♠ ♦s ❡❧❡♠❡♥t♦s ♣❛r❡s ❞❡Z10✳ ❙❡ s♦♠❛r♠♦s ❞♦✐s ♥ú♠❡r♦s ♣❛r❡s ❛r❜✐✲

trár✐♦s✱ ♦ r❡s✉❧t❛❞♦s é ♣❛r✱ ❡ q✉❛♥❞♦ r❡❞✉③✐♠♦s ♦ r❡s✉❧t❛❞♦ mod 10✱ ❛ r❡s♣♦st❛ ❛✐♥❞❛ é

♣❛r✳ ❱❡♠♦s q✉❡ 0_∈H ❡ q✉❡ ♦s ✐♥✈❡rs♦s ❞❡ 0,2,4,6,8 sã♦0,8,6,4,2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

P♦rt❛♥t♦✱ H é ✉♠ s✉❜❣r✉♣♦ ❞❡ (Z₁₀,_⊕)

❯♠ ❡❧❡♠❡♥t♦ ❞❡ G❞❛ ❢♦r♠❛ am1 1 ∗am

2

2 ∗...∗amkk✱ ♦♥❞❡ ai ∈X✱ mi ∈Z é ❝❤❛♠❛❞♦

♣❛❧❛✈r❛ ❡♠X✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❡st❛s ♣❛❧❛✈r❛s ❡♠X é ✉♠ s✉❜❣r✉♣♦ ❞❡G✱ ❝❤❛♠❛❞♦

s✉❜❣r✉♣♦ ❣❡r❛❞♦ ♣♦r X✳ ❙❡ G é ❡st❡ s✉❜❣r✉♣♦✱ ❞✐③❡♠♦s q✉❡ G é ❣❡r❛❞♦ ♣♦r X ❡

❞❡♥♦t❛♠♦s ♣♦r G=_hX_i ❡ X é ❝❤❛♠❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❣❡r❛❞♦r❡s✳

❉❡✜♥✐çã♦ ✶✳✻✳ ❙❡❥❛ a_∈G✱ ♦♥❞❡ (G,_∗) é ✉♠ ❣r✉♣♦✳ ❆ ♦r❞❡♠ ❞❡ a é ❛ ♦r❞❡♠ ❞❡ _ha_i✱

♦✉ ❡q✉✐✈❛❧❡♥t❡ ♥♦ ❝❛s♦ ✜♥✐t♦✱ ❛ ♦r❞❡♠ ❞❡ a é ♠❡♥♦r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n t❛❧ q✉❡ an_=e✳

❉❡✜♥✐çã♦ ✶✳✼✳ ❙❡❥❛ (G,_∗) ✉♠ ❣r✉♣♦✳ ❯♠ ❡❧❡♠❡♥t♦ a _∈ G é ❝❤❛♠❛❞♦ ✉♠ ❣❡r❛❞♦r

❞❡ G s❡✱ ❡ s♦♠❡♥t❡ s❡✱ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ G ♣♦❞❡ s❡r ❡①♣r❡ss♦ ❡♠ t❡r♠♦s ❞❡ a ❡ a−1 ✉t✐❧✐③❛♥❞♦ ❛♣❡♥❛s ❛ ♦♣❡r❛çã♦ ∗✳ ◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ (G,_∗) é ✉♠ ❣r✉♣♦ ❝í❝❧✐❝♦✳

❖s ❣r✉♣♦s ❞❡ ♦r❞❡♠ ✜♥✐t❛ sã♦ ❝❤❛♠❛❞♦s ❞❡ gruposf initos✱ ❡ sã♦ ♦s ♠❛✐s ✐♥t❡r❡s✲

s❛♥t❡s ♥♦ ❡st✉❞♦ ❞❛ ❝r✐♣t♦❣r❛✜❛✳

Pr♦♣♦s✐çã♦ ✶✳✸✳ ❙❡❥❛♠ (G,_∗) ✉♠ ❣r✉♣♦ ✜♥✐t♦ ❞❡ ♦r❞❡♠ n+ 1 ❡ g _∈ G✳ ❊♥tã♦✱ ♣❛r❛

❛❧❣✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n✱ t❡♠♦s

g−1 _{=g∗g∗g∗...∗g}

| {z }

nvezes

✳

❚❡♦r❡♠❛ ✶✳✶✳ ❚♦❞♦ ❣r✉♣♦ ❝í❝❧✐❝♦ é ❛❜❡❧✐❛♥♦✳

❚❡♦r❡♠❛ ✶✳✷✳ ❚♦❞♦ s✉❜❣r✉♣♦ ❞❡ ✉♠ ❣r✉♣♦ ❝í❝❧✐❝♦ é ❝í❝❧✐❝♦✳

❉❡✜♥✐çã♦ ✶✳✽✳ ❙❡❥❛ (G,_∗) ✉♠ ❣r✉♣♦✳ ❉✐③❡♠♦s q✉❡ G é ✉♠ ❣r✉♣♦ ❞❡ t♦rçã♦ s❡ t♦❞♦

❡❧❡♠❡♥t♦ ❞❡Gé ❞❡ ♦r❞❡♠ ✜♥✐t❛✳ ❙❡ ❛♣❡♥❛s ❛ ✐❞❡♥t✐❞❛❞❡e_∈Gt❡♠ ♦r❞❡♠ ✜♥✐t❛✱ ❡♥tã♦ G é ❧✐✈r❡ ❞❡ t♦rçã♦✳

❱❡❥❛♠♦s ❛❣♦r❛ ❛❧❣✉♠❛s r❡❧❛çõ❡s ❡♠ ✉♠ ❣r✉♣♦ ✜♥✐t♦✱ ✉t✐❧✐③❛♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥✲ t♦s ❞♦ ❣r✉♣♦ ❡ ❞♦ s✉❜❣r✉♣♦✳ ■st♦ t♦r♥❛ ♠❛✐s ❢á❝✐❧ ❛ t❛r❡❢❛ ❞❡ ❞❡t❡r♠✐♥❛r ♦s s✉❜❣r✉♣♦s ❞❡ ✉♠ ❞❛❞♦ ❣r✉♣♦ ✜♥✐t♦✳

❚❡♦r❡♠❛ ✶✳✸✳ ✭▲❛❣r❛♥❣❡✮ ❙❡❥❛(H,_∗)✉♠ s✉❜❣r✉♣♦ ❞❡ ✉♠ ❣r✉♣♦ ✜♥✐t♦ (G,_∗)✱ ❡ s❡❥❛♠

➪❧❣❡❜r❛ ✶✷

❚❡♦r❡♠❛ ✶✳✹✳ ❚♦❞♦ ❣r✉♣♦ ✜♥✐t♦ ❞❡ ♦r❞❡♠ ♣r✐♠❛ é ❝í❝❧✐❝♦✳

❚❡♦r❡♠❛ ✶✳✺✳ ❙❡ G é ✉♠ ❣r✉♣♦ ❝í❝❧✐❝♦ ❞❡ ♦r❞❡♠ n✱ ❡♥tã♦ ❛ ♦r❞❡♠ ❞❡ q✉❛❧q✉❡r ❡❧❡✲

♠❡♥t♦ a_∈G ❞✐✈✐❞❡ n✳

❈♦r♦❧ár✐♦ ✶✳✶✳ ❙❡ G é ✉♠ ❣r✉♣♦ ✜♥✐t♦ ❡ a_∈G✱ ❡♥tã♦ a|G|_=e

❉❡✜♥✐çã♦ ✶✳✾✳ ❙❡❥❛♠ ♦s ❣r✉♣♦s (G,_∗) ❡ (H,_•)✳ ❯♠❛ ❢✉♥çã♦ f : G _→ H é ✉♠

✐s♦♠♦r✜s♠♦ ✭❞❡ ❣r✉♣♦s✮ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ f é ❜✐❥❡t✐✈❛ ❡ s❡ ✈❡r✐✜❝❛

∀g, h_∈G, f(g_∗h) =f(g)_•f(h)✳

❙❡ ❡①✐st❡ ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡G♣❛r❛H✱ ❞✐③❡♠♦s q✉❡ Gé ✐s♦♠♦r❢♦ ❛H ❡ ❡s❝r❡✈❡♠♦s G∼=H✳

❚❡♦r❡♠❛ ✶✳✻✳ ❙❡❥❛ (G,_∗)✉♠ ❣r✉♣♦ ❝í❝❧✐❝♦ ✜♥✐t♦ ❡ n =_|G_|✳ ❊♥tã♦✱ (G,_∗) é ✐s♦♠♦r❢♦

❛ (Zn,⊕)✳

✶✳✶✳✷ ❆♥é✐s

❆ ♥♦çã♦ ❞❡ ❛♥❡❧ s✉r❣✐✉ ❛ ♣❛rt✐r ❞❛ s✐st❡♠❛t✐③❛çã♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♥♦ sé❝✉❧♦ ❳■❳✱ ♠❛s ❛♣❡♥❛s ♥♦ sé❝✉❧♦ ❳❳ ❛s ♣r♦♣r✐❡❞❛❞❡s ❢♦r❛♠ ♦r❣❛♥✐③❛❞❛s ♥❛ ❢♦r♠❛ ❞❛s ❞❡✜♥✐çõ❡s q✉❡ sã♦ ✉t✐❧✐③❛❞❛s ❤♦❥❡✳

❉❡✜♥✐çã♦ ✶✳✶✵✳ ❉✐③✲s❡ q✉❡ ✉♠ ❝♦♥❥✉♥t♦ A ❝♦♠ ❛s ♦♣❡r❛çõ❡s ❜✐♥ár✐❛s ❞❡ ❛❞✐çã♦ (+)

❡ ♠✉❧t✐♣❧✐❝❛çã♦ (.) é ✉♠ ❛♥❡❧ s❡✿

✶✳ (A,+) é ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✳

• ❆ss♦❝✐❛t✐✈✐❞❛❞❡✿ ◗✉❛✐sq✉❡r q✉❡ s❡❥❛♠ a, b, c_∈A t❡♠♦s q✉❡ a+ (b+c) = (a+b) +c❀

• ❊❧❡♠❡♥t♦ ♥❡✉tr♦✿ ❊①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ 0_∈A t❛❧ q✉❡ ♣❛r❛ t♦❞♦ a_∈A✱ a+ 0 = 0 +a=a❀

