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❙tr❡❛♠ ❝✐♣❤❡rs

❆♥♥❡ ❈❛♥t❡❛✉t ❆♥♥❡✳❈❛♥t❡❛✉t❅✐♥r✐❛✳❢r ❤tt♣✿✴✴✇✇✇✲r♦❝q✳✐♥r✐❛✳❢r✴s❡❝r❡t✴❆♥♥❡✳❈❛♥t❡❛✉t✴ ❙✉♠♠❡r ❙❝❤♦♦❧✱ ➆✐❜❡♥✐❦✱ ❏✉♥❡ ✷✵✶✹

❖✉t❧✐♥❡

• ❇❛s✐❝ ♣r✐♥❝✐♣❧❡
• ●❡♥❡r❛❧ ♠♦❞❡❧ ❢♦r ❛ P❘◆●
• ●❡♥❡r✐❝ ❛tt❛❝❦s
• ▼❛✐♥ ❢❛♠✐❧✐❡s ♦❢ P❘◆●

✶

❙tr❡❛♠ ❝✐♣❤❡rs ✈s ❜❧♦❝❦ ❝✐♣❤❡rs ❬❍❛♥❞❜♦♦❦ ♦❢ ❝r②♣t♦❪ ❇❧♦❝❦ ❝✐♣❤❡r✿ ❛ ❢❛♠✐❧② ♦❢ ♣❡r♠✉t❛t✐♦♥s ♦♣❡r❛t✐♥❣ ♦♥ ❧❛r❣❡ ❜❧♦❝❦s ✭✻✹ ♦r ✶✷✽ ❜✐ts✮✱ ❞❡♣❡♥❞✐♥❣ ♦♥ ❛ ❦❡②✳ ❙tr❡❛♠ ❝✐♣❤❡r✿ ❛♥ ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ ✇❤✐❝❤ ❡♥❝r②♣ts ✐♥❞✐✈✐❞✉❛❧ ❞✐❣✐ts ✭✉s✉❛❧❧② ❜✐ts ♦r ❜②t❡s✮ ♦❢ ❛ ♣❧❛✐♥t❡①t ♦♥❡ ❛t ❛ t✐♠❡✱ ✉s✐♥❣ ❛ tr❛♥s❢♦r✲ ♠❛t✐♦♥ ✇❤✐❝❤ ✈❛r✐❡s ✇✐t❤ t✐♠❡✳ ❘❡♠❛r❦✿ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ ❞✐✛❡r❡♥t ♥❛t✉r❡✿ str❡❛♠ ❝✐♣❤❡rs ♦♣❡r✲ ❛t❡ ♦♥ ✈❛r✐❛❜❧❡✲❧❡♥❣t❤ ♠❡ss❛❣❡s ✇❤✐❧❡ ❜❧♦❝❦ ❝✐♣❤❡rs ♦♣❡r❛t❡ ♦♥ ✜①❡❞✲ ❧❡♥❣t❤ ♠❡ss❛❣❡s✳ ❜❧♦❝❦ ❝✐♣❤❡rs ♠✉st ❜❡ ✉s❡❞ ✇✐t❤ ❛ ♠♦❞❡ ♦❢ ♦♣❡r❛t✐♦♥ ✭❡✳❣✳✱ ❈❇❈✮✳ ❙♦♠❡ ♠♦❞❡s ♦❢ ♦♣❡r❛t✐♦♥ ❜✉✐❧❞ ❛ str❡❛♠ ❝✐♣❤❡r ❢r♦♠ ❛ ❜❧♦❝❦ ❝✐♣❤❡r ✭❡✳❣✳ ❆❊❙✲❈❚❘✮✦

✷

❙tr❡❛♠ ❝✐♣❤❡rs ✈s ❜❧♦❝❦ ❝✐♣❤❡rs ❬❍❛♥❞❜♦♦❦ ♦❢ ❝r②♣t♦❪ ❇❧♦❝❦ ❝✐♣❤❡r✿ ❛ ❢❛♠✐❧② ♦❢ ♣❡r♠✉t❛t✐♦♥s ♦♣❡r❛t✐♥❣ ♦♥ ❧❛r❣❡ ❜❧♦❝❦s ✭✻✹ ♦r ✶✷✽ ❜✐ts✮✱ ❞❡♣❡♥❞✐♥❣ ♦♥ ❛ ❦❡②✳ ❙tr❡❛♠ ❝✐♣❤❡r✿ ❛♥ ❡♥❝r②♣t✐♦♥ s❝❤❡♠❡ ✇❤✐❝❤ ❡♥❝r②♣ts ✐♥❞✐✈✐❞✉❛❧ ❞✐❣✐ts ✭✉s✉❛❧❧② ❜✐ts ♦r ❜②t❡s✮ ♦❢ ❛ ♣❧❛✐♥t❡①t ♦♥❡ ❛t ❛ t✐♠❡✱ ✉s✐♥❣ ❛ tr❛♥s❢♦r✲ ♠❛t✐♦♥ ✇❤✐❝❤ ✈❛r✐❡s ✇✐t❤ t✐♠❡✳ ❘❡♠❛r❦✿

• tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ ❞✐✛❡r❡♥t ♥❛t✉r❡✿

str❡❛♠ ❝✐♣❤❡rs ♦♣❡r❛t❡ ♦♥ ✈❛r✐❛❜❧❡✲❧❡♥❣t❤ ♠❡ss❛❣❡s ✇❤✐❧❡ ❜❧♦❝❦ ❝✐♣❤❡rs ♦♣❡r❛t❡ ♦♥ ✜①❡❞✲ ❧❡♥❣t❤ ♠❡ss❛❣❡s✳

• ❜❧♦❝❦ ❝✐♣❤❡rs ♠✉st ❜❡ ✉s❡❞ ✇✐t❤ ❛ ♠♦❞❡ ♦❢ ♦♣❡r❛t✐♦♥ ✭❡✳❣✳✱ ❈❇❈✮✳

❙♦♠❡ ♠♦❞❡s ♦❢ ♦♣❡r❛t✐♦♥ ❜✉✐❧❞ ❛ str❡❛♠ ❝✐♣❤❡r ❢r♦♠ ❛ ❜❧♦❝❦ ❝✐♣❤❡r ✭❡✳❣✳ ❆❊❙✲❈❚❘✮✦

✸

❆❞❞✐t✐✈❡ s②♥❝❤r♦♥♦✉s str❡❛♠ ❝✐♣❤❡rs ❚❤❡ ❦❡②str❡❛♠ ✐s ❛ ♣s❡✉❞♦✲r❛♥❞♦♠ s❡q✉❡♥❝❡ ❞❡r✐✈❡❞ ❢r♦♠ ❛ ✭s❤♦rt✮ s❡❝r❡t ❦❡②✳

❢

❣❡♥❡r❛t♦r

✲ ✻ ✲ ✚✙ ✛✘

+ si

❦❡②str❡❛♠ ♣❧❛✐♥t❡①t

ci K❂ ✐♥✐t✐❛❧✐③❛t✐♦♥ mi

❝✐♣❤❡rt❡①t

✹

❆❞✈❛♥t❛❣❡s ♦❢ str❡❛♠ ❝✐♣❤❡rs

• ♥♦ ❜✉✛❡r✐♥❣❀
• ♣r❡❝♦♠♣✉t❛t✐♦♥ ✐s ♣♦ss✐❜❧❡❀
• ♥♦ ♣❛❞❞✐♥❣ ✭✐♠♣♦rt❛♥t ✐❢ s❤♦rt ♣❛❝❦❡ts✮❀
• ♥♦ ❡rr♦r✲♣r♦♣❛❣❛t✐♦♥✳

❲❤❡♥ ❞♦ ✇❡ ✉s❡ ❛ str❡❛♠ ❝✐♣❤❡r❄

• ❧♦✇✲❜❛♥❞✇✐❞t❤ ❝♦♠♠✉♥✐❝❛t✐♦♥s
• ♥♦✐s② tr❛♥s♠✐ss✐♦♥s
• ✐♥ ♠♦st ❛♣♣❧✐❝❛t✐♦♥s✳✳✳✳

✺

▼♦❞❡❧ ❢♦r ❛ ♣s❡✉❞♦✲r❛♥❞♦♠ ❣❡♥❡r❛t♦r

✻

Ps❡✉❞♦✲r❛♥❞♦♠ ❣❡♥❡r❛t♦r ❢♦r ❛❞❞✐t✐✈❡ str❡❛♠ ❝✐♣❤❡rs ❉❡✜♥✐t✐♦♥✳ ❋✐♥✐t❡✲st❛t❡ ❛✉t♦♠❛t♦♥ ✇❤✐❝❤ ♣r♦❞✉❝❡s ✐♥ ❛ ❞❡t❡r♠✐♥✐st✐❝ ✇❛② ❛ ❧♦♥❣ s❡q✉❡♥❝❡ s ❢r♦♠ ❛ ✭s❤♦rt✮ s❡❡❞ s✉❝❤ t❤❛t✱ ❢♦r ❛♥ ❛❞✈❡rs❛r② ✇❤♦ ❦♥♦✇s ❡✈❡r②t❤✐♥❣ ❡①❝❡♣t t❤❡ s❡❡❞✱ ✐t ✐s ✐♠♣♦ss✐❜❧❡ t♦ ❞✐st✐♥❣✉✐s❤

s ❢r♦♠ ❛ r❛♥❞♦♠ s❡q✉❡♥❝❡ ✇✐t❤ ❛ s✐❣♥✐✜❝❛♥t❧② ❧♦✇❡r ❝♦♠♣❧❡①✐t② t❤❛♥

❛♥ ❡①❤❛✉st✐✈❡ s❡❛r❝❤ ❢♦r t❤❡ s❡❡❞✳

✼

◆♦t t♦ ❜❡ ❝♦♥❢✉s❡❞ ✇✐t❤ ❘❛♥❞♦♠ ♥✉♠❜❡r ❣❡♥❡r❛t♦r✿

• ❚❤❡r♠❛❧ ♥♦✐s❡ ❢r♦♠ ❛ r❡s✐st♦r✱
• ❖❜s❡r✈❛t✐♦♥s ♦❢ s♦♠❡ ♣❤②s✐❝❛❧ ❡✈❡♥ts ❛✈❛✐❧❛❜❧❡ t♦ t❤❡ s♦❢t✇❛r❡✱

