❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛
■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠
▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧
■◆❚❘❖❉❯➬➹❖ ❆ ❙■❙❚❊▼❆❙
❈❘■P❚❖●❘➪❋■❈❖❙ ❊ ❖ ❯❙❖ ❉❊
●❊❘❆❉❖❘❊❙ ❉❊ ❙❊◗❯✃◆❈■❆❙ ❉❊
◆Ú▼❊❘❖❙ ❆▲❊❆❚Ó❘■❖❙ ❊
P❙❊❯❉❖✲❆▲❊❆❚Ó❘■❖❙
❊❉❯❆❘❉❖ ❈❮❈❊❘❖ ❱■❊■❘❆ ❇❖❘●❊❙ ❏❯◆■❖❘
❇r❛sí❧✐❛
Ficha catalográfica elaborada pela Biblioteca Central da Universidade de Brasília. Acervo 1019073.
Bo r ges Jun i o r , Edua r do Cí ce r o V i e i r a .
B732 i I n t r odução a s i s t emas c r i p t og r á f i cos e o uso de ge r ado r es de sequênc i as de núme r os a l ea t ó r i os e pseudo - a l ea t ó r i os / Edua r do Cí ce r o Vi e i r a Bo r ges Jun i o r . - - 2014 .
54 f . : i l . ; 30 cm.
Di sse r t ação (mes t r ado ) - Un i ve r s i dade de Br as í l i a , I ns t i t u t o de Ci ênc i as Exa t as , Depa r t amen t o de Ma t emá t i ca , 2014 .
Or i en t ação : Ru i Se ime t z . I nc l u i b i b l i og r a f i a .
1 . Cr i p t og r a f i a - Ma t emá t i ca . 2 . Va r i áve i s a l ea t ó r i as . I . Se ime t z , Ru i . I I . T í t u l o .
❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛
■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠
▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧
■◆❚❘❖❉❯➬➹❖ ❆ ❙■❙❚❊▼❆❙
❈❘■P❚❖●❘➪❋■❈❖❙ ❊ ❖ ❯❙❖ ❉❊
●❊❘❆❉❖❘❊❙ ❉❊ ❙❊◗❯✃◆❈■❆❙ ❉❊
◆Ú▼❊❘❖❙ ❆▲❊❆❚Ó❘■❖❙ ❊
P❙❊❯❉❖✲❆▲❊❆❚Ó❘■❖❙
❆✉t♦r✿ ❊❞✉❛r❞♦ ❈í❝❡r♦ ❱✐❡✐r❛ ❇♦r❣❡s ❏✉♥✐♦r ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❘✉✐ ❙❡✐♠❡t③
▼♦♥♦❣r❛✜❛ ❛♣r❡s❡♥t❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛ Pr♦✜ss✐♦♥❛❧ ♣❡❧♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛✳
❇r❛sí❧✐❛
❊❉❯❆❘❉❖ ❈❮❈❊❘❖ ❱■❊■❘❆ ❇❖❘●❊❙ ❏❯◆■❖❘
■◆❚❘❖❉❯➬➹❖ ❆ ❙■❙❚❊▼❆❙ ❈❘■P❚❖●❘➪❋■❈❖❙ ❊ ❖ ❯❙❖ ❉❊ ●❊❘❆❉❖❘❊❙ ❉❊ ❙❊◗❯✃◆❈■❆❙ ❉❊ ◆Ú▼❊❘❖❙ ❆▲❊❆❚Ó❘■❖❙ ❊ P❙❊❯❉❖✲❆▲❊❆❚Ó❘■❖❙
▼♦♥♦❣r❛✜❛ ❛♣r❡s❡♥t❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛ Pr♦✜ss✐♦♥❛❧ ♣❡❧♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛✳
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛
❇r❛sí❧✐❛
❆❣r❛❞❡❝✐♠❡♥t♦s
➚ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ✭❙❇▼✮ ♣♦r ♠❡✐♦ ❞♦ P❘❖❋▼❆❚✱ ♣♦rq✉❡ ♣r♦♣✐❝✐❛ ✉♠❛ ♠❡❧❤♦r ❝❛♣❛❝✐t❛çã♦ ❞♦s ❡❞✉❝❛❞♦r❡s ♥♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛✳
❆♦s ♣r♦❢❡ss♦r❡s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛ ✭❯♥❇✮ q✉❡ ♣❛rt✐❝✐♣❛r❛♠ ❞♦ ❝✉rs♦ ❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ✉♠❛ ♠❡❧❤♦r ❢♦r♠❛çã♦ ❞♦s ❛❧✉♥♦s✳
❆♦ ♣r♦❢❡ss♦r ❉r✳ ❘✉✐ ❙❡✐♠❡t③✱ ❝♦♦r❞❡♥❛❞♦r ❞♦ P❘❖❋▼❆❚ ♥❛ ❯♥❇✱ ♣❡❧❛ ❞❡❞✐❝❛✲ çã♦ ♥❛ ❝♦♥❞✉çã♦ ❞♦ ❝✉rs♦✱ ❜❡♠ ❝♦♠♦ ♥❛ ❝♦❧❛❜♦r❛çã♦✱ ♣❛❝✐ê♥❝✐❛ ❡ ❡♥s✐♥❛♠❡♥t♦s q✉❡ ♣❡r♠✐t✐r❛♠ ❛ ❡❧❛❜♦r❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❆♦s ♠❡✉s ❛♠✐❣♦s ▲♦♣❡s✱ ❉❛✈✐❞ ❡ ❋❡❧✐♣❡✱ ♣❡❧❛ ❛♠✐③❛❞❡✱ ❡ ♣r❡s❡♥ç❛ ♥♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s✳
❆♦s ❝♦❧❡❣❛s ❞❡ t✉r♠❛✱ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ♦s ❛♠✐❣♦s ❞♦ ❈♦❧é❣✐♦ Pó❞✐♦♥✱ ♣❡❧♦ ❛✉①í❧✐♦ ❡ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❛♦ ❧♦♥❣♦ ❞❡ t♦❞♦ ♦ ❝✉rs♦✳
❆♦s ❢❛♠✐❧✐❛r❡s✱ ♣❛✐s✱ ✐r♠ã♦✱ ❡s♣♦s❛✱ ♣❡❧♦ ❛♣♦✐♦✱ ❝♦♠♣r❡❡♥sã♦ ❡ ✐♥❝❡♥t✐✈♦ ♣❛r❛ q✉❡ t✉❞♦ ❞❡ss❡ ❝❡rt♦✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ ✐r❡♠♦s ❛❜♦r❞❛r ❛❧❣✉♥s tó♣✐❝♦s r❡❢❡r❡♥t❡s à ❝r✐♣t♦❣r❛✜❛ ❡❧❡♠❡♥t❛r✱ ❞❡ s✐st❡♠❛s ❝r✐♣t♦❣rá✜❝♦s s✐♠étr✐❝♦s ❡ ❛ss✐♠étr✐❝♦s✱ ❛♣r❡s❡♥t❛❞♦s ❞❡ ♠♦❞♦ ❛ ❡①♣♦r ❛ ✐♥t❡r♣r❡t❛çã♦ ♠❛t❡♠át✐❝❛ ❛❝❡r❝❛ ❞♦s ♣r✐♥❝✐♣❛✐s ♠ét♦❞♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛ ❡♥❝r✐♣t❛çã♦ ❞❡ ❞❛❞♦s✳
❆❜str❛❝t
■♥ t❤✐s ♣❛♣❡r ✇❡ ❞✐s❝✉ss s♦♠❡ t♦♣✐❝s r❡❧❛t❡❞ t♦ ❡❧❡♠❡♥t❛r② ❝r②♣t♦❣r❛♣❤②✱ s②♠♠❡tr✐❝ ❛♥❞ ❛s②♠♠❡tr✐❝ ❝r②♣t♦s②st❡♠s✱ ♣r❡s❡♥t❡❞ t♦ ❡①♣♦s❡ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ✐♥t❡r♣r❡t❛t✐♦♥ ❛❜♦✉t t❤❡ ♠❛✐♥ ♠❡t❤♦❞s ✉s❡❞ ❢♦r ❞❛t❛ ❡♥❝r②♣t✐♦♥✳
❆❧s♦ ✐♥ t❤✐s ✇♦r❦ ❛r❡ ♣r❡s❡♥t❡❞ st✉❞✐❡s ♦♥ t❤❡ ♠❛❥♦r ♠❡t❤♦❞s ♦❢ r❛♥❞♦♠ ♥✉♠❜❡r ❣❡♥❡r❛t♦rs ❛♥❞ ❛✉①✐❧✐❛r② ❣❡♥❡r❛t♦rs t♦ t❤❡s❡ ♠❡t❤♦❞s✱ ❛s ✇❡❧❧ ❛s t❤❡✐r ♥❡❡❞ ❛♥❞ r❡❧❛t✐♦♥ ✇✐t❤ ❡♥❝r②♣t✐♦♥✳ ❚❤❡s❡ ❛✉①✐❧✐❛r② st✉❞✐❡s✱ ❛s ✇✐❧❧ ❜❡ s❤♦✇♥✱ ♣r♦♠♦t❡ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❛ ♠♦r❡ r♦❜✉st s②st❡♠✱ ♠♦r❡ ❞✐✣❝✉❧t t♦ ❜r❡❛❦ t❤❡ ❝♦❞❡✳
❙✉♠ár✐♦
❘❡s✉♠♦ ✼
❆❜str❛❝t ✽
✶ ■♥tr♦❞✉çã♦ ✶✶
✶✳✶ ❈❖◆❚❊❳❚❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✶✳✷ ❖❇❏❊❚■❱❖❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✷ ❈❘■P❚❖●❘❆❋■❆ ✶✷
✷✳✶ ❙■❙❚❊▼❆❙ ❈❘■P❚❖●❘➪❋■❈❖❙ ❙■▼➱❚❘■❈❖❙ ❊ ❆❙❙■▼➱❚❘■❈❖❙ ✳ ✶✸
