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❙tr❡❛♠ ❝✐♣❤❡rs ■

❚❤♦♠❛s ❏♦❤❛♥ss♦♥

❉❡♣t✳ ♦❢ ❊■❚✱ ▲✉♥❞ ❯♥✐✈❡rs✐t②✱ P✳❖✳ ❇♦① ✶✶✽✱ ✷✷✶ ✵✵ ▲✉♥❞✱ ❙✇❡❞❡♥ t❤♦♠❛s❅❡✐t✳❧t❤✳s❡

▼❛② ✶✻✱ ✷✵✶✶

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❖✉t❧✐♥❡✿

• ■♥tr♦❞✉❝t✐♦♥ t♦ str❡❛♠ ❝✐♣❤❡rs
• ❉✐st✐♥❣✉✐s❤❡rs
• ❇❛s✐❝ ❝♦♥str✉❝t✐♦♥s ♦❢ ❞✐st✐♥❣✉✐s❤❡rs
• ❱❛r✐♦✉s t②♣❡s ♦❢ ❞✐st✐♥❣✉✐s❤✐♥❣ ❛tt❛❝❦s
• ❡❙❚❘❊❆▼ ❛♥❞ t✇♦ ❝✐♣❤❡rs ❢r♦♠ t❤❡ ♣♦rt❢♦❧✐♦

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

■♥tr♦❞✉❝t✐♦♥ t♦ str❡❛♠ ❝✐♣❤❡rs

• ❙tr❡❛♠ ❝✐♣❤❡rs ❛r❡ ✐♠♣♦rt❛♥t ✐♥ ❝r②♣t♦❣r❛♣❤② s✐♥❝❡ t❤❡② ❢♦r♠

♦♥❡ ♦❢ t❤❡ t✇♦ ♣♦ss✐❜❧❡ ✇❛②s t♦ ♣r♦✈✐❞❡ s②♠♠❡tr✐❝ ❡♥❝r②♣t✐♦♥✳

• ❙tr❡❛♠ ❝✐♣❤❡rs ❡♥❝r②♣t ✐♥❞✐✈✐❞✉❛❧ ❝❤❛r❛❝t❡rs ♦❢ ❛ ♣❧❛✐♥t❡①t

♠❡ss❛❣❡ ♦♥❡ ❜② ♦♥❡✱ ✉s✐♥❣ ❛♥ ❡♥❝r②♣t✐♦♥ tr❛♥s❢♦r♠❛t✐♦♥ t❤❛t ✈❛r✐❡s ✇✐t❤ t✐♠❡✳

• ❙tr❡❛♠ ❝✐♣❤❡rs ❛r❡ ❣❡♥❡r❛❧❧② ❢❛st❡r t❤❛♥ ❜❧♦❝❦ ❝✐♣❤❡rs ✐♥

❤❛r❞✇❛r❡✱ ❛♥❞ ❤❛✈❡ ❧❡ss ❝♦♠♣❧❡① ❤❛r❞✇❛r❡ ❝✐r❝✉✐tr②✳ ❚❤❡② ❛❧s♦ ❤❛✈❡ s♦♠❡ ♦t❤❡r ♥✐❝❡ ❢❡❛t✉r❡s t❤❛t ✐♥ s♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ✭t②♣✐❝❛❧❧② ❝♦♠♠✉♥✐❝❛t✐♦♥s ❛♣♣❧✐❝❛t✐♦♥s✮ t❡♥❞ t♦ ❜❡ q✉✐t❡ ✐♠♣♦rt❛♥t✱ ❧✐❦❡ ❧✐♠✐t❡❞ ❜✉✛❡r✐♥❣✱ ❧✐♠✐t❡❞ ❡rr♦r ♣r♦♣❛❣❛t✐♦♥✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

■♥tr♦❞✉❝t✐♦♥ t♦ str❡❛♠ ❝✐♣❤❡rs

• ❚❤❡r❡ ✐s ❛ ❧♦t ♦❢ t❤❡♦r❡t✐❝❛❧ ❦♥♦✇❧❡❞❣❡ ♦♥ str❡❛♠ ❝✐♣❤❡rs✱ ❛♥❞

✈❛r✐♦✉s ❞❡s✐❣♥ ♣r✐♥❝✐♣❧❡s ❢♦r str❡❛♠ ❝✐♣❤❡rs ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞ ❛♥❞ ❡①t❡♥s✐✈❡❧② ❛♥❛❧②③❡❞✳

• ❘❡❝❡♥t❧②✱ ✇❡ ❤❛✈❡ s❡❡♥ ❛ ❧♦t ♦❢ ❢✉❧❧②✲s♣❡❝✐✜❡❞ str❡❛♠ ❝✐♣❤❡r

♣r♦♣♦s❛❧s t❤r♦✉❣❤ s❡✈❡r❛❧ ❞❡s✐❣♥ ♣r♦❥❡❝ts✱ ❡✳❣✳ ◆❊❙❙■❊✱ ❡❙❚❘❊❆▼✳

• ■♥ ❛❞❞✐t✐♦♥✱ ♠❛♥② ♣r♦♣r✐❡t❛r② ❛♥❞ ❝♦♥✜❞❡♥t✐❛❧ str❡❛♠ ❝✐♣❤❡rs

❛r❡ ✉s❡❞ ✐♥ ♣r❛❝t✐❝❡✳ ❙♦♠❡ ❝✐♣❤❡rs ❤❛✈❡ ✐♥✐t✐❛❧❧② ❜❡❡♥ ❝♦♥✜❞❡♥t✐❛❧ ❜✉t ❧❛t❡r ❜❡❡♥ ❧❡❛❦❡❞ t♦ t❤❡ ♣✉❜❧✐❝✱ ❡✳❣✳✱ ❆✺ ❛♥❞ ❘❈✹✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

■♥tr♦❞✉❝t✐♦♥ t♦ str❡❛♠ ❝✐♣❤❡rs

• ▼♦st str❡❛♠ ❝✐♣❤❡r ❝♦♥str✉❝t✐♦♥s ✉s❡ ❛ ♣s❡✉❞♦✲r❛♥❞♦♠

❦❡②str❡❛♠ ❣❡♥❡r❛t♦r✱ ♦r s✐♠♣❧② ❛ ❣❡♥❡r❛t♦r✱ t♦ ♣r♦❞✉❝❡ ❛ ❧♦♥❣ s❡q✉❡♥❝❡ ♦❢ ❜✐♥❛r② s②♠❜♦❧s✳

• ❚❤❡ s❡❝✉r✐t② ♦❢ ❛ str❡❛♠ ❝✐♣❤❡r ✐s ❝❧♦s❡❧② ❝♦♥♥❡❝t❡❞ t♦ ❤♦✇

✇❡❧❧ t❤✐s s❡q✉❡♥❝❡ ♦❢ ❜✐ts r❡s❡♠❜❧❡s ❛ tr✉❧② r❛♥❞♦♠ s❡q✉❡♥❝❡✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❙tr❡❛♠ ❝✐♣❤❡rs

• ❈♦♥s✐❞❡r ❛ ❜✐♥❛r② ❛❞❞✐t✐✈❡ str❡❛♠ ❝✐♣❤❡r✳ ❚❤❡ ♦✉t♣✉t s❡q✉❡♥❝❡

♦❢ t❤❡ ❦❡②str❡❛♠ ❣❡♥❡r❛t♦r✱ ③ = ③✶, ③✷, . . . ✐s ❛❞❞❡❞ ❜✐t✇✐s❡ t♦ t❤❡ ♣❧❛✐♥t❡①t s❡q✉❡♥❝❡ ♠ = ♠✶, ♠✷, . . .✱ ♣r♦❞✉❝✐♥❣ t❤❡ ❝✐♣❤❡rt❡①t ❝ = ❝✶, ❝✷, . . .✳

• ❚❤❡ ❦❡②str❡❛♠ ❣❡♥❡r❛t♦r ✐s ✐♥✐t✐❛❧✐③❡❞ t❤r♦✉❣❤ ❛ s❡❝r❡t ❦❡② ❑✳

♠ ✲ ✲ ❄

❦❡②str❡❛♠ ❣❡♥❡r❛t♦r ♠✶, ♠✷, . . . ❝✶, ❝✷, . . . ③✶, ③✷, . . .

❋✐❣✉r❡✿ ❆ ❜✐♥❛r② ❛❞❞✐t✐✈❡ str❡❛♠ ❝✐♣❤❡r

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❆tt❛❝❦s

• ❆ ❦♥♦✇♥✲♣❧❛✐♥t❡①t ❛tt❛❝❦ ✭♦r ❝❤♦s❡♥✲♣❧❛✐♥t❡①t ♦r

❝❤♦s❡♥✲❝✐♣❤❡rt❡①t✮ ✐s ❡q✉✐✈❛❧❡♥t t♦ ❤❛✈✐♥❣ ❛❝❝❡ss t♦ t❤❡ ❦❡②str❡❛♠ ③ = ③✶, ③✷, . . . , ③◆✳

• ❉❡s✐❣♥ ❣♦❛❧✿ ❡✣❝✐❡♥t❧② ♣r♦❞✉❝❡ r❛♥❞♦♠✲❧♦♦❦✐♥❣ s❡q✉❡♥❝❡s

t❤❛t ❛r❡ ✏✐♥❞✐st✐♥❣✉✐s❤❛❜❧❡✑ ❢r♦♠ tr✉❧② r❛♥❞♦♠ s❡q✉❡♥❝❡s✳

♠ ✲ ✲ ❄

❦❡②str❡❛♠ ❣❡♥❡r❛t♦r ♠✶, ♠✷, . . . ❝✶, ❝✷, . . . ③✶, ③✷, . . .