❡✳❣✳✱ ✴❞❡✈✴r❛♥❞♦♠✴ r❛♥❞✭✮ ✐♥ t❤❡ ❆◆❙■✴■❙❖ ❈ st❛♥❞❛r❞✿ ✐♥t r❛♥❞✭✈♦✐❞✮ ✴✴ ❘❆◆❉❴▼❆❳ ❛ss✉♠❡❞ t♦ ❜❡ ✸✷✼✻✼ ④ ♥❡①t ❂ ♥❡①t ✯ ✶✶✵✸✺✶✺✷✹✺ ✰ ✶✷✸✹✺❀ r❡t✉r♥ ✭✉♥s✐❣♥❡❞ ✐♥t✮✭♥❡①t✴✻✺✺✸✻✮ ✪ ✸✷✼✻✽❀ ⑥

✽

• ❡♥❡r❛❧ ❝♦♥str✉❝t✐♦♥

✐♥✐t✐❛❧✐③❛t✐♦♥

❏ ❏ ❏ ❏ ❏ ❏ ❫ ✡ ✡ ✡ ✡ ✡ ✡ ✢

x0

❄

s❡❝r❡t ❦❡② k ❜✐ts ♣✉❜❧✐❝ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥t❡r♥❛❧ st❛t❡ ✐♥✐t✐❛❧ st❛t❡ n ❜✐ts

✾

• ❡♥❡r❛❧ ❝♦♥str✉❝t✐♦♥

✐♥✐t✐❛❧✐③❛t✐♦♥

❏ ❏ ❏ ❏ ❏ ❏ ❫ ✡ ✡ ✡ ✡ ✡ ✡ ✢

x0

• ❅

❅ ❄ ❄ ❄ ❄ ❄ ❄ ✲ ❄

s❡❝r❡t ❦❡② k ❜✐ts ♣✉❜❧✐❝ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥t❡r♥❛❧ st❛t❡ ✜❧t❡r

f

❦❡②str❡❛♠

s0

n ❜✐ts

✶✵

• ❡♥❡r❛❧ ❝♦♥str✉❝t✐♦♥

✫✪ ✬✩

Φ

✫✪ ✬✩

Φ

✐♥✐t✐❛❧✐③❛t✐♦♥

❏ ❏ ❏ ❏ ❏ ❏ ❫ ✡ ✡ ✡ ✡ ✡ ✡ ✢

x1

• ❅

❅ ❄ ❄ ❄ ❄ ❄ ❄ ✲ ✲

x0

• ❅

❅ ❄ ❄ ❄ ❄ ❄ ❄ ✲ ✲ ❄

s❡❝r❡t ❦❡② k ❜✐ts ♣✉❜❧✐❝ ✐♥✐t✐❛❧ ✈❛❧✉❡ ✐♥t❡r♥❛❧ st❛t❡ ✜❧t❡r ❦❡②str❡❛♠

f f s0 s1

· · n ❜✐ts tr❛♥s✐t✐♦♥

✶✶

• ❡♥❡r✐❝ ❛tt❛❝❦s

✶✷

❉✐✛❡r❡♥t t②♣❡s ♦❢ ❛tt❛❝❦s ❆tt❛❝❦❡r t②♣❡s✿ ❝✐♣❤❡rt❡①t ♦♥❧②❀ ❦♥♦✇♥ ♣❧❛✐♥t❡①t ✭♦r ❝❤♦s❡♥ ♣❧❛✐♥t❡①t✴❝✐♣❤❡rt❡①t✮❀ r❡❧❛t❡❞ ■❱s❀ ❝❤♦s❡♥ ■❱✳ ❆tt❛❝❦❡r ❣♦❛❧s✿

• ❑❡② r❡❝♦✈❡r②❀
• ■♥✐t✐❛❧✲st❛t❡ r❡❝♦✈❡r② ✭♦♥❧② ❢♦r t❤❡ ❝✉rr❡♥t ■❱✮❀
• ◆❡①t✲❜✐t ♣r❡❞✐❝t✐♦♥❀
• ❉✐st✐♥❣✉✐s❤❡r ✭❡✳❣✳✱ ❢♦r ❝❤❡❝❦✐♥❣ ✇❤❡t❤❡r s♦♠❡ ❡❛✈❡s❞r♦♣♣❡❞

❝✐♣❤❡rt❡①t ❝♦rr❡s♣♦♥❞s t♦ ❛ ❣✐✈❡♥ ♣❧❛✐♥t❡①t✮✳✳✳ ❚❤❡ ❧❛st t✇♦ ❛tt❛❝❦s ❛r❡ ❡q✉✐✈❛❧❡♥t ❬❨❛♦ ✽✷❪✳