✷✳✷ ❆▲●❖❘■❚▼❖❙ ❉❊ ❙■❙❚❊▼❆❙ ❙■▼➱❚❘■❈❖❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✷✳✷✳✶ ❖◆❊✲❚■▼❊✲P❆❉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✷✳✷✳✷ ❉❊❙ ✭❉❆❚❆ ❊◆❈❘❨P❚■❖◆ ❙❚❆◆❉❆❘❉✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✷✳✷✳✸ ❚❘■P▲❊ ❉❊❙ ✭✸❉❊❙✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✷✳✷✳✹ ❘❈✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✷✳✸ ❆▲●❖❘■❚▼❖❙ ❉❊ ❙■❙❚❊▼❆❙ ❆❙❙■▼➱❚❘■❈❖❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✷✳✸✳✶ ❉■❋❋■❊✲❍❊▲▲▼❆◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✷✳✸✳✷ ❘❙❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷✳✹ ❆❚❆◗❯❊❙ ❆❖❙ ❙■❙❚❊▼❆❙ ❈❘■P❚❖●❘➪❋■❈❖❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✷✳✺ ❯❚■▲■❩❆➬➹❖ ❉❊ ◆Ú▼❊❘❖❙ ❆▲❊❆❚Ó❘■❖❙ ❊▼ ❈❘■P❚❖●❘❆❋■❆ ✷✻
✸ ●❊❘❆❉❖❘❊❙ ❉❊ ◆Ú▼❊❘❖❙ ❆▲❊❆❚Ó❘■❖❙ ✷✻
✸✳✶ ❋❖◆❚❊❙ ❆▲❊❆❚Ó❘■❆❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✸✳✶✳✶ ▲❆◆➬❆▼❊◆❚❖ ❉❊ ▼❖❊❉❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✸✳✶✳✷ ❘❯❮❉❖ ❉❖ ❉■❖❉❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✸✳✶✳✸ ▼❆❚❊❘■❆▲ ❘❆❉■❖❆❚■❱❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✸✳✶✳✹ ❚✶✷❘◆● ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✹
●❊❘❆❉❖❘❊❙ ❉❊ ◆Ú▼❊❘❖❙ P❙❊❯❉❖✲❆▲❊❆❚Ó❘■❖❙
✷✾✹✳✶ ●❊❘❆➬➹❖ ❉❊ ❙❊◗Ü✃◆❈■❆❙ ❉❊ ◆Ú▼❊❘❖❙ ❆▲❊❆❚Ó❘■❖❙ ❯❚■▲■✲ ❩❆◆❉❖ ❙❖❋❚❲❆❘❊❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✹✳✷ ❆▲●❖❘■❚▼❖❙ ❉❊ ●❊❘❆➬➹❖ ❉❊ ❙❊◗Ü✃◆❈■❆❙ ❉❊ ◆Ú▼❊❘❖❙ P❙❊❯❉❖✲❆▲❊❆❚Ó❘■❖❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✹✳✷✳✶ ▼➱❚❖❉❖ ❈❖◆●❘❯❊◆❚❊ ▲■◆❊❆❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✹✳✷✳✷ ▲❋❙❘ ✭▲■◆❊❆❘ ❋❊❊❉❇❆❈❑ ❙❍■❋❚ ❘❊●■❙❚❊❘✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✹✳✷✳✸ ●❊❘❆❉❖❘ ●❊❋❋❊ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✺ ❚❘❆◆❙P❖❙■➬➹❖ ❉■❉➪❚■❈❆ ✸✺
✺✳✶ ❙❯●❊❙❚Õ❊❙ ❉❊ ❆❯▲❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✺✳✶✳✶ ❉■❋❋■❊✲❍❊▲▼❆◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✺✳✶✳✷ ❘❙❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✺✳✷ ❚❘❆❇❆▲❍❖❙ ❋❯❚❯❘❖❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽
✺✳✸ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
❆P✃◆❉■❈❊ ✺✷
✶ ■♥tr♦❞✉çã♦
✶✳✶ ❈❖◆❚❊❳❚❖
❆ s❡❣✉r❛♥ç❛ ❞♦s s✐st❡♠❛s ❞❡ ✐♥❢♦r♠❛çã♦ é ♦❜t✐❞❛ ♣♦r ♠❡✐♦ ❞❡ s✐st❡♠❛s ❝r✐♣t♦✲ ❣rá✜❝♦s ❞❡st✐♥❛❞♦s ❛ ♠♦❞✐✜❝❛r ♦ ❝ó❞✐❣♦ ❡s❝r✐t♦ ♣❛r❛ ♦✉tr♦ ♥ã♦ ❧❡❣í✈❡❧ ❛ q✉❡♠ ♥ã♦ ❢♦r ❡♥❞❡r❡ç❛❞❛ t❛❧ ✐♥❢♦r♠❛çã♦✳
❊st❛ s❡❣✉r❛♥ç❛ ♦❜t✐❞❛ ♣♦r ♠✉✐t♦s s✐st❡♠❛s ❝r✐♣t♦❣rá✜❝♦s ❞❡♣❡♥❞❡ ❞❛ ❣❡r❛çã♦ ❞❡ ❞❛✲ ❞♦s ♥ã♦ ❝♦rr❡❧❛t♦s✱ ✐♥❞❡♣❡♥❞❡♥t❡s✱ q✉❡ ♥ã♦ s❡ ♣♦❞❡♠ ♣r❡✈❡r✳ ❈♦♠♦ ❡①❡♠♣❧♦s ♣♦❞❡♠ s❡r ❝✐t❛❞♦s ❛ s❡q✉ê♥❝✐❛ ❝❤❛✈❡ ❞♦ ♦♥❡✲t✐♠❡✲♣❛❞✱ ❛ ❝❤❛✈❡ ❞♦ ❛❧❣♦r✐t♠♦ ❞❡ ❡♥❝r✐♣t❛çã♦ ❉❊❙✱ ♦s ♣r✐♠♦s ♣ ❡ q ❞♦ ❛❧❣♦r✐t♠♦ ❘❙❆ ❡ ❞♦s ❡sq✉❡♠❛s ❞❡ ❛ss✐♥❛t✉r❛s ❞✐❣✐t❛✐s✳
❊♠ t♦❞♦s ❡ss❡s ❝❛s♦s ♦s ❞❛❞♦s ❞❡✈❡♠ s❡r ❛❧❡❛tór✐♦s ❡ s✉❛ q✉❛♥t✐❞❛❞❡ ❞❡✈❡ s❡r ❣r❛♥❞❡ ♦ s✉✜❝✐❡♥t❡✱ ❞❡ ♠❛♥❡✐r❛ q✉❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ s❡ q✉❡❜r❛r ♦ ❝ó❞✐❣♦ s❡❥❛ ♠í♥✐♠❛ ❡ q✉❡ ❛ s✉❛ ♦❜t❡♥çã♦ ♥ã♦ s❡❥❛ tã♦ ❝✉st♦s❛✳
❊♠ t❡r♠♦s ♣rát✐❝♦s✱ ♣♦❞❡✲s❡ ♣❡r❝❡❜❡r q✉❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝❤❛✈❡s ♣♦ssí✈❡✐s ♣❛r❛ ♦ ❉❊❙ é ❞❡256✱ s❡ ✉♠❛ ❝❤❛✈❡ ❦ ❢♦r s❡❧❡❝✐♦♥❛❞❛ ❞❡ ✉♠ ❣❡r❛❞♦r ✈❡r❞❛❞❡✐r❛♠❡♥t❡ ❛❧❡❛tór✐♦✳
❯♠ ✐♥✐♠✐❣♦ ♣♦❞❡ t❡♥t❛r ❛té 255 ❝❤❛✈❡s ❛♥t❡s ❞❡ ❞❡s❝♦❜r✐r✱ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ 1
2✱ ❛
❝❤❛✈❡ ❝♦rr❡t❛ ❦✳ ▼❛s s❡ ❡st❛ t✐✈❡r s✐❞♦ ❝r✐❛❞❛ ❞❛ ❡①♣❛♥sã♦ ❞❡ ✶✻ ❜✐ts ✈❡r❞❛❞❡✐r❛♠❡♥t❡ ❛❧❡❛tór✐♦s ❡♠ ✺✻ ❜✐ts✱ ♣♦r ✉♠ ❣❡r❛❞♦r ♣s❡✉❞♦✲❛❧❡❛tór✐♦✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ t❡♥t❛t✐✈❛s ❝❛✐ ♣❛r❛ ✷✶✺✳ ■st♦ ❞❡♠♦♥str❛ ♦ ❞✉❛❧ ❡♥tr❡ s❡❣✉r❛♥ç❛ ❡ ❢❛❝✐❧✐❞❛❞❡ ❞❡ ✐♠♣❧❡♠❡♥t❛çã♦✳ ❆ss✐♠ ♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ é ❞❡s❡♥✈♦❧✈✐❞♦ ♥♦ â♠❜✐t♦ ❞❛ ❛♥á❧✐s❡ ♠❛t❡♠át✐❝❛ ❞♦s ♣r✐♥❝✐♣❛✐s ♠ét♦❞♦s ❝r✐♣t♦❣rá✜❝♦s✱ s✐♠étr✐❝♦s ❡ ❛ss✐♠étr✐❝♦s✱ ❜❡♠ ❝♦♠♦ ❞❛ ❡①♣♦s✐çã♦ ❞❡ ❛❧❣✉♥s ❞♦s s✐st❡♠❛s ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛ ❣❡r❛çã♦ ❞❡ ♥ú♠❡r♦s ❛❧❡❛tór✐♦s ❡ ♣s❡✉❞♦✲❛❧❡❛tór✐♦s✳
✶✳✷ ❖❇❏❊❚■❱❖❙
❖ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❞♦ tr❛❜❛❧❤♦ é ✐♥tr♦❞✉③✐r ❛ ❝r✐♣t♦❣r❛✜❛ ❝♦♠♦ ✉♠ t♦❞♦✱ ❜❡♠ ❝♦♠♦ ❡①♣♦r ❛❧❣✉♥s ❞♦s ♠ét♦❞♦s ❝r✐♣t♦❣rá✜❝♦s✱ ✉♠❛ ✈❡③ q✉❡ ❡st❡s sã♦ ❛♣❧✐❝❛çõ❡s ❞✐r❡t❛s ❡ ♣rát✐❝❛s ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s ❡ ❝♦♠♣✉t❛❝✐♦♥❛✐s ❡❧❡♠❡♥t❛r❡s✳
❊♠ s❡❣✉♥❞♦ ♣❧❛♥♦ ♣♦❞❡ s❡r ❞✐t♦ q✉❡ ❡st❡ tr❛❜❛❧❤♦ ❡s❝❧❛r❡❝❡ ♦ ❧❡✐t♦r ❡♠ r❡❧❛çã♦ às ✉t✐❧✐③❛çõ❡s ❡ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ❣❡r❛çã♦ ❞❡ s❡q✉ê♥❝✐❛s ❞❡ ♥ú♠❡r♦s ❛❧❡❛tór✐♦s ♣❛r❛ ❛ ❝r✐♣t♦❣r❛✜❛✱ s❡♠ ❝♦♥t❛r ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ❝r✐♣t♦❣r❛✜❛ ❡♠ s✐✳
✷ ❈❘■P❚❖●❘❆❋■❆
❆ ❝r✐♣t♦❣r❛✜❛ ❡stá ✐♥t✐♠❛♠❡♥t❡ ❧✐❣❛❞❛ ❛♦ ❝✉♥❤♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❞✐s❝✉t✐r ❛❧❣✉♥s ❞♦s ♣r✐♥❝✐♣❛✐s ♠ét♦❞♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛ ❡♥❝r✐♣t❛çã♦ ❞❡ ♠❡♥s❛❣❡♥s✱ s❡❥❛♠ ❡st❡s ❞❡ ❝r✐♣t♦❣r❛✜❛ s✐♠étr✐❝❛✱ ♦✉ ❛ss✐♠étr✐❝❛✱ ❜❡♠ ❝♦♠♦ ❛♣r❡s❡♥t❛r ❛❧❣✉♥s ♠❡❝❛♥✐s♠♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛ ❣❡r❛çã♦ ❞❡ s❡q✉ê♥❝✐❛s ❞❡ ♥ú♠❡r♦s ❛❧❡❛tór✐♦s ❡ ♣s❡✉❞♦✲❛❧❡❛tór✐♦s✳