❋✐❣✉r❡✿ ❆ ❜✐♥❛r② ❛❞❞✐t✐✈❡ str❡❛♠ ❝✐♣❤❡r

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❆tt❛❝❦s

❚✇♦ ♠❛✐♥ t②♣❡s ♦❢ ❛tt❛❝❦s✿

• ❑❡② r❡❝♦✈❡r② ❛tt❛❝❦✿ ❊✈❡ tr✐❡s t♦ r❡❝♦✈❡r t❤❡ ✈❛❧✉❡ ♦❢ t❤❡

s❡❝r❡t ❦❡② ❑✳

• ❉✐st✐♥❣✉✐s❤✐♥❣ ❛tt❛❝❦✿ ❊✈❡ tr✐❡s t♦ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r ❛ ❣✐✈❡♥

s❡q✉❡♥❝❡ ③ = ③✶, ③✷, . . . , ③◆ ✐s ❧✐❦❡❧② t♦ ❤❛✈❡ ❜❡❡♥ ❣❡♥❡r❛t❡❞ ❢r♦♠ t❤❡ ❝♦♥s✐❞❡r❡❞ str❡❛♠ ❝✐♣❤❡r ♦r ✇❤❡t❤❡r ✐t ✐s ❥✉st ❛ tr✉❧② r❛♥❞♦♠ s❡q✉❡♥❝❡✳ ■❢ ❛ ❞✐st✐♥❣✉✐s❤❡r✱ ✐✳❡✳✱ ❛ ❜♦① ✭❛❧❣♦r✐t❤♠✮ t❤❛t ❝❛♥ ❝♦rr❡❝t❧② ❛♥s✇❡r t❤❡ ❛❜♦✈❡ q✉❡st✐♦♥ ✇✐t❤ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t②✱ ❝❛♥ ❜❡ ❜✉✐❧t✱ ✇❡ ❤❛✈❡ ❛ ❞✐st✐♥❣✉✐s❤✐♥❣ ❛tt❛❝❦✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

▼♦❞❡❧ ♦❢ ❛ str❡❛♠ ❝✐♣❤❡r

• ❘❡q✉✐r❡♠❡♥ts ✐♥ ♠❛♥② r❡❝❡♥t ❛♣♣❧✐❝❛t✐♦♥s ❤❛✈❡ ❛s❦❡❞ ❢♦r ❛

♠♦❞✐✜❡❞ ♠♦❞❡❧ ♦❢ ❛ str❡❛♠ ❝✐♣❤❡r ✐♥❝❧✉❞✐♥❣ ❛ ♣✉❜❧✐❝ ♣❛r❛♠❡t❡r ❝❛❧❧❡❞ ■❱ ✭✐♥✐t✐❛❧ ✈❛❧✉❡✮ ♦r ♥♦♥❝❡ ✭♥✉♠❜❡r ✉s❡❞ ♦♥❝❡✮✳

• ❆ ❣❡♥❡r❛t♦r t❛❦❡s t✇♦ ✐♥♣✉t ♣❛r❛♠❡t❡rs✱ ♦♥❡ ❦❡② ❑ ❛♥❞ ♦♥❡

♣✉❜❧✐❝ ♣❛r❛♠❡t❡r ■❱✱ ❛♥❞ ♣r♦❞✉❝❡s ❛♥ ❛r❜✐tr❛r② ❧♦♥❣ ❦❡②str❡❛♠ s❡q✉❡♥❝❡ ③✳ key k z IV PUBLIC!

❋✐❣✉r❡✿ ❆ ❦❡②str❡❛♠ ❣❡♥❡r❛t♦r ✐♥✐t✐❛❧✐③❡❞ ❜② ❛ ❦❡② ❛♥❞ ❛♥ ■❱ ✈❛❧✉❡

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❆ ❣❡♥❡r❛t♦r ✐♥ t❛❜❧❡ ❢♦r♠

• ❡♥❡r❛t♦r = ❛ t❛❜❧❡ ✐♥❞❡①❡❞ ❜② (❑, ■❱ ) ❝♦♥t❛✐♥✐♥❣ ③✳

■♥✐t✐❛❧ ✈❛❧✉❡ ■❱ ❑❡② ❑ ❑❡②str❡❛♠ s❡q✉❡♥❝❡ ③ ✵✵✳ ✳ ✳ ✵✵✵ ✵✵✳ ✳ ✳ ✵✵✵ ✶✶✵✶✵✶✵✶✵✶✶✶✵✶✵✶✵✶✵✶✵✵ ✵✵✳ ✳ ✳ ✵✵✵ ✵✵✳ ✳ ✳ ✵✵✶ ✶✶✶✵✶✵✶✵✶✵✶✵✶✶✵✵✵✵✵✶✵✶ ✵✵✳ ✳ ✳ ✵✵✵ ✵✵✳ ✳ ✳ ✵✶✵ ✵✵✶✵✶✶✵✶✵✶✵✶✶✵✶✵✶✵✵✶✵✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✵✵✳ ✳ ✳ ✵✵✶ ✵✵✳ ✳ ✳ ✵✵✵ ✶✵✶✵✵✵✶✵✶✵✶✵✶✵✶✵✶✶✵✶✵✶ ✵✵✳ ✳ ✳ ✵✵✶ ✵✵✳ ✳ ✳ ✵✵✶ ✵✵✶✵✶✶✶✵✶✵✶✵✶✵✶✵✶✵✵✶✶✶ ✵✵✳ ✳ ✳ ✵✵✶ ✵✵✳ ✳ ✳ ✵✶✵ ✶✵✶✵✶✶✵✶✵✶✵✶✵✶✵✵✵✵✶✶✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✳ ✳ ✳ ✶✶✶ ✵✵✳ ✳ ✳ ✵✵✵ ✵✶✵✶✵✶✵✶✵✶✶✵✶✵✵✶✵✵✵✶✵✵ ✶✶✳ ✳ ✳ ✶✶✶ ✵✵✳ ✳ ✳ ✵✵✶ ✵✶✵✶✶✶✶✶✶✶✶✶✵✵✵✵✵✶✵✶✶✵ ✶✶✳ ✳ ✳ ✶✶✶ ✵✵✳ ✳ ✳ ✵✶✵ ✶✵✶✶✵✶✵✶✶✶✵✶✵✶✵✵✵✵✶✶✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✳ ✳ ✳ ✶✶✶ ✶✶✳ ✳ ✳ ✶✶✶ ✵✵✶✵✶✶✶✶✶✶✶✵✶✵✵✶✵✵✵✶✶✶

❋✐❣✉r❡✿ ❱✐s✉❛❧✐③✐♥❣ t❤❡ ❣❡♥❡r❛t♦r ❛s ❛ ❤✉❣❡ t❛❜❧❡

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❘❡♠❛r❦s ♦♥ t❤❡ ♠♦❞❡❧

• ❚❤❡ ♦♣t✐♠❛❧ ❞❡s✐❣♥ ♦❢ ❛ ❣❡♥❡r❛t♦r ✭✐❞❡❛❧ ❣❡♥❡r❛t♦r✮✱ ✇♦✉❧❞ ❜❡

✐❢ ❡✈❡r② ❡♥tr② ✐♥ t❤❡ t❛❜❧❡ ✇❛s ❣❡♥❡r❛t❡❞ tr✉❧② ❛t r❛♥❞♦♠ ✭✉♥✐❢♦r♠❧②✮✳

• ❲❡ ❤❛✈❡ s♦♠❡ ❣❡♥❡r✐❝ ❛tt❛❝❦s ♦♥ t❤❡ ✐❞❡❛❧ ❣❡♥❡r❛t♦r✳ ❋♦r

❡①❛♠♣❧❡✱ ❛♥ ❡①❤❛✉st✐✈❡ ❦❡② s❡❛r❝❤ ✇♦✉❧❞ r❡q✉✐r❡ t❡st✐♥❣ ❛❧❧ t❤❡ ❦❡②s ❛♥❞ ❝❤❡❝❦✐♥❣ ✇❤❡t❤❡r ❛ ❝❤♦s❡♥ ❦❡② ❣❡♥❡r❛t❡s t❤❡ ❣✐✈❡♥ ♦✉t♣✉t✳

• ❚❤❡ ❞❡s✐❣♥ ♣r♦❜❧❡♠ ✐s t❤❡♥ ❡ss❡♥t✐❛❧❧② t♦ ❝♦♥str✉❝t ❛ ❣❡♥❡r❛t♦r

t❤❛t ✐♥ ❛❧❧ ❛s♣❡❝ts ✐♠♣❧❡♠❡♥ts ❛♥ ✐❞❡❛❧ ❣❡♥❡r❛t♦r✱ ❧❡❛✈✐♥❣ ♦♥❧② t❤❡ ❣❡♥❡r✐❝ ❛tt❛❝❦s ❧✐❦❡ ❡①❤❛✉st✐✈❡ ❦❡② s❡❛r❝❤ ❢♦r t❤❡ ❝r②♣t❛♥❛❧②st✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❊①❛♠♣❧❡s ♦❢ ✐♥s❡❝✉r❡ ❣❡♥❡r❛t♦rs