✶✸

P❡r✐♦❞ ♦❢ t❤❡ s❡q✉❡♥❝❡ ♦❢ ✐♥t❡r♥❛❧ st❛t❡s ❋♦r ❛♥② ✐♥✐t✐❛❧ x0✱ (xt)t≥0 = (Φt(x0))t≥0 s❤♦✉❧❞ ❤❛✈❡ ❛ ❤✐❣❤ ♣❡r✐♦❞✳ ❋✉♥❝t✐♦♥❛❧ ❣r❛♣❤ ♦❢ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✿

Φ : {1, . . . , 20} → {1, . . . , 20} x → (x − 1)2 + 2 mod 20 + 1

✶✹

❋✉♥❝t✐♦♥❛❧ ❣r❛♣❤ ♦❢ ❛ r❛♥❞♦♠ ♠❛♣♣✐♥❣ ❬❋❧❛❥♦❧❡t ❖❞❧②③❦♦ ✽✾❪ ❊①❛♠♣❧❡ ❢r♦♠ ❬◗✉✐sq✉❛t❡r ❉❡❧❡s❝❛✐❧❧❡ ✽✼❪✿ ❖♥❡ ✏❣✐❛♥t ❝♦♠♣♦♥❡♥t✑ ✇✐t❤ ❧❡♥❣t❤ O(

√ N) ✇❤❡r❡ N ✐s t❤❡ s✐③❡ ♦❢

t❤❡ ✐♥♣✉t✴♦✉t♣✉t s❡t✳

O( √ N) ♣♦✐♥ts ✐♥ ❛ ❝②❝❧❡ ✭❡♥tr♦♣② ♦❢ t❤❡ st❛t❡ ❛❢t❡r s❡✈❡r❛❧ ✐t❡r❛t✐♦♥s✮✳

✶✺

❈❤♦♦s✐♥❣ t❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ Φ ❚✇♦ ♣♦ss✐❜✐❧✐t✐❡s✿

• ❈❤♦♦s❡ ❛ r❛♥❞♦♠✲❧♦♦❦✐♥❣ ♠❛♣♣✐♥❣✴♣❡r♠✉t❛t✐♦♥ ♦♣❡r❛t✐♥❣ ♦♥ ❛

❧❛r❣❡ ✐♥t❡r♥❛❧ st❛t❡✿ t❤❡ ♣❡r✐♦❞ ♦❢ (xt)t≥0 ✐s ❡①♣❡❝t❡❞ t♦ ❜❡ ❝❧♦s❡ t♦ 2

n 2✳ ❙❤♦rt ❝②❝❧❡s ❡①✐st ❜✉t ❛r❡ ✉♥❧✐❦❡❧② t♦ ♦❝❝✉r✳ ❊❣✿ ❘❈✹✳

• ❈❤♦♦s❡ ❛ ♣❡r♠✉t❛t✐♦♥ ✇✐t❤ s♦♠❡ ❦♥♦✇♥ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦♣❡rt✐❡s

♦♣❡r❛t✐♥❣ ♦♥ ❛ s♠❛❧❧ ✐♥t❡r♥❛❧ st❛t❡✿ t❤❡ ♣❡r✐♦❞ ♦❢ (xt)t≥0 ❝❛♥ ❜❡ ♣r♦✈❡❞ t♦ ❜❡ ❝❧♦s❡ t♦ 2n✳ ❙❤♦rt ❝②❝❧❡s ❛r❡ ❛✈♦✐❞❡❞✳ ❊❣✿ ❝♦✉♥t❡r✱ ▲❋❙❘✳

✶✻

❚✐♠❡✲♠❡♠♦r②✲❞❛t❛ tr❛❞❡✲♦✛ ❬●♦❧✐❝ ✾✺❪❬❇❛❜❜❛❣❡ ✾✺❪ ❋♦r ❛ str❡❛♠ ❝✐♣❤❡r✿

F : {0, 1}n → {0, 1}n x0 ✭✐♥✐t✐❛❧ st❛t❡✮ → s0, s1, . . . , sn−1

■♠♣r♦✈❡♠❡♥t✿ ■❢ ✇❡ ♥❡❡❞ t♦ ✜♥❞ ❛ ♣r❡✐♠❛❣❡ ❢♦r ❛ s✐♥❣❧❡ y ❛♠♦♥❣ s❡✈❡r❛❧ ♦♥❡s✱ t❤❡ tr❛❞❡✲♦✛ ❝❛♥ ✐♥✈♦❧✈❡ t❤❡ ❛♠♦✉♥t ♦❢ ❞❛t❛✳ ■❢ D ❝♦♥s❡❝✉t✐✈❡ ❜✐ts ♦❢ t❤❡ ❦❡②str❡❛♠ ❛r❡ ❦♥♦✇♥✱ ✇❡ ❣❡t (D − n + 1) ❢r❛♠❡s ♦❢ n ❜✐ts✿ yt = st, st+1 . . . st+n−1 ❢♦r 0 ≤ t ≤ D − n✳ ❚✐♠❡ ❂ D ♠❡♠♦r② ❂ ♣r❡❝♦♠♣✉t❛t✐♦♥ ❂ M = 2n