❖ ❡♥✈♦❧✈✐♠❡♥t♦ ❞♦s ♥ú♠❡r♦s ❛❧❡❛tór✐♦s ♥♦s ♣r♦❝❡ss♦s ❝r✐♣t♦❣rá✜❝♦s✱ ❛❝✐♠❛ ❞❡ t✉❞♦✱ ❝♦♥tr✐❜✉✐ ♣❛r❛ ❛ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ❞❛ ❝♦♠✉♥✐❝❛çã♦✱ t❡♥t❛ ❡❧✐♠✐♥❛r ❛s ❝❤❛♥❝❡s ❞❡ r♦♠♣✐✲ ♠❡♥t♦ ❞❡ ❝ó❞✐❣♦ ♣♦r ♣❡ss♦❛s ❛❧❤❡✐❛s ❡ q✉❛♥❞♦ ✐ss♦ ♥ã♦ ❢♦r ♣♦ssí✈❡❧✱ ❛❞✐❛ ❛♦ ♠á①✐♠♦ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛ ✈❡r❞❛❞❡✐r❛ ✐♥❢♦r♠❛çã♦ tr❛♥s♠✐t✐❞❛✱ ❛té q✉❡ ♦✉tr♦ ♠❡✐♦ ♠❛✐s ❝♦♥✜á✈❡❧ s❡❥❛ ❡st❛❜❡❧❡❝✐❞♦✳ ❆ ❝r✐♣t♦❣r❛✜❛ s❡ ❝♦♠♣r♦♠❡t❡ ❛ ❢♦r♥❡❝❡r s❡r✈✐ç♦s ❞❡ s❡❣✉r❛♥ç❛ ❞❛ ✐♥❢♦r♠❛çã♦✳ ❆❧❣✉♥s ❞❡❧❡s sã♦✿
• ❙❡r✈✐ç♦ ❝♦♥✜❞❡♥❝✐❛❧✿ ❝♦♥✜❛♥ç❛ ❡♠ q✉❡ ❛ ♠❡♥s❛❣❡♠ ♣♦ss❛ s❡r ❧✐❞❛ ❛♣❡♥❛s ♣❡❧♦
r❡❝❡♣t♦r ❞❡s❡❥❛❞♦✳
• ❆✉t❡♥t✐❝❛çã♦ ❞❛ ♦r✐❣❡♠✿ ❣❛r❛♥t✐❛ s♦❜r❡ q✉❡♠ ♦r✐❣✐♥♦✉ ❛ ♠❡♥s❛❣❡♠✳
• ■♥t❡❣r✐❞❛❞❡✿ ❝♦♥✜❛♥ç❛ ❡♠ q✉❡ ❛ ♠❡♥s❛❣❡♠ ♥ã♦ t❡♥❤❛ s✐❞♦ ♠♦❞✐✜❝❛❞❛ ❞❡s❞❡ ♦
♠♦♠❡♥t♦ ❞❛ ❝r✐❛çã♦✳
• ◆ã♦✲r❡tr❛t❛çã♦✿ q✉❡♠ ♦r✐❣✐♥♦✉ ❛ ♠❡♥s❛❣❡♠ ♥ã♦ ♣♦❞❡ ♥❡❣❛r tê✲❧♦ ❢❡✐t♦✳
P♦r ❡st❛s q✉❡stõ❡s✱ é ✈á❧✐❞♦ s❛❧✐❡♥t❛r ❛s♣❡❝t♦s ❞❡ ❝♦♠♣r♦♠❡t✐♠❡♥t♦ ❞❡ ❝ó❞✐❣♦ ❡ ♣♦ssí✈❡✐s ❛t❛q✉❡s ❛♦s ❝ó❞✐❣♦s ❡st❛❜❡❧❡❝✐❞♦s✳ ❊st❡s ❡s❝❧❛r❡❝✐♠❡♥t♦s ❝♦♥st❛♠ ♥♦ ✜♥❛❧ ❞❡st❡ ❝❛♣ít✉❧♦✳
✷✳✶ ❙■❙❚❊▼❆❙ ❈❘■P❚❖●❘➪❋■❈❖❙ ❙■▼➱❚❘■❈❖❙ ❊ ❆❙❙■✲
▼➱❚❘■❈❖❙
❊①✐st❡♠ ❞♦✐s t✐♣♦s ❞❡ ❝r✐♣t♦❣r❛✜❛✱ ❛ ❝r✐♣t♦❣r❛✜❛ s✐♠étr✐❝❛ ❡ ❛ ❛ss✐♠étr✐❝❛✳
❆ ❝r✐♣t♦❣r❛✜❛ s✐♠étr✐❝❛ ♦♣❡r❛ ❝♦♠ ❛ ♣r❡♠✐ss❛ ❞❡ q✉❡ ❛ ❝❤❛✈❡ é ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ ✉♥✐❝❛♠❡♥t❡ ❞♦ ❡♠✐ss♦r ❡ ❞♦ r❡❝❡♣t♦r✳
◆❛ ❝r✐♣t♦❣r❛✜❛ ❛ss✐♠étr✐❝❛✱ ♦✉ ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛✱ ❝❛❞❛ ✉♠ ❞♦s ✉s✉ár✐♦s ❞♦ ♠❡✐♦ ❝r✐♣t♦❣rá✜❝♦ ♣♦ss✉✐ ✉♠ ♠❛t❡r✐❛❧ ❝❤❛✈❡✱ ❞✐✈✐❞✐❞♦ ❡♠ ❞✉❛s ♣♦rçõ❡s✿ ❛ ❝♦♠♣♦♥❡♥t❡ ♣ú✲ ❜❧✐❝❛ ❡ ❛ ❝♦♠♣♦♥❡♥t❡ ♣r✐✈❛❞❛✱ ❝❛❞❛ ✉♠❛ ❣❡r❛♥❞♦ ✉♠❛ r❡s♣❡❝t✐✈❛ tr❛♥s❢♦r♠❛çã♦✳
◆❛ ❝r✐♣t♦❣r❛✜❛ s✐♠étr✐❝❛ ❛♣❧✐❝❛✲s❡ ✉♠ ❝♦♥❥✉♥t♦ ♣ré✲❞❡t❡r♠✐♥❛❞♦ ✭♣❡❧❛ ❝❤❛✈❡✮ ❞❡ ❝✐❢r❛s ❛♦ t❡①t♦ ❞✐r❡t♦✳ ❙❡❥❛ ❡❧❛ ❛ tr❛♥s❢♦r♠❛çã♦ ✐♥✈❡rtí✈❡❧ ❋✱ ❛♣❧✐❝❛❞❛ à ♠❡♥s❛❣❡♠ ❳ ✉t✐❧✐③❛♥❞♦✲s❡ ❛ ❝❤❛✈❡ ❩✳ ❉❡ss❛ ❢♦r♠❛ t❡♠✲s❡✿
Y =F(X, Z) = FZ(X)
◆❡st❛ s✐t✉❛çã♦✱ ♦ r❡s✉❧t❛❞♦ ❨ é ❛ ♠❡♥s❛❣❡♠ ❝r✐♣t♦❣r❛❢❛❞❛✳
◆❛ ❞❡❝r✐♣t❛çã♦ ❝❛❜❡ ❛♦ ❡❧❡♠❡♥t♦ r❡❝❡♣t♦r ❞❛ ♠❡♥s❛❣❡♠ ❛♣❧✐❝❛r ❛ tr❛♥s❢♦r♠❛çã♦ ✐♥✈❡rs❛ ❡♠ ❨ ❞❡ ♠♦❞♦ ❛ s❡ t❡r ♦ t❡①t♦ ✐♥✐❝✐❛❧✳ P❛r❛ ✐ss♦ ❢❛③✲s❡✿
F−1(Y, Z) =F−1
Z (Y) = F−
1
Z [FZ(X)] =X
❆ ❝❤❛✈❡ ❩ é r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❡s❝♦❧❤❛ ❞♦ ♠♦❞♦ ♣❛rt✐❝✉❧❛r ❞❡ s❡ ❢❛③❡r t❛♥t♦ ❛ ❡♥❝r✐♣✲ t❛çã♦ ❝♦♠♦ ❛ ❞❡❝r✐♣t❛çã♦✳
❊ss❛ ❝❤❛✈❡ ❞❡✈❡ s❡r ❡♥tr❡❣✉❡ ♣♦r ✉♠ ❝❛♥❛❧ s❡❣✉r♦✱ ❝♦♥s✐❞❡r❛♥❞♦✲s❡ q✉❡ ❛ s❡❣✉r❛♥ç❛ ❞♦ s✐st❡♠❛ ❡stá ❡♠❜❛s❛❞❛ ♥❛ ♥❛t✉r❡③❛ s❡❝r❡t❛ ❞❛ ❝❤❛✈❡✳
◆❛ ❝r✐♣t♦❣r❛✜❛ ❛ss✐♠étr✐❝❛✱ ♦✉ ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛✱ ❝❛❞❛ ✉♠ ❞♦s ✉s✉ár✐♦s ❞♦ ♠❡✐♦ ❝r✐♣✲ t♦❣rá✜❝♦ ♣♦ss✉✐ ✉♠ ♠❛t❡r✐❛❧ ❝❤❛✈❡✱ ❞✐✈✐❞✐❞♦ ❡♠ ❞✉❛s ♣♦rçõ❡s✿ ❛ ❝♦♠♣♦♥❡♥t❡ ♣ú❜❧✐❝❛ ❡ ❛ ❝♦♠♣♦♥❡♥t❡ ♣r✐✈❛❞❛✱ ❝❛❞❛ ✉♠❛ ❣❡r❛♥❞♦ ✉♠❛ r❡s♣❡❝t✐✈❛ tr❛♥s❢♦r♠❛çã♦✳ ❙✉❛ ✜♥❛❧✐❞❛❞❡ é ❛ ❝♦♠✉♥✐❝❛çã♦ ❞❡ ✉s✉ár✐♦s ❡♠ ✉♠ ❝❛♥❛❧ ✐♥s❡❣✉r♦✱ ❝♦♠ ❞✐str✐❜✉✐çã♦ ❞❡ ♠út✉❛s ❝❤❛✈❡s✳
❙❡✉ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❜ás✐❝♦ r❡q✉❡r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ tr❛♥s❢♦r♠❛çõ❡s ✐♥✈❡rtí✈❡✐s ✭❛❧❣♦✲ r✐t♠♦s✮✳
❆ss✐♠✱ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❞❛s ❝♦♠♣♦♥❡♥t❡s ♣ú❜❧✐❝❛ ❡ ♣r✐✈❛❞❛ ❞❛s ❝❤❛✈❡s✱ ❣❡r❛♥❞♦ ❛ ❝❤❛✈❡ ❩✱ é ❡st❛❜❡❧❡❝✐❞♦✱ ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦ à ❝r✐♣t♦❣r❛✜❛ s✐♠étr✐❝❛✱ ♦ ♣r♦❝❡ss♦ ❞❡ ❡♥❝r✐♣t❛çã♦✱EZ✱ ❡ ❞❡❝r✐♣t❛çã♦✱ DZ✳
EZ :fZ(x) = y
DZ :fZ−1(y) = x
P❛r❛ q✉❡ ♦ ♣r♦❝❡ss♦ ♦❝♦rr❛✱ ❝♦♠♦ ❡s♣❡r❛❞♦✱ ❞❡ ❢♦r♠❛ á❣✐❧ ❡ s❡❣✉r❛✱ ❞❡✈❡✲s❡ ❣❛r❛♥t✐r q✉❡✿
• fZ(x) s❡❥❛ ❢❛❝✐❧♠❡♥t❡ ❝♦♠♣✉t❛❞❛❀
• f−1
Z (y) s❡❥❛ ❡①tr❡♠❛♠❡♥t❡ ❞✐❢í❝✐❧ ❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ♣♦r ✉♠ ✐♥✐♠✐❣♦❀
• f−1
Z (y)s❡❥❛ ❢á❝✐❧ ❞❡ s❡r ❡s❝r✐t❛ ♣❛r❛ ♦ r❡❝❡♣t♦r ❡♠ ♣♦ss❡ ❞❛ ❝❤❛✈❡✱ ♣♦ré♠ q✉❡ s❡❥❛ ✐♠♣r❛t✐❝á✈❡❧ ❛ ♣❛rt✐r ❞❡❧❛ ❡♥❝♦♥tr❛r ❛ ❝❤❛✈❡ ♣r✐✈❛❞❛✳
❆♣ós ❡ss❛ ❡①♣❧✐❝❛çã♦ s♦❜r❡ ♦s t✐♣♦s ❞❡ s✐st❡♠❛s ❝r✐♣t♦❣rá✜❝♦s ❞❡s❝r❡✈❡♠♦s ♦s ❛❧❣♦✲ r✐t♠♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛ ❝❛❞❛ s✐st❡♠❛✳
✷✳✷ ❆▲●❖❘■❚▼❖❙ ❉❊ ❙■❙❚❊▼❆❙ ❙■▼➱❚❘■❈❖❙
✷✳✷✳✶ ❖◆❊✲❚■▼❊✲P❆❉
▼ét♦❞♦ ❝r✐❛❞♦ ❡♠ ✶✾✶✼ ♣♦r ●✐❧❜❡rt ❙❛♥❞❢♦r❞ ❱❡r♥❛♠ ✭❝♦♠ ♦ ♥♦♠❡ ❞❡ ❈✐❢r❛ ❞❡ ❱❡r✲ ♥❛♠✮ ❡ ❛♣❡r❢❡✐ç♦❛❞♦ ♣♦r ❏♦s❡♣❤ ▼❛✉❜♦r❣♥❡✳
❯♠❛ ❝❤❛✈❡ é ❣❡r❛❞❛ ♥❛ ❢♦r♠❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❜✐ts ❣❡r❛❞♦s ❛❧❡❛t♦r✐❛♠❡♥t❡✳ ❈♦♠ ❡st❛ ❝❤❛✈❡ é r❡❛❧✐③❛❞❛ ❛ ♦♣❡r❛çã♦ ♦✉ ❡①❝❧✉s✐✈♦ ✭①♦r✮ ❡♥tr❡ ❛ ❝❤❛✈❡ ❡ ❛ ♠❡♥s❛❣❡♠✳ ❆ r❡s♣♦st❛ ❛ ❡st❛ ♦♣❡r❛çã♦ ❝♦♥st✐t✉✐ ❛ ♠❡♥s❛❣❡♠ ❝✐❢r❛❞❛✳
P❛r❛ r❡❝✉♣❡r❛r ❛ ♠❡♥s❛❣❡♠ ♦r✐❣✐♥❛❧ ❜❛st❛ ❡❢❡t✉❛r ♦ ♥♦✈❛♠❡♥t❡ ❛ ♦♣❡r❛çã♦ ♦✉ ❡①✲ ❝❧✉s✐✈♦ ✭①♦r✮ ❞❛ ❝✐❢r❛ ❝♦♠ ❛ ❝❤❛✈❡✳
❊ss❡ ♠ét♦❞♦ é ❜❡♠ s✐♠♣❧❡s ❡ ❡✜❝✐❡♥t❡✱ ❞❡s❞❡ q✉❡ ❛ ❝❤❛✈❡ s❡❥❛ tã♦ ❧♦♥❣❛ q✉❛♥t♦ ❛ ♠❡♥s❛❣❡♠ ❡ q✉❡ ❛ ❝❤❛✈❡ s❡❥❛ r❡❛❧♠❡♥t❡ ❛❧❡❛tór✐❛✳ ❊ss❛s ❝♦♥❞✐çõ❡s✱ s♦❜r❡t✉❞♦ ❡♠ r❛③ã♦ ❞❛ ❞✐✜❝✉❧❞❛❞❡ ❞❡ s❡ ❣❡r❛r ♥ú♠❡r♦s ❛❧❡❛tór✐♦s✱ ♥❡♠ s❡♠♣r❡ ♣♦❞❡♠ s❡r ❛t❡♥❞✐❞❛s✱ ♦ q✉❡ r❡str✐♥❣❡ s❡✉ ✉s♦✳
✷✳✷✳✷ ❉❊❙ ✭❉❆❚❆ ❊◆❈❘❨P❚■❖◆ ❙❚❆◆❉❆❘❉✮