❊✈❡♥ ✐❢ s♦♠❡ ♣s❡✉❞♦✲r❛♥❞♦♠ ❣❡♥❡r❛t♦rs ♠❛② ❜❡ s✉✐t❛❜❧❡ ❢♦r s✐♠✉❧❛t✐♦♥ ♣✉r♣♦s❡s✱ t❤❡② ❝❛♥ ❜❡ ❝♦♠♣❧❡t❡❧② ✐♥s❡❝✉r❡ ✐♥ ❛ ❝r②♣t♦❣r❛♣❤✐❝ s❡♥s❡✳

• ❖✉t♣✉t ♦❢ ❛ ❧✐♥❡❛r ❢❡❡❞❜❛❝❦ s❤✐❢t r❡❣✐st❡r✳ ❚❤❡ ❦❡② ❞❡t❡r♠✐♥❡s

❛ st❛rt✐♥❣ st❛t❡ (s✶, s✷, . . . , s▲)✱ ❛ s❡q✉❡♥❝❡ ✐s ❞❡✜♥❡❞ ❜② s✐ = ▲

❥=✶ ❝❥s✐−❥ ❢♦r ✐ > ▲✱ ❛♥❞ t❤❡ ♣s❡✉❞♦✲r❛♥❞♦♠ s❡q✉❡♥❝❡ ✐s

❣✐✈❡♥ ❜② ③ = s▲+✶, s▲+✷, . . .✳

• ❱❛r✐♦✉s ✈❡rs✐♦♥s ♦❢ t❤❡ ❧✐♥❡❛r ❝♦♥❣r✉❡♥t✐❛❧ ❣❡♥❡r❛t♦r✳
• ❡♥❡r❛t♦rs t❤❛t ✐♥ s♦♠❡ ❢♦r♠ ✉s❡ t❤❡ r❡❝✉rr❡♥❝❡

s✐+✶ = ❛s✐ + ❜ (♠♦❞ ♠), ✇❤❡r❡ ♥♦✇ ❛, ❜, s✐ ∈ Z♠✱ ✐ = ✶, ✷, . . .✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❉❡✜♥✐♥❣ ❛ ❉✐st✐♥❣✉✐s❤❡r

• ❘♦✉❣❤ ❞❡s❝r✐♣t✐♦♥✿ ❛ ❞✐st✐♥❣✉✐s❤❡r ❢♦r ❛ ❣❡♥❡r❛t♦r ❳ ✐s ❣✐✈❡♥ ❛s

❢♦❧❧♦✇s✳ ▲❡t ❉(③) ❜❡ ❛♥ ❛❧❣♦r✐t❤♠ t❤❛t t❛❦❡s ❛s ✐♥♣✉t ❛ ❧❡♥❣t❤ ◆ s❡q✉❡♥❝❡ ③ ❛♥❞ ❛s ♦✉t♣✉t ❣✐✈❡s ♦♥❡ ♦✉t ♦❢ t✇♦ ♣♦ss✐❜❧❡ ❛♥s✇❡rs✱ ❡✐t❤❡r ✏❳✑ ♦r ✏❘❆◆❉❖▼✑✳

• ❚❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t ❉(③) ❝♦rr❡❝t❧② ❞❡t❡r♠✐♥❡s t❤❡ ♦r✐❣✐♥ ♦❢ ③

✐s ✇r✐tt❡♥ (✶ + ε)/✷✳ ■❢ ε ✐s ♥♦t ✈❡r② ❝❧♦s❡ t♦ ③❡r♦ ✇❡ s❛② t❤❛t ❉(③) ✐s ❛ ❞✐st✐♥❣✉✐s❤❡r ❢♦r ❣❡♥❡r❛t♦r✳

• ❚❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ ❛ ❞✐st✐♥❣✉✐s❤❡r ❉✱ ❆❞✈❉✱ ❛s

❆❞✈❉ = |P(❉(③) = ❳|③ ❣❡♥❡r❛t❡❞ ❜② ❳)−P(❉(③) = ❳|③ tr✉❧② r❛♥❞♦♠) ❚❤❡ ❛❞✈❛♥t❛❣❡ ✐s ❆❞✈❉ = |ε|✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❉✐✛❡r❡♥t ❉✐st✐♥❣✉✐s❤✐♥❣ ❆tt❛❝❦ ❙❝❡♥❛r✐♦s

• ❛ s✐♥❣❧❡ ❦❡②str❡❛♠ ✭❢r♦♠ ❦♥♦✇♥ ♦r ❝❤♦s❡♥ ■❱✮✱

s❡✈❡r❛❧ ❦❡②str❡❛♠s ❢r♦♠ ❞✐✛❡r❡♥t ❦♥♦✇♥ ✈❛❧✉❡s ♦❢ ■❱✱ s❡✈❡r❛❧ ❦❡②str❡❛♠s ❢r♦♠ ❞✐✛❡r❡♥t ❝❤♦s❡♥ ✈❛❧✉❡s ♦❢ ■❱✳

• ❉ r❡❝❡✐✈❡s ♠ ❞✐✛❡r❡♥t ❦❡②str❡❛♠s ③✶, ③✷, .., ③♠ ❣❡♥❡r❛t❡❞ ❢r♦♠

♠ ❞✐✛❡r❡♥t ■❱ ✈❛❧✉❡s ■❱✶, ■❱✷, . . . , ■❱♠✳ ❲r✐t❡ ❩ =      ③✶ ③✷ ✳ ✳ ✳ ③♠      =      ③✶,✶ ③✶,✷ . . . ③✶,◆ ③✷,✶ ③✷,✷ . . . ③✷,◆ ✳ ✳ ✳ ③♠,✶ ③♠,✷ . . . ③♠,◆      . ❙♣❡❝✐❛❧ ❛tt❡♥t✐♦♥ t♦ t❤❡ t✇♦ s♣❡❝✐❛❧ ❝❛s❡s ♠ = ✶✱ ❛♥❞ ◆ = ✶✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❇❛s✐❝ ❝♦♥str✉❝t✐♦♥s ♦❢ ❞✐st✐♥❣✉✐s❤❡rs

• ❆ ✜rst ❛♥❞ ✈❡r② ❜❛s✐❝ ❛♣♣r♦❛❝❤ ✇♦✉❧❞ t❤❡♥ ❜❡ t♦ ❛♣♣❧② ✈❛r✐♦✉s

st❛t✐st✐❝❛❧ t❡sts ♦♥ t❤❡ r❡❝❡✐✈❡❞ ❦❡②str❡❛♠ ❩ ✭◆■❙❚ st❛t✐st✐❝❛❧ t❡sts✱ ❉■❊❍❆❘❉✱ ✳✳✳✮

• ❚❤❡s❡ ❛♣♣r♦❛❝❤❡s ♠❛② ❞❡t❡❝t st❛t✐st✐❝❛❧ ✇❡❛❦♥❡ss❡s ✐♥ s♦♠❡

✇❡❛❦ ❣❡♥❡r❛t♦rs ❜✉t t❤❡② ❛r❡ ♥♦t ✈❡r② ♣♦✇❡r❢✉❧ ✐♥ ❣❡♥❡r❛❧✳

• ❙tr♦♥❣❡r ❛tt❛❝❦s ❝❛♥ ❜❡ ❛❝❤✐❡✈❡❞ ✐❢ ✇❡ t❛❦❡ t❤❡ ✐♥t❡r♥❛❧

str✉❝t✉r❡ ♦❢ t❤❡ ❝✐♣❤❡r ✐♥t♦ ❛❝❝♦✉♥t ✇❤❡♥ ✇❡ ❞❡s✐❣♥ ❛ ❞✐st✐♥❣✉✐s❤❡r✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❇❛s✐❝ ❝♦♥str✉❝t✐♦♥s ♦❢ ❞✐st✐♥❣✉✐s❤❡rs

• ❚r② t♦ ❞❡t❡❝t s♦♠❡ st❛t✐st✐❝❛❧ ❞❡✈✐❛t✐♦♥ ✐♥ t❤❡ ❦❡②str❡❛♠ ❩

❜❛s❡❞ ♦♥ s♦♠❡ ✐♥t❡r♥❛❧ r❡❧❛t✐♦♥s❤✐♣✳

• ❍♦✇❡✈❡r✱ s②♠❜♦❧s ✐♥ ❩ ✭♦r ❡✈❡♥ s♠❛❧❧ ❜❧♦❝❦s ♦❢ s②♠❜♦❧s✮ ✇✐❧❧

♦❢t❡♥ ❜❡ ✈❡r② ❝❧♦s❡ t♦ t❤❡ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥✳

• ■♥st❡❛❞✱ t❤❡ ✐♥t❡r♥❛❧ r❡❧❛t✐♦♥s❤✐♣ ♦❢t❡♥ ❣✐✈❡s ❞❡♣❡♥❞❡♥❝❡

❛♠♦♥❣ ❞✐✛❡r❡♥t ③✐,❥ s②♠❜♦❧s t❤❛t ❝❛♥ ❜❡ ❢❛r ❛♣❛rt ✐♥ t✐♠❡✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❚r❛♥s❢♦r♠✐♥❣ ❦❡②str❡❛♠ ✐♥t♦ s❛♠♣❧❡s

• ❙♦ ✐t ✐s ♥❛t✉r❛❧ t❤❛t ✇❡ tr❛♥s❢♦r♠ ♦✉r ❦❡②str❡❛♠ ❩ ✐♥t♦ ❛ ♥❡✇

s❡q✉❡♥❝❡ ♦❢ s②♠❜♦❧s✱ ❝❛❧❧❡❞ s❛♠♣❧❡s✱ ❞❡♥♦t❡❞ ❜② ❳ = ①✶, ①✷, . . .✳ ■♥ ❣❡♥❡r❛❧✱ t❤✐s ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ ❛❧♠♦st ❛♥② ✇❛②✱ ①✐ = ❋(✐, ❩), ✐ = ✶, ✷, . . . ✇❤❡r❡ ❋ ✐s s♦♠❡ ❢✉♥❝t✐♦♥✳