D ✳

❲❡ ❣❡t ❛♥ ❛tt❛❝❦ ✇✐t❤ ❞❛t❛✴t✐♠❡✴♠❡♠♦r② ❝♦♠♣❧❡①✐t② 2

n 2✳

✶✼

❘❡s✐st✐♥❣ t❤❡ ♠❛✐♥ ❣❡♥❡r✐❝ ❛tt❛❝❦s

• ❚❤❡ ✐♥t❡r♥❛❧ st❛t❡ ♠✉st ❜❡ ❛t ❧❡❛st t✇✐❝❡ ❧❛r❣❡r t❤❛♥ t❤❡ ❦❡②✳
• ❊✐t❤❡r t❤❡ ✐♥t❡r♥❛❧ st❛t❡ s❤♦✉❧❞ ❜❡ ❧❛r❣❡ ✇✐t❤ ❛ r❛♥❞♦♠✲❧♦♦❦✐♥❣

♥❡①t✲st❛t❡ ♠❛♣♣✐♥❣ Φ✱ ♦r ✐t ♠✉st ❜❡ ❣✉❛r❛♥t❡❡❞ t❤❛t Φ ❤❛s ♥♦ s❤♦rt ❝②❝❧❡s✳

• ❚❤❡ ❣❡♥❡r❛t♦r ♠✉st ♣❛ss t❤❡ st❛t✐st✐❝❛❧ t❡sts✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡

✜❧t❡r✐♥❣ ❢✉♥❝t✐♦♥ f ♠✉st ❜❡ ❜❛❧❛♥❝❡❞✳

• ❆t ❧❡❛st ♦♥❡ ❢✉♥❝t✐♦♥ ❛♠♦♥❣ Φ ❛♥❞ f ♠✉st ❜❡ ♥♦♥❧✐♥❡❛r✳

✶✽

▼❛✐♥ ❢❛♠✐❧✐❡s ♦❢ ❣❡♥❡r❛t♦rs

✶✾

❇❧♦❝❦✲❝✐♣❤❡r ❜❛s❡❞ P❘◆● ❈♦✉♥t❡r ♠♦❞❡ ✭❈❚❘✮ K K K K c = 0 c = 1 c = 2 c = 3 E E E E ■❱ s0 . . . sn−1 sn . . . s2n−1 s2n . . . s3n−1 s3n . . . s4n−1 . . . ❈❤♦s❡♥✲■❱ ❞✐st✐♥❣✉✐s❤✐♥❣ ❛tt❛❝❦✿ ✐❢ ✇❡ ❡♥❝r②♣t m = (m0, m1) ✇✐t❤

(K, IV ) ❛♥❞ m′ = (m1, m2) ✇✐t❤ (K, IV + 1)✱ ✇❡ ❣❡t t✇♦ ✐❞❡♥t✐❝❛❧

❝✐♣❤❡rt❡①t ❜❧♦❝❦s✱ ♥❛♠❡❧② c1 = c′

0✳

✷✵

❇❧♦❝❦✲❝✐♣❤❡r ❜❛s❡❞ P❘◆● ▼♦❞✐✜❡❞ ❝♦✉♥t❡r ♠♦❞❡ ✭▼✐❧❡♥❛❣❡ ✐♥ ❯▼❚❙✮ ■❱ K E K E K K K c = 0 c = 1 c = 2 c = 3 E E E s0 . . . sn−1 sn . . . s2n−1 s2n . . . s3n−1 s3n . . . s4n−1 . . . ❉✐st✐♥❣✉✐s❤❡r ✇✐t❤ ❝♦♠♣❧❡①✐t② O

• 2

n 2

• ✇❤❡r❡ n ✐s t❤❡ ❜❧♦❝❦ s✐③❡✳

✷✶

❉❡❞✐❝❛t❡❞ P❘◆● ❚②♣✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s✿

• ❤✐❣❤ t❤r♦✉❣❤♣✉t ✐♥ s♦❢t✇❛r❡ ✭❢❛st❡r t❤❛♥ t❤❡ ❆❊❙✮❀
• ❧♦✇✲❝♦st ❤❛r❞✇❛r❡ ✭❤❡❛✈❡❧② r❡str✐❝t❡❞ ❣❛t❡ ❝♦✉♥t ♦r ♣♦✇❡r✮✳

✷✷

▲❋❙❘

st+L−1

✲ st+L−2 ✲ ✫✪ ✬✩

c2

✫✪ ✬✩

cL

✫✪ ✬✩

+

✫✪ ✬✩

+

✫✪ ✬✩

c1

✫✪ ✬✩

cL−1

✫✪ ✬✩

+

✲

st st+1

❄ ❄ ✲ ✲ ❄ ❄ ✛ ✛ ✛ ✲ ❄ ❄ ❄

✳✳✳ ♦✉t♣✉t st+L

c1, . . . , cL ❛r❡ t❤❡ ❜✐♥❛r② ❢❡❡❞❜❛❝❦ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ▲❋❙❘✳

❚❤❡ ❜✐♥❛r② s❡q✉❡♥❝❡ ♣r♦❞✉❝❡❞ ❜② t❤❡ ▲❋❙❘ s❛t✐s✜❡s ❛ ❧✐♥❡❛r r❡❝✉rr❡♥❝❡ r❡❧❛t✐♦♥ ♦❢ ❞❡❣r❡❡ L✿

st+L =

L

• i=1

cist+L−i, ∀t ≥ 0 .