❈r✐❛❞♦ ♣❡❧❛ ■❇▼ ❡♠ ✶✾✼✵✱ ❡st❡ ❛❧❣♦r✐t♠♦ tr❛❜❛❧❤❛ ❝♦♠ ❜❧♦❝♦s ❞❡ ✻✹ ❜✐ts ❞❡ ❡♥tr❛❞❛ ❡ ❞❡ s❛í❞❛✱ ❝❤❛✈❡s ❞❡ ✺✻ ❜✐ts ❝♦♠ ♣❛r✐❞❛❞❡✱ ❡ é ❢✉♥❞❛♠❡♥t❛❧♠❡♥t❡ ❜❛s❡❛❞♦ ♥❛ ♣❡r♠✉t❛✲ çã♦ ❞❡ ❡❧❡♠❡♥t♦s✳ ➱ ❢á❝✐❧ ❞❡ ✐♠♣❧❡♠❡♥t❛r✱ ♠❛s é ✈✉❧♥❡rá✈❡❧ ❛ ❛t❛q✉❡s ❞❡ ❢♦rç❛ ❜r✉t❛ ❡ ❛ tr❛♥s♠✐ssã♦ ❞❛ ❝❤❛✈❡✱ ❝♦♠♦ é ❝♦♠✉♠ ♥♦s ❛❧❣♦r✐t♠♦s ❞❡ ❝r✐♣t♦❣r❛✜❛ s✐♠étr✐❝❛✱ ♥❡✲ ❝❡ss✐t❛ ❞❡ ♠á①✐♠❛ s❡❣✉r❛♥ç❛✳ ❊ss❛ ❝❤❛✈❡ é ♦ ❝♦♠♣♦♥❡♥t❡ ❛❧❡❛tór✐♦ ❞♦ ❛❧❣♦r✐t♠♦✳ ❖ ♠❡❝❛♥✐s♠♦ ❞♦ ❉❊❙ ❝♦♥s✐st❡ ❡♠ ❞✐✈✐❞✐r ❛ ♠❡♥s❛❣❡♠ ✭❜❧♦❝♦ ❞❡ ✻✹ ❜✐ts✮ ❡♠ ❞✉❛s ♣❛rt❡s
✭✸✷ ❜✐ts ♣❛r❛ ❝❛❞❛ ❧❛❞♦✮✱ ❛ ♣❛rt❡ ❡sq✉❡r❞❛ ▲ ❡ ❛ ♣❛rt❡ ❞✐r❡✐t❛ ❘✳ ❆ ❝❤❛✈❡ ❝♦♥s✐st❡ ❡♠ ✻✹ ❜✐ts✱ ♠❛s ❛♣❡♥❛s ✺✻ sã♦ ✉t✐❧✐③❛❞♦s ❡❢❡t✐✈❛♠❡♥t❡ ♥♦ ❛❧❣♦r✐t♠♦✱ s❡♥❞♦ ♦s ✽ ❜✐ts r❡st❛♥t❡s ✉s❛❞♦s s♦♠❡♥t❡ ♣❛r❛ ❝❤❡❝❛r ♣❛r✐❞❛❞❡ s❡♥❞♦ ❞❡s❝❛rt❛❞♦s ❞❡♣♦✐s✳
◆♦ ♣r♦❝❡ss♦ ❞❡ ❡♥❝r✐♣t❛çã♦✱ ♦s ❧❛❞♦s✱ ❡sq✉❡r❞♦ ❡ ❞✐r❡✐t♦✱ ♣❡r♠✉t❛♠ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
• Li =Ri−1 ✐ ✈❛r✐❛♥❞♦ ❞❡ ✶ ❛ ✶✻✱ ✉♠❛ ✈❡③ q✉❡ ♦❝♦rr❡rã♦ ✶✻ ✐t❡r❛çõ❡s✳
• Ri =Li−1⊕F(Ri−1, Zi)é ✉♠❛ ❢✉♥çã♦ ✐♥✈❡rtí✈❡❧ ♣ré✲❞❡✜♥✐❞❛✱ ❞❡♥♦♠✐♥❛❞❛ ❢✉♥çã♦
❞❡ ❋❡✐st❡❧✳
❖ sí♠❜♦❧♦ ⊕ s✐❣♥✐✜❝❛ ❛ ♦♣❡r❛çã♦ ❞❡ ♦✉ ❡①❝❧✉s✐✈♦ ✭①♦r✮✳
❆ ❢✉♥çã♦ ❋ ❛t✉❛ ♥❛ ♠❡t❛❞❡ ❞♦ ❜❧♦❝♦✱ ❥✉♥t♦ ❛ ✉♠❛ ❝❤❛✈❡✳ ❖ r❡s✉❧t❛❞♦ é ❝♦♠❜✐♥❛❞♦ à ♦✉tr❛ ♠❡t❛❞❡ ❞♦ ❜❧♦❝♦✱ ❡ ❛s ♠❡t❛❞❡s tr♦❝❛♠ ❞❡ ✧❧❛❞♦✧✳
❊ss❡ ♣r♦❝❡ss♦ ❞❡ ❝r✉③❛r ❡ ❛❧t❡r♥❛r é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❡sq✉❡♠❛ ❞❡ ❋❡✐st❡❧✱ ❡ ❛ss❡❣✉r❛ q✉❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❡♥❝r✐♣t❛r ❡ ❞❡❝r✐♣t❛r s❡❥❛♠ s✐♠✐❧❛r❡s✳ ❆ ú♥✐❝❛ ❞✐❢❡r❡♥ç❛ é q✉❡ ❛s ❝❤❛✈❡s ❞❡ ❝❛❞❛ ❡stá❣✐♦ ❞❡✈❡rã♦ s❡r ❛♣❧✐❝❛❞❛s ❡♠ ♦r❞❡♠ ✐♥✈❡rs❛ ♣❛r❛ ❞❡❝r✐♣t❛r✳
❆ ❝❤❛✈❡ Zi ❞❡ ✹✽✲❜✐ts ❛t✉❛ ❡♠ Ri−1 ❡①♣❛♥❞✐❞♦ t❛♠❜é♠ ♣❛r❛ ✹✽ ❜✐ts ♣❛r❛ r❡❛❧✐③❛r ❛
❢✉♥çã♦ q✉❡ ❡❢❡t✉❛rá ✉♠ ①♦r ❝♦♠ ♦ ❧❛❞♦ ❡sq✉❡r❞♦✱ r❡s✉❧t❛♥❞♦ ❡♠Ri ✳ ❆ ❡①♣❛♥sã♦ ❞❡ é ❢❡✐t❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
❙❡❥❛Ri ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❜✐ts✿
Ri =r1r2r3r4r5r6r7r8...r29r30r31r32
❊st❛ s❡q✉ê♥❝✐❛ é ❞✐✈✐❞✐❞❛ ❡♠ ❜❧♦❝♦s ❞❡ ✹ ❜✐ts r❡s✉❧t❛♥❞♦ ❡♠Ri✬✿
R∗
i =r1r2r3r4 r5r6r7r8 ... r29r30r31r32
❊♥tã♦✱ à ❢r❡♥t❡ ❞❡ ❝❛❞❛ ❜❧♦❝♦ é r❡♣❡t✐❞♦ ♦ ❜✐t ❛♥t❡r✐♦r ❡ ❛♦ ✜♥❛❧✱ ♦ ♣♦st❡r✐♦r✳
R′i =r32r1r2r3r4r5 r4r5r6r7r8r9 ... r28r29r30r31r32r1
❉❡ss❛ ❢♦r♠❛✱ ♦ r❡s✉❧t❛❞♦R′
❉❡♣♦✐s ❞❡ r❡❛❧✐③❛❞♦ ♦ ①♦r ❡♥tr❡ ❛ ❝❤❛✈❡ ❡ ♦ ❜❧♦❝♦ ❡①♣❛♥❞✐❞♦✱ ♦s ✹✽ ❜✐ts r❡s✉❧t❛♥t❡s ♣❛ss❛♠ ♣♦r ✉♠ ♣r♦❝❡ss♦ ❞❡ s✉❜st✐t✉✐çã♦✳ ❊ss❛s s✉❜st✐t✉✐çõ❡s sã♦ ❢❡✐t❛s ❡♠ ✽ ✧❝❛✐✲ ①❛s✧❝♦♠ ✻ ❜✐ts ❞❡ ❡♥tr❛❞❛ ❡ ✹ ❞❡ s❛í❞❛✱ ❝❛❞❛ ❝❛✐①❛✳ ➱ ❢❡✐t♦ ✐ss♦ ♣❛r❛ q✉❡ ❡st❡ r❡s✉❧t❛❞♦ r❡❝✉♣❡r❡ ♦ t❛♠❛♥❤♦ ❞❡ ✸✷ ❜✐ts✳
❖ ♣r♦❝❡ss♦ ❞❡ ❞❡❝r✐♣t❛çã♦ é ❢❡✐t♦ s❡❣✉✐♥❞♦ ♦s ♣❛ss♦s ✐♥✈❡rs♦s ❛♦ ❞❛ ❡♥❝r✐♣t❛çã♦✳ ❆t✉❛❧♠❡♥t❡✱ ❡st❡ ♣r♦❝❡ss♦ ♥ã♦ é ♠❛✐s ✉t✐❧✐③❛❞♦✱ ♣♦ré♠ ❡❧❡ s❡r✈❡ ❞❡ ❜❛s❡ ♣❛r❛ ❛ ❛♣❧✐❝❛çã♦ ❞❡ ✈❡rsõ❡s ♠❛✐s ♠♦❞❡r♥❛s✱ ❝♦♠♦ ♦ tr✐♣❧❡ ❉❊❙✱ ♠❛✐s s❡❣✉r♦✳
✷✳✷✳✸ ❚❘■P▲❊ ❉❊❙ ✭✸❉❊❙✮
❖ ❚r✐♣❧❡ ❉❊❙ é ✉♠❛ ❝✐❢r❛ ❞❡ ❜❧♦❝♦s q✉❡ t❡♠ ❝♦♠♦ ❜❛s❡ ♦ ❉❊❙✳
❆ s✉❛ ✈❛♥t❛❣❡♠ ❡♠ r❡❧❛çã♦ ❛♦ ❉❊❙ é q✉❡ ♦ t❛♠❛♥❤♦ ❞❛ ❝❤❛✈❡ ❢♦✐ ❛✉♠❡♥t❛❞♦ s❡♠ ♣r❡❝✐s❛r ♠♦❞✐✜❝❛r ♦ ❛❧❣♦r✐t♠♦✳
❯t✐❧✐③❛ três ❡t❛♣❛s✱ ❡ ♦♣❡r❛ ❝♦♠ três ❝❤❛✈❡s✿ K1, K2, K3✳
❊st❡ ❛❧❣♦r✐t♠♦ r❡❛❧✐③❛❞❛ ❛ ❝♦♠♣♦s✐çã♦ ❞❛ ❢✉♥çã♦ ❉❊❙ às três ❝❤❛✈❡s✱ ♦✉ s❡❥❛✿
C = 3DES(M) =DES{K3, DES[K2, DES(K1, M)]}
◆❡ss❡ ❝❛s♦✱ ▼ é ♦ ❜❧♦❝♦ ❞❡ ♠❡♥s❛❣❡♠ ❡ ❈ é ♦ r❡s✉❧t❛❞♦ ❝✐❢r❛❞♦✳
❆❜❛✐①♦ ❡stá r❡♣r❡s❡♥t❛❞♦ ♦ ❞✐❛❣r❛♠❛ ❞❡ ❢✉♥❝✐♦♥❛♠❡♥t♦ s✐♠♣❧✐✜❝❛❞♦ ❞♦ ❚r✐♣❧♦ ❉❊❙✳
✷✳✷✳✹ ❘❈✹
❋✉♥❝✐♦♥❛ ❝♦♠ ❜❛s❡ ❡♠ ❝✐❢r❛s s❡q✉❡♥❝✐❛✐s✱ ❝♦♠ ♦♣❡r❛çõ❡s ♦r✐❡♥t❛❞❛s ❛ ❜②t❡s✱ s❡✉ ❛❧❣♦r✐t♠♦ ❢❛③ ✉s♦ ❞❡ ♣❡r♠✉t❛çõ❡s ♣s❡✉❞♦✲❛❧❡❛tór✐❛s ❞❛ ❝❤❛✈❡✳ ❋❛③ ♣❛rt❡ ❞❡ ♣r♦t♦❝♦❧♦s
✉s✉❛✐s ❞❡ s❡❣✉r❛♥ç❛✱ ❝♦♠♦ ♦ ❙❙▲✱ q✉❡ ♣r♦t❡❣❡ ♦ trá❢❡❣♦ ❞❡ ✐♥❢♦r♠❛çõ❡s ♥❛ ■♥t❡r♥❡t✱ ❡ ♦ ❲❊P✱ q✉❡ ❢❛③ ♣❛rt❡ ❞❡ r❡❞❡ ✇✐r❡❧❡ss✳
❙❡✉ ♠❡❝❛♥✐s♠♦ é s❡♠❡❧❤❛♥t❡ ❛♦ ❞♦ ♦♥❡✲t✐♠❡✲♣❛❞✱ ✉♠❛ ♦♣❡r❛çã♦ ①♦r é ❛♣❧✐❝❛❞❛ ❡♥tr❡ ✉♠❛ s❡q✉ê♥❝✐❛ ♣s❡✉❞♦✲❛❧❡❛tór✐❛ ❝❤❛✈❡❛❞❛ ❡ ♦s ❞❛❞♦s✱ ♠❡s♠❛ ❝❤❛✈❡ ♣❛r❛ ❝✐❢r❛r ❡ ❞❡❝✐❢r❛r✳ ❙❡✉ t❛♠❛♥❤♦ ❞❡ ❝ó❞✐❣♦ é 1/10 ❞♦ ❉❊❙✳
❆ ❝❤❛✈❡✱ q✉❡ é ❢♦r♠❛❞❛ ♣♦r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❜✐ts✱ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ t❡①t♦ ❛ s❡r ❝✐❢r❛❞♦✳
❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❧✐t❡r❛t✉r❛ ❛t✉❛❧✱ ♥ã♦ ❢♦✐ ❡♥❝♦♥tr❛❞♦ ♥❡♥❤✉♠ ❛t❛q✉❡ ♣rát✐❝♦ ❛♦ ❛❧❣♦r✐t♠♦ ❞❛ s❡q✉ê♥❝✐❛ ♣s❡✉❞♦✲❛❧❡❛tór✐❛✱ ♠❛s ❛ ♠❛♥❡✐r❛ ❡♠ q✉❡ ❡❧❡ é ✐♥✐❝✐❛❧✐③❛❞♦ é ❝♦♥s✐❞❡r❛❞❛ ✉♠❛ ❢♦r♠❛ ✐♥s❡❣✉r❛✳
❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ q✉❡ ❢♦✐ ✈✐st♦✱ ♦ ♣r♦❝❡ss♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❡st❡ ❛❧❣♦r✐t♠♦ ❡stá ♥❛ ♦❜t❡♥çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❛❧❡❛tór✐♦✳ ❉❡♣♦✐s ❞❡ss❡ ♣r♦❝❡ss♦✱ ♦ ❛❧❣♦r✐t♠♦ s❡ r❡❞✉③ à r❡❛❧✐③❛çã♦ ❞❛ ❢✉♥çã♦ ①♦r ❝♦♠ ♦ t❡①t♦ ♣r✐♥❝✐♣❛❧✳
✷✳✸ ❆▲●❖❘■❚▼❖❙ ❉❊ ❙■❙❚❊▼❆❙ ❆❙❙■▼➱❚❘■❈❖❙
✷✳✸✳✶ ❉■❋❋■❊✲❍❊▲▲▼❆◆
❊st❡ ❛❧❣♦r✐t♠♦ ❢❛③ ♣❛rt❡ ❞♦ ♣r♦t♦❝♦❧♦ ✧❦❡② ❡①❝❤❛♥❣❡✧✱ q✉❡ ♣❡r♠✐t❡ q✉❡ ❞✉❛s ♣❛rt❡s✱ q✉❡ ♥✉♥❝❛ ❤❛✈✐❛♠ t✐❞♦ ❝♦♥t❛t♦✱ ❝♦♠♣❛rt✐❧❤❡♠ ✉♠❛ ❝❤❛✈❡ s❡❝r❡t❛ ❛tr❛✈és ❞❡ ✉♠ ♠❡✐♦ ✐♥s❡❣✉r♦ ❞❡ ❝♦♠✉♥✐❝❛çã♦✳ ❊ss❛ ❝❤❛✈❡✱ ♣♦st❡r✐♦r♠❡♥t❡✱ ♣♦❞❡ ❡♥❝r✐♣t❛r ♠❡♥s❛❣❡♥s✱ ❡s✲ t❛❜❡❧❡❝❡♥❞♦ ✉♠ s✐st❡♠❛ s✐♠étr✐❝♦ ❞❡ ❝r✐♣t♦❣r❛✜❛✳