• ❲✐t❤ ❛ ❣✐✈❡♥ s❛♠♣❧❡ s❡q✉❡♥❝❡✱ ✇❡ ✇♦✉❧❞ ✜♥❛❧❧② tr② t♦

❞✐st✐♥❣✉✐s❤ ✐❢ ❳ ❜❡❤❛✈❡s ❛s ✐❢ ❣❡♥❡r❛t❡❞ ❢r♦♠ ❛ tr✉❧② r❛♥❞♦♠ ❩ ♦r ♥♦t✳

• ▲✐♥❡❛r ❞✐st✐♥❣✉✐s❤❡rs✱ t❤❡ s❛♠♣❧❡s ❛r❡ s❡❧❡❝t❡❞ ❛s ❧✐♥❡❛r

❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❦❡②str❡❛♠ ❜✐ts✳ ❯s✉❛❧❧②✱ t❤❡ s❛♠♣❧❡s ❛r❡ r❡❣❛r❞❡❞ ❛s ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ t❤❡ ❞✐st✐♥❣✉✐s❤❡r ❡①❛♠✐♥❡s ✇❤❡t❤❡r t❤❡ s❛♠♣❧❡ ✈❛❧✉❡s ❛r❡ ❝♦♥s✐st❡♥t ✇✐t❤ ❛ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ♦r ♥♦t✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❚r❛♥s❢♦r♠✐♥❣ ❦❡②str❡❛♠ ✐♥t♦ s❛♠♣❧❡s

• ❈♦♥❝❧✉❞✐♥❣✱ t❤❡ ❝❤❛❧❧❡♥❣❡ ❢♦r t❤❡ ❛❞✈❡rs❛r② ✐s t♦ s♦♠❡❤♦✇ ✜♥❞

❛ s✉✐t❛❜❧❡ ✇❛② t♦ tr❛♥s❢♦r♠ t❤❡ ❦❡②str❡❛♠s t♦ ❛ s❛♠♣❧❡ s❡q✉❡♥❝❡ ❳✳

• ❖♥❝❡ t❤❡ s❛♠♣❧❡ s❡q✉❡♥❝❡ ✐s ❣✐✈❡♥✱ ✇❡ ❛♣♣❧② st❛t✐st✐❝❛❧ t♦♦❧s

t♦ ❞❡t❡r♠✐♥❡ ✇❤✐❝❤ ❞✐str✐❜✉t✐♦♥ t❤❡ s❛♠♣❧❡ s❡q✉❡♥❝❡ ❢♦❧❧♦✇s✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❍②♣♦t❤❡s✐s ❚❡st✐♥❣

❚✇♦ ❝❛s❡s✿

• ❲❡ ✇❛♥t t♦ ❞❡t❡r♠✐♥❡ ✐❢ ❛♥ ♦❜s❡r✈❡❞ s❡q✉❡♥❝❡ ✐s ❞✐str✐❜✉t❡❞

❛❝❝♦r❞✐♥❣ t♦ ♦♥❡ ♦❢ t✇♦ ❦♥♦✇♥ ❞✐str✐❜✉t✐♦♥s✱ ✉s✉❛❧❧② t❤❡ ❝✐♣❤❡r ❞✐str✐❜✉t✐♦♥ ❛♥❞ t❤❡ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥✳

• ❲❡ ✇❛♥t t♦ ❞❡t❡r♠✐♥❡ ✐❢ ❛♥ ♦❜s❡r✈❡❞ s❡q✉❡♥❝❡ ✐s ❧✐❦❡❧② t♦ ❜❡

❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ♦♥❡ ❦♥♦✇♥ ❞✐str✐❜✉t✐♦♥✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❚❤❡ ❈❛s❡ ❲❤❡♥ ❇♦t❤ ❉✐str✐❜✉t✐♦♥s ❆r❡ ❑♥♦✇♥

❚❤❡ ♦♣t✐♠❛❧ ❤②♣♦t❤❡s✐s t❡st ✐s ❣✐✈❡♥ ❜②✿

▲❡♠♠❛ ✭◆❡②♠❛♥✲P❡❛rs♦♥✮

▲❡t ❳✶, ❳✷, . . . , ❳♥ ❜❡ ❞r❛✇♥ ✐✳✐✳❞✳ ❛❝❝♦r❞✐♥❣ t♦ ♠❛ss ❢✉♥❝t✐♦♥ P♦❜s✳ ❈♦♥s✐❞❡r t❤❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❤②♣♦t❤❡s❡s P♦❜s = P✵ ✈s✳ P♦❜s = P✶✳ ❋♦r ❚ ≥ ✵ ❞❡✜♥❡ ❛ r❡❣✐♦♥ A♥(❚) = P✵(①✶, ①✷, . . . , ①♥) P✶(①✶, ①✷, . . . , ①♥) > ❚

• .

▲❡t α♥ = P♥

✵ (A❝ ♥(❚)) ❛♥❞ β♥ = P♥ ✶ (A♥(❚)) ❜❡ t❤❡ ❡rr♦r

♣r♦❜❛❜✐❧✐t✐❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❞❡❝✐s✐♦♥ r❡❣✐♦♥ A♥✳ ▲❡t B♥ ❜❡ ❛♥② ♦t❤❡r ❞❡❝✐s✐♦♥ r❡❣✐♦♥ ✇✐t❤ ❛ss♦❝✐❛t❡❞ ❡rr♦r ♣r♦❜❛❜✐❧✐t✐❡s α∗ ❛♥❞ β∗✳ ■❢ α∗ ≤ α✱ t❤❡♥ β∗ ≥ β✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❚❤❡ ❈❛s❡ ❲❤❡♥ ❇♦t❤ ❉✐str✐❜✉t✐♦♥s ❆r❡ ❑♥♦✇♥

❆ss✉♠✐♥❣ t❤❛t ❛❧❧ s❛♠♣❧❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t t❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ A♥(❚) = ♥

• ✐=✶

❧♦❣ P✵(①✐) P✶(①✐)

• > ❧♦❣ ❚
• .

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❙♦♠❡ ❦♥♦✇♥ ❢❛❝ts

• ❚❤❡r❡ ❡①✐st ❛s②♠♣t♦t✐❝ ❡①♣r❡ss✐♦♥s ❢♦r t❤❡ ❡rr♦r ♣r♦❜❛❜✐❧✐t✐❡s✳
• ❇✐♥❛r② ❞✐str✐❜✉t✐♦♥s✿ ❚❤❡ ❜✐❛s ♦❢ ❛ ❞✐str✐❜✉t✐♦♥ ε ✐s ❞❡✜♥❡❞ ❛s

Pr(❳ = ✵) = ✵.✺(✶ + ε). ✭✶✮ ❋♦r ❦ ❜✐♥❛r② ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s ❳✶, ❳✷, . . . , ❳❦✱ t❤❡ ❜✐❛s εt♦t ♦❢ t❤❡ s✉♠ ✐s ❣✐✈❡♥ ❜② εt♦t = ε❦. ✭✷✮

• ❲❤❡♥ α ❛♥❞ β ❛r❡ ❛❜♦✉t ❡q✉❛❧✱ ❛ ❞✐st✐♥❣✉✐s❤❡r ♥❡❡❞s r♦✉❣❤❧②

♥ ≈ ✶ ε✷ ✭✸✮ s❛♠♣❧❡s t♦ ❞❡t❡r♠✐♥❡ ✐❢ ❛♥ ♦❜s❡r✈❡❞ ❞✐str✐❜✉t✐♦♥ ✐s t❤❡ ❝✐♣❤❡r ❞✐str✐❜✉t✐♦♥ ♦r t❤❡ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❚❤❡ ❈❛s❡ ❲❤❡♥ ❖♥❡ ❉✐str✐❜✉t✐♦♥ ■s ❑♥♦✇♥

■❢ ✇❡ ❝❛♥ ♥♦t ✜♥❞ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❝✐♣❤❡r✳

• ❆ ❝❤✐✲sq✉❛r❡ t❡st ❝❛♥ ❜❡ ✉s❡❞ t♦ ❞❡t❡r♠✐♥❡ ✐❢ ❛♥ ♦❜s❡r✈❡❞

❞✐str✐❜✉t✐♦♥ ✐s ❧✐❦❡❧② t♦ ❢♦❧❧♦✇ ♦♥❡ ❣✐✈❡♥ ❞✐str✐❜✉t✐♦♥✳ ❍✵✿ P❳ = P✵ ❍✶✿ P❳ = P✵✳

• ▲❡t ❖(①) ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ♦✉t❝♦♠❡s ♦❢ ① ∈ X ✐♥ t❤❡

♦❜s❡r✈❡❞ s❡q✉❡♥❝❡ ❛♥❞ ❧❡t t❤❡ ❡①♣❡❝t❡❞ ♥✉♠❜❡r ♦❢ ♦✉t❝♦♠❡s ♦❢ ① ∈ X ❛❝❝♦r❞✐♥❣ t♦ P✵ ❜❡ ❞❡♥♦t❡❞ ❊(①)✳ ❚❤❡ ❞✐str✐❜✉t✐♦♥ ◗ =