✷✸

P❡r✐♦❞ ♦❢ t❤❡ s❡q✉❡♥❝❡ ❆♥② s❡q✉❡♥❝❡ ❣❡♥❡r❛t❡❞ ❜② ❛♥ ▲❋❙❘ ♦❢ ❧❡♥❣t❤ L ✐s ✉❧t✐♠❛t❡❧② ♣❡r✐♦❞✐❝✱ ✐✳❡✳✱ t❤❡ s❡q✉❡♥❝❡ ♦❜t❛✐♥❡❞ ❜② ✐❣♥♦r✐♥❣ ❛ ❝❡rt❛✐♥ ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ✐s ♣❡r✐♦❞✐❝✱ ❛♥❞ ✐ts ♣❡r✐♦❞ ✐s ❛t ♠♦st 2L − 1✳ ▼♦r❡♦✈❡r✱ ✐❢ cL = 1✱ t❤❡ ▲❋❙❘ ✐s ♥♦♥✲s✐♥❣✉❧❛r✱ ❛♥❞ ✐t ♣r♦❞✉❝❡s ♣❡r✐♦❞✐❝ s❡q✉❡♥❝❡s✳ ❋❡❡❞❜❛❝❦ ♣♦❧②♥♦♠✐❛❧✿

P (X) = 1 +

L

• i=1

ciXi .

❉❡✜♥✐t✐♦♥✳ ❚❤❡ ♠✐♥✐♠❛❧ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❛ s❡q✉❡♥❝❡ (st)t≥0 ✐s t❤❡ ❢❡❡❞❜❛❝❦ ♣♦❧②♥♦♠✐❛❧ ✇✐t❤ t❤❡ ❧♦✇❡st ♣♦ss✐❜❧❡ ❞❡❣r❡❡ ❢♦r ❛♥ ▲❋❙❘ ✇❤✐❝❤ ❝❛♥ ❣❡♥❡r❛t❡ t❤❡ s❡q✉❡♥❝❡✳

✷✹

▲❋❙❘s ✇✐t❤ ♠❛①✐♠❛❧ ♣❡r✐♦❞ Pr♦♣♦s✐t✐♦♥✳ ❚❤❡ ❧❡❛st ♣❡r✐♦❞ ♦❢ ❛ ❧✐♥❡❛r r❡❝✉rr✐♥❣ s❡q✉❡♥❝❡ ✐s ❡q✉❛❧ t♦ t❤❡ ♦r❞❡r ♦❢ ✐ts ♠✐♥✐♠❛❧ ♣♦❧②♥♦♠✐❛❧✱ ✐✳❡✳✱ t❤❡ ❧❡❛st ♣♦s✐t✐✈❡ ✐♥t❡❣❡r

e ❢♦r ✇❤✐❝❤ P0(X) ❞✐✈✐❞❡s Xe + 1✳

❚❤❡♥✱ ❛ s❡q✉❡♥❝❡ ❤❛s ♠❛①✐♠❛❧ ♣❡r✐♦❞ 2deg P0 − 1 ✐❢ ❛♥❞ ♦♥❧② ✐❢ P0 ✐s ❛ ♣r✐♠✐t✐✈❡ ♣♦❧②♥♦♠✐❛❧✳ ❙✐♠✐❧❛r t♦ ❛ ❝♦✉♥t❡r✿ ❚❤❡ s❡q✉❡♥❝❡s ♣r♦❞✉❝❡❞ ❜② ❛♥ ▲❋❙❘ ♦❢ ❧❡♥❣t❤ L ✇✐t❤ ♣r✐♠✐t✐✈❡ ❢❡❡❞❜❛❝❦ ♣♦❧②♥♦♠✐❛❧ P ❛r❡ ♦❢ t❤❡ ❢♦r♠

{x0, αx0, α2x0, α3x0, ....}, ✇✐t❤ x0 ∈ GF (2L)∗

✇❤❡r❡ α ✐s ❛ r♦♦t ♦❢ P ✱ ✐✳❡✳✱

{αi, αi+1 mod (2L−1), αi+2 mod (2L−1), ....}, ✇✐t❤ 0 ≤ i ≤ 2L − 2

✷✺

▲❋❙❘s ✇✐t❤ ♠❛①✐♠❛❧ ♣❡r✐♦❞ Pr♦♣♦s✐t✐♦♥✳ ❚❤❡ ❧❡❛st ♣❡r✐♦❞ ♦❢ ❛ ❧✐♥❡❛r r❡❝✉rr✐♥❣ s❡q✉❡♥❝❡ ✐s ❡q✉❛❧ t♦ t❤❡ ♦r❞❡r ♦❢ ✐ts ♠✐♥✐♠❛❧ ♣♦❧②♥♦♠✐❛❧✱ ✐✳❡✳✱ t❤❡ ❧❡❛st ♣♦s✐t✐✈❡ ✐♥t❡❣❡r