❖ ❛❧❣♦r✐t♠♦ ❉✐✣❡✲❍❡❧❧♠❛♥ ❢♦✐ ✐♥✈❡♥t❛❞♦ ❡♠ ✶✾✼✻✱ s❡♥❞♦ ♦ ♣r✐♠❡✐r♦ ❛❧❣♦r✐t♠♦ ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛✳
❊st❡ ❛❧❣♦r✐t♠♦ é ❢✉♥❞❛♠❡♥t❛❧♠❡♥t❡ ❜❛s❡❛❞♦ ♥❛ ❞✐✜❝✉❧❞❛❞❡ ❞❡ s❡ ❝❛❧❝✉❧❛r ❧♦❣❛r✐t♠♦s ❞✐s❝r❡t♦s✳ ❆ ❢✉♥çã♦ ❧♦❣❛r✐t♠♦ ❞✐s❝r❡t♦ é ❛ ✐♥✈❡rs❛ ❞❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ♠♦❞✉❧❛r✱ ❞❡✜♥✐❞❛ ❝♦♠♦ y=gx(mod p) ✱ ♦♥❞❡ p❡ g sã♦ ♣r✐♠♦s✳
❊♠ s❡✉ ❢✉♥❝✐♦♥❛♠❡♥t♦✱ ♣❛r❛ ❛ ❞✐str✐❜✉✐çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❞✐s❝r❡t❛ t❡♠♦s✿
y ≡gx( mod p), ♣❛r❛ 1≤x≤p−1
◆❡ss❡ ❝❛s♦✱ ❣ t❛♠❜é♠ é ❝❤❛♠❛❞♦ ❞❡ ♣r✐♠✐t✐✈❛ ♥♦ ♠ó❞✉❧♦ ❞❡ ♣✳ ❖ r❡t♦r♥♦ ❞❡ ① é ❝❛❧❝✉❧❛❞♦ ❝♦♥❢♦r♠❡ ❛❜❛✐①♦✿
x≡(logdgy)( mod p), ♣❛r❛ 1≤y≤p−1
❖s ✉s✉ár✐♦s tê♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ❣ ❡ ♣✳
❖ ✉s✉ár✐♦ ✐ s❡❧❡❝✐♦♥❛ ✉♠ ♥ú♠❡r♦ ❛❧❡❛tór✐♦xi ❞♦ ❝♦♥❥✉♥t♦ {1,2, ..., p} q✉❡ s❡rá s✉❛ ❝❤❛✈❡ ♣r✐✈❛❞❛✳ ❉❡ss❛ ❢♦r♠❛✱ ❛ ✐♥❢♦r♠❛çã♦ q✉❡ s❡rá ❣✉❛r❞❛❞❛ ♥✉♠ ❞✐r❡tór✐♦ ♣ú❜❧✐❝♦ ❝♦♠ ♦s ❞❛❞♦s ❞♦ ✉s✉ár✐♦ ✐ é ❞❛❞❛ ♣♦r
yi ≡gxi( modp)
❖✉tr♦ ✉s✉ár✐♦ ❥ r❡❛❧✐③❛ ♦ ♠❡s♠♦ ♣r♦❝❡❞✐♠❡♥t♦✳
❆ ❝♦♠✉♥✐❝❛çã♦ ❡♥tr❡ ✐ ❡ ❥ é ❢❡✐t❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
❖ ✉s✉ár✐♦ ✐ ❛❞q✉✐r❡ ✐♥❢♦r♠❛çõ❡s ❞❡ ❥✱ yj✱ ❞♦ ❞✐r❡tór✐♦ ♣ú❜❧✐❝♦✱ ♣❛r❛ ❢♦r♠❛r✱ ❛ ♣❛rt✐r ❞❡ s✉❛ ❝❤❛✈❡ ♣r✐✈❛❞❛ ✐♥❞✐✈✐❞✉❛❧✱ xi ✱ ❛ ❝❤❛✈❡ ♣r✐✈❛❞❛ ❝♦♠✉♠ ❛ ❛♠❜♦s✱ kij ♦✉ kji ❛ ♣❛rt✐r ❞❛s ♦♣❡r❛çõ❡s ❛ s❡❣✉✐r✿
kij ≡[yj]xi( mod p)
kij ≡[(gxj)] xi
( modp)
kij ≡gxj.xi( mod p)
❆♥❛❧♦❣❛♠❡♥t❡✱ ❛ ♣❛rt✐r ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❡①♣♦♥❡♥❝✐❛çã♦ ♠♦❞✉❧❛r✱ t❡♠✲s❡ q✉❡
kij ≡kji
❆ ú❧t✐♠❛ ❡q✉❛çã♦ s✐❣♥✐✜❝❛ q✉❡ ✐ ❡ ❥ ❡♠ ❝♦♠✉♠ ❛❝♦r❞♦ ❞❡✜♥❡♠ kij ♦✉ kji ❝♦♠♦ ❛ ❝❤❛✈❡ s❡❝r❡t❛ ❡♠ s❡✉ s✐st❡♠❛ ❝r✐♣t♦❣rá✜❝♦ ❞❡ ❝♦♠✉♥✐❝❛çã♦✳
P❛r❛ ✉♠ ✐♥✐♠✐❣♦ ❝r✐♣t♦❛♥❛❧✐st❛ q✉❡ t❡♥❤❛ ❛❝❡ss♦ à ✐♥❢♦r♠❛çã♦ ❞❡ ✐ ❡ ❥ ❝♦♥t✐❞❛ ♥♦ ❞✐r❡tór✐♦✱ s❡r✐❛ ♥❡❝❡ssár✐♦ ❡♥❝♦♥tr❛r ❛ ❝❤❛✈❡
kij ≡
h
(yj)logdyi
i
( modp)
❆ ♥❡❝❡ss✐❞❛❞❡ ❞♦ ❝á❧❝✉❧♦ ❞❡st❡ ❧♦❣❛r✐t♠♦ ❞✐s❝r❡t♦ ♠♦❞✉❧❛r✱ q✉❛♥❞♦ s❡ t❡♠ ✉♠ ✈❛❧♦r ❞❡ ♣ ❡❧❡✈❛❞♦ ✭t✐♣✐❝❛♠❡♥t❡ s✉♣❡r✐♦r ❛ ✶✵✵ ❝❛s❛s ❞❡❝✐♠❛✐s✮✱ t♦r♥❛ ✐♠♣r❛t✐❝á✈❡❧ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛ ❝❤❛✈❡ s❡❝r❡t❛ ❝♦♠✉♠ ❞❡ ✐ ❡ ❥✱ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❛s ❝❛♣❛❝✐❞❛❞❡s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❛t✉❛✐s✳
➱ ♣♦ssí✈❡❧ ♣❡r❝❡❜❡r✱ q✉❡ ❡st❡ ❛❧❣♦r✐t♠♦✱ ❜❡♠ ❝♦♠♦ ♦ ❘❙❆ ❡ ♦ ❞❡ ❊❧ ●❛♠❛❧✱ ❞✐s✲ ❝✉t✐❞♦s ❛ s❡❣✉✐r✱ ♣❛r❛ s✉❛ ❡✜❝✐ê♥❝✐❛ ❡♠ t❡r♠♦s ❞❡ s❡❣✉r❛♥ç❛✱ ♥❡❝❡ss✐t❛♠ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s ❝♦♠ ✉♠ ♥ú♠❡r♦ ❡❧❡✈❛❞♦ ❞❡ ❝❛s❛s ❞❡❝✐♠❛✐s✳ ❚❛❧ ❝❛r❛❝t❡ríst✐❝❛ t❡♠ ❛❝❡♥t✉❛❞♦ ❛✐♥❞❛ ♠❛✐s ❛ ❞✐s❝✉ssã♦ ❡ ♦ ✐♥t❡r❡ss❡ ❛ r❡s♣❡✐t♦ ❞♦ ❡st✉❞♦ ❡ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ ❞♦s ♥ú♠❡r♦s ♣r✐♠♦s✳
❯♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❛✉t❡♥t✐❝❛çõ❡s ❝r✐♣t♦❣rá✜❝❛s ✐♥❝♦r♣♦r❛ ♦ ❉✐✣❡✲❍❡❧❧♠❛♥ ❝♦♠♦ ❛❧❣♦r✐t♠♦ ✐♥✐❝✐❛❧✱ s♦❜r❡t✉❞♦ ❝♦♠♦ ❜❛s❡ ♣❛r❛ ❛ tr♦❝❛ ❞❡ ❝❤❛✈❡s ❡♥tr❡ ❞♦✐s ♦✉ ♠❛✐s ♣♦♥t♦s ❞❡ ❝♦♠✉♥✐❝❛çã♦✱ ❞❡ ♠♦❞♦ q✉❡ s❡ ♣♦ss❛ ❡st❛❜❡❧❡❝❡r ✉♠❛ ❝♦♠✉♥✐❝❛çã♦ s❡❣✉r❛ ♠út✉❛✳
✷✳✸✳✷ ❘❙❆
❖ ❘❙❆ é ✉♠ ❛❧❣♦r✐t♠♦ ❞❡ ❡♥❝r✐♣t❛çã♦ ❞❡ ❞❛❞♦s✱ q✉❡ ❞❡✈❡ ♦ s❡✉ ♥♦♠❡ ❛ três ♣r♦❢❡s✲ s♦r❡s ❞♦ ■♥st✐t✉t♦ ▼■❚ ✭▼❛ss❛❝❤✉ss❡ts ■♥st✐t✉t❡ ♦❢ ❚❡❝♥♦❧♦❣②✮✱ ♦s q✉❛✐s sã♦ ❢✉♥❞❛❞♦r❡s ❞❛ ❛t✉❛❧ ❡♠♣r❡s❛ ❘❙❆ ❉❛t❛ ❙❡❝✉r✐t②✱ ■♥❝❀ ❘♦♥ ❘✐✈❡st✱ ❆❞✐ ❙❤❛♠✐r ❡ ▲❡♥ ❆❞❧❡♠❛♥✱ q✉❡ ✐♥✈❡♥t❛r❛♠ ❡st❡ ❛❧❣♦r✐t♠♦✳
❆ s❡❣✉r❛♥ç❛ ❞♦ ❛❧❣♦r✐t♠♦ s❡ ❜❛s❡✐❛ ♣❡❧♦ ❢❛t♦ ❞❛ ❢❛t♦r❛çã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ❣r❛♥❞❡ s❡r ♠✉✐t♦ ❞✐❢í❝✐❧✱ ❡♠ t❡r♠♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s✱ ✉♠❛ ✈❡③ q✉❡ ❛✐♥❞❛ ♥ã♦ s❡ ❡st❛❜❡❧❡❝❡✉ ✉♠ ♠é✲ t♦❞♦ ❝❛♣❛③ ❞❡ r❡❛❧✐③❛r ❡st❛ ♦♣❡r❛çã♦ ❞❡ ♠♦❞♦ ♠❛✐s ❡✜❝✐❡♥t❡ q✉❡ ♦s ♠ét♦❞♦s ✉s✉❛❧♠❡♥t❡ ❛♣r❡s❡♥t❛❞♦s✳ ❖ ❛❧❣♦r✐t♠♦ ✉s❛ ❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ♥❛ ❢♦r♠✉❧❛çã♦ ♠❛t❡♠át✐❝❛✳
P❡❧♦s ♣r♦❝❡ss♦s ♠❛t❡♠át✐❝♦s q✉❡ ❛ ♠❡♥s❛❣❡♠ ❞❡✈❡ ♣❛ss❛r ❡ ♣❡❧❛ ♣ró♣r✐❛ ❝❛r❛❝t❡rís✲ t✐❝❛ ❞❡ ❛❧❣♦r✐t♠♦ ❛ss✐♠étr✐❝♦✱ ♦ ♣r♦❝❡ss♦ ✉s✉❛❧♠❡♥t❡ s❡ t♦r♥❛ ❧❡♥t♦✱ t✐♣✐❝❛♠❡♥t❡ ❝❡r❝❛ ❞❡ ✶✵✵ ✈❡③❡s ♠❛✐s ❧❡♥t♦ q✉❡ ♦ ❉❊❙✳
➱ ♥❡❝❡ssár✐❛ ❛ ❝♦♠♣✉t❛çã♦ ❞♦s s❡❣✉✐♥t❡s ♣❛râ♠❡tr♦s✿
• ❙❡❧❡çã♦ ❞❡ ♣r✐♠♦s✿ ♣ ❡ q❀
• ❘❡❛❧✐③❛✲s❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞♦s ♣r✐♠♦s✿ ♣❛râ♠❡tr♦ n=p.q✳
❆ ♣❛rt✐r ❞❡st❡s ♣❛râ♠❡tr♦s✱ ❛♣❧✐❝❛✲s❡ ❛ ❢✉♥çã♦ϕ ❞❡ ❊✉❧❡r ❛♦ ✈❛❧♦r ♥✳
❉❡✈❡ s❡r ❧❡♠❜r❛❞♦ q✉❡ ❛ ❢✉♥çã♦ϕ(n)❞❡ ❊✉❧❡r r❡t♦r♥❛ ♦ ♥ú♠❡r♦ ❞❡ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s
♠❡♥♦r❡s q✉❡ ♥ ❝♦♠ ♦s q✉❛✐s ♥ é ♣r✐♠♦ ❡♥tr❡ s✐✱ ♦✉ s❡❥❛✱ ❝♦♠ ♦s q✉❛✐s ♥ ♥ã♦ ♣♦ss✉✐ ❞✐✈✐s♦r❡s ❝♦♠✉♥s✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❛r❛ ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ q✉❛❧q✉❡r✱ ♣✱ ❡st❡ ♥ú♠❡r♦ s❡r✐❛ ♣r✐♠♦ ❡♥tr❡ s✐ ❝♦♠ t♦❞♦s ♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ❛tép−1✱ ❛ss✐♠✱ ♦ r❡t♦r♥♦ ❞❛ ❢✉♥çã♦ s❡r✐❛
ϕ(p) =p−1