• ❊❧❡♠❡♥t♦ ✐♥✈❡rs♦✿ ❉❛❞♦ a _∈ A✱ ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ ₋a _∈ A✱ ❝❤❛♠❛❞♦

✐♥✈❡rs♦ ❞❡ a✱ t❛❧ q✉❡

a+ (₋a) = (₋a) +a = 0❀

• ❈♦♠✉t❛t✐✈✐❞❛❞❡✿ P❛r❛ t♦❞♦ a, b_∈A✱ t❡♠♦s q✉❡ a+b =b+a✳

✷✳ (A, .) é ✉♠ s❡♠✐❣r✉♣♦✱ ♦✉ s❡❥❛✱ ❡♠ A é ✈á❧✐❞❛ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

• ❋❡❝❤❛♠❡♥t♦✿ P❛r❛ t♦❞♦ a, b_∈A✱ a.b_∈A❀

• ❆ss♦❝✐❛t✐✈✐❞❛❞❡✿ ◗✉❛✐sq✉❡r q✉❡ s❡❥❛♠ a, b, c_∈A t❡♠♦s q✉❡

➪❧❣❡❜r❛ ✶✸

✸✳ ❊♠ ❆✱ ❛ ♦♣❡r❛çã♦ ✏✳✑ é ❞✐str✐❜✉t✐✈❛ ❡♠ r❡❧❛çã♦ ❛ ✏✰✑✱ ♦✉ s❡❥❛✿ • ◗✉❛✐sq✉❡r q✉❡ s❡❥❛♠ a, b, c_∈A t❡♠♦s q✉❡

à ❡sq✉❡r❞❛ (a+b).c=a.c+b.c

à ❞✐r❡✐t❛ a.(b+c) = a.b+a.c

❉❡✜♥✐çã♦ ✶✳✶✶✳ ❙❡ ❛❧é♠ ❞❡ss❛s ♣r♦♣r✐❡❞❛❞❡s A s❛t✐s❢❛③ t❛♠❜é♠ ❡ss❛s ♦✉tr❛s ❞✉❛s✿

• ❊❧❡♠❡♥t♦ ♥❡✉tr♦ ♥❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♦✉ ❊❧❡♠❡♥t♦ ✐❞❡♥t✐❞❛❞❡✿ ❊①✐st❡ ✉♠ ❡❧❡♠❡♥t♦

1_∈A t❛❧ q✉❡ ♣❛r❛ t♦❞♦ a_∈A✱ a.1 = 1.a=a❀

• ❈♦♠✉t❛t✐✈✐❞❛❞❡ ♥❛ ♠✉❧t✐♣❧✐❝❛çã♦✿ P❛r❛ t♦❞♦ a, b_∈A✱ t❡♠♦s q✉❡ a.b=b.a

❡♥tã♦ ❞✐③❡♠♦s q✉❡ A é ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❝♦♠ ✐❞❡♥t✐❞❛❞❡✳

❉❡✜♥✐çã♦ ✶✳✶✷✳ ❙❡❥❛ A ✉♠ ❛♥❡❧✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❡❧❡♠❡♥t♦ a _∈A✱ a ₆= 0✱ é ❞✐✈✐s♦r

❞❡ ③❡r♦ s❡ ❡①✐st❡ b_∈A✱ b₆= 0✱t❛❧ q✉❡ a.b= 0_∈A✳

❉❡✜♥✐çã♦ ✶✳✶✸✳ ❙❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❝♦♠ ✐❞❡♥t✐❞❛❞❡A♥ã♦ ♣♦ss✉✐ ❞✐✈✐s♦r❡s ❞❡ ③❡r♦✱

❡♥tã♦ A é ❞✐t♦ ❛♥❡❧ ❞❡ ✐♥t❡❣r✐❞❛❞❡✳

Pr♦♣♦s✐çã♦ ✶✳✹✳ ✭▲❡✐ ❞♦ ❝❛♥❝❡❧❛♠❡♥t♦✮✿ ❙❡❥❛ A ✉♠ ❛♥❡❧ ❞❡ ✐♥t❡❣r✐❞❛❞❡✱ t❡♠♦s q✉❡

❞❛❞♦s a, b, c_∈A ❝♦♠ c₆= 0✱ ❡♥tã♦✿

a.c=b.c_⇒a=b c.a=c.b_⇒a=b

✶✳✶✳✸ ❈♦r♣♦s

❉❡✜♥✐çã♦ ✶✳✶✹✳ ❙❡❥❛ A ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❝♦♠ ✐❞❡♥t✐❞❛❞❡✳ ❉✐③❡♠♦s q✉❡ a _∈ A∗ _é

✐♥✈❡rsí✈❡❧ s❡ ❡①✐st❡ a−1 _{∈A✱ t❛❧ q✉❡}

a.a−1 _=a−1_{.a= 1}

❙❡ ♣❛r❛ t♦❞♦a _∈A∗ _{❡①✐st✐r a}−1_{✱ ❞✐③❡♠♦s q✉❡ A é ✉♠ ❝♦r♣♦✳}

❚❡♦r❡♠❛ ✶✳✼✳ ❚♦❞♦ ❝♦r♣♦ é ✉♠ ❛♥❡❧ ❞❡ ✐♥t❡❣r✐❞❛❞❡✱ ♦✉ s❡❥❛✱ ❞❛❞♦ ✉♠ ❝♦r♣♦ K ❡ a_∈K✱ a₆= 0✱ ❡♥tã♦ a ♥ã♦ é ❞✐✈✐s♦r ❞❡ ③❡r♦✳

❚❡♦r❡♠❛ ✶✳✽✳ ❚♦❞♦ ❛♥❡❧ ❞❡ ✐♥t❡❣r✐❞❛❞❡ ❝♦♠ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❡❧❡♠❡♥t♦s é ✉♠ ❝♦r♣♦✳

❚❡♦r❡♠❛ ✶✳✾✳ ❚❡♠♦s q✉❡ a _∈ Zn é ✐♥✈❡rsí✈❡❧ ♣❛r❛ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱

mdc(a, n) = 1

❊❧❡♠❡♥t♦s ❞❡ ❚❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ✶✹

✶✳✶✳✹ ❆♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s

❙❡❥❛ (A,+, .) ✉♠ ❛♥❡❧✳ ❯♠ ♣♦❧✐♥ô♠✐♦ ♥✉♠❛ ✈❛r✐á✈❡❧ s♦❜r❡ A é ✉♠❛ s❡q✉ê♥❝✐❛

(a0, a1, ..., an, ...)✱ ♦♥❞❡ai ∈A♣❛r❛ t♦❞♦ í♥❞✐❝❡ ❡ ♦♥❞❡ai 6= 0s♦♠❡♥t❡ ♣❛r❛ ✉♠ ♥ú♠❡r♦

✜♥✐t♦ ❞❡ í♥❞✐❝❡s✳ ❉❡♥♦t❛♠♦s ♣♦r A[X] ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣♦r t❛✐s ♣♦❧✐♥ô♠✐♦s✳

❚❡♦r❡♠❛ ✶✳✶✶✳ ❙❡❥❛ A ✉♠ ❛♥❡❧✳ ❚❡♠♦s q✉❡✿

✶✳ ❙❡ A é ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❡♥tã♦ A[X] t❛♠❜é♠ é ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦✳

✷✳ ❙❡ A é ✉♠ ❛♥❡❧ ❝♦♠ ✐❞❡♥t✐❞❛❞❡ ❡♥tã♦ A[X] t❛♠❜é♠ é ✉♠ ❛♥❡❧ ❝♦♠ ✐❞❡♥t✐❞❛❞❡✳

✸✳ ❙❡ A é ✉♠ ❛♥❡❧ ❞❡ ✐♥t❡❣r✐❞❛❞❡ ❡♥tã♦ A[X] t❛♠❜é♠ é ✉♠ ❛♥❡❧ ❞❡ ✐♥t❡❣r✐❞❛❞❡✳

❉❡✜♥✐çã♦ ✶✳✶✺✳ ❙❡❥❛ A ✉♠ ❛♥❡❧ ❡ s❡❥❛ f(X) := a0+a1X +...+anXn ∈ A[X] ❝♦♠

an 6= 0✳ ❖ ♥ú♠❡r♦n s❡ ❝❤❛♠❛ ♦ ❣r❛✉ ❞❡f(X)❡ ❞❡♥♦t❛♠♦s ♣♦rgr(f(X))✳ ❖ ❝♦❡✜❝✐❡♥t❡

an é ❝❤❛♠❛❞♦ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r ❞❡ f(X)✳ ◗✉❛♥❞♦ ♦ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r ❢♦r ✐❣✉❛❧ ❛ ✶✱ ♦

♣♦❧✐♥ô♠✐♦ é ❞✐t♦ ♠ô♥✐❝♦✳

❚❡♦r❡♠❛ ✶✳✶✷✳ ❆❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ❙❡❥❛ A ✉♠ ❝♦r♣♦✳ ❉❛❞♦s ❞♦✐s ♣♦❧✐♥ô♠✐♦s p(X) ❡ q(X) ❡♠ A[X]✱ ❡♥tã♦ ❡①✐st❡♠ ❞♦✐s ú♥✐❝♦s ♣♦❧✐♥ô♠✐♦s t(X) ❡ r(X) ❡♠ A[X]

t❛✐s q✉❡ p(X) = t(X).q(X) +r(X)✱ ♦♥❞❡ r(X) = 0 ♦✉ gr(r(X))< gr(q(X))✳

❉❡✜♥✐çã♦ ✶✳✶✻✳ ❯♠ ♣♦❧✐♥ô♠✐♦ p(X)_∈A[X] é ❞✐t♦ ✐rr❡❞✉tí✈❡❧ s♦❜r❡ A s❡ s❡♠♣r❡ q✉❡ p(X) =a(X).b(X)✱ ❝♦♠ a(X), b(X)_∈A[X]✱ ❡♥tã♦ gr(a(X)) = 0 ♦✉ gr(b(X)) = 0✱ ♦✉