• ①∈X

(❖(①) − ❊(①))✷ ❊(①) ✭✹✮ ❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ❜② t❤❡ ❝❤✐✲sq✉❛r❡ ❞✐str✐❜✉t✐♦♥✱ χ✷

r ✇✐t❤ r

❜❡✐♥❣ t❤❡ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❆ Pr❛❝t✐❝❛❧ ❙✐t✉❛t✐♦♥

❆ ❞✐st✐♥❣✉✐s❤❡r ✐s ✉s❡❞ t♦ ❞❡r✐✈❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ♣❧❛✐♥t❡①t✳

• ❆❧✐❝❡ ❛♥❞ ❇♦❜ ✐s ❝♦♠♠✉♥✐❝❛t✐♥❣ ♦✈❡r ❛♥ ✐♥s❡❝✉r❡ ❝❤❛♥♥❡❧✳ ❚❤❡

❛❞✈❡rs❛r② ❊✈❡ ✐s ❛❜❧❡ t♦ ♣❛ss✐✈❡❧② ❡❛✈❡s❞r♦♣ t❤❡ ❝❤❛♥♥❡❧✳

• ❆❧✐❝❡ s❡♥❞s ❛ ♠❡ss❛❣❡ ▼ = ♠✶, ♠✷, . . . , ♠◆ t♦ ❇♦❜✳

❊✈❡ ❦♥♦✇s t❤❛t t❤❡ ❞❛t❛ s❡♥t ✐s ❡✐t❤❡r ▼✶ = ♠✶✶, ♠✶✷, . . . , ♠✶◆ ♦r ▼✷ = ♠✷✶, ♠✷✷, . . . , ♠✷◆✳ ❚❤❡ ❝✐♣❤❡rt❡①t ✐s ❈ = ❝✶, ❝✷, . . . , ❝◆ ❛♥❞ ❣✐✈❡♥ ❜② ❝✐ = ♠✐ ⊕ ③✐, ✶ ≤ ✐ ≤ ◆. ✭✺✮ ✇❤❡r❡ ③✐ ✐s t❤❡ ❦❡②str❡❛♠✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❆ Pr❛❝t✐❝❛❧ ❙✐t✉❛t✐♦♥

❆tt❛❝❦ s❝❡♥❛r✐♦✿ ❊✈❡✬s t❛s❦ ✐s t♦ ❞❡t❡r♠✐♥❡ ✐❢ ▼ = ▼✶ ♦r ▼ = ▼✷✳

• ❇② ①♦r✐♥❣ t❤❡ ❝✐♣❤❡rt❡①t ❈ ✇✐t❤ ▼✶ ❊✈❡ ✇✐❧❧ ❣❡t ❛ ❦❡②str❡❛♠

❫ ③ = ❈ ⊕ ▼✶✳

• ■❢ ✐♥❞❡❡❞ ▼ = ▼✶✱ t❤❡♥ ❫

③ ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❝✐♣❤❡r ❞✐str✐❜✉t✐♦♥ s✐♥❝❡ ˆ ③✐ = ❝✐ ⊕ ♠✶✐ = ♠✶✐ ⊕ ③✐ ⊕ ♠✶✐ = ③✐, ✭✻✮

• ■❢ ▼ = ▼✷✱ t❤❡♥ ❫

③ ✐s ✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞ s✐♥❝❡ ˆ ③✐ = ❝✐ ⊕ ♠✶✐ = ♠✷✐ ⊕ ③✐ ⊕ ♠✶✐, ✭✼✮ ❢♦r ✶ ≤ ✐ ≤ ◆✱ ❛ss✉♠✐♥❣ t❤❛t ▼✶ ⊕ ▼✷ ✐s ✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

• ❡♥❡r✐❝ ❛tt❛❝❦s ♦♥ ❇❧♦❝❦ ❈✐♣❤❡rs ✐♥ ❖❋❇ ❛♥❞ ❈❚❘ ♠♦❞❡
• ❡♥❡r✐❝ ❞✐st✐♥❣✉✐s❤✐♥❣ ❛tt❛❝❦s ❛♣♣❧② t♦ ♠❛♥② ❝♦♠♠♦♥ ♠♦❞❡s ♦❢

♦♣❡r❛t✐♦♥s ♦❢ ❜❧♦❝❦ ❝✐♣❤❡rs ✭❤❡r❡ ❖❋❇ ♠♦❞❡ ❛♥❞ ❝♦✉♥t❡r ♠♦❞❡✮✳

• ❊❑(①) ✐s t❤❡ ❜❧♦❝❦ ❝✐♣❤❡r ❡♥❝r②♣t✐♦♥ ❢✉♥❝t✐♦♥✱

❇ ❂ t❤❡ ❜❧♦❝❦ s✐③❡ ✐♥ ❜✐ts✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❖✉t♣✉t ❢❡❡❞❜❛❝❦ ♠♦❞❡

❖❋❇ t✉r♥s ❛♥② ❜❧♦❝❦ ❝✐♣❤❡r ✐♥t♦ ❛ s②♥❝❤r♦♥♦✉s str❡❛♠ ❝✐♣❤❡r✳ ❚❤❡ ❇✲❜✐t ❦❡②str❡❛♠ ✇♦r❞s (③✶, ③✷, ③✸ . . .) ❛r❡ ❣❡♥❡r❛t❡❞ ❜② r❡♣❡❛t❡❞❧② ❡♥❝r②♣t✐♥❣ ❛ ❇✲❜✐t ■❱✳ ▲❡t ③✵ = ■❱ ✱ t❤❡♥ ③✐ = ❊❑(③✐−✶), ✐ ≥ ✶.

• ❙✐♥❝❡ ❛ ❜❧♦❝❦ ❝✐♣❤❡r ❞❡✜♥❡s ❛ ♣❡r♠✉t❛t✐♦♥ ♦✈❡r ❛❧❧ ❇✲❜✐t

❜❧♦❝❦s✱ ✇❡ ❡①♣❡❝t t❤❡ ❛✈❡r❛❣❡ ♣❡r✐♦❞ ♦❢ t❤❡ ❦❡②str❡❛♠ t♦ ❜❡ ✐♥ t❤❡ ♦r❞❡r ♦❢ ✷❇−✶ ❜❧♦❝❦s✳

• ■❢ t❤❡r❡ ✐s ❛ ❝♦❧❧✐s✐♦♥✱ t❤❡♥ ✇❡ ❦♥♦✇ t❤❛t ❛❧❧ s✉❜s❡q✉❡♥t ❜❧♦❝❦s

✇✐❧❧ ❜❡ t❤❡ s❛♠❡✳ ■✳❡✳✱ ✐❢ ③✐ = ③❥ (✐ = ❥)✱ t❤❡♥ ③✐+❦ = ③❥+❦ (❦ ≥ ✵)✳

• ❚❤❡ ❜✐rt❤❞❛② ♣❛r❛❞♦①✿ ✐♥ ❛ tr✉❧② r❛♥❞♦♠ s❡q✉❡♥❝❡ ✇❡ ❡①♣❡❝t

t♦ ✜♥❞ ❛ ❝♦❧❧✐s✐♦♥ ❛❢t❡r ♦❜s❡r✈✐♥❣ ✷❇/✷ ❇✲❜✐t ❜❧♦❝❦s✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❉✐st✐♥❣✉✐s❤❡r ❢♦r ❖❋❇ ♠♦❞❡

■♥♣✉t✭③✶, ③✷, . . . , ③✷❇/✷✮ ✐❢ ✭③✐ = ③❥ ❛♥❞ ③✐+✶ = ③❥+✶ ❢♦r s♦♠❡ ✐ = ❥✮ r❡t✉r♥ ❘❛♥❞♦♠ ❡❧s❡ r❡t✉r♥ ❖❋❇ ▼♦❞❡

❋✐❣✉r❡✿ ❉✐st✐♥❣✉✐s❤❡r ❢♦r ❖❋❇ ♠♦❞❡

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❈♦✉♥t❡r ▼♦❞❡

■♥ ❝♦✉♥t❡r ♠♦❞❡ ✭❈❚❘✮✱ t❤❡ ❇✲❜✐t ❦❡②str❡❛♠ ✇♦r❞s (③✶, ③✷, ③✸ . . .) ❛r❡ ❣❡♥❡r❛t❡❞ ❜② ❡♥❝r②♣t✐♥❣ ❛♥ ✐♥❝r❡♠❡♥t✐♥❣ ❝♦✉♥t❡r✱ ✐✱ ③✐ = ❊❑(■❱ ||✐), ✇❤❡r❡ ❛||❜ ❞❡♥♦t❡s str✐♥❣ ❝♦♥❝❛t❡♥❛t✐♦♥ ♦❢ ❜✐t str✐♥❣s ❛ ❛♥❞ ❜✳

• ❙✐♥❝❡ ❛ ❝♦✉♥t❡r ✐s ✉s❡❞✱ ❛♥❞ ❛ ❜❧♦❝❦ ❝✐♣❤❡r t♦❣❡t❤❡r ✇✐t❤ t❤❡

❦❡② ❞❡✜♥❡s ❛ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥✱ ❛ ❦❡②str❡❛♠ ❜❧♦❝❦ ✇✐❧❧ ♥❡✈❡r r❡♣❡❛t ✭❛s ❧♦♥❣ ❛s t❤❡ ❝♦✉♥t❡r ✐s ♥♦t r❡♣❡❛t❡❞✮✳