❖✉tr❛ ♣r♦♣r✐❡❞❛❞❡ q✉❡ ❞❡✈❡ s❡r ❞❡st❛❝❛❞❛ é ❛ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ r❡❢❡r❡♥t❡ ❛ ♥ú♠❡r♦s ♣r✐♠♦s ❡♥tr❡ s✐✱ ♦✉ s❡❥❛✱ s❡ ❞♦✐s ♥ú♠❡r♦s ❛ ❡ ❜ sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✱ ❛ ❢✉♥çã♦ ❞❡ ❊✉❧❡r ❛♣❧✐❝❛❞❛ ❛♦ s❡✉ ♣r♦❞✉t♦ s❡r✐❛ ❞❛❞❛ ♣♦r
ϕ(a.b) =ϕ(a).ϕ(b)
❈❧❛r♦ q✉❡ ❞♦✐s ♥ú♠❡r♦s ♣r✐♠♦s ❞✐st✐♥t♦s s❡rã♦ ♣r✐♠♦s ❡♥tr❡ s✐✱ ♣♦rt❛♥t♦✱ ♣❛r❛ ♦ ✈❛❧♦r ❞❛ ❢✉♥çã♦ ❞❡ ❊✉❧❡r ❛♣❧✐❝❛❞❛ ❛♦ ♣❛râ♠❡tr♦ ♥ ❝✐t❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ t❡♠✲s❡
ϕ(n) = ϕ(p.q) = (p−1).(q−1)
❖ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r ❞✐③ q✉❡✱ ♣❛r❛ t♦❞♦ ❛ ❡ ♥ ♣r✐♠♦s ❡♥tr❡ s✐✱ t❡♠✲s❡ q✉❡
aϕ(n) ≡1( mod n)
❖❜s❡r✈❡✲s❡ q✉❡ ♣❛r❛ ❛ ♣❛rt✐❝✉❧❛r✐③❛çã♦ ❞❡ n=p♣r✐♠♦✱ t❡♠✲s❡ ♦ P❡q✉❡♥♦ ❚❡♦r❡♠❛
❞❡ ❋❡r♠❛t✱ ♦✉ s❡❥❛
aϕ(p) ≡ap−1 ≡1( mod p)⇒ap ≡a( modp)
❆ s❡❣✉✐r✱ sã♦ ❡st❛❜❡❧❡❝✐❞♦s ♦s ♣❛râ♠❡tr♦s ❡ ❡ ❞✿
• ❊s❝♦❧❤❡✲s❡ ✉♠ ✈❛❧♦r ✭t❛♠❜é♠ ❞❡♥♦♠✐♥❛❞♦ ❡①♣♦❡♥t❡ ❞❡ ❡♥❝r✐♣t❛çã♦✮ e < ϕ(n)
✭✐♥t❡✐r♦ ♣♦s✐t✐✈♦✮✱ t❛❧ q✉❡ e ❡ ϕ(n) s❡❥❛♠ ♣r✐♠♦s ❡♥tr❡ s✐✳
• ❊♥❝♦♥tr❛✲s❡ ♦ ✈❛❧♦r ✭♣♦r s✉❛ ✈❡③✱ ✉s✉❛❧♠❡♥t❡ ❞❡♥♦♠✐♥❛❞♦ ❡①♣♦❡♥t❡ ❞❡ ❞❡❝r✐♣t❛✲
çã♦✮ d < ϕ(n) t❛❧ q✉❡
e.d≡1( mod ϕ(n))
❊st❡ ♥ú♠❡r♦ ❞ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞♦ ✉t✐❧✐③❛♥❞♦✲s❡ ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s ❡st❡♥❞✐❞♦✱ ❡ r❡♣r❡s❡♥t❛ ♦ ✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ♠♦❞✉❧❛r ❞❡ ❡✳
❆ ❡♥❝r✐♣t❛çã♦ ♥♦ ❛❧❣♦r✐t♠♦ é ❢❡✐t❛ ♣❡❧❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛çã♦ ❞✐s❝r❡t❛✱ ♦✉ ♠♦❞✉❧❛r✳ ❙❡❥❛ ① ❛ ♠❡♥s❛❣❡♠ ❡ ② ♦ t❡①t♦ ❝✐❢r❛❞♦✳ ➱ ❡st❛❜❡❧❡❝✐❞❛ ❛ ♦♣❡r❛çã♦
y≡xe( mod n)
❆ ❝❤❛✈❡ ♣ú❜❧✐❝❛✱ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ ❣❡r❛❧✱ ❝♦♥s✐st❡ ♥♦ ♣❛r (n, e) ❡ ❛ ❝❤❛✈❡ ♣r✐✈❛❞❛✱
❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ r❡str✐t♦ às ♣❛rt❡s ❝♦♠✉♥✐❝❛♥t❡s✱ ♥♦ ♣❛r ❞❡ ♣r✐♠♦s (n, d)✳ ✳ ❆ ❝❤❛✈❡
♣r✐✈❛❞❛ ❞❡✈❡ s❡r✱ ♥❛t✉r❛❧♠❡♥t❡✱ ♠❛♥t✐❞❛ ❡♠ s✐❣✐❧♦✱ ❜❡♠ ❝♦♠♦ ♦s ✈❛❧♦r❡s ❞❡ ♣✱ q ❡ϕ(n)✱
✉♠❛ ✈❡③ q✉❡ ❛ ♣❛rt✐r ❞❡st❡s ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞♦ ♦ ✈❛❧♦r ❞❡ ❞✳
❆♣ós ♦ r❡❝❡❜✐♠❡♥t♦ ❞❛ ♠❡♥s❛❣❡♠ ②✱ q✉❡ ♣♦❞❡ s❡r tr❛♥s♠✐t✐❞❛ ♣♦r ✉♠ ♠❡✐♦ ✐♥s❡❣✉r♦ ❞❡ ❝♦♠✉♥✐❝❛çã♦✱ ❛ ♣❛rt❡ r❡❝❡♣t♦r❛ ❞❛ ♠❡♥s❛❣❡♠✱ ❛ q✉❡♠ ❡st❛ s❡ ❞❡st✐♥❛✱ r❡❛❧✐③❛ ❛ ❞❡❝r✐♣t❛çã♦ ❛tr❛✈és ❞❛ ♦♣❡r❛çã♦
yd≡(xe)d≡xed( modn)
❆ ♣❛rt✐r ❞❛ r❡❧❛çã♦ ♣ré✲❡st❛❜❡❧❡❝✐❞❛ ❡♥tr❡ ❡ ❡ ❞✱ t❡♠✲s❡
e.d≡1( mod ϕ(n))⇔e.d =Q.ϕ(n) + 1; Q∈Z
❆♣❧✐❝❛♥❞♦✲s❡ ❡st❡ r❡s✉❧t❛❞♦ ♥❛ ❡q✉❛çã♦ ❛♥t❡r✐♦r✱ t❡♠✲s❡
yd ≡xed( modn)
≡xQ.ϕ(n)+1( modn)
≡nh xϕ(n)Qi
.xo( mod n)
P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r✿
xϕ(n) ≡1( mod n)
x≡yd( mod n)
❉❡ss❛ ❢♦r♠❛✱ ❝♦♠ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ✭♣r✐✈❛❞♦✮ ❞❡ ❞✱ ♦ ❞❡st✐♥❛tár✐♦ ❝♦♥s❡❣✉❡ r❡t♦r♥❛r à ♠❡♥s❛❣❡♠ ♦r✐❣✐♥❛❧✳
❆❥✉st❛♥❞♦✲s❡ ❞❡ ♠❛♥❡✐r❛ ✉♠ ♣♦✉❝♦ ❞✐st✐♥t❛✱ ♦ ❛❧❣♦r✐t♠♦ ❘❙❆ t❛♠❜é♠ ♣♦❞❡ s❡r ✉s❛❞♦ ❡♠ ❛ss✐♥❛t✉r❛s ❞✐❣✐t❛✐s✱ ♥❛s q✉❛✐s ❞❡s❡❥❛✲s❡ ❛ss❡❣✉r❛r ❛ ✐❞❡♥t✐❞❛❞❡ ❞♦ ❡♠✐ss♦r ❞❛ ♠❡♥s❛❣❡♠✳
❖ ❘❙❆ s♦❢r❡ ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❛t❛q✉❡ ❞♦s t✐♣♦s ❢♦rç❛ ❜r✉t❛✱ ❛t❛q✉❡s ♠❛t❡♠át✐❝♦s ❡ ❛t❛q✉❡s ❞❡ t❡♠♣♦r✐③❛çã♦✳ ❊st❡s ú❧t✐♠♦s✱ ♠❛✐s s♦✜st✐❝❛❞♦s✱ s❡ ❜❛s❡✐❛♠ ❡♠ ❞✐❢❡r❡♥ç❛s ❞❡ t❡♠♣♦ ♥♦ ❝á❧❝✉❧♦ ❞❡ ✉♠❛ ♦♣❡r❛çã♦ ✭❡①♣♦♥❡♥❝✐❛çã♦ ♠♦❞✉❧❛r✱ ♣♦r ❡①❡♠♣❧♦✮ ❝♦♠ ❞✐❢❡r❡♥t❡s ♦♣❡r❛♥❞♦s✳ ❉❡♥tr❡ ❛s ❢♦r♠❛s ❞❡ s❡ ❡✈✐t❛r ❡st❡ ❛t❛q✉❡✱ ♣♦❞❡ s❡ r❡❛❧✐③❛r ❛ ❡①♣♦♥❡♥❝✐❛çã♦ ♥✉♠ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ ❝♦♥st❛♥t❡ ♦✉ ❛♣❧✐❝❛r ❛tr❛s♦s ❛❧❡❛tór✐♦s✳
✷✳✹ ❆❚❆◗❯❊❙ ❆❖❙ ❙■❙❚❊▼❆❙ ❈❘■P❚❖●❘➪❋■❈❖❙
❈r✐♣t♦❛♥á❧✐s❡ é ♦ ♥♦♠❡ ❞❛❞♦ à ♠❡t♦❞♦❧♦❣✐❛ q✉❡ ❡st✉❞❛ ❛s ❢♦r♠❛s ❞❡ s❡ ❞❡s❢❛③❡r ♦s ❡❢❡✐t♦s ❞❛ ❝r✐♣t♦❣r❛✜❛✱ q✉❡❜r❛♥❞♦✲s❡ ❝✐❢r❛s ♦✉ ❢♦r❥❛♥❞♦✲s❡ s✐♥❛✐s q✉❡ s❡❥❛♠ ❛❝❡✐t♦s ❝♦♠♦ ❣❡♥✉í♥♦s✳
❆❧❣✉♥s t✐♣♦s ❞❡ ❛t❛q✉❡s ❞❡ ✉♠ ❝r✐♣t♦❛♥❛❧✐st❛ sã♦ ❞❡s❝r✐t♦s ❛ s❡❣✉✐r✳
◆♦ ❛t❛q✉❡ ❞♦ t✐♣♦ ♥♦r♠❛❧ ✭❝✐♣❤❡rt❡①t✲♦♥❧②✮✱ ♦ ✐♥✐♠✐❣♦ só t❡♠ ❛❝❡ss♦ à ♣❛rt❡ ♦✉ t♦❞♦ ❞♦ ❝r✐♣t♦❣r❛♠❛✱ ♦✉ ♠❡♥s❛❣❡♠ ❝✐❢r❛❞❛✳ ❚♦❞♦ s✐st❡♠❛ ❝r✐♣t♦❣rá✜❝♦ ❞❡✈❡ r❡s✐st✐r ❛ ♣❡❧♦ ♠❡♥♦s ❡ss❡ t✐♣♦ ❞❡ ❛t❛q✉❡✳
❯♠❛ té❝♥✐❝❛ ✉s✉❛❧ é ♦ ❡st✉❞♦ ❞❛ ❢r❡q✉ê♥❝✐❛ ❞❡ ❝❛r❛❝t❡r❡s✱ ♣♦r ❡①❡♠♣❧♦✱ ♥❛ ❧í♥❣✉❛ ♣♦rt✉❣✉❡s❛ ❛ ❢r❡q✉ê♥❝✐❛ ❡♠ q✉❡ ♦❝♦rr❡ ❛ ❧❡tr❛ ✧❛✧é ♠❛✐♦r q✉❡ ❛ ❢r❡q✉ê♥❝✐❛ ❞❡ ✧②✧✳ ❈♦♠ ❛ ❝♦♠♣❛r❛çã♦ ❞❛s ❢r❡q✉ê♥❝✐❛s ❞♦s ❝❛r❛❝t❡r❡s ❝✐❢r❛❞♦s✱ s❡♥❞♦ ❛ ♠❡♥s❛❣❡♠ r❡❛❧✐③❛❞❛ ♥✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❧í♥❣✉❛✱ é ♣♦ssí✈❡❧ ✐♥❢❡r✐r ❛❧❣✉♠❛ ✐♥❢♦r♠❛çã♦ ❞♦ t❡①t♦ ❝✐❢r❛❞♦✳
◆♦ ❛t❛q✉❡ à ✐♥❢♦r♠❛çã♦ ❝♦♥❤❡❝✐❞❛ ✭❦♥♦✇♥✲♣❧❛✐♥t❡①t✮ ♦ ✐♥✐♠✐❣♦ t❡♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ❛❧❣✉♥s ❝r✐♣t♦❣r❛♠❛s ❡ s✉❛s ❝♦rr❡s♣♦♥❞❡♥t❡s ♠❡♥s❛❣❡♥s ♦r✐❣✐♥❛✐s✱ ❢♦r♠❛❞♦s ♣❡❧❛ ❝❤❛✈❡ ❛t✉❛❧✳
❖ ❛t❛q✉❡ à ✐♥❢♦r♠❛çã♦ ❡s❝♦❧❤✐❞❛ ✭❝❤♦s❡♥✲♣❧❛✐♥t❡①t✮ é ❝❛r❛❝t❡r✐③❛❞♦ ♣❡❧♦ ❢❛t♦ ❞♦ ✐♥✐♠✐❣♦ s❡r ❝❛♣❛③ ❞❡ s✉❜♠❡t❡r q✉❛❧q✉❡r ♠❡♥s❛❣❡♠ à ❝r✐♣t♦❣r❛✜❛ ❡ r❡❝❡❜❡r ♦ ❝r✐♣✲ t♦❣r❛♠❛ ❝♦rr❡t♦ ❝♦♠ ❛ ❝❤❛✈❡ ❛t✉❛❧ ✉t✐❧✐③❛❞❛✳