s❡❥❛✱ a(X) ♦✉ b(X) é ✉♠❛ ❝♦♥st❛♥t❡✳

❉❡✜♥✐çã♦ ✶✳✶✼✳ ❙❡❥❛ A ✉♠ ❛♥❡❧ ❡ s❡❥❛ p(X)_∈A[X]✳ ❉✐r❡♠♦s q✉❡ α_∈A é ✉♠❛ r❛✐③

❞❡ p(X) s❡ p(α) = 0✳

❚❡♦r❡♠❛ ✶✳✶✸✳ ❙❡❥❛ A✉♠ ❝♦r♣♦✳ ❚❡♠♦s q✉❡ s❡p(X)_∈A[X]é ✉♠ ♣♦❧✐♥ô♠✐♦ ❞❡ ❣r❛✉

n✱ ❡♥tã♦ p(X) ♣♦ss✉✐✱ ♥♦ ♠á①✐♠♦✱ n r❛í③❡s ❞✐st✐♥t❛s ❡♠ A✳

Pr♦♣♦s✐çã♦ ✶✳✺✳ ❙❡❥❛♠ ❆ ✉♠ ❛♥❡❧✱ p(X) _∈ A[X] ❡ α _∈ A✳ ❊♥tã♦ f(α) = 0 s❡ ❡

s♦♠❡♥t❡ s❡ ❡①✐st❡ ✉♠ ♣♦❧✐♥ô♠✐♦ t(X)_∈A[X] t❛❧ q✉❡ f(X) = (X₋α).t(X)✳

❉❡✜♥✐çã♦ ✶✳✶✽✳ ❙❡❥❛♠A✉♠ ❛♥❡❧✱p(X)_∈A[X]✱α _∈A✱ ❡ ✉♠ ✐♥t❡✐r♦s_≥1✳ ❉✐③❡♠♦s

q✉❡α é ✉♠❛ r❛✐③ ❞❡ f(X) ❞❡ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ s s❡ (X₋α)s _{❞✐✈✐❞❡ p(X)♠❛s (X−α)}s+1 ♥ã♦ ❞✐✈✐❞❡ p(X)✳

✶✳✷ ❊❧❡♠❡♥t♦s ❞❡ ❚❡♦r✐❛ ❞♦s ♥ú♠❡r♦s

✶✳✷✳✶ ❆❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s

❊❧❡♠❡♥t♦s ❞❡ ❚❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ✶✺

❉❡✜♥✐çã♦ ✶✳✶✾✳ ❙❡❥❛♠ a, b ❡ n > 1 ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❉✐r❡♠♦s q✉❡ a é ❝♦♥❣r✉❡♥t❡

❝♦♠ b ♠ó❞✉❧♦ n q✉❛♥❞♦ a₋b ❢♦r ❞✐✈ís✐✈❡❧ ♣♦r n✳ ❉❡♥♦t❛♠♦s ♣♦r a_≡b mod n✳

❆ss✐♠✱ ❛ ❝♦♥❣r✉ê♥❝✐❛a_≡b modn é ✈❡r❞❛❞❡✐r❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ a mod n=b mod n✳

♦✉ s❡❥❛✱ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ✐♥t❡✐r❛ ❞❡a ♣♦rn é ✐❣✉❛❧ ❛♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ✐♥t❡✐r❛ ❞❡b♣♦r n✳

❚❡♦r❡♠❛ ✶✳✶✹✳ ❙❡❥❛♠ a, b, c❡ n ♥ú♠❡r♦s ✐♥t❡✐r♦s ❝♦♠n₆= 0✳ ❙❡a_≡b modn✱ ❡♥tã♦

✶✳ a+c_≡b+c mod n

✷✳ a_×c_≡b_×c modn

✸✳ a₋c_≡b₋c modn

✹✳ a_≡b+cn mod n

❈♦r♦❧ár✐♦ ✶✳✷✳ ◗✉❛♥❞♦ a ❡ n ❢♦r❡♠ ♣r✐♠♦s ❡♥tr❡ s✐ ❡ ac_≡b modn✱ ❡♥tã♦ c_≡ba−1

mod n✳

❉❡✜♥✐çã♦ ✶✳✷✵✳ P❛r❛ ❝❛❞❛ ♥ú♠❡r♦ ♥❛t✉r❛❧ n❞❡✜♥✐♠♦sϕ(n)✱ ❛ ❢✉♥çã♦ ❞❡ ❊✉❧❡r✱ ❝♦♠♦

s❡♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s q✉❡ ♥ã♦ ❡①❝❡❞❡♠ n ❡ sã♦ ♣r✐♠♦s ❝♦♠ n✳

❊①❡♠♣❧♦ ✶✳✺✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ r❡st♦s ♠ó❞✉❧♦ ✷✽ é ❢♦r♠❛❞♦ ♣♦r t♦❞♦s ♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s

a ♠❡♥♦r❡s q✉❡ ✷✽ t❛✐s q✉❡ ♠✳❞✳❝(28, a) = 1✳

▲♦❣♦ t❡♠♦s ♦ ❝♦♥❥✉♥t♦ {1,3,5,9,11,13,15,17,19,23,25,27_}✱ ❡ ϕ(28) é ❞❛❞♦ ♣❡❧❛

♦r❞❡♠ ❞❡st❡ ❝♦♥❥✉♥t♦✱ ♦✉ s❡❥❛✱ ✶✷✳

Pr♦♣♦s✐çã♦ ✶✳✻✳ ❙❡❥❛ n ✉♠ ✐♥t❡✐r♦ ❝♦♠ n_≥2✳ ❊♥tã♦

|Z∗_n|=ϕ(n)

❚❡♦r❡♠❛ ✶✳✶✺✳ ❙❡❥❛ m ❡ n ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s t❛✐s q✉❡ ♠✳❞✳❝(m, n)❂✶✱ ❡♥tã♦✿

ϕ(mn) =ϕ(m)ϕ(n)✳

❊❧❡♠❡♥t♦s ❞❡ ❚❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ✶✻

✶✳✷✳✷ ❖s t❡♦r❡♠❛s ❞❡ ❊✉❧❡r✱ ❋❡r♠❛t ❡ ❲✐❧s♦♥

❚❡♦r❡♠❛ ✶✳✶✼✳ ✭❚❡♦r❡♠❛ ❞❡ ❲✐❧s♦♥✮ ❙❡pé ✉♠ ♣r✐♠♦✱ ❡♥tã♦(p₋1)! _{≡ −}1 mod p✳

❱❡❥❛♠♦s ✉♠ ❡①❡♠♣❧♦ ❡①tr❛í❞♦ ❞❛ r❡❢❡rê♥❝✐❛ ❬✶❪✳

❊①❡♠♣❧♦ ✶✳✻✳ ❉❡♥tr♦ ♦s ♥ú♠❡r♦s 1,2, ...,12 s♦♠❡♥t❡ ♦s ♥ú♠❡r♦s ✶ ❡ ✶✷ sã♦ ♦s s❡✉s

♣ró♣r✐♦s ✐♥✈❡rs♦s ♠ó❞✉❧♦ ✶✸ ♣♦✐s1_≡1 mod 13❡12_{≡ −}1_≡12 mod 13✳ ▼❛s ❞❡♠❛✐s

♥ú♠❡r♦s ✭ ✷✱✸✱✹✱✺✱✻✱✼✱✽✱✾✱✶✵✮ t❡♠ ✐♥✈❡rs♦s ❧✐st❛❞♦s ❛❜❛✐①♦ ❞♦✐s ❛ ❞♦✐s✿

2_×7_≡13 , 3_×9_≡13, 4_×10_≡13 , 5_×8_≡13, 6_×11_≡13✳

❆♦ ♠✉❧t✐♣❧✐❝❛r ❡st❛s ❝♦♥❣r✉ê♥❝✐❛s ♠❡♠❜r♦ ❛ ♠❡♠❜r♦✱ t❡♠♦s✿

2_×3_×4_×5_×6_×7_×8_×9_×10_×11_≡1 mod 13

❡ ♠✉❧t✐♣❧✐❝❛♥❞♦ ♦s ❞♦✐s ❧❛❞♦s ♣♦r ✶✷✱ t❡r❡♠♦s✿

2_×3_×4_×5_×6_×7_×8_×9_×10_×11_×12_≡1_×12 mod 13

❡ ♣♦rt❛♥t♦✱ ❝♦♠♦ 12_{≡ −}1 mod 13✱ t❡♠♦s ✜♥❛❧♠❡♥t❡ (13₋1)!_{≡ −}1 mod 13

❚❡♦r❡♠❛ ✶✳✶✽✳ ✭P❡q✉❡♥♦ t❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✮ ❙❡❥❛ p ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ ❡ s❡❥❛ a

✉♠ ✐♥t❡✐r♦✳ ❊♥tã♦✱

ap _{≡a mod p}

❚❡♦r❡♠❛ ✶✳✶✾✳ ✭❊✉❧❡r✮ ❙❡❥❛♠ n ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❡ a ✉♠ ✐♥t❡✐r♦ r❡❧❛t✐✈❛♠❡♥t❡

♣r✐♠♦ ❝♦♠ n✳ ❊♥tã♦

aϕ(n)_{≡1 mod n}

✶✳✷✳✸ ❊q✉❛çõ❡s ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❡ ♦ ❚❡♦r❡♠❛ ❝❤✐♥ês ❞♦ r❡st♦

✐♥❝ó❣♥✐t❛ x✿

ax _≡b mod n

◗✉❛♥❞♦ mdc(a, n) = 1✱ ✉♠❛ s♦❧✉çã♦ q✉❡ ♣❡rt❡♥❝❡ ❛♦ ✐♥t❡r✈❛❧♦ 0_≤x0 < n é

x0 =a−1b mod n✱

♦♥❞❡ a−1 _{é ♦ ✐♥✈❡rs♦ ❞❡a ♠ó❞✉❧♦ n✳ ❆ s♦❧✉çã♦ ♥ã♦ é ✉♥✐❝❛ ♣♦✐s x}

0 +qn t❛♠❜é♠ s❡rá s♦❧✉çã♦✱ ♣❛r❛ q✉❛❧q✉❡r ♥ú♠❡r♦ ✐♥t❡✐r♦ q✳

❊❧❡♠❡♥t♦s ❞❡ ❚❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ✶✼

❊st❛ ❝♦♥❣r✉ê♥❝✐❛ t❡rá ✉♠❛ s♦❧✉çã♦ s❡ ❡ só s❡d ❞✐✈✐❞✐r b✳ ◆❡st❡ ❝❛s♦✱

_a d x_≡ b d mod n d

❈♦♠♦ a/d ❡ n/d sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✱ x0 = (a/d)−1 mod n/d é ✉♠❛ s♦❧✉çã♦ ❞❡st❛ ❡q✉❛çã♦ ❡

x0, x0+

_n

d

, x0+ 2

_n

d

, ..., x0+ (d−1)