• ❇② ♦❜s❡r✈✐♥❣ ✷❇/✷ ❦❡②str❡❛♠ ❜❧♦❝❦s✱ ✇❡ ❝❛♥ ❞❡❝✐❞❡ ✐❢ t❤❡

s❡q✉❡♥❝❡ ✐s r❛♥❞♦♠ ♦r ❣❡♥❡r❛t❡❞ ❜② ❛ ❜❧♦❝❦ ❝✐♣❤❡r ✐♥ ❝♦✉♥t❡r ♠♦❞❡✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❉✐st✐♥❣✉✐s❤❡r ❢♦r ❈♦✉♥t❡r ♠♦❞❡

■♥♣✉t✭③✶, ③✷, . . . , ③✷♥/✷✮ ✐❢ ✭③✐ = ③❥ ❢♦r s♦♠❡ ✐ = ❥✮ r❡t✉r♥ ❘❛♥❞♦♠ ❡❧s❡ r❡t✉r♥ ❈♦✉♥t❡r ▼♦❞❡

❋✐❣✉r❡✿ ❉✐st✐♥❣✉✐s❤❡r ❢♦r ❈♦✉♥t❡r ♠♦❞❡

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❘❡✢❡❝t✐♦♥s

• ❚❤❡ ❛♠♦✉♥t ♦❢ ❦❡②str❡❛♠ ♥❡❡❞❡❞ ✐♥ t❤❡ ❞✐st✐♥❣✉✐s❤❡r ✐s

✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ s✐③❡ ♦❢ t❤❡ ❦❡②✳

• ❆❊❙ ❞❡✜♥❡s ❛ ❜❧♦❝❦ s✐③❡ ♦❢ ✶✷✽ ❜✐ts✱ ❜✉t t❤❡ ❦❡② ❝❛♥ ❜❡

❝❤♦s❡♥ ❢r♦♠ t❤❡ s❡t {✶✷✽, ✶✾✷, ✷✺✻}✳ ❚❤❡ ❛❜♦✈❡ ❞✐st✐♥❣✉✐s❤❡rs ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ ❆❊❙ ✉s✐♥❣ ❛❜♦✉t ✷✻✹ ❦❡②str❡❛♠ ❜❧♦❝❦s✱

• ❋♦r ✻✹ ❜✐t ❜❧♦❝❦ s✐③❡ ✭❉❊❙✮ t❤✐s ❝❛♥ ❜❡ ❛ ♣r❛❝t✐❝❛❧ ♣r♦❜❧❡♠✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

▲✐♥❡❛r ❞✐st✐♥❣✉✐s❤✐♥❣ ❛tt❛❝❦s

❆ s❡q✉❡♥❝❡ ♦❢ s❛♠♣❧❡s ❛s ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❦❡②str❡❛♠ ❜✐ts✳

• ❯s✉❛❧❧② t✐♠❡✲✐♥✈❛r✐❛♥t✱ ✐✳❡✳✱

①t =

❦

• ❥=✵

❝❥③t+❥, ❢♦r s♦♠❡ ❦ ❛♥❞ t = ✶, ✷, . . .✳ ❚❤❡ s❛♠♣❧❡s ①t ❛r❡ ❝♦♥s✐❞❡r❡❞ ❛s ✐✐❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ P♦❜s✳

• ❋✐♥❞✐♥❣ ❣♦♦❞ ❧✐♥❡❛r ❞✐st✐♥❣✉✐s❤❡rs r❡s❡♠❜❧❡s ❛ ❧♦t ❧✐♥❡❛r

❝r②♣t❛♥❛❧②s✐s ♦❢ ❜❧♦❝❦ ❝✐♣❤❡rs ❛s ✐♥✈❡♥t❡❞ ❜② ▼❛ts✉✐✳

• ▲✐♥❡❛r✐③❡ t❤❡ ❝✐♣❤❡r ❜② r❡♣❧❛❝✐♥❣ s♦♠❡ ♥♦♥❧✐♥❡❛r ❜❧♦❝❦s ✇✐t❤

❧✐♥❡❛r ♦♥❡s✳

• ❋✐♥❞ ❛ ❧✐♥❡❛r r❡❧❛t✐♦♥s❤✐♣ ❛♠♦♥❣ ❦❡②str❡❛♠ s②♠❜♦❧s✱ ✇❤❡r❡ t❤❡

r❡❧❛t✐♦♥s❤✐♣ ✐♥✈♦❧✈❡s ❛s ❢❡✇ ❛♣♣r♦①✐♠❛t❡❞ ❜❧♦❝❦s ❛s ♣♦ss✐❜❧❡✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❉✐st✐♥❣✉✐s❤❡rs ❢♦r ❛rr❛②✲❜❛s❡❞ str❡❛♠ ❝✐♣❤❡rs

▼❛♥② s♦❢t✇❛r❡✲♦r✐❡♥t❡❞ str❡❛♠ ❝✐♣❤❡rs ❛r❡ ✉s✐♥❣ ❧❛r❣❡ ❛rr❛②s ❛♥❞ ❛♣♣❧② ❛ s❧♦✇ ❝♦♥t✐♥♦✉s ✉♣❞❛t❡ ✭❘❈✹✮✳

• ❊①❛♠♣❧❡s✿ P②✲❢❛♠✐❧②❀ ❍❈✲✶✷✽ ❛♥❞ ❍❈✲✷✺✻❀ ▼❯●■❀ ❙❝r❡❛♠✱

❘❈✹✳

• ❙[] ❞❡♥♦t❡s ❛♥ ❛rr❛② ❙[✵], ❙[✶], . . . ❙[❧]✳
• ❇❡t✇❡❡♥ s✉❝❝❡ss✐✈❡ ♦✉t♣✉ts t❤❡ ❛rr❛② ✐s ✉♣❞❛t❡❞ ❛s

❙[](t) = ●(❙[](t − ✶)), ✇❤❡r❡ ● ✐s s♦♠❡ ✉♣❞❛t✐♥❣ ❢✉♥❝t✐♦♥✳ ❆♥ ♦✉t♣✉t s②♠❜♦❧ ✐s t❤❡♥ ❣❡♥❡r❛t❡❞ ❛t t✐♠❡ t ❛s ③t = ❋(❙[](t)), ✇❤❡r❡ ❋ ✐s s♦♠❡ ❢✉♥❝t✐♦♥✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❆ ❜❛s✐❝ ❛tt❛❝❦ str❛t❡❣②

❉❡t❡❝t s♦♠❡ ❞❡♣❡♥❞❡♥❝❡ ♦r st❛t✐st✐❝❛❧ ❞❡✈✐❛t✐♦♥ ✐♥ t❤❡ ✉♣❞❛t❡ ♦❢ t❤❡ ❛rr❛② t❤❛t ✇✐❧❧ ❜❡ ✈✐s✐❜❧❡ ✐♥ t❤❡ ❦❡②str❡❛♠ s❡q✉❡♥❝❡✳

• ❈♦♥s✐❞❡r t✇♦ ❞✐✛❡r❡♥t ❜✉t r❡❧❛t❡❞ ❡✈❡♥ts ❊❩ ❛♥❞ ❊❙✱ ✇❤❡r❡ ❊❩

✐s s♦♠❡ ❡✈❡♥t r❡❧❛t❡❞ t♦ t❤❡ ❦❡②str❡❛♠ ❛♥❞ ❊❙ ✐s s♦♠❡ ❡✈❡♥t r❡❧❛t❡❞ t♦ t❤❡ ❛rr❛② ❙[]✳

• ❋♦r ❡①❛♠♣❧❡✱ ✐❢ ❡✈❡♥t ❊❙ ♦❝❝✉rs t❤❡♥ ❊❩ ♦❝❝✉rs ✇✐t❤

♣r♦❜❛❜✐❧✐t② ✶✱ ✐✳❡✳✱ P(❊❩|❊❙) = ✶✳ ❍♦✇❡✈❡r✱ ✐❢ ❡✈❡♥t ❊❙ ❞♦❡s ♥♦t ♦❝❝✉r t❤❡♥ ✇❡ ❛ss✉♠❡ P(❊❩|❊ ❈

❙ ) = P❯(❊❩)✳

• ■♥ t❤✐s ✇❛② ✇❡ ❝❛♥ ❞❡t❡❝t ❛ ❜✐❛s s✐♥❝❡

P(❊❩) = P(❊❩|❊❙) · P(❊❙) + P(❊❩|❊ ❈

❙ ) · P(❊ ❈ ❙ )

= ✶ · P(❊❙) + P❯(❊❩) · P(❊ ❈

❙ )

= (✶ − P❯(❊❩))P(❊❙) + P❯(❊❩).