❖ ♦❜❥❡t✐✈♦ ❞❡ss❡ ❛t❛q✉❡ é ♦❜t❡r ❛❧❣✉♠❛ ✐♥❢♦r♠❛çã♦ q✉❡ r❡❞✉③❛ ❛ s❡❣✉r❛♥ç❛ ❞♦ ❡sq✉❡♠❛ ❝r✐♣t♦❣rá✜❝♦✱ ❡ ♥❛ ♣✐♦r ❞❛s ❤✐♣ót❡s❡s✱ r❡✈❡❧❛r ♦ ❡sq✉❡♠❛ ❞❡ ♣r♦❞✉çã♦ ❞❛ ❝❤❛✈❡✳
❊ss❡ t✐♣♦ ❞❡ ❛t❛q✉❡ t❡♠ ♠❛✐♦r ✐♠♣♦rtâ♥❝✐❛ q✉❛♥❞♦ s❡ tr❛t❛ ❞❡ ❝r✐♣t♦❣r❛✜❛ ❛ss✐♠é✲ tr✐❝❛✱ ♦♥❞❡ ❛ ❝❤❛✈❡ é ♣ú❜❧✐❝❛ ❡ ✉♠ ✐♥✐♠✐❣♦ ♣♦❞❡ ❡♥❝r✐♣t❛r q✉❛❧q✉❡r ♠❡♥s❛❣❡♠ ❞❡ s❡✉ ✐♥t❡r❡ss❡✳
❖ ❛t❛q✉❡ ❞♦ t✐♣♦ ✐♥❢♦r♠❛çã♦ ❝✐❢r❛❞❛ ❡s❝♦❧❤✐❞❛ ✭❝❤♦s❡♥✲❝✐♣❤❡rt❡①t✮ é ✉♠ t✐♣♦ ❞❡ ❛t❛q✉❡ ♥♦ q✉❛❧ ♦ ✐♥✐♠✐❣♦ ♣♦❞❡ ❡s❝♦❧❤❡r ✉♠ ❝r✐♣t♦❣r❛♠❛ ❛r❜✐trár✐♦ ❡ ♦❜t❡r ♦ r❡s✉❧t❛❞♦ ❝♦rr❡t♦ ♣❛r❛ s✉❛ ❞❡❝r✐♣t❛çã♦✳ ❆❧❣✉♥s ❛❧❣♦r✐t♠♦s ❢❛♠♦s♦s ♥❛ s✉❛ ✈❡rsã♦ ♠❛✐s ❛♥t✐❣❛ ❡r❛♠ s✉s❝❡tí✈❡✐s ❛ ❡st❡ ❛t❛q✉❡✱ ❝♦♠♦ ♦ ❘❙❆✳
◆♦ ❛t❛q✉❡ ❞❡ ❢♦rç❛ ❜r✉t❛ ✭❜r✉t❡✲❢♦r❝❡✮ ♦ ✐♥✐♠✐❣♦ t❡♥t❛ t♦❞♦s ♦s ✈❛❧♦r❡s ♣♦ssí✈❡✐s ❞❡ ❝❤❛✈❡✳ ❊st❡ t✐♣♦ ❞❡ ❛t❛q✉❡ só é ❡①❡❝✉t❛❞♦ q✉❛♥❞♦ ♦s ♦✉tr♦s ♥ã♦ ❢♦r❡♠ ❡✜❝❛③❡s✳
◆❡ss❡ t✐♣♦ ❞❡ ❛t❛q✉❡✱ ♦ ♥ú♠❡r♦ ❡s♣❡r❛❞♦ ❞❡ t❡♥t❛t✐✈❛s ❛♥t❡s ❞❛ ❝❤❛✈❡ ❝♦rr❡t❛ s❡r ❡♥❝♦♥tr❛❞❛ é ❛ ♠❡t❛❞❡ ❞♦ t❛♠❛♥❤♦ ❞❛ ❝❤❛✈❡✳ ❈✐❢r❛s s✐♠étr✐❝❛s ❝♦♠ ❝❤❛✈❡s ❞❡ t❛♠❛♥❤♦ s✉♣❡r✐♦r ❛ ✻✹ ❜✐ts ❥á ❢♦r❛♠ q✉❡❜r❛❞❛s ❛tr❛✈és ❞❡ ❛t❛q✉❡s ❞❡ ❢♦rç❛ ❜r✉t❛✳ ❆ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ q✉❡❜r❛ ♣♦r ❡ss❡ ❛t❛q✉❡ ❛✉♠❡♥t❛ ❝♦♠ ♦ ♣❛ss❛r ❞♦ t❡♠♣♦ ❡ ❛ ❡✈♦❧✉çã♦ t❡❝♥♦❧ó❣✐❝❛✱ ❝♦♠ s✉❛ ❢♦rç❛ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❝❛❞❛ ✈❡③ ♠❛✐s rá♣✐❞❛✳
❆❧❣✉♥s t✐♣♦s ❞❡ ❡♥❝r✐♣t❛çã♦ ♥ã♦ ♣♦❞❡♠ s❡r q✉❡❜r❛❞♦s ♣❡❧♦ ♠ét♦❞♦ ❞❛ ❢♦rç❛ ❜r✉t❛✱ ❞❡✈✐❞♦ às s✉❛s ♣r♦♣r✐❡❞❛❞❡s ♠❛t❡♠át✐❝❛s✳ ❯♠ ❡①❡♠♣❧♦ é ♦ ♦♥❡✲t✐♠❡✲♣❛❞✱ ♦♥❞❡ ❝❛❞❛ ❜✐t ♣♦ss✉✐ s❡✉ ❝♦rr❡s♣♦♥❞❡♥t❡ ❜✐t ❝❤❛✈❡✳ ❊♠ r❡❧❛çã♦ ❛♦ ❛t❛q✉❡ ❞❡ ❢♦rç❛ ❜r✉t❛✱ ♦ ♦♥❡✲t✐♠❡✲ ♣❛❞ ❛♣❡♥❛s r❡✈❡❧❛ ✉♠❛ ❞❡❝♦❞✐✜❝❛çã♦ ❝♦rr❡t❛✱ ♠❛s q✉❛❧q✉❡r ♦✉tr❛ ♣♦ssí✈❡❧ ❝♦♠❜✐♥❛çã♦ ❞❡ ❜✐ts ♣♦❞❡ r❡s✉❧t❛r ♥❛ ♠❡s♠❛ ❝♦❞✐✜❝❛çã♦✱ ♦✉ s❡❥❛✱ ♥ã♦ é ♣♦ssí✈❡❧ ❞✐st✐♥❣✉✐r ✉♠❛ ❝❤❛✈❡ ✈❡r❞❛❞❡✐r❛ ❞❡ ♦✉tr❛ ❢❛❧s❛ q✉❡ r❡s✉❧t❡ ♥♦ ♠❡s♠♦ t❡①t♦ ❝✐❢r❛❞♦✳
❆ s❡❣✉✐r ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ❞❛s ❛♣❧✐❝❛çõ❡s ❞✐r❡t❛s ❞❡ s❡q✉ê♥❝✐❛s ❞❡ ♥ú♠❡r♦s ❛❧❡❛tór✐♦s ❡♠ ❝r✐♣t♦❣r❛✜❛✳
✷✳✺ ❯❚■▲■❩❆➬➹❖ ❉❊ ◆Ú▼❊❘❖❙ ❆▲❊❆❚Ó❘■❖❙ ❊▼ ❈❘■P✲
❚❖●❘❆❋■❆
❉❡♥tr❡ ❛s ❛♣❧✐❝❛çõ❡s q✉❡ ♦s ♥ú♠❡r♦s ❛❧❡❛tór✐♦s ♣♦❞❡♠ t❡r✱ ♥♦ q✉❡ s❡ r❡❢❡r❡ ❛♦ ❝❛♠♣♦ ❞❛ ❝r✐♣t♦❣r❛✜❛✱ ♣♦❞❡✲s❡ ❡♥✉♠❡r❛r✿
• ❈❤❛✈❡s ❞❡ s❡çõ❡s ❡ ❞❡ ♠❡♥s❛❣❡♥s ❞❡ ❝✐❢r❛s s✐♠étr✐❝❛s✱ ❝♦♠♦ ♦ ♦♥❡✲t✐♠❡✲♣❛❞✱ ♦
❉❊❙✱ ♦ ✸❉❊❙✳
• ◆ú♠❡r♦s q✉❡ s❡ ❝♦♠❜✐♥❛♠ ❝♦♠ s❡♥❤❛s✱ ♣❛r❛ ❝♦♥❢✉♥❞✐r ♣r♦❣r❛♠❛s ❞❡ ✈ír✉s q✉❡
❞❡❞✉③❡♠ s❡♥❤❛s✳
• ■♥✐❝✐❛❧✐③❛çã♦ ❞❡ ✈❡t♦r❡s ❞❡ ❝✐❢r❛s ❞❡ ❜❧♦❝♦s✳ • ❈❤❛✈❡s ❛❧❡❛tór✐❛s r❡❧❛❝✐♦♥❛❞❛s ❛ ♣r♦t♦❝♦❧♦s✳
❉❡ss❛ ♠❛♥❡✐r❛✱ ♥♦ ✐♥t✉✐t♦ ❞❡ ❛✉♠❡♥t❛r ❛ s❡❣✉r❛♥ç❛ ❞❡ s✐st❡♠❛s ❝r✐♣t♦❣rá✜❝♦s✱ ❞❡✈❡✲ s❡ ❛♥❛❧✐s❛r ❞❡ ♠❛♥❡✐r❛ ♠❛✐s ❛t❡♥❝✐♦s❛ ❛❧❣✉♠❛s ❞❛s ♠❛♥❡✐r❛s ❝♦♥❤❡❝✐❞❛s ❞❡ s❡ ❣❡r❛r ♥ú♠❡r♦s ❛❧❡❛tór✐♦s✱ ❡ ❞❡ q✉❡ ❢♦r♠❛ ❡st❡s ♣♦❞❡♠ s❡r ❛♣❧✐❝❛❞♦s ❡♠ s✐st❡♠❛s ♣rát✐❝♦s✳ ❊st❛ ❛♥á❧✐s❡ é ♦ ♦❜❥❡t♦ ❞❡ ❞✐s❝✉ssã♦ ❞♦ ❝❛♣ít✉❧♦ ❛ s❡❣✉✐r✳
✸ ●❊❘❆❉❖❘❊❙ ❉❊ ◆Ú▼❊❘❖❙ ❆▲❊❆❚Ó❘■❖❙
❆♣❡s❛r ❞❡ t❡r s❡ ❢❛❧❛❞♦ ❡♠ ♥ú♠❡r♦ ❛❧❡❛tór✐♦✱ ❡st❡ t❡r♠♦ só ❛❞q✉✐r❡ s❡♥t✐❞♦ q✉❛♥❞♦ ✈✐st♦ ❞❡♥tr♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦✱ ✉♠❛ ✈❡③ q✉❡ ♥ã♦ é ♣♦ssí✈❡❧ ❞✐③❡r s❡ ✉♠ ♥ú♠❡r♦ é ♠❛✐s ♦✉
♠❡♥♦s ❛❧❡❛tór✐♦ s❡♠ ❝♦♥❤❡❝❡r ❛ s❡q✉ê♥❝✐❛ ❛ q✉❡ ❡❧❡ ♣❡rt❡♥❝❡ ❡ ♦ ♥ú♠❡r♦ ❞❡ ❛♠♦str❛s ♥❡ss❛ s❡q✉ê♥❝✐❛✳
❆ ❣❡r❛çã♦ ❛❧❡❛tór✐❛ ❡♠ ❝r✐♣t♦❣r❛✜❛ ❝♦♥✜❣✉r❛ ✉♠❛ ❞❛s ❢♦♥t❡s ❞❡ s❡❣r❡❞♦ q✉❡ ❝♦♥st❛ ♥♦s s✐st❡♠❛s ❝r✐♣t♦❣rá✜❝♦s✱ ♣♦ré♠✱ ❛❧é♠ ❞✐ss♦✱ ❞❡✈❡✲s❡ ✉s❛r ✉♠ ❜♦♠ ❛❧❣♦r✐t♠♦ ❞❡ ❡♥❝r✐♣t❛çã♦ ❡ ♣r♦❝❡ss♦s r♦❜✉st♦s ❞❡ ♠❛♥❡❥♦ ❞❡ ❝❤❛✈❡s ♣❛r❛ ❢♦rt❛❧❡❝❡r ❛✐♥❞❛ ♠❛✐s ♦s ❛❧❣♦r✐t♠♦s✳
◆❛ ❣❡r❛çã♦ ❞❡ ❝❤❛✈❡s✱ ❛ ❜✉s❝❛ ♣♦r s❡q✉ê♥❝✐❛s ❛❧❡❛tór✐❛s é ♠✉✐t♦ ✐♠♣♦rt❛♥t❡✱ ♣r✐♥❝✐✲ ♣❛❧♠❡♥t❡ ♥♦ s❡♥t✐❞♦ ❞❡ ♣r♦❞✉çã♦ ❞❡ ❝❤❛✈❡s ❢♦rt❡s✳ ❆ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ s❡r ❣❡r❛❞❛ ✉♠❛ ❝❤❛✈❡ ❢r❛❝❛ é ♣❡q✉❡♥❛✳ ◆♦ ❛❧❣♦r✐t♠♦ ❉❊❙ ✭❉❛t❛ ❊♥❝r✐♣t✐♦♥ ❙t❛♥❞❛r❞✮ ♣♦r ❡①❡♠♣❧♦ ♣♦ss✉✐ ✶✻ ❝❤❛✈❡s ❢r❛❝❛s ❞❡ ✉♠ t♦t❛❧ ❞❡256✳ ▼❛s ❥á q✉❡ ❛ ❝❤❛✈❡ ♥ã♦ é ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦
❣❡r❛❧✱ ❛ ❢r❛q✉❡③❛ ❞❛ ❝❤❛✈❡ ♥ã♦ ❣❡r❛ t❛♥t♦ ♣❡r✐❣♦ ❞❡ ❡♥❢r❛q✉❡❝✐♠❡♥t♦ ❞♦ s✐st❡♠❛ ❝r✐♣✲ t♦❣rá✜❝♦✱ ❛ ♥ã♦ s❡r q✉❡ ❤❛❥❛ ❢❛❧❤❛ ♥♦ s✐st❡♠❛ ❞❡ tr❛♥s♠✐ssã♦ ❞♦s ❞❛❞♦s✳ ❆ ❣❡r❛çã♦ ❞❡ ❝❤❛✈❡s ❛❧❡❛tór✐❛s ♣❛r❛ ♦s s✐st❡♠❛s ❝r✐♣t♦❣rá✜❝♦s ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛ é ♠❛✐s ❞✐❢í❝✐❧✱ ♣♦✐s ❛❧é♠ ❞❡ ✉♠ r✐❣♦r ♠❛✐♦r ❡♠ r❡❧❛çã♦ à ❝♦♥✜r♠❛çã♦ ❞❡ ❛❧❡❛t♦r✐❡❞❛❞❡✱ ❡st❛s ❝❤❛✈❡s ♣♦ss✉❡♠ ❝❛r❛❝t❡ríst✐❝❛s ♠❛t❡♠át✐❝❛s ❡s♣❡❝í✜❝❛s✳
❆❧❣✉♠❛s ❢♦♥t❡s ❢ís✐❝❛s ❞❡ ❣❡r❛❞♦r❡s ❛❧❡❛tór✐♦s sã♦ ❞❡s❝r✐t❛s ❛ s❡❣✉✐r✳
✸✳✶ ❋❖◆❚❊❙ ❆▲❊❆❚Ó❘■❆❙
✸✳✶✳✶ ▲❆◆➬❆▼❊◆❚❖ ❉❊ ▼❖❊❉❆
❉❡ ♠❛♥❡✐r❛ ♠❛✐s t❡ór✐❝❛ ❛té ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ③❡r♦s ❡ ✉♥s ♦❜t✐❞❛ ♣❡❧♦s ❧❛♥ç❛♠❡♥t♦s ❞❡ ✉♠❛ ♠♦❡❞❛ ❝✉❥❛ ❢❛❝❡ ❝❛r❛ t❡♥❤❛ ❛♣❛r❡❝✐❞♦ ❜❛st❛ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❡ ✉♠❛ s❡q✉ê♥✲ ❝✐❛✱ ❛ss✉♠✐♥❞♦ q✉❡ ❡st❛ ♠♦❡❞❛ ♥ã♦ s❡❥❛ ✈✐❝✐❛❞❛✱ ♦✉ s❡❥❛✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❡♠ ✐♥✜♥✐t♦s ❧❛♥ç❛♠❡♥t♦s é ✐❣✉❛❧ ❛ 1/2 ✳ ➱ ❝❧❛r♦ q✉❡ ❡st❛ ❢♦r♠❛ ❞❡ ❣❡r❛çã♦ ❞❡ ❜✐ts ❛❧❡❛tór✐♦s ♥ã♦