_n

d

sã♦ t♦❞❛s ❛s s♦❧✉çõ❡s ❡♠Z_n ❞❛ ❡q✉❛çã♦ ♦r✐❣✐♥❛❧ ax_≡b modn✳

❚❡♦r❡♠❛ ✶✳✷✵✳ ✭❚❡♦r❡♠❛ ❝❤✐♥ês ❞♦ r❡st♦✮ ❙❡❥❛♠ m1✱ m2✱✳✳✳✱mk✱ ♥ú♠❡r♦s ✐♥t❡✐r♦s

❞♦✐s ❛ ❞♦✐s ♣r✐♠♦s ❡♥tr❡ s✐ ❡ m=m1m2...mk✳ ❉❛❞♦s ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s a1✱ a2✱✳✳✳✱ak✱

♦ s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛s ♥❛ ✐♥❝ó❣♥✐t❛ x

          

x _≡ a1 mod m1

x _≡ a2 mod m2 ✳✳✳ ✳✳✳

x _≡ ak modmk

♣♦ss✉✐ s♦❧✉çã♦ ❡ é ❞❛❞❛ ♣♦r✿

x0 =a1M1N1+a2M2N2+...+akMkNk

♦♥❞❡ Mi =m/mi ❡ Ni =Mi−1 modmi✱ ♣❛r❛ i= 1,2, ..., k✳

❊st❛ s♦❧✉çã♦ ♥ã♦ é ú♥✐❝❛ ♣♦✐s ♣❛r❛ q✉❛❧q✉❡r ✐♥t❡✐r♦ q✱ ♦ ♥ú♠❡r♦ y _≡ x0+qm t❛♠❜é♠ é s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❡ ❞✉❛s s♦❧✉çõ❡s q✉❛✐sq✉❡r ❞✐❢❡r❡ ♣♦r ✉♠ ♠ú❧t✐♣❧♦ ❞❡ m✳

❊①❡♠♣❧♦ ✶✳✼✳ ❱❛♠♦s ❞❡t❡r♠✐♥❛r ❛s s♦❧✉çõ❡s ❞❡

  

 

x _≡ 1 mod 15

x _{≡ −}1 mod 8

x _≡ 2 mod 13

◆❡st❡ s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛s✱ a1 = 1✱ m1 = 15✱ a2 =−1✱ m2 = 8✱a3 = 2 ❡m1 = 13 ❡ ❝❛❧❝✉❧❛♠♦s ♦ ♣r♦❞✉t♦ m=m1m2m3 = 15.8.13 = 1560 ♣❛r❛ ❡♥❝♦♥tr❛r ♦s ♥ú♠❡r♦s✿

  

 

M1 = m/m1 = 1560/15 = 104

M2 = m/m2 = 1560/8 = 195

M3 = m/m3 = 1560/13 = 120

❡ s❡✉s ✐♥✈❡rs♦ ♠♦❞✉❧❛r❡s

  

 

N1 = M1−1 mod 15 = 104−1 mod 15 = 14

N2 = M2−1 mod 8 = 195−1 mod 8 = 3

N3 = M3−1 mod 13 = 120−1 mod 13 = 9

❊❧❡♠❡♥t♦s ❞❡ ❚❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ✶✽

x=a1M1N1+a2M2N2+a3M3N3

= 1.104.14₋1.195.3 + 2.120.9 = 3031

❘❡❞✉③✐♥❞♦ ♠ó❞✉❧♦ m✱ s❡❣✉❡ q✉❡3031_≡1471 mod 1560é ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ♦ s✐st❡♠❛

❞❛❞♦✳

✶✳✷✳✹ ❘❡sí❞✉♦s q✉❛❞rát✐❝♦s

❯♠ ❞♦s ♣r✐♠❡✐r♦s r❡s✉❧t❛❞♦s ❞❛ ❚❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ♠♦❞❡r♥❛ ❢♦✐ ❛ ▲❡✐ ❞❡ ❘❡❝✐♣r♦✲ ❝✐❞❛❞❡ ◗✉❛❞rát✐❝❛ ❝♦♥❥❡❝t✉r❛❞❛ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ♣♦r ❊✉❧❡r ❡ ▲❡❣❡♥❞r❡ ♥❛ ♣r✐♠❡✐r❛ ♠❡t❛❞❡ ❞♦ sé❝✉❧♦ ❳❱■■■✱ ♣♦ré♠ ❡❧❡s só ♦❜t✐✈❡r❛♠ ❛ ❞❡♠♦♥str❛çã♦ ♣❛r❛ ❝❛s♦s ♣❛rt✐❝✉❧❛✲ r❡s✳ ❊♠ ✶✼✾✻ ●❛✉ss ❞❡✉ ❛ ♣r✐♠❡✐r❛ ❞❡♠♦♥str❛çã♦ ❞❛ ▲❡✐ ❞❡ ❘❡❝✐♣r♦❝✐❞❛❞❡ ◗✉❛❞rát✐❝❛ ❡ ❞✉r❛♥t❡ s✉❛ ✈✐❞❛ ❡♥❝♦♥tr♦✉ ♦✉tr❛s ❞❡♠♦♥str❛çõ❡s ❞❡ss❡ tr❛❜❛❧❤♦✳ ❚r❛t❛♥❞♦✲s❡ ❞❡ r❡sí❞✉♦s q✉❛❞rát✐❝♦s✱ ❛s ✐❞❡✐❛s ❞❡s❡♥✈♦❧✈✐❞❛s ♣❛r❛ s♦❧✉❝✐♦♥á✲❧❛s sã♦ ❞❡ ❣r❛♥❞❡ r✐q✉❡③❛s ❡♠ ✐♥❢♦r♠❛çõ❡s ❛r✐t♠ét✐❝❛s ❡ ❞❡ ♦❢❡r❡❝✐♠❡♥t♦ ❞❡ ♣r♦❜❧❡♠❛s ❛ s❡r❡♠ s♦❧✉❝✐♦♥❛❞♦s✳

❆ ♣❛❧❛✈r❛ ✏r❡sí❞✉♦ q✉❛❞rát✐❝♦✑ é ❛tr✐❜✉í❞♦ ❛ ✉♠ ✐♥t❡✐r♦ a q✉❡ s❛t✐s❢❛③ ❛ ❡q✉❛çã♦ x2 _{≡a mod n✳ ❋♦r♠❛❧♠❡♥t❡✿}

❉❡✜♥✐çã♦ ✶✳✷✶✳ ❉✐③❡♠♦s q✉❡ a é r❡sí❞✉♦ q✉❛❞rát✐❝♦ ♠ó❞✉❧♦ n s❡✱ s❡ s♦♠❡♥t❡ s❡✱ ❛

❡q✉❛çã♦ x2 _{≡a mod n ♣♦ss✉✐ s♦❧✉çã♦✳ ❈❛s♦ ♥ã♦ ♣♦ss✉❛ s♦❧✉çã♦ ❞✐③❡♠♦s q✉❡ a ♥ã♦ é} r❡sí❞✉♦ q✉❛❞rát✐❝♦ ♠ó❞✉❧♦ n✳

❊①❡♠♣❧♦ ✶✳✽✳ ❙❡ ✜①❛r♠♦s n = 7 ❡♥tã♦ ❛ ❝❧❛ss❡ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ ✼ ♣♦ss✉✐ ❡①❛t❛✲

♠❡♥t❡ ✼ ❡❧❡♠❡♥t♦s ❞♦s q✉❛✐s ✸ ❡❧❡♠❡♥t♦s sã♦ q✉❛❞r❛❞♦s✱ ❛ s❛❜❡r✿ 1 = 12_{,4 = 2}2_{,9 = 3}2_✱ ✭32 _{= 9 ≡ 2 mod 7✮✳ P♦rt❛♥t♦✱ ♦ ✐♥t❡✐r♦ ✷ é r❡sí❞✉♦ q✉❛❞rát✐❝♦ ♠ó❞✉❧♦ ✼✱ ❡♥✲} q✉❛♥t♦ ✺ ♥ã♦ é r❡sí❞✉♦ q✉❛❞rát✐❝♦ ♠ó❞✉❧♦ ✼✱ ♣♦✐s ♥❡♥❤✉♠ ❞♦s ❡❧❡♠❡♥t♦s s♦ ❝♦♥❥✉♥t♦ {1,2,3,4,5,6_} s❛t✐s❢❛③ ❛ ❡q✉❛çã♦ x2 _{≡5 mod 7✳}

❖s ✐♥t❡✐r♦s ❝✉❥❛s r❛í③❡s q✉❛❞r❛❞❛s sã♦ ❡❧❛s ♣ró♣r✐❛s✱ ♥ú♠❡r♦s ✐♥t❡✐r♦s r❡❝❡❜❡♠ ♦ ♥♦♠❡ ❞❡ q✉❛❞r❛❞♦s ♣❡r❢❡✐t♦s

Pr♦♣♦s✐çã♦ ✶✳✼✳ ❙❡❥❛♠ p✉♠ ♥ú♠❡r♦ ♣r✐♠♦ ❡ a_∈Z_p✳ ❊♥tã♦✱ a t❡♠ ♥♦ ♠á①✐♠♦ ❞✉❛s

r❛í③❡s q✉❛❞r❛❞❛s ❡♠ Z_p✳

Pr♦♣♦s✐çã♦ ✶✳✽✳ ❙❡❥❛ p ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✱ ❝♦♠ p _≡ 3 mod 4✳ ❙❡❥❛ a _∈ Zp ✉♠

r❡sí❞✉♦ q✉❛❞rát✐❝♦✳ ❊♥tã♦✱ ❛s r❛í③❡s q✉❛❞r❛❞❛s ❞❡ a ❡♠ Zp sã♦

±a(p+1)/4 _{mod p✳}

❊①❡♠♣❧♦ ✶✳✾✳ ✶✼ é ✉♠ r❡sí❞✉♦ q✉❛❞rát✐❝♦ mod 59✱ ♣♦✐s ✺✾ é ♣r✐♠♦ ❡59_≡3 mod 4✱

❧♦❣♦

1759+1/4 _{= 17}15_{= 28}

❊❧❡♠❡♥t♦s ❞❡ ❚❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ✶✾

✶✳✷✳✺ ❙í♠❜♦❧♦ ❞❡ ▲❡❣❡♥❞r❡

❉❡✜♥✐çã♦ ✶✳✷✷✳ ❙❡❥❛ p ✉♠ ✐♥t❡✐r♦ ♣r✐♠♦ í♠♣❛r✳ P❛r❛ ✉♠ ✐♥t❡✐r♦ a✱ ❞❡✜♥✐♠♦s ♦

sí♠❜♦❧♦ ❞❡ ▲❡❣❡♥❞r❡ ♣♦r

a p =     

0 se p_|a,

1 se a e res´ ´iduo quadr´atico modulo p,´

−1 se a e n´ ˜ao res´iduo quadr´atico modulo p.´

❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦✱ ❞❡s❝♦❜❡rt♦ ♣♦r ❊✉❧❡r✱ ♣❡r♠✐t❡ ❝❛❧❝✉❧❛r ♦ sí♠❜♦❧♦ ❞❡ ▲❡❣❡♥❞r❡✳ ❚❡♦r❡♠❛ ✶✳✷✶✳ ❈r✐tér✐♦ ❞❡ ❊✉❧❡r ❙❡❥❛♠ a ✉♠ ✐♥t❡✐r♦ ❡ p ✉♠ ♣r✐♠♦✱ t❛✐s q✉❡