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❆ ❈❤♦s❡♥✲■❱ ❉✐st✐♥❣✉✐s❤❡r ✲ ❜❛s✐❝s

• ♥✲✈❛r✐❛❜❧❡ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ❢ ✐♥ ❆◆❋ ❢♦r♠✿
• ❛♥ ❡♥tr② ✐♥ t❤❡ tr✉t❤ t❛❜❧❡ ✐s ❞❡♥♦t❡❞ ❢ (✈) ✇✐t❤

✈ = (✈✶, ✈✷, . . . , ✈♥)✳

• ❚❤❡r❡ ❛r❡ ❡✣❝✐❡♥t ✇❛②s t♦ ❝♦♠♣✉t❡ t❤❡ ❆◆❋ ❢r♦♠ t❤❡ tr✉t❤

t❛❜❧❡✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❚❤❡ ❞✲♠♦♥♦♠✐❛❧ ❚❡st

✭❙❛❛r✐♥❡♥✮

• ❚❤❡ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ❛s

③ = ❢ (✐✈✵, ✐✈✶, . . . , ✐✈♥−✶), ♥ ❜✐ts ♦❢ t❤❡ ■❱ ❛r❡ ✉s❡❞ ❛s ✐♥♣✉t ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡ ♦✉t♣✉t ✐s ♦♥❡ ✭✜rst✮ ❜✐t ♦❢ t❤❡ ❦❡②str❡❛♠✳

• ❚❤❡ ❦❡② ❛♥❞ t❤❡ r❡♠❛✐♥✐♥❣ ❜✐ts ♦❢ t❤❡ ■❱ ❛r❡ ❦❡♣t ❝♦♥st❛♥t✳
• ❈♦♠♣✉t❡ t❤❡ ❆◆❋ ♦❢ ❢ ✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❚❤❡ ❞✲♠♦♥♦♠✐❛❧ ❚❡st

• ■♥ ❛ ❞✲♠♦♥♦♠✐❛❧ t❡st t❤❡ ❛✐♠ ✐s t♦ ❝♦✉♥t t❤❡ ♥✉♠❜❡r ♦❢

♠♦♥♦♠✐❛❧s ✐♥ t❤❡ ❆◆❋ ♦❢ ❞❡❣r❡❡ ❞✳

• ■❢ t❤❡ ♦❜s❡r✈❡❞ ♥✉♠❜❡r ♦❢ ❞✲♠♦♥♦♠✐❛❧s s✐❣♥✐✜❝❛♥t❧② ❞❡✈✐❛t❡s

❢r♦♠ ✶

✷

♥

❞

• ✱ t❤❡ ❡①♣❡❝t❡❞ ❝❛s❡✱ ✇❡ ❝❛♥ ❞✐st✐♥❣✉✐s❤ t❤❡ ❝✐♣❤❡r

❢r♦♠ r❛♥❞♦♠ ✭P❡❛rs♦♥✬s ❝❤✐✲sq✉❛r❡ t❡st✮✳

• ❇r♦❦❡ s❡✈❡r❛❧ ❡❙❚❘❊❆▼ ❝❛♥❞✐❞❛t❡s ✐♥ t❤✐s ✇❛②✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❆ ●❡♥❡r❛❧ ❆♣♣r♦❛❝❤

• P ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ❜② ✉s✐♥❣ ❛ ❞✐✛❡r❡♥t ✈❛❧✉❡ ❢♦r t❤❡ ❝♦♥st❛♥t

❜✐ts ✐♥ t❤❡ ■❱ ❢♦r ❡❛❝❤ ♣♦❧②♥♦♠✐❛❧✳ ❚❤❡ ♦❝❝✉rr❡♥❝❡ ♦❢ ❡❛❝❤ ♠♦♥♦♠✐❛❧ ❝❛♥ ❜❡ ❝♦✉♥t❡❞ ✐♥❞✐✈✐❞✉❛❧❧②✳

• ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ♠♦♥♦♠✐❛❧ ♦❢ ♠❛①✐♠❛❧ ❞❡❣r❡❡✳ ❚❤✐s ♠♦♥♦♠✐❛❧

✇✐❧❧ ♥♦t ♦❝❝✉r ✉♥❧❡ss ❛❧❧ t❤❡ ❝♦♥s✐❞❡r❡❞ ■❱ ❜✐ts ❤❛✈❡ ❜❡❡♥ ♣r♦♣❡r❧② ♠✐①❡❞✳

• ■ts ❝♦❡✣❝✐❡♥t ✐s ❝❛❧❝✉❧❛t❡❞ ❛s t❤❡ ❳❖❘ ♦❢ ❛❧❧ ✈❛❧✉❡s ✐♥ t❤❡

tr✉t❤ t❛❜❧❡✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❚❤❡ ♠❛① ❞❡❣r❡❡ t❡st

✬ ✫ ✩ ✪

❢♦r ❥ = ✶, . . . , P ❢♦r ✐✈ = ✶, . . . , ✷♥ − ✶ ■♥✐t✐❛❧✐③❡ ❝✐♣❤❡r ✇✐t❤ ✐✈ ③ = ✜rst ❦❡②str❡❛♠ ❜✐t ❛❢t❡r ✐♥✐t✐❛❧✐③❛t✐♦♥ ❛ = ❛ ⊕ ③ ❡♥❞ ❢♦r ✐❢❛ = ✶ ♦♥❡s✰✰ ❡♥❞ ❢♦r ✐❢ ♦♥❡s= ✵ ♦r ♦♥❡s= P r❡t✉r♥ ❝✐♣❤❡r ❡❧s❡ r❡t✉r♥ r❛♥❞♦♠

❋✐❣✉r❡✿ ❚❤❡ ♠❛①✐♠❛❧ ❞❡❣r❡❡ t❡st

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❚❤❡ ❡❙❚❘❊❆▼ ♣r♦❥❡❝t

• ❡❙❚❘❊❆▼ ✲ ❛♥ ❡✈❛❧✉❛t✐♦♥ ♣r♦❥❡❝t ✇✐t❤✐♥ ❊❈❘❨P❚✱ t♦ ❝♦♠❡

✉♣ ✇✐t❤ ❛ ♣♦rt❢♦❧✐♦ ♦❢ ♥❡✇ ❛♥❞ ♣r♦♠✐s✐♥❣ str❡❛♠ ❝✐♣❤❡rs✳ Pr❡✈✐♦✉s ♣r♦❥❡❝ts✿ ❆❊❙ ❝♦♠♣❡t✐t✐♦♥✱ ◆❊❙❙■❊✱ ✳✳✳

• ❡❙❚❘❊❆▼ ✇❛s ❞❡❝✐❞❡❞ t♦ ❜❡ ♠♦r❡ r❡s❡❛r❝❤ ♦r✐❡♥t❡❞✱ ❡✳❣✳✱

❛❧❧♦✇✐♥❣ ❞❡s✐❣♥❡rs t♦ ♠♦❞✐❢②✳

• ✷✵✵✹ ✲ ✷✵✵✽✳ ❚❤❡ ❡❙❚❘❊❆▼ P♦rt❢♦❧✐♦ ✐s ❛♥♥♦✉♥❝❡❞ ✐♥ ✷✵✵✽✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❚❤❡ ❡❙❚❘❊❆▼ ♣r♦❥❡❝t

• P❘❖❋■▲❊ ✶✳ ❙tr❡❛♠ ❝✐♣❤❡rs ❢♦r s♦❢t✇❛r❡ ❛♣♣❧✐❝❛t✐♦♥s ✇✐t❤

❤✐❣❤ t❤r♦✉❣❤♣✉t r❡q✉✐r❡♠❡♥ts✳ ✭✷✸ s✉❜♠✐ss✐♦♥s✮

• P❘❖❋■▲❊ ✷✳ ❙tr❡❛♠ ❝✐♣❤❡rs ❢♦r ❤❛r❞✇❛r❡ ❛♣♣❧✐❝❛t✐♦♥s ✇✐t❤

r❡str✐❝t❡❞ r❡s♦✉r❝❡s s✉❝❤ ❛s ❧✐♠✐t❡❞ st♦r❛❣❡✱ ❣❛t❡ ❝♦✉♥t✱ ❛♥❞✴♦r ♣♦✇❡r ❝♦♥s✉♠♣t✐♦♥✳ ✭✷✺ s✉❜♠✐ss✐♦♥s✮

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❚❤❡ ❡❙❚❘❊❆▼ ♣♦rt❢♦❧✐♦

• Pr♦✜❧❡ ✶ ❙❖❋❚❲❆❘❊✿

❍❈✲✶✷✽✱ ❘❛❜❜✐t✱ ❙❛❧s❛✷✵✴✶✷✱ ❙❖❙❊▼❆◆❯❑

• Pr♦✜❧❡ ✷ ❍❆❘❉❲❆❘❊✿
• r❛✐♥ ✈✶✱ ▼■❈❑❊❨ ✈✷✱ ❚r✐✈✐✉♠

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❍❈✲✶✷✽

• ■♥t❡r♥❛❧ st❛t❡✿ ❚✇♦ t❛❜❧❡s P ❛♥❞ ◗✳ ❊❛❝❤ ❝♦♥t❛✐♥s ✺✶✷ ✇♦r❞s✳
• ❣✶(①, ②, ③)

= ((① ≫ ✶✵) ⊕ (③ ≫ ✷✸)) + (② ≫ ✽) ❣✷(①, ②, ③) = ((① ≪ ✶✵) ⊕ (③ ≪ ✷✸)) + (② ≪ ✽) ❤✶(①) = ◗[①✵] + ◗[✷✺✻ + ①✷] ❤✷(①) = P[①✵] + P[✷✺✻ + ①✷] ✇❤❡r❡ ① = ①✸||①✷||①✶||①✵✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❍❈✲✶✷✽

❍❈✲✶✷✽ ❑❡②str❡❛♠ ●❡♥❡r❛t✐♦♥ ■♥♣✉t✿ ❚❛❜❧❡s P ❛♥❞ ◗✱ ❡❛❝❤ ❝♦♥t❛✐♥✐♥❣ ✺✶✷ ✇♦r❞s✳ ❖✉t♣✉t✿ ❑❡②str❡❛♠ ✇♦r❞s s✐ ❢♦r ✐ = ✵, ✶, . . .✳ ✐ = ✵❀ r❡♣❡❛t ✭✉♥t✐❧ ❡♥♦✉❣❤ ❦❡②str❡❛♠ ❜✐ts ❛r❡ ❣❡♥❡r❛t❡❞✮ ④ ❥ = ✐ ♠♦❞ ✺✶✷❀ ✐❢ ((✐ ♠♦❞ ✶✵✷✹) < ✺✶✷) ④ P[❥] ✰❂ ❣✶(P[❥ ⊟ ✸], P[❥ ⊟ ✶✵], P[❥ ⊟ ✺✶✶])❀ s✐ = ❤✶(P[❥ ⊟ ✶✷]) ⊕ P[❥]❀ } ❡❧s❡ ④ ◗[❥] ✰❂ ❣✷(◗[❥ ⊟ ✸], ◗[❥ ⊟ ✶✵], ◗[❥ ⊟ ✺✶✶])❀ s✐ = ❤✷(◗[❥ ⊟ ✶✷]) ⊕ ◗[❥]❀ } ✐ ✰❂ ✶❀ }