e ❢♦r ✇❤✐❝❤ P0(X) ❞✐✈✐❞❡s Xe + 1✳

❚❤❡♥✱ ❛ s❡q✉❡♥❝❡ ❤❛s ♠❛①✐♠❛❧ ♣❡r✐♦❞ 2deg P0 − 1 ✐❢ ❛♥❞ ♦♥❧② ✐❢ P0 ✐s ❛ ♣r✐♠✐t✐✈❡ ♣♦❧②♥♦♠✐❛❧✳ ❙✐♠✐❧❛r t♦ ❛ ❝♦✉♥t❡r✿ ❚❤❡ s❡q✉❡♥❝❡s ♣r♦❞✉❝❡❞ ❜② ❛♥ ▲❋❙❘ ♦❢ ❧❡♥❣t❤ L ✇✐t❤ ♣r✐♠✐t✐✈❡ ❢❡❡❞❜❛❝❦ ♣♦❧②♥♦♠✐❛❧ P ❛r❡ ♦❢ t❤❡ ❢♦r♠

{x0, αx0, α2x0, α3x0, ....}, ✇✐t❤ x0 ∈ GF (2L)∗

✇❤❡r❡ α ✐s ❛ r♦♦t ♦❢ P ✱ ✐✳❡✳✱

{αi, αi+1 mod (2L−1), αi+2 mod (2L−1), ....}, ✇✐t❤ 0 ≤ i ≤ 2L − 2

✷✻

❚❤❡ ✜❧t❡r ❣❡♥❡r❛t♦r f st ut

ut+γ1 ut+γ2ut+γ3 . . . ut+γn

✲ ✻ ✻ ✻ ✻ ✻ ✻ ✲ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ ✻

st = f(ut+γ1, ut+γ2, . . . , ut+γn), ∀t ≥ 0

✇❤❡r❡ (ut)t≥0 ✐s t❤❡ ▲❋❙❘ s❡q✉❡♥❝❡✱ f ✐s ❛ ❜❛❧❛♥❝❡❞ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ♦❢ n ✈❛r✐❛❜❧❡s✱ n ≤ L✱ ❛♥❞ (γi)1≤i≤n ✐s ❛ ❞❡❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ ♥♦♥♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs✳

✷✼

▲✐♥❡❛r ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ✜❧t❡r ❣❡♥❡r❛t♦r ❉❡✜♥✐t✐♦♥✳ ❋♦r ❛ s❡♠✐✲✐♥✜♥✐t❡ s❡q✉❡♥❝❡ s = (st)t≥0✱ t❤❡ ❧✐♥❡❛r ❝♦♠✲ ♣❧❡①✐t② Λ(s) ✐s t❤❡ s♠❛❧❧❡st ✐♥t❡❣❡r Λ s✉❝❤ t❤❛t s ❝❛♥ ❜❡ ❣❡♥❡r❛t❡❞ ❜② ❛♥ ▲❋❙❘ ♦❢ ❧❡♥❣t❤ Λ✱ ❛♥❞ ✐s ∞ ✐❢ ♥♦ s✉❝❤ ▲❋❙❘ ❡①✐sts✳ ▲♦✇❡r ❜♦✉♥❞ ♦♥ t❤❡ ❧✐♥❡❛r ❝♦♠♣❧❡①✐t② ❬❘✉❡♣♣❡❧ ✽✻❪✿ ✐❢ L ✐s ❛ ❧❛r❣❡ ♣r✐♠❡✱

Λ(s) ≥ L d

• ❢♦r ♠♦st ✜❧t❡r✐♥❣ ❢✉♥❝t✐♦♥s ✇✐t❤ ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡ d✳

→ ❚❤❡ ❞❡❣r❡❡ ♦❢ f s❤♦✉❧❞ ❜❡ ❛s ❤✐❣❤ ❛s ♣♦ss✐❜❧❡✳ ▼❛② ❜❡ ✈✉❧♥❡r❛❜❧❡ t♦ ❛❧❣❡❜r❛✐❝ ❛tt❛❝❦s ❛♥❞ ✈❛r✐❛♥ts ❬❈♦✉rt♦✐s ▼❡✐❡r ✵✸❪

✷✽

❚❤❡ ❝♦♠❜✐♥❛t✐♦♥ ❣❡♥❡r❛t♦r

✲ ❄ ❄ ❄ ✲ ❄ ❄ ❄ ✲ ❄ ❄ ✲ ✲ ✲ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✲

✳ ✳ ✳ u1

t

u2

t

un

t

f st ❚❤❡ ♦✉t♣✉ts ♦❢ n ▲❋❙❘s ❛r❡ ❝♦♠❜✐♥❡❞ ❜② ❛ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ♦❢ n ✈❛r✐❛❜❧❡s✿

st = f(u1

t, u2 t, . . . , un t )