♠❞❝(a, p) = 1✳ ❊♥tã♦

ap−21 ≡

a p

mod p✳

❊①❡♠♣❧♦ ✶✳✶✵✳ 4(11−1)/2 _{≡1 mod 11❡ 4∈}

a2

p

= 1❀ 7(11−1)/2 _{≡10≡ −1 mod 11❡ 7∈Q11✳}

❊♠ ♣❛rt✐❝✉❧❛r 1 p = 1 −1 p

= (₋1)(p−1)/2✳ P♦rt❛♥t♦

−1_∈Qp s❡ p= 1 mod 4❡ −1∈Qp s❡p= 3 mod 4✳

❚❡♦r❡♠❛ ✶✳✷✷✳ ❙❡❥❛ p ✉♠ ♣r✐♠♦✳ P❡❧♦ ❝r✐tér✐♦ ❞❡ ❊✉❧❡r ❡ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞♦ sí♠❜♦❧♦

❞❡ ▲❡❣❡♥❞r❡✱ s❡❣✉❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿ ✶✳ ❙❡ a _≡b mod p✱ ❡♥tã♦

a p = b p ✳ ✷✳ a2 p = 1 ✸✳ ab p = a p b p ✹✳ −1 p

= (₋1)p−21 =

(

1 se p_≡1 mod 4,

−1 se p_≡3 mod 4.

✺✳ ❙❡ p é ♣r✐♠♦✱ ❡♥tã♦

2

p

= (₋1)p

2

−1 8

✻✳ ▲❡✐ ❞❛ r❡❝✐♣r♦❝✐❞❛❞❡ q✉❛❞rát✐❝❛ ✭▲❡❣❡♥❞r❡✲●❛✉ss✮✿ ❙❡ p ❡ q sã♦ ♣r✐♠♦s í♠♣❛r❡s✱

❡♥tã♦ p q q p

= (₋1)p−21.

q−1 2 =

(

1 se p_≡1 mod 4 ou q_≡1 mod 4

✷ ❈r✐♣t♦❣r❛✜❛

✷✳✶ ❯♠ ♣♦✉❝♦ ❞❡ ❍✐stór✐❛

❆t✉❛❧♠❡♥t❡ é ❝♦♠✉♠ r❡❛❧✐③❛r ❝♦♠♣r❛s ♣❡❧❛ ✐♥t❡r♥❡t✱ ♣♦ré♠ ♥ã♦ ✐♠❛❣✐♥❛♠♦s ♦ q✉❛♥t♦ ✉♠❛ s✐♠♣❧❡s ❝♦♠♣r❛ ♣♦❞❡ tr❛③❡r ❞❡ ❝r✐♣t♦❣r❛✜❛ ❡♠ s❡✉s ❜❛st✐❞♦r❡s✳ ❙✉♣♦✲ ♥❤❛ q✉❡ ❆❧✐❝❡ ❞❡s❡❥❛ ❢❛③❡r ✉♠❛ ❝♦♠♣r❛ ♣❡❧❛ ✐♥t❡r♥❡t✳ P❛r❛ ✐st♦✱ ❡❧❛ ✈✐s✐t❛ ✉♠ s✐t❡ ❞❡ ❝♦♠♣r❛s✱ ❢❛③ s❡✉ ♣❡❞✐❞♦ ❡ ♣❛r❛ ♣❛❣á✲❧♦✱ ✐♥tr♦❞✉③ ♦ ♥ú♠❡r♦ ❞♦ s❡✉ ❝❛rtã♦ ❞❡ ❝ré❞✐t♦✳ ❙❛❜❡♠♦s q✉❡ é ❞❡ ❣r❛♥❞❡ ♣❡r✐❣♦ q✉❡ ♦✉tr❛s ♣❡ss♦❛s s❛✐❜❛♠ ♦ ♥ú♠❡r♦ ❞♦ s❡✉ ❝❛rtã♦ ❞❡ ❝ré❞✐t♦ ❛ ♥ã♦ s❡r ♦ ❢♦r♥❡❝❡❞♦r✱ q✉❡ ❝❤❛♠❛r❡♠♦s ❞❡ ❇♦❜✳ ◗✉❛♥❞♦ ❆❧✐❝❡ ❛❝✐♦♥❛ ♦ ❜♦tã♦ ❙❊◆❉✱ ❛ ✐♥❢♦r♠❛çã♦ ❝♦♥✜❞❡♥❝✐❛❧✱ q✉❡ é ♦ ♥ú♠❡r♦ ❞♦ ❝❛rtã♦✱ ♣❡r❝♦rr❡ ✉♠ tr❛❥❡t♦ ❛té ♦ ❢♦r♥❡❝❡❞♦r ♣❛ss❛♥❞♦ ♣♦r ✈ár✐♦s ❝♦♠♣✉t❛❞♦r❡s✱ ❝♦♠♦ ♦ ❝♦♠♣✉t❛❞♦r ❞♦ s❡✉ ♣r♦✈❡❞♦r ❞❡ ✐♥t❡r♥❡t✱ q✉❡ t❡♠ ❝♦♠♦ ♦♣❡r❛❞♦r ❊✈❛✳ ▼❛s ❝♦♠♦ ❣❛r❛♥t✐r q✉❡ ❊✈❛ ♥ã♦ ✐rá ❞❡s❝♦❜r✐r ♦ ♥ú♠❡r♦ ❞♦ ❝❛rtã♦ ❞❡ ❝ré❞✐t♦ ❞❡ ❆❧✐❝❡❄ ❆❧✐❝❡ ♣♦❞❡ ❝❡❞❡r ❡st❛ ✐♥❢♦r♠❛çã♦ ❛♦ ❇♦❜ ❝♦♠ s❡❣✉r❛♥ç❛❄ ❙❡rá ❡st❡ ✉♠ ♣r♦❜❧❡♠❛ s♦♠❡♥t❡ ❞❛ ❛t✉❛❧✐❞❛❞❡❄

❆ s❡❣✉r❛♥ç❛ ♥❛ tr❛♥s♠✐ssã♦ ❞❡ ✐♥❢♦r♠❛çã♦ t❡♠ r❡❧❛t♦s ❛♥t✐❣♦s✱ ❡ ✉♠ ❞♦s ♣r✐♠❡✐r♦s ❞❛t❛♠ ❞❡ ❍❡ró❞♦t♦✱ ♦ ✏♣❛✐ ❞❛ ❤✐stór✐❛✑✱ s❡❣✉♥❞♦ ♦ ✜❧ós♦❢♦ ❡ ❡st❛❞✐st❛ r♦♠❛♥♦ ❈í❝❡r♦✳ ❍❡ró❞♦t♦ ❡s❝r❡✈❡✉ ❆s ❤✐stór✐❛s q✉❡ ❞❡s❝r❡✈✐❛♠ ♦s ❝♦♥✢✐t♦s ❡♥tr❡ ❛ ●ré❝✐❛ ❡ ❛ Pérs✐❛✱ ♦❝♦rr✐❞♦s ♥♦ sé❝✉❧♦ V ❛✳❈✳✱ ❡ ♥❡❧❡ ❡r❛ r❡❧❛t❛❞♦ q✉❡ ❛ ❡s❝r✐t❛ s❡❝r❡t❛ s❛❧✈♦✉ ❛ ●ré❝✐❛

❞❡ s❡r ❝♦♥q✉✐st❛❞❛ ♣❡❧♦ ❧í❞❡r ♣❡rs❛ ❳❡r①❡s✳ ◆❡st❛ ♦❝❛s✐ã♦ ❤❛✈✐❛ ✉♠ ❣r❡❣♦ ❝❤❛♠❛❞♦ ❉❡♠❛r❛t♦✱ q✉❡ ❡①✐❧❛❞♦ ❞❛ ●ré❝✐❛✱ ✈✐✈✐❛ ♥✉♠❛ ❝✐❞❛❞❡ ❞❛ Pérs✐❛✱ ❡ t❡st❡♠✉♥❤♦✉ ♦s ♣❧❛♥♦s ❞❡ ❳❡r①❡s ♣❛r❛ ♣❛r❛ ❞❡str✉✐r ❛ ●ré❝✐❛✳ P❛r❛ ❡✈✐t❛r ✐ss♦ ❡❧❡ r❛s♣♦✉ ❛ ❝❡r❛ ❞❡ ✉♠ ♣❛r ❞❡ t❛❜✉❧❡t❛s ❞❡ ♠❛❞❡✐r❛ ❡s❝r❡✈❡♥❞♦ s♦❜r❡ ❡❧❛s ♦s ♣❧❛♥♦s ❞❡ ❳❡r①❡s✱ ❝♦❜r✐✉✲❛s ❝♦♠ ❝❡r❛ ♥♦✈❛♠❡♥t❡ ❡ ❡♥✈✐♦✉ à ●ré❝✐❛✳ ❈♦♠ t❛❧ ✐♥❢♦r♠❛çã♦ ♦s ❣r❡❣♦s s❡ ♣r❡♣❛r❛♠ ♣❛r❛ ❛ ❝❤❡❣❛❞❛ ❞❛ ❢r♦t❛ ❞❡ ❳❡r①❡s✱ ❡ s✉r♣r❡❡♥❞❡♥❞♦✲♦✱ ❝♦♥s❡❣✉✐r❛♠ ❡s❝❛♣❛r ❞❡ ✉♠❛ ♣♦ssí✈❡❧ ❞♦♠✐♥❛çã♦ ♣❡rs❛✳