t❡♠ ♥❡♥❤✉♠ ✈❛❧♦r ♣rát✐❝♦✱ ❛té ♣♦rq✉❡ ❛ t❛①❛ ❞❡ ❣❡r❛çã♦ ♥ã♦ é s✉✜❝✐❡♥t❡ ❡ ❡st❛ ❢♦r♠❛ é ✐♥❡✜❝✐❡♥t❡✳
✸✳✶✳✷ ❘❯❮❉❖ ❉❖ ❉■❖❉❖
❖ r✉í❞♦ ❞❡ ✉♠ ❞✐♦❞♦ ♣♦❞❡ s❡r✈✐r ♣❛r❛ ♣r♦❞✉③✐r s❡q✉ê♥❝✐❛s ❞❡ ♥ú♠❡r♦s ❛❧❡❛tór✐♦s✳ ■ss♦ ❛❝♦♥t❡❝❡ ♣♦r ❝❛✉s❛ ❞❡ ❞❡❢❡✐t♦s ✐♥❡r❡♥t❡s ♥♦s ♠❛t❡r✐❛✐s s❡♠✐❝♦♥❞✉t♦r❡s ❞♦ ❞✐♦❞♦ ❡ s❡✉ ❝♦♠♣♦rt❛♠❡♥t♦ ♥❛ r❡❣✐ã♦ ❧✐♠✐t❡ ❞❡ ♣♦❧❛r✐③❛çã♦✳
❆ ú♥✐❝❛ ♠❛♥❡✐r❛ ❞❡ s❛❜❡r s❡ ❡st❛ ❢♦r♠❛ ❞❡ ❣❡r❛çã♦ r❡❛❧♠❡♥t❡ ♣r♦♠♦✈❡ ♥ú♠❡r♦s ❛❧❡❛tór✐♦s é r❡❛❧✐③❛♥❞♦ t❡st❡s ❡st❛tíst✐❝♦s ❞❡ ❛❧❡❛t♦r✐❡❞❛❞❡✳ ❊♠ ❣❡r❛❧✱ ♦s ❞✐♦❞♦s sã♦ ❢♦♥t❡s ❞❡ s❡q✉ê♥❝✐❛s ❜✐♥ár✐❛s q✉❡ t❡♠ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡st❛tíst✐❝❛ ✉♥✐❢♦r♠❡✱ ♦ q✉❡ ♣r♦♠♦✈❡ ✉♠❛ ♠❛♥❡✐r❛ ❡✜❝✐❡♥t❡ ❞❡ ❣❡r❛çã♦ ❞❡ s❡q✉ê♥❝✐❛s ❛❧❡❛tór✐❛s✱ ♣♦ré♠ ♥❡♠ s❡♠♣r❡ ❛ t❛①❛s q✉❡ ❛t❡♥❞❛♠ ❞❡t❡r♠✐♥❛❞♦s s✐st❡♠❛s✳
✸✳✶✳✸ ▼❆❚❊❘■❆▲ ❘❆❉■❖❆❚■❱❖
❖s ♠❛t❡r✐❛✐s r❛❞✐♦❛t✐✈♦s t❛♠❜é♠ ♣♦❞❡♠ ❢✉♥❝✐♦♥❛r ❝♦♠♦ ✉♠ ❣❡r❛❞♦r ❞❡ ♥ú♠❡r♦s ❛❧❡❛tór✐♦s✱ ✉♠❛ ✈❡③ q✉❡ ❛ r❛❞✐♦❛t✐✈✐❞❛❞❡ ❞❡❝❛✐ ❛❧❡❛t♦r✐❛♠❡♥t❡✳
❯♠ ❝♦♥t❛❞♦r ●❡✐❣❡r✱ ❝♦♥❡❝t❛❞♦ ❛ ✉♠ ❝♦♠♣✉t❛❞♦r✱ ♣♦❞❡ s❡r ✉s❛❞♦ ♣❛r❛ tr❛♥s❢♦r♠❛r ❡st❡s ❞❛❞♦s ❡♠ ✐♥❢♦r♠❛çã♦ út✐❧ ❡ ❛❞♠✐♥✐strá✈❡❧ ❡♠ ❝r✐♣t♦❣r❛✜❛✳
❊ss❛ ❣❡r❛çã♦ é r❛r❛ ❡ ❝❛r❛ ❞❡✈✐❞♦ ❛♦s ❝✉st♦s ❞❡ ♠❛♥✉t❡♥çã♦ ❡ s❡❣✉r❛♥ç❛ ❞♦s ❞✐s♣♦✲ s✐t✐✈♦s ♥♦r♠❛❧♠❡♥t❡ ✉t✐❧✐③❛❞♦s✳
✸✳✶✳✹ ❚✶✷❘◆●
❖ ❚✶✷❘◆● é ✉♠ ❞✐s♣♦s✐t✐✈♦ ❝❛♣❛③ ❞❡ ❣❡r❛r ♥ú♠❡r♦s ❛❧❡❛tór✐♦s ❛tr❛✈és ❞❡ ✉♠ ❤❛r❞✇❛r❡✱ q✉❡ ♣♦❞❡ s❡r ❝♦♥❡❝t❛❞♦ à ✉♠❛ ♣♦rt❛ s❡r✐❛❧ ❞❡ ❝♦♠♣✉t❛❞♦r✳ ❆ ♦r✐❣❡♠ ❞❛ s✉❛ ❛❧❡❛t♦r✐❡❞❛❞❡ ♥❛ ❣❡r❛çã♦ é ❞❡✈✐❞♦ ❛♦ r✉í❞♦ ❜r❛♥❝♦✱ ♣r♦♣r✐❡❞❛❞❡ ❞♦s s❡♠✐❝♦♥❞✉t♦r❡s ✐♥st❛❧❛❞♦s ♥♦ ❝✐r❝✉✐t♦✳
P♦❞❡ s❡r ✉s❛❞♦ ❡♠ s❡♥❤❛s✱ ❝❤❛✈❡s ❝r✐♣t♦❣rá✜❝❛s✱ ♦♥❡✲t✐♠❡✲♣❛❞s✱ ❡t❝✳
❚❡♠ ✉♠❛ t❛①❛ ❞❡ ❣❡r❛çã♦ ❞❡ ✻✵ ❜②t❡s❄s❡❣✉♥❞♦✱ ♦ q✉❡ é ❝♦♥s✐❞❡r❛❞♦ ♠✉✐t♦ ❧❡♥t♦ ❡♠ ❛♣❧✐❝❛çõ❡s ♣rát✐❝❛s✳
❈♦♠♦ ❛s ❢♦♥t❡s ✈❡r❞❛❞❡✐r❛♠❡♥t❡ ❛❧❡❛tór✐❛s sã♦ ♦r✐❣✐♥❛❞❛s ✉s✉❛❧♠❡♥t❡ ♣♦r ❢❡♥ô✲ ♠❡♥♦s ❢ís✐❝♦s✱ ❡ ❡st❡s sã♦ ❞❡ ❞✐❢í❝✐❧ ❝❛♣t❛çã♦ ❡ tr❛♥s❢♦r♠❛çã♦ ❡♠ ❞❛❞♦s ✉t✐❧✐③á✈❡✐s ❛ ✉♠❛ ❢r❡q✉ê♥❝✐❛ ❛❧t❛ ♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ❣❛r❛♥t✐r ❛ ❝♦♠✉♥✐❝❛çã♦ ❡✜❝✐❡♥t❡✱ ♦♣t❛✲s❡ ♣♦r ✉t✐❧✐③❛r ❡ss❡s r❛r♦s ♥ú♠❡r♦s ❛❧❡❛tór✐♦s ❝♦♠♦ s❡♠❡♥t❡s ❞❡ ❣❡r❛❞♦r❡s ❞❡ ♥ú♠❡r♦s ♣s❡✉❞♦✲ ❛❧❡❛tór✐♦s✳ ❊st❛s s❡q✉ê♥❝✐❛s sã♦ s✉✜❝✐❡♥t❡♠❡♥t❡ ❛❧❡❛tór✐❛s ❞❡s❞❡ q✉❡ ❛ s❡♠❡♥t❡ ♥ã♦ s❡❥❛ ❝♦♥❤❡❝✐❞❛✳ ❆ ❛♣r❡s❡♥t❛çã♦ ❞❡ ❛❧❣✉♥s ❞♦s ♠ét♦❞♦s ✉t✐❧✐③❛❞♦s ♣♦r ❡ss❡s ❣❡r❛❞♦r❡s ❞❡ ♥ú♠❡r♦s ♣s❡✉❞♦✲❛❧❡❛tór✐♦s é ♦ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦ ❞♦ ❝❛♣ít✉❧♦ ❛ s❡❣✉✐r✳
✹
●❊❘❆❉❖❘❊❙ ❉❊ ◆Ú▼❊❘❖❙ P❙❊❯❉❖✲❆▲❊❆❚Ó❘■❖❙
❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ ✉s♦ ❡s♣❡❝í✜❝♦ ❞❛s s❡q✉ê♥❝✐❛s ❞❡ ♥ú♠❡r♦s ♣s❡✉❞♦✲❛❧❡❛tór✐♦s✱ ❡st❛s ♣♦ss✉❡♠ ❞✐✈❡rs❛s ❞❡✜♥✐çõ❡s✱ ❛❧❣✉♠❛s ♠❛✐s r❡str✐t✐✈❛s ❡ ♦✉tr❛s ♠❛✐s ❛❜r❛♥❣❡♥t❡s✳ ❆q✉✐ s❡ ❛♣r❡s❡♥t❛♠ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❛❜r❛♥❣❡♥t❡s✳
❯♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s ♣s❡✉❞♦✲❛❧❡❛tór✐♦s s❡ ❝❛r❛❝t❡r✐③❛ ♣♦r s❡r ✉♠❛ s❡q✉ê♥✲ ❝✐❛ ❝✉❥♦ ♣❡rí♦❞♦ é ❧♦♥❣♦ ♦ ❜❛st❛♥t❡ ♣❛r❛ q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛ ✜♥✐t❛ r❡t✐r❛❞❛ ❞❡st❛✱ ❞❡ ❝♦♠♣r✐♠❡♥t♦ r❛③♦á✈❡❧ ✭❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ✉s♦ q✉❡ s❡rá ❢❡✐t♦ ❞❡❧❛✮ ♥ã♦ s❡❥❛ ♣❡r✐ó❞✐❝❛✳
P❛r❛ ♦ ♣r♦♣ós✐t♦ ❞❡ ❛✉①í❧✐♦ ❛♦s ♣r♦❝❡ss♦s ❝r✐♣t♦❣rá✜❝♦s✱ ♦ ❣❡r❛❞♦r ❞❡ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s ♣s❡✉❞♦✲❛❧❡❛tór✐♦s ❞❡✈❡ t❡r ❛ s❡❣✉✐♥t❡ ❝❛r❛❝t❡ríst✐❝❛✿ ❛ss❡♠❡❧❤❛r✲s❡ ❛ ✉♠❛ s❡q✉ê♥❝✐❛ ❛❧❡❛tór✐❛✱ ♦✉ s❡❥❛✱ ❡st❛ s❡q✉ê♥❝✐❛ ❞❡✈❡ ♣❛ss❛r ❡♠ t♦❞♦s ♦s t❡st❡s ❡st❛tíst✐❝♦s ❞❡ ❛❧❡❛t♦r✐❡❞❛❞❡ ♣♦ssí✈❡✐s✳
❚♦❞♦s ♦s ❣❡r❛❞♦r❡s ♣s❡✉❞♦✲❛❧❡❛tór✐♦s sã♦ ♣❡r✐ó❞✐❝♦s✱ ♠❛s ❡st❡s ♣❡rí♦❞♦s ♣♦❞❡♠ ❝❤❡❣❛r ❛2256❜✐ts ♦✉ ♠❛✐s✱ ❞❛í ❛ s✉❛ ✉t✐❧✐❞❛❞❡ ❡♠ ❝r✐♣t♦❣r❛✜❛✳ ❙❡♥❞♦ ♠❛✐s ♣rát✐❝♦ ❞❡
s❡ ✐♠♣❧❡♠❡♥t❛r ❡ ♣♦ss✉✐♥❞♦ ✉♠ ♣❡rí♦❞♦ tã♦ ❧♦♥❣♦ ❝♦♠♦ ❡st❡✱ ❛✉♠❡♥t❛✲s❡ ❛ ❢❛❝✐❧✐❞❛❞❡ ❞❡ ♦❜t❡♥çã♦ ❡ ❞❡ ❣❡r❡♥❝✐❛♠❡♥t♦ ❞❡ ❝❤❛✈❡s ♣s❡✉❞♦✲❛❧❡❛tór✐❛s✱ ❝❛r❛❝t❡ríst✐❝❛s ❞♦s ♣r♦❝❡ss♦s ❝r✐♣t♦❣rá✜❝♦s✳
❉♦s ♠ét♦❞♦s ❞❡ ❣❡r❛çã♦ ❞❡ ❝❤❛✈❡s ✉t✐❧✐③❛❞♦s ❡♠ ❝r✐♣t♦❣r❛✜❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❞♦ t✐♣♦ s✐♠étr✐❝❛✱ ♦s ❣❡r❛❞♦r❡s ❞❡ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s ♣s❡✉❞♦✲❛❧❡❛tór✐♦s sã♦ ❛q✉❡❧❡s ❞❡ ❛♣❧✐❝❛çã♦ ♠❛✐s ✐♠♣♦rt❛♥t❡✳ P❛r❛ s❡r ❝r✐♣t♦❣r❛✜❝❛♠❡♥t❡ s❡❣✉r❛✱ ✉♠❛ s❡q✉ê♥❝✐❛ ❛❧❡❛tór✐❛ ❣❡r❛❞❛ ❞❡✈❡ t❡r ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
• ❙❡r ✐♠♣r❡✈✐sí✈❡❧ − ❉❡ ♠♦❞♦ q✉❡ s❡❥❛ ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡ ✐♠♣r❛t✐❝á✈❡❧ ♣r❡✈❡r
♦ ♣ró①✐♠♦ ❜✐t ❛❧❡❛tór✐♦ ❞❛ s❡q✉ê♥❝✐❛✱ ❝♦♥❤❡❝❡♥❞♦ t♦❞♦ ♦ ❛❧❣♦r✐t♠♦✱ ❤❛r❞✇❛r❡ ❣❡r❛❞♦r ♦✉ t♦❞❛ ❛ s❡q✉ê♥❝✐❛ ❛♥t❡r✐♦r ❞❡ ❜✐ts✳
• ◆ã♦ ♣♦❞❡ s❡r r❡♣r♦❞✉③✐❞❛ ✜❡❧♠❡♥t❡ − ❙❡ ✉♠ ❣❡r❛❞♦r ❢♦r ✐♥✐❝✐❛❞♦ ❝♦♠ ❛ ♠❡s♠❛
❡♥tr❛❞❛ ❥á ✉t✐❧✐③❛❞❛✱ ❛ s❛í❞❛ ❞❡✈❡ s❡r ✉♠❛ s❡q✉ê♥❝✐❛ ❛❧❡❛tór✐❛ ❝♦♠♣❧❡t❛♠❡♥t❡ ❞❡s❝♦rr❡❧❛t❛❞❛ ❞❛ ♣r✐♠❡✐r❛✳
▼❡s♠♦ ♣♦ss✉✐♥❞♦ ❞✐✈❡rs❛s ✈❛♥t❛❣❡♥s✱ ❡st❡ ♠ét♦❞♦ ❞❡ ❣❡r❛çã♦ ❞❡ s❡q✉ê♥❝✐❛s só ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ❡♠ ❝♦♥❥✉♥t♦ ❝♦♠ ♦ ♠ét♦❞♦ ✐♥t❡✐r❛♠❡♥t❡ ❛❧❡❛tór✐♦✳