❈♦rr❡❧❛t✐♦♥ ❛tt❛❝❦s ❬❙✐❡❣❡♥t❤❛❧❡r ✽✺❪ ❛♥❞ ♠❛♥② ✈❛r✐❛♥ts ❬▼❡✐❡r✲❙t❛✛❡❧❜❛❝❤✽✽❪

✷✾

▲❋❙❘ ✇✐t❤ ✐rr❡❣✉❧❛r ❝❧♦❝❦✐♥❣ ❚❤❡ ❣❡♥❡r❛t♦r ✐s ❝♦♠♣♦s❡❞ ♦❢ ♦♥❡ ♦r s❡✈❡r❛❧ ▲❋❙❘s✳ ❙♦♠❡ ▲❋❙❘ ❜✐ts ❞❡❝✐❞❡ ✇❤✐❝❤ ▲❋❙❘ t♦ ❝❧♦❝❦ ❛♥❞ ❤♦✇ ♦❢t❡♥✳ ❚❤❡ s❤r✐♥❦✐♥❣ ❣❡♥❡r❛t♦r ❬❈♦♣♣❡rs♠✐t❤✲❑r❛✇❝③②❦✲▼❛♥s♦✉r ✾✸❪

❤ ❝

▲❋❙❘ ❆ ▲❋❙❘ ❇

✲ ✻ ✲ ❚ ❚ ❚ ❚

st ■❢ ▲❋❙❘ ❇ ♦✉t♣✉ts ✵✱ t❤❡ ♦✉t♣✉t ❜✐t ♦❢ ▲❋❙❘ ❆ ✐s ❞✐s❝❛r❞❡❞✳

Λ(s) ≥ LA2LB−2 .

✸✵

❆✺✴✶ ✭●❙▼ str❡❛♠ ❝✐♣❤❡r✮ ✸ ▲❋❙❘s ♦❢ ❧❡♥❣t❤s ✶✾✱ ✷✷ ❛♥❞ ✷✸✳

❥

+

❥

+

❥

+

❥

+

❥

+

❥

+

❥

+

✒✑ ✓✏

+

✛ ❄ ❄ ❄ ✛ ✛ ✛ ❄ ✛ ❄ ✛ ❄ ❄ ✛ ✲ ✲ ✲ ❈ ❈ ❈ ❈ ❈ ❈ ❈❈ ❲ ✲ ✄ ✄ ✄ ✄ ✄ ✄ ✄✄ ✗ ✡ ☛ ✠ ✟

b

✟ ✟ ✟ ✟ ✙ ✻ ✲ st ❄

→❚❤❡ 64✲❜✐t ✐♥t❡r♥❛❧ st❛t❡ ♠❛❦❡s ✐t ✈✉❧♥❡r❛❜❧❡ t♦ ❚▼❉❚❖✳

✸✶

❚❤❡ ❡❙❚❘❊❆▼ ♣r♦❥❡❝t ✭✷✵✵✹✲✷✵✵✽✮ ❧❛✉♥❝❤❡❞ ❜② t❤❡ ❊✉r♦♣❡❛♥ ♥❡t✇♦r❦ ♦❢ ❡①❝❡❧❧❡♥❝❡ ❊❈❘❨P❚ ❤tt♣✿✴✴✇✇✇✳❡❝r②♣t✳❡✉✳♦r❣✴str❡❛♠✴ s♦❢t✇❛r❡ ❛♣♣❧✐❝❛t✐♦♥s ❤❛r❞✇❛r❡ ❛♣♣❧✐❝❛t✐♦♥s ❍❈✲✶✷✽

• r❛✐♥ ✈✶

❘❛❜❜✐t ▼■❈❑❊❨ ✷✳✵ ❙❛❧s❛✷✵✴✶✷ ❚r✐✈✐✉♠ ❙❖❙❊▼❆◆❯❑

✸✷

• r❛✐♥ ✈✶ ❬❍❡❧❧ ❏♦❤❛♥ss♦♥ ▼❡✐❡r❪

✸✸

❈♦♥❝❧✉s✐♦♥s ❇② ❞❡❢❛✉❧t✿ ✉s❡ ❆❊❙✲❈❚❘ ✭■❱✲r❡❧❛t❡❞ ❛tt❛❝❦s✱ ❞✐st✐♥❣✉✐s❤❡r ♦❢ ❝♦♠♣❧❡①✐t② 264✮✳ ❖t❤❡r✇✐s❡✿

• ✐♥ s♦❢t✇❛r❡✿ t❛❜❧❡✲❞r✐✈❡♥ ❣❡♥❡r❛t♦rs ✭❍❈✲✶✷✽✱ ♥♦t ❘❈✹✦✮

▲❋❙❘s ♦✈❡r GF (232) ✭❙◆❖❲ ✷✳✵✳✳✳✮

• ✐♥ ❤❛r❞✇❛r❡✿ ▲❋❙❘ ❛♥❞ ◆▲❋❙❘✲❜❛s❡❞ ❞❡s✐❣♥s✳

✸✹