❖✉tr♦ r❡❧❛t♦ ♥♦s ❝♦♥t❛ q✉❡ ♦s ❛♥t✐❣♦s ❝❤✐♥❡s❡s ❡s❝r❡✈✐❛♠ ♠❡♥s❛❣❡♥s ❡♠ s❡❞❛ ✜♥❛✱ q✉❡ ❡r❛ ❛♠❛ss❛❞❛ ❛té ❢♦r♠❛r ✉♠❛ ♣❡q✉❡♥❛ ❜♦❧❛ ❡ ❛ ❝♦❜r✐❛♠ ❝♦♠ ❝❡r❛✳ ❊st❛ ❡r❛ ❡♥❣♦❧✐❞❛ ♣♦r ✉♠ ♠❡♥s❛❣❡✐r♦ q✉❡ ❧❡✈❛✈❛ ❛ ♠❡♥s❛❣❡♠ ❛té s❡✉ ❞❡st✐♥♦✳ ❯♠ ♦✉tr♦ ❡①❡♠♣❧♦ ❢♦✐ ❞♦ ❝✐❡♥t✐st❛ ●✐♦✈❛♥✐ P♦rt❛✱ ♥♦ sé❝✉❧♦ XV I q✉❡ ❞❡s❝r❡✈❡✉ ❝♦♠♦ ❡s❝♦♥❞❡r ✉♠❛ ♠❡♥s❛❣❡♠

❞❡♥tr♦ ❞❡ ✉♠ ♦✈♦ ❝♦③✐❞♦ ❢❛③❡♥❞♦ ✉♠❛ t✐♥t❛ ❝♦♠ ✉♠❛ ♦♥ç❛ ❞❡ ❛❧✉♠❡ ❡ ✉♠ q✉❛rt✐❧❤♦ ❞❡ ✈✐♥❛❣r❡✱ ♦♥❞❡ ❛ s♦❧✉çã♦ ♣❡♥❡tr❛✈❛ ♥❛ ❝❛s❝❛ ❞♦ ♦✈♦ ❡ ❞❡✐①❛ ❛ ♠❡♥s❛❣❡♠ s♦❜r❡ ❛ ❝❧❛r❛ ❡♥❞✉r❡❝✐❞❛✳ ❖ ❞❡st✐♥❛tár✐♦ ♣❛r❛ ❧❡r ❛♣❡♥❛s r❡t✐r❛✈❛ ❛ ❝❛s❝❛✳

❈♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❡ ❝r✐♣t♦❣r❛✜❛ ✷✶

❖ ♣r✐♠❡✐r♦ ❞♦❝✉♠❡♥t♦ q✉❡ ✉s♦✉ ✉♠❛ ❝✐❢r❛ ❞❡ s✉❜st✐t✉✐çã♦ ♣❛r❛ ✜♥s ♠✐❧✐t❛r❡s ❛♣❛✲ r❡❝❡ ❡♠ ❘♦♠❛ ♥❛s ❣✉❡rr❛s ❞❡ ●á❧✐❛ ❞❡ ❏ú❧✐♦ ❈és❛r✱ ❝❤❛♠❛❞❛s ❞❡ ❝✐❢r❛s ❞❡ ❈és❛r✱ q✉❡ ❝♦♥s✐st✐❛ ❡♠ s✉❜st✐t✉✐r ❛s ❧❡tr❛s ❞♦ ❛❧❢❛❜❡t♦ r♦♠❛♥♦ ♣♦r ❧❡tr❛s ❣r❡❣❛s ❡ ❛ss✐♠ ❛ ♠❡♥✲ s❛❣❡♠ ✜❝♦✉ ✐♥❝♦♠♣r❡❡♥sí✈❡❧ ♣❛r❛ ♦ ✐♥✐♠✐❣♦✳

❆✐♥❞❛ ♥❛ ❛♥t✐❣✉✐❞❛❞❡✱ ♦s ❡st✉❞✐s♦s ár❛❜❡s ✐♥✈❡♥t❛r❛♠ ❛ ❈r✐♣t♦á♥❛❧✐s❡✱ ❝✐ê♥❝✐❛ q✉❡ ♣❡r♠✐t❡ ❞❡❝✐❢r❛r ✉♠❛ ♠❡♥s❛❣❡♠ s❡♠ ❝♦♥❤❡❝❡r ❛ ❝❤❛✈❡✳ ❊❧❡s ✉t✐❧✐③❛✈❛♠ ✉♠ ❛❧❢❛❜❡t♦ ❝✐❢r❛❞♦✱ q✉❡ ❡r❛ ✉♠ s✐♠♣❧❡s r❡❛rr❛♥❥♦ ❞❡ ❛❧❢❛❜❡t♦ ❝✐❢r❛❞♦✱ ❝♦♥❤❡❝✐❞♦ t❛♠❜é♠ ❝♦♠♦ ❝✐❢r❛ ❞❡ s✉❜st✐t✉✐çã♦ ♠♦♥♦❛❧❢❛❜ét✐❝❛✱ q✉❡ ❝♦♥s✐t❡ ❡♠ s✉❜st✐t✉✐r ❝❛❞❛ ❧❡tr❛ ♣♦r ✉♠ sí♠❜♦❧♦✳ ❊st❡ ♠ét♦❞♦ ❛❝❛❜♦✉ s❡♥❞♦ ✈✉❧♥❡rá✈❡❧✱ ♣♦✐s ❝♦♥❤❡❝❡♥❞♦ ♦ ✐❞✐♦♠❛ ♦ q✉❛❧ ❛ ♠❡♥s❛❣❡♠ ❢♦✐ ❡s❝r✐t❛✱ ❡r❛♠ ✉t✐❧✐③❛❞♦s ♠ét♦❞♦s ❡st❛tíst✐❝♦s ✉t✐❧✐③❛♥❞♦ ❞❛s ❧❡tr❛s ♠❛✐s ❢r❡q✉❡♥t❡s ♥♦ ✐❞✐♦♠❛ ❡ ❝♦♠♣❛r❛♥❞♦ ❝♦♠ ♦ sí♠❜♦❧♦ q✉❡ ♠❛✐s ❛♣❛r❡❝❡ ♥❛ ♠❡♥s❛❣❡♠ ❝♦❞✐✜❝❛❞❛ ❞❡ ❢♦r♠❛ s✉❝❡ss✐✈❛ ❛té ❛ ♠❡♥s❛❣❡♠ s❡r ❞❡s❝♦❜❡rt❛✳

❆s té❝♥✐❝❛s ✉t✐❧✐③❛❞❛s ❞❡ ♦❝✉❧t❛çã♦ ✉t✐❧✐③❛❞❛s ♥❡st❡s ❢❛t♦s sã♦ t✐♣♦s ❞❡ ❡st❡❣❛♥♦✲ ❣r❛✜❛✱ q✉❡ é ✉♠ ♥♦♠❡ ❞❡r✐✈❛❞♦ ❞❛s ♣❛❧❛✈r❛s ❣r❡❣❛s st❡❣❛♥♦s✱ q✉❡ s✐❣♥✐✜❝❛ ❝♦❜❡rt♦✱ ❡ ❣r❛♣❤❡✐♥✱ q✉❡ s✐❣♥✐✜❝❛ ❡s❝r❡✈❡r✳ ❈♦♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❡st❡❣❛♥♦❣r❛✜❛✱ ❤♦✉✈❡ ❛ ❡✈♦❧✉çã♦ ❞❛ ❈r✐♣t♦❣r❛✜❛✱ ❝✉❥♦ ♦❜❥❡t✐✈♦ ♥ã♦ é ♦❝✉❧t❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ❛ ♠❡♥s❛❣❡♠✱ ❡ s✐♠✱ ❡s❝♦♥❞❡r ♦ s❡✉ s✐❣♥✐✜❝❛❞♦ ✉t✐❧✐③❛♥❞♦ ❞❡ ✉♠ ♣r♦❝❡ss♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❡♥❝r✐♣t❛çã♦✱ ♦♥❞❡ ♦ t❡①t♦ é ♠✐st✉r❛❞♦ ❞❡ ❛❝♦r❞♦ ❝♦♠ ✉♠ ♣❛râ♠❡tr♦ ❡s♣❡❝í✜❝♦✳

◆❛ s❡❣✉♥❞❛ ❣✉❡rr❛ ♠✉♥❞✐❛❧✱ ❛ ❢♦r♠❛ ❞❡ ❡st❡❣❛♥♦❣r❛✜❛ q✉❡ s❡ t♦r♥♦✉ ♣♦♣✉❧❛r ❢♦✐ ♦ ♠✐❝r♦♣♦♥t♦✳ ❆❣❡♥t❡s ❛❧❡♠ã❡s ♦♣❡r❛♥❞♦ ♥❛ ❆♠ér✐❝❛ ▲❛t✐♥❛✱ r❡❞✉③✐❛♠ ❢♦t♦❣r❛✜❝❛♠❡♥t❡ ✉♠❛ ♣á❣✐♥❛ ❞❡ t❡①t♦ ❛té tr❛♥s❢♦r♠á✲❧❛ ♥✉♠ ♣♦♥t♦ ❝♦♠ ♠❡♥♦s ❞❡ ✉♠ ♠✐❧í♠❡tr♦ ❞❡ ❞✐â♠❡tr♦✳ ❖ ♠✐❝r♦♣♦♥t♦ ❡r❛ ♦❝✉❧t❛❞♦ s♦❜r❡ ♦ ♣♦♥t♦ ✜♥❛❧ ❞❡ ✉♠❛ ❝❛rt❛ ❛♣❛r❡♥t❡♠❡♥t❡ ✐♥♦❢❡♥s✐✈❛ ❡ ♣❛r❛ s❛❜❡r ♦ ❝♦♥t❡ú❞♦ ❡r❛ ♥❡❝❡ssár✐❛ ✉♠❛ ❧✉♣❛✳