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❲✉✬s ❞✐st✐♥❣✉✐s❤✐♥❣ ❛tt❛❝❦

• P ✐s ✉♣❞❛t❡❞ ❛s

P[✐ ♠♦❞ ✺✶✷] ✰❂ ❣✶(P[✐ ⊟ ✸], P[✐ ⊟ ✶✵], P[✐ ⊟ ✺✶✶]) ❇✉t✱ s✐ = ❤✶(P[✐ ⊟ ✶✷]) ⊕ P[✐ ♠♦❞ ✺✶✷]✳ ❋♦r ♠♦st ✐✱ s✐ ⊕ ❤✶(③✐) = (s✐−✶✵✷✹ ⊕ ❤′

✶(③✐−✶✵✷✹)) +

✭✽✮ ❣✶(s✐−✸ ⊕ ❤✶(③✐−✸), s✐−✶✵ ⊕ ❤✶(③✐−✶✵), s✐−✶✵✷✸ ⊕ ❤′

✶(③✐−✶✵✷✸))

• ❤✶(①) ❛♥❞ ❤′

✶(①) ❞✐✛❡r❡♥t ❢✉♥❝t✐♦♥s❀

③❥ ❞❡♥♦t❡s t❤❡ P[❥ ⊟ ✶✷] ❛t t❤❡ ❥✲t❤ st❡♣✳

• ❋♦r t❤❡ ❧❡❛st s✐❣♥✐✜❝❛♥t ❜✐t✱

[s✐]✵ ⊕ [s✐−✸]✶✵ ⊕ [s✐−✶✵]✽ ⊕ [s✐−✶✵✷✸]✷✸ ⊕ [s✐−✶✵✷✹]✵ = ✭✾✮ [❤✶(③✐)]✵ ⊕ [❤✶(③✐−✸)]✶✵ ⊕ [❤✶(③✐−✶✵)]✽ ⊕ [❤′

✶(③✐−✶✵✷✸)]✷✸ ⊕ [❤′ ✶(③✐−✶✵✷✹)]✵

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❲✉✬s ❞✐st✐♥❣✉✐s❤✐♥❣ ❛tt❛❝❦

• ▲♦♦❦✐♥❣ ❛t t✐♠❡ ✐ ❛♥❞ ❥✱ ✐ = ❥✱ ✇❤❡r❡

✶✵✷✹ × α + ✶✵ ≤ ✐, ❥ < ✶✵✷✹ × α + ✺✶✶ [s✐]✵ ⊕ [s✐−✸]✶✵ ⊕ [s✐−✶✵]✽ ⊕ [s✐−✶✵✷✸]✷✸ ⊕ [s✐−✶✵✷✹]✵ = [s❥]✵ ⊕ [s❥−✸]✶✵ ⊕ [s❥−✶✵]✽ ⊕ [s❥−✶✵✷✸]✷✸ ⊕ [s❥−✶✵✷✹]✵ ✭✶✵✮ ✇❤✐❝❤ ❤♦❧❞s ✐❢ ❛♥❞ ♦♥❧② ✐❢ [❤✶(③✐)]✵ ⊕ [❤✶(③✐−✸)]✶✵ ⊕ [❤✶(③✐−✶✵)]✽ ⊕ [❤′

✶(③✐−✶✵✷✸)]✷✸ ⊕ [❤′ ✶(③✐−✶✵✷✹)]✵ =

[❤✶(③❥)]✵ ⊕ [❤✶(③❥−✸)]✶✵ ⊕ [❤✶(③❥−✶✵)]✽ ⊕ [❤′

✶(③❥−✶✵✷✸)]✷✸ ⊕ [❤′ ✶(③❥−✶✵✷✹)]✵

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❲✉✬s ❞✐st✐♥❣✉✐s❤✐♥❣ ❛tt❛❝❦

• ❚❤❛t ❡q✉❛t✐♦♥ ❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ❛s

❍(❛✶) = ❍(❛✷), ✭✶✶✮ ✇❤❡r❡ ❍ ❞❡♥♦t❡s ❛ r❛♥❞♦♠ s❡❝r❡t ✽✵✲❜✐t✲t♦✲✶✲❜✐t ❙✲❜♦①✱ ❛✶ ❛♥❞ ❛✷ ❛r❡ t✇♦ ✽✵✲❜✐t r❛♥❞♦♠ ✐♥♣✉ts✱ ❛✶ = ③✐||③✐−✸||③✐−✶✵||③✐−✶✵✷✸||③✐−✶✵✷✹ ✭✶✷✮ ❛✷ = ③❥||③❥−✸||③❥−✶✵||③❥−✶✵✷✸||③❥−✶✵✷✹,

❚❤❡♦r❡♠

▲❡t ❍ ❜❡ ❛♥ ♠✲❜✐t✲t♦✲♥✲❜✐t ❙✲❜♦① ❛♥❞ ❛❧❧ t❤♦s❡ ♥✲❜✐t ❡❧❡♠❡♥ts ❛r❡ r❛♥❞♦♠❧② ❣❡♥❡r❛t❡❞✱ ✇❤❡r❡ ♠ ≥ ♥✳ ▲❡t ❛✶ ❛♥❞ ❛✷ ❜❡ t✇♦ ♠✲❜✐t r❛♥❞♦♠ ✐♥♣✉ts t♦ ❍✳ ❚❤❡♥ ❍(❛✶) = ❍(❛✷) ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✷−♠ + ✷−♥ − ✷−♠−♥✳ ❚❤✉s✱ ✭❄❄✮ ❤♦❧❞s ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶

✷ + ✷−✽✶✳ ◆✉♠❜❡r ♦❢

s❛♠♣❧❡s ♥❡❡❞❡❞ ≈ ✹ε−✷✱ s♦ ✷✶✻✹ s✉❝❤ s❛♠♣❧❡s✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❲✉✬s ❞✐st✐♥❣✉✐s❤✐♥❣ ❛tt❛❝❦

• ❙❡✈❡r❛❧ ❛tt❡♠♣ts ❤❛✈❡ ❜❡❡♥ ♠❛❞❡ t♦ ✐♠♣r♦✈❡ t❤✐s ❜❛s✐❝ ✐❞❡❛✳
• ❙♦♠❡ ✐♠♣r♦✈❡♠❡♥ts ❤❛✈❡ ❜❡❡♥ ❢♦✉♥❞✱ ❜✉t ♥♦ ❛tt❛❝❦ ❜❡❧♦✇

❝♦♠♣❧❡①✐t② ✷✶✷✽ ❤❛✈❡ ❜❡❡♥ ❢♦✉♥❞✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❚r✐✈✐✉♠

• ❡①tr❡♠❡❧② s✐♠♣❧❡ ❤❛r❞✇❛r❡ ❞❡s✐❣♥
• t❤❡ ♠♦st ❝❤❛❧❧❡♥❣✐♥❣ ❞❡s✐❣♥ ✐♥ t❤❡ ❡❙❚❘❊❆▼ ♣♦rt❢♦❧✐♦
• ❆ ✷✽✽✲❜✐t ✐♥t❡r♥❛❧ st❛t❡ (s✶, s✷, . . . , s✷✽✽) ❛♥❞ ❛ ✈❡r② s✐♠♣❧❡

✉♣❞❛t❡✴♦✉t♣✉t ❢✉♥❝t✐♦♥✳

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■

❚r✐✈✐✉♠

❚r✐✈✐✉♠ ❑❡②str❡❛♠ ●❡♥❡r❛t✐♦♥ ■♥♣✉t✿ ❙t❛t❡ (s✶, s✷, . . . , s✷✽✽) ❖✉t♣✉t✿ ❑❡②str❡❛♠ ❜✐ts ③✐ ❢♦r ✐ = ✶, ✷, . . .✳ ❢♦r ✐ = ✶t♦ ◆ ❞♦ t✶ ← s✻✻ + s✾✸❀ t✷ ← s✶✻✷ + s✶✼✼❀ t✸ ← s✷✹✸ + s✷✽✽❀ ③✐ ← t✶ + t✷ + t✸ t✶ ← t✶ + s✾✶ · s✾✷ + s✶✼✶; t✷ ← t✷ + s✶✼✺ · s✶✼✻ + s✷✻✹; t✸ ← t✸ + s✷✽✻ · s✷✽✼ + s✻✾; (s✶, s✷, . . . , s✾✸) ← (t✸, s✶, . . . , s✾✷) (s✾✹, s✾✺, . . . , s✶✼✼) ← (t✶, s✾✹, . . . , s✶✼✻)❀ (s✶✼✽, s✶✼✾, . . . , s✷✽✽) ← (t✷, s✶✼✽, . . . , s✷✽✼)❀ ❡♥❞ ❢♦r

❚❤♦♠❛s ❏♦❤❛♥ss♦♥ ❙tr❡❛♠ ❝✐♣❤❡rs ■