◆❡st❛ ♠❡s♠❛ é♣♦❝❛ ♦s ❜r✐tâ♥✐❝♦s ❝♦♥str✉ír❛♠ ♦ ♣r✐♠❡✐r♦ ❝♦♠♣✉t❛❞♦r ♣r♦❣r❛♠á✈❡❧ q✉❡ ❞❡❝✐❢r❛✈❛ ❛ ❝✐❢r❛ ❛❧❡♠ã ▲♦r❡♥③ ✉s❛❞❛ ♣❛r❛ ❡st❛❜❡❧❡❝❡r ❛ ❝♦♠✉♥✐❝❛çã♦ ❡♥tr❡ ❍✐t❧❡r ❡ s❡✉s ❣❡♥❡r❛✐s✳ ❊r❛ ♦ ✐♥í❝✐♦ ❞❛ ❝r✐♣t♦❣r❛✜❛ ♠♦❞❡r♥❛✳

✷✳✷ ❈♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❡ ❝r✐♣t♦❣r❛✜❛

❆ ♣❛❧❛✈r❛ ❝r✐♣t♦❣r❛✜❛ ✈❡♠ ❞♦ ❣r❡❣♦ ❝r②♣t♦s q✉❡ s✐❣♥✐✜❝❛ ♦❝✉❧t♦✱ s❡❝r❡t♦ ❡ ❡st✉❞❛ ♠ét♦❞♦s ♣❛r❛ ❝♦❞✐✜❝❛r ✉♠❛ ♠❡♥s❛❣❡♠ ♦♥❞❡ ❛♣❡♥❛s ♦ ❞❡st✐♥❛tár✐♦ ❧❡❣ít✐♠♦ t❡♠ ❢❡rr❛✲ ♠❡♥t❛s s✉✜❝✐❡♥t❡s ♣❛r❛ ✐♥t❡r♣r❡tá✲❧❛✳

❙❡❥❛♠ ❝♦♠♦ ❛♥t❡s✱ ❆❧✐❝❡ ♦ r❡♠❡t❡♥t❡ ❞❛ ♠❡♥s❛❣❡♠✱ ❇♦❜ ♦ ❞❡st✐♥át❛r✐♦ ❡ ❊✈❛ ✉♠❛ ✐♥✈❛s♦r❛✳ ❆ ♣❛rt✐r ❞❡ ✉♠ ❛❧❣♦r✐t♠♦ ❝r✐♣t♦❣rá✜❝♦ ❞❡ ❆❧✐❝❡ ✉t✐❧✐③❛♥❞♦ ✉♠❛ ❝❤❛✈❡ ❝r✐♣t♦✲ ❣r❛❢❛k ❡ ✉♠❛ ♠❡♥s❛❣❡♠ x✱ ♦❜t❡♠✲s❡ ✉♠❛ ♦✉tr❛ ♠❡♥s❛❣❡♠ y=fK(x)✱ q✉❡ ❝❤❛♠❛♠♦s

❞❡ ♠❡♥s❛❣❡♠ ❝r✐♣t♦❣r❛❢❛❞❛✳ ❆ ♠❡♥s❛❣❡♠ ❝r✐♣t♦❣r❛❢❛❞❛ y é ❡♥✈✐❛❞❛ ♣❛r❛ ❇♦❜ ♦♥❞❡ y é ❞❡❝r✐♣t♦❣r❛❢❛❞❛ ♣❡❧♦ ❛❧❣♦r✐t♠♦ ✐♥✈❡rs♦ f_K−1(y) ♦❜t❡♥❞♦✲s❡ x s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❇♦❜

❝♦♥❤❡❝❡ ❛ ❝❤❛✈❡ k✳

❈r✐♣t♦❣r❛✜❛ ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛ ♦✉ ❛ss✐♠étr✐❝❛ ✷✷

❞❡♥❞♦ ❞❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❞♦ ❛❧❣♦r✐t♠♦ ✉t✐❧✐③❛❞♦✳

✷✳✸ ❈r✐♣t♦❣r❛✜❛ ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛ ♦✉ ❛ss✐♠étr✐❝❛

■♠❛❣✐♥❡ q✉❡ ❆❧✐❝❡ ❞❡s❡❥❛ ❡♥✈✐❛r ✉♠❛ ♠❡♥s❛❣❡♠ ♣❡ss♦❛❧ ❡ ❛❧t❛♠❡♥t❡ s❡❝r❡t❛ ♣❛r❛ ❇♦❜✳ ❊❧❛ ❝♦❧♦❝❛ s✉❛ ❝❛rt❛ s❡❝r❡t❛ ❡♠ ✉♠❛ ❝❛✐①❛ ❞❡ ❢❡rr♦ ❝♦♠ ✉♠ ❝❛❞❡❛❞♦ ❡ ❡♥✈✐❛ ♣❛r❛ ❇♦❜✳ ❊st❡ ❝♦❧♦❝❛ ✉♠ ♦✉tr♦ ❝❛❞❡❛❞♦ ❡ ❡♥✈✐❛ ♣❛r❛ ❆❧✐❝❡ ♥♦✈❛♠❡♥t❡✳ ❆♦ r❡❝❡❜❡r✱ ❆❧✐❝❡ r❡t✐r❛ s❡✉ ❝❛❞❡❛❞♦ ❝♦❧♦❝❛❞♦ ✐♥✐❝✐❛❧♠❡♥t❡ ❡ r❡❡♥✈✐❛ ♣❛r❛ ❇♦❜✱ q✉❡ ❛❣♦r❛ ♣♦❞❡ ❛❜rí✲❧❛ ❡ ❧❡r ❛ ❝❛rt❛ ♣♦✐s ❛ ❝❛✐①❛ ❡stá tr❛♥❝❛❞❛ ❛♣❡♥❛s ♣❡❧♦ s❡✉ ❝❛❞❡❛❞♦✳ ❖❜s❡r✈❛♠♦s ❛q✉✐ q✉❡ é ♣♦ssí✈❡❧ r❡❛❧✐③❛r ❛ tr♦❝❛ ❞❡ ❝❤❛✈❡s ❛tr❛✈és ❞❡ ✉♠ ❝❛♥❛❧ ✐♥s❡❣✉r♦ ❥á q✉❡ t♦❞♦s ♣♦❞❡♠ s❛❜❡r ❞♦ tr❛♥s♣♦rt❡ ❞❛ ❝❛✐①❛ ❞❡ ❢❡rr♦✱ ♣♦ré♠ ♥✐♥❣✉é♠ ❛❧é♠ ❞❡ ❆❧✐❝❡ ❡ ❇♦❜ ♣♦❞❡♠ ❞❡s❝♦❜r✐r ♦ q✉❡ ❡st❛✈❛ ❡s❝r✐t♦ ♥❛ ❝❛rt❛ ♥♦ ✐♥t❡r✐♦r ❞❛ ❝❛✐①❛✳ ❊st❡ é ♦ ♣r✐♥❝í♣✐♦ ✉t✐❧✐③❛❞♦ ♥❛ tr♦❝❛ ❞❡ ❝❤❛✈❡s ❞❡ ❉✐✣❡✲❍❡❧❧♠❛♥ q✉❡ ❛❜r✐✉ ♣♦rt❛s ♣❛r❛ ♦ ❡st✉❞♦ ❞❛ ❝r✐♣t♦❣r❛✜❛ ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛✳

✷✳✸✳✶ Pr♦❜❧❡♠❛ ❞♦ ❧♦❣❛r✐t♠♦ ❞✐s❝r❡t♦ ✭P▲❉✮

❆♥t❡s ❞❡ ❝♦♥❤❡❝❡r♠♦s ♠❡❧❤♦r ❛ ❝r✐♣t♦❣r❛✜❛ ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛✱ ✈❛♠♦s ❝♦♥❤❡❝❡r ♦ q✉❡ é ♦ ♣r♦❜❧❡♠❛ ❞♦ ❧♦❣❛r✐t♠♦ ❞✐s❝r❡t♦ ✭P▲❉✮✱ ❥á q✉❡ ❛ ♠❡s♠❛ ❡stá ❜❛s❡❛❞❛ ♥❡st❡ ♣r♦❜❧❡♠❛✳

❉❡✜♥✐çã♦ ✷✳✶✳ ❙❡❥❛ ✉♠ ❣r✉♣♦ G ❡ y, α _∈G t❛❧ q✉❡ y é ♣♦tê♥❝✐❛ ❞❡ α✳ ❉✐③❡♠♦s q✉❡

♦ ❧♦❣❛r✐t♠♦ ❞✐s❝r❡t♦ ❞❡ y ♥❛ ❜❛s❡ α é ♦ ♠❡♥♦r ✐♥t❡✐r♦ ♥ã♦ ♥❡❣❛t✐✈♦ x t❛❧ q✉❡ αx _{= y✱}

❞❡♥♦t❛❞♦ ♣♦r logαy=x✳

❆ss✐♠ ♦ ♣r♦❜❧❡♠❛ ❞♦ ❧♦❣❛r✐t♠♦ ❞✐s❝r❡t♦ ❝♦♥s✐st❡ ❡♠ ❣❛r❛♥t✐r q✉❡ ♥ã♦ é ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛r t❛❧ x ❡♠ t❡♠♣♦ ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡ r❛③♦á✈❡❧✳

✷✳✸✳✷ ❆ tr♦❝❛ ❞❡ ❝❤❛✈❡s ❉✐✣❡✲❍❡❧❧♠❛♥

❖ ♣r♦❜❧❡♠❛ s♦❜r❡ tr♦❝❛ ❞❡ ✐♥❢♦r♠❛çõ❡s ❡♥tr❡ ❆❧✐❝❡ ❡ ❇♦❜ ❡♠ ✉♠ ❝❛♥❛❧ ✐♥s❡❣✉r♦✱ ❡①❡♠♣❧✐✜❝❛❞♦ ♥♦ ✐♥í❝✐♦ ❞❡st❡ ❝❛♣ít✉❧♦✱ ❢♦✐ ✉♠ ❞♦s q✉❡st✐♦♥❛♠❡♥t♦s q✉❡ ❡st✐♠✉❧♦✉ ❛ ❝r✐❛çã♦ ❞❛ tr♦❝❛ ❞❡ ❝❤❛✈❡s ♣ú❜❧✐❝❛s ♣♦r ❲❤✐t✜❡❧❞ ❉✐✣❡ ❡ ▼❛rt✐♥ ❍❡❧❧♠❛♥ ❝♦♠ ❝♦♥tr✐✲ ❜✉✐çõ❡s ❞❡ ❘❛❧♣❤ ▼❡r❦❧❡ ❡♠ ✶✾✼✻✳

❱❡❥❛♠♦s ❝♦♠♦ ❆❧✐❝❡ ❡ ❇♦❜ ❝♦♠❜✐♥❛♠ ✉♠❛ ❝❤❛✈❡ s❡❝r❡t❛ k ✉t✐❧✐③❛♥❞♦ ♦ ♠ét♦❞♦ ❞❡

❉✐✣❡✲❍❡❧❧♠❛♥✳

✶✳ ❆❧✐❝❡ ❡ ❇♦❜ ❡s❝♦❧❤❡♠ ♣✉❜❧✐❝❛♠❡♥t❡ ✉♠ ♣r✐♠♦p❡α_∈Z∗_p t❛❧ q✉❡ m.d.c(α, p) = 1❀

✷✳ ❆❧✐❝❡ ❡s❝♦❧❤❡ ❛❧❡❛t♦r✐❛♠❡♥t❡ ✉♠ ♥ú♠❡r♦ a t❛❧ q✉❡ 1 _≤ a _≤ p₋2 ❡ ❡♥✈✐❛ ♣❛r❛

❇♦❜ ♦ ♥ú♠❡r♦ m=αa _{mod p❀}

❈r✐♣t♦❣r❛✜❛ ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛ ♦✉ ❛ss✐♠étr✐❝❛ ✷✸

✹✳ ❆❧✐❝❡ ❝❛❧❝✉❧❛ k =na_{= (α}b₎a _modp❀

✺✳ ❇♦❜ ❝❛❧❝✉❧❛k =mb _{= (α}a₎b _modp✳

◆♦ ✜♠ ❞❡st❡ ♣r♦❝❡ss♦✱ ❆❧✐❝❡ ❡ ❇♦❜ ❝♦♠♣❛rt✐❧❤❛♠ ❞❛ ❝❤❛✈❡k❡ ❛s ♠❡♥s❛❣❡♥s ❝✐❢r❛❞❛s

❝♦♠ ❡st❛ ❝❤❛✈❡ ♣♦❞❡rã♦ s❡r ❞❡❝✐❢r❛❞❛s ❛♣❡♥❛s ❝♦♠ ❛ ❝❤❛✈❡ ♣r✐✈❛❞❛ ❝♦rr❡s♣♦♥❞❡♥t❡✳ ❊①❡♠♣❧♦ ✷✳✶✳ ❙❡❥❛ ♦ ❣r✉♣♦ Z17 ❡ α = 3✳ ❆❧✐❝❡ ❡s❝♦❧❤❡ a = 7 ❡ ❝❛❧❝✉❧❛ m = 37 ≡ 11

mod 17 ❡ ❡♥✈✐❛ ♦ r❡s✉❧t❛❞♦ ♣❛r❛ ❇♦❜✳ ❇♦❜ ❡s❝♦❧❤❡ b = 4 ❡ ❝❛❧❝✉❧❛ n = 34 _{≡ 18}

mod 17 ❡ ❡♥✈✐❛ ♦ r❡s✉❧t❛❞♦ n ♣❛r❛ ❆❧✐❝❡✳ P♦r s✉❛ ✈❡③✱ ❆❧✐❝❡ ❝❛❧❝✉❧❛ n7 _{≡ 4 mod 17 ❡} ❇♦❜ ❝❛❧❝✉❧❛ m4 _{≡ 4 mod 17✳ ❉❡st❛ ❢♦r♠❛ ❛ ❝❤❛✈❡ s❡❝r❡t❛ k ❡♥❝♦♥tr❛❞❛ ♣❡❧♦ ♠ét♦❞♦} ❞❡ ❉✐✣❡✲❍❡❧❧♠❛♥ é 4✳

❱❛♠♦s ♣❡♥s❛r ❛❣♦r❛ ❡♠ ✉♠❛ t❡r❝❡✐r❛ ♣❡ss♦❛✱ ❊✈❛✱ q✉❡ ❝♦♥❤❡❝❡ p ❡ α✱ ♣♦✐s ❡st❡s

sã♦ ❡s❝♦❧❤✐❞♦s ♣✉❜❧✐❝❛♠❡♥t❡ ❡ ❛✐♥❞❛ ❝♦♥s❡❣✉❡ ❞❡s❝♦❜r✐r n ❡ m✱ ♣♦✐s ❡st❡s ✈❛❧♦r❡s sã♦

tr❛♥s♠✐t✐❞♦s ❡♠ ✉♠ ❝❛♥❛❧ ✐♥s❡❣✉r♦✱ ♦ q✉❡ ❢❛❝✐❧✐t❛ ❡ss❡ ❛❝❡ss♦✳ P♦❞❡r✐❛ ❊✈❛ ❞❡s❝♦❜r✐r q✉❛❧ ❛ ❝❤❛✈❡ s❡❝r❡t❛ ❞❡ ❆❧✐❝❡ ❡ ❇♦❜❄

❊st❛ q✉❡stã♦ ❡♥❝♦♥tr❛ s✉❛ r❡s♣♦st❛ ♥❛ ❣❛r❛♥t✐❛ ❞❡ s❡❣✉r❛♥ç❛ ❞♦ ♠ét♦❞♦✱ ❜❛s❡❛❞❛ ♥❛ ❞✐✜❝✉❧❞❛❞❡ ❞❡ s♦❧✉çã♦ ❞♦ Pr♦❜❧❡♠❛ ❞♦ ▲♦❣❛r✐t♠♦ ❉✐s❝r❡t♦✳ P❛r❛ ❡♥❝♦♥tr❛r ❛ ❝❤❛✈❡ s❡❝r❡t❛✱ ❊✈❛ ♣r❡❝✐s❛ ❡♥❝♦♥tr❛r b❛trá✈❡s ❞♦ ❧♦❣❛r✐t♠♦ ❞✐s❝r❡t♦ ❞❡n ♥❛ ❜❛s❡α❡ ❝❛❧❝✉❧❛r k = mb_{✳ ▲♦❣♦✱ q✉❛♥t♦ ♠❛✐♦r ❛ ❞✐✜❝✉❧❞❛❞❡ ❡♠ r❡s♦❧✈❡r ❡st❡ ❛❧❣♦r✐t♠♦✱ ♥ã♦ t❡rá ❝♦♠♦}

✉♠ ❡s♣✐ã♦ ❝♦♠♦ ❊✈❛ ❞❡s❝♦❜r✐r ❛ ❝❤❛✈❡ s❡❝r❡t❛ ❛trá✈❡s ❞❛s ✐♥❢♦r♠❛çõ❡s ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ ♣ú❜❧✐❝♦✳

❆té ♦s ❞✐❛s ❛t✉❛✐s ♥ã♦ ❢♦✐ ♣r♦✈❛❞♦ q✉❡ s❡ ❊✈❛ ❞❡s❝♦❜r✐r ♦ ♣r♦❜❧❡♠❛ ❞♦ ❧♦❣❛r✐t♠♦ ❞✐s❝r❡t♦✱ ❡❧❛ ❝♦♥s❡❣✉✐rá ❝❛❧❝✉❧❛r ❞❡ ❢♦r♠❛ ❡✜❝✐❡♥t❡ ❧♦❣❛r✐t♠♦s ❞✐s❝r❡t♦s modp✱ ❡ ♣♦r

✐ss♦ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ♦ ♠ét♦❞♦ ❞❡ tr♦❝❛ ❞❡ ❝❤❛✈❡s ❞❡ ❉✐✣❡✲❍❡❧❧♠❛♥ é ❛✐♥❞❛ s❡❣✉r♦ ♥❛ ❛t✉❛❧✐❞❛❞❡✳

✷✳✸✳✸

❖ ♠ét♦❞♦ ❞❡ ❊❧●❛♠❛❧

❖ ♠ét♦❞♦ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ❚❛❤❡r ❊❧●❛♠❛❧ ❡♠ ✶✾✽✺ é ❜❛s❡❛❞♦ t❛♠❜é♠ ♥❛ ❞✐✜❝✉❧✲ ❞❛❞❡ ❞❡ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞♦ ❧♦❣❛r✐t♠♦ ❞✐s❝r❡t♦✳ ❯t✐❧✐③❛♠♦s ❣❡r❛❧♠❡♥t❡ ❡st❡ ♠ét♦❞♦ s♦❜r❡ ♦ ❣r✉♣♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦Z∗_p✱ ♠❛s ♣♦❞❡♠♦s ❣❡♥❡r❛❧✐③á✲❧♦ ♣❛r❛ q✉❛❧q✉❡r ❣r✉♣♦ ❝í❝❧✐❝♦

✜♥✐t♦G✱ ❞❡s❞❡ q✉❡ ❛s ♦♣❡r❛çõ❡s s♦❜r❡ ❡st❡ s❡❥❛♠ ❞❡ ❢á❝✐❧ ❛♣❧✐❝❛çã♦ ❡ q✉❡ s❡❥❛ ✐♥tr❛tá✈❡❧

♦ ♣r♦❜❧❡♠❛ ❞♦ ❧♦❣❛r✐t♠♦ ❞✐s❝r❡t♦ ♥♦ ❣r✉♣♦ G✳

P❛r❛ ❛ ❣❡r❛çã♦ ❞❡ ❝❤❛✈❡s✱ ❆❧✐❝❡ ❡s❝♦❧❤❡ ✉♠ ❣r✉♣♦ ❝í❝❧✐❝♦ ✜♥✐t♦ G ❝♦♠ ♦r❞❡♠ n ❡

❣❡r❛❞♦r g ❡ s❡❧❡❝✐♦♥❛ ✉♠ ✐♥t❡✐r♦ a t❛❧ q✉❡ 1 _≤ a _≤ n₋2 ❡ ❝❛❧❝✉❧❛ A = ga_{✳ ❆ss✐♠}

é ♣r♦❞✉③✐❞❛ ❛ ❝❤❛✈❡ ♣ú❜❧✐❝❛ (A, g)✱ ♦♥❞❡ ❞❡✈❡✲s❡ ❧❡♠❜r❛r q✉❡ ♦ ❣r✉♣♦ G é ❝♦♥❤❡❝✐❞♦✳

❚❡♠♦s ❛ ♣❛rt✐r ❞❛í ✉♠ Pr♦❜❧❡♠❛ ❞❡ ▲♦❣❛r✐t♠♦ ❉✐s❝r❡t♦✱ ❥á q✉❡ a é ❝♦♥❤❡❝✐❞♦ ❛♣❡♥❛s

♣♦r ❆❧✐❝❡✳