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❖♥ t❤❡ ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ t❤❡ ●●❍✶✸ ▼✉❧t✐❧✐♥❡❛r ▼❛♣ ❛♥❞ s♦♠❡ ❱❛r✐❛♥ts

▲é♦ ❉✉❝❛s✶✱ ❆❧✐❝❡ P❡❧❧❡t✲▼❛r②✷

✶❈r②♣t♦❧♦❣② ●r♦✉♣✱ ❈❲■✱ ❆♠st❡r❞❛♠ ✷▲■P✱ ❊◆❙ ❞❡ ▲②♦♥

❆s✐❛❝r②♣t ✷✵✶✽

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶ ✴ ✷✷

❲❤❛t ✐s t❤✐s t❛❧❦ ❛❜♦✉t❄

❖❜❥❡❝t✐✈❡✿ ❆♥❛❧②s❡ t❤❡ st❛t✐st✐❝❛❧ ❧❡❛❦ ♦❢ t❤❡ ●●❍✶✸ ♠✉❧t✐❧✐♥❡❛r ♠❛♣ ❉❡s❝r✐♣t✐♦♥ ♦❢ ❛ s✐♠♣❧❡ s❡tt✐♥❣ ✉s✐♥❣ t❤❡ ●●❍✶✸ ♠❛♣ ❆♥❛❧②s❡ ♦❢ t❤❡ st❛t✐st✐❝❛❧ ❧❡❛❦ ✐♥ t❤✐s s✐♠♣❧❡ s❡tt✐♥❣

❋♦r ✹ ❞✐✛❡r❡♥t ✈❛r✐❛♥ts ♦❢ t❤❡ ●●❍✶✸ ♠❛♣

Pr♦♣♦s✐t✐♦♥ ♦❢ ❛ ❝♦✉♥t❡r♠❡❛s✉r❡

• ●❍✶✸✿ ●❛r❣✱ ●❡♥tr② ❛♥❞ ❍❛❧❡✈✐✳ ❈❛♥❞✐❞❛t❡ ♠✉❧t✐❧✐♥❡❛r ♠❛♣s ❢r♦♠ ✐❞❡❛❧ ❧❛tt✐❝❡s✱

❊✉r♦❝r②♣t✳

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷ ✴ ✷✷

❲❤❛t ✐s t❤✐s t❛❧❦ ❛❜♦✉t❄

❖❜❥❡❝t✐✈❡✿ ❆♥❛❧②s❡ t❤❡ st❛t✐st✐❝❛❧ ❧❡❛❦ ♦❢ t❤❡ ●●❍✶✸ ♠✉❧t✐❧✐♥❡❛r ♠❛♣ ❉❡s❝r✐♣t✐♦♥ ♦❢ ❛ s✐♠♣❧❡ s❡tt✐♥❣ ✉s✐♥❣ t❤❡ ●●❍✶✸ ♠❛♣ ❆♥❛❧②s❡ ♦❢ t❤❡ st❛t✐st✐❝❛❧ ❧❡❛❦ ✐♥ t❤✐s s✐♠♣❧❡ s❡tt✐♥❣

❋♦r ✹ ❞✐✛❡r❡♥t ✈❛r✐❛♥ts ♦❢ t❤❡ ●●❍✶✸ ♠❛♣

Pr♦♣♦s✐t✐♦♥ ♦❢ ❛ ❝♦✉♥t❡r♠❡❛s✉r❡

• ●❍✶✸✿ ●❛r❣✱ ●❡♥tr② ❛♥❞ ❍❛❧❡✈✐✳ ❈❛♥❞✐❞❛t❡ ♠✉❧t✐❧✐♥❡❛r ♠❛♣s ❢r♦♠ ✐❞❡❛❧ ❧❛tt✐❝❡s✱

❊✉r♦❝r②♣t✳

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷ ✴ ✷✷

❲❤❛t ✐s t❤✐s t❛❧❦ ❛❜♦✉t❄

❖❜❥❡❝t✐✈❡✿ ❆♥❛❧②s❡ t❤❡ st❛t✐st✐❝❛❧ ❧❡❛❦ ♦❢ t❤❡ ●●❍✶✸ ♠✉❧t✐❧✐♥❡❛r ♠❛♣ ❉❡s❝r✐♣t✐♦♥ ♦❢ ❛ s✐♠♣❧❡ s❡tt✐♥❣ ✉s✐♥❣ t❤❡ ●●❍✶✸ ♠❛♣ ❆♥❛❧②s❡ ♦❢ t❤❡ st❛t✐st✐❝❛❧ ❧❡❛❦ ✐♥ t❤✐s s✐♠♣❧❡ s❡tt✐♥❣

◮ ❋♦r ✹ ❞✐✛❡r❡♥t ✈❛r✐❛♥ts ♦❢ t❤❡ ●●❍✶✸ ♠❛♣

Pr♦♣♦s✐t✐♦♥ ♦❢ ❛ ❝♦✉♥t❡r♠❡❛s✉r❡

• ●❍✶✸✿ ●❛r❣✱ ●❡♥tr② ❛♥❞ ❍❛❧❡✈✐✳ ❈❛♥❞✐❞❛t❡ ♠✉❧t✐❧✐♥❡❛r ♠❛♣s ❢r♦♠ ✐❞❡❛❧ ❧❛tt✐❝❡s✱

❊✉r♦❝r②♣t✳

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷ ✴ ✷✷

❲❤❛t ✐s t❤✐s t❛❧❦ ❛❜♦✉t❄

❖❜❥❡❝t✐✈❡✿ ❆♥❛❧②s❡ t❤❡ st❛t✐st✐❝❛❧ ❧❡❛❦ ♦❢ t❤❡ ●●❍✶✸ ♠✉❧t✐❧✐♥❡❛r ♠❛♣ ❉❡s❝r✐♣t✐♦♥ ♦❢ ❛ s✐♠♣❧❡ s❡tt✐♥❣ ✉s✐♥❣ t❤❡ ●●❍✶✸ ♠❛♣ ❆♥❛❧②s❡ ♦❢ t❤❡ st❛t✐st✐❝❛❧ ❧❡❛❦ ✐♥ t❤✐s s✐♠♣❧❡ s❡tt✐♥❣

◮ ❋♦r ✹ ❞✐✛❡r❡♥t ✈❛r✐❛♥ts ♦❢ t❤❡ ●●❍✶✸ ♠❛♣

Pr♦♣♦s✐t✐♦♥ ♦❢ ❛ ❝♦✉♥t❡r♠❡❛s✉r❡

• ●❍✶✸✿ ●❛r❣✱ ●❡♥tr② ❛♥❞ ❍❛❧❡✈✐✳ ❈❛♥❞✐❞❛t❡ ♠✉❧t✐❧✐♥❡❛r ♠❛♣s ❢r♦♠ ✐❞❡❛❧ ❧❛tt✐❝❡s✱

❊✉r♦❝r②♣t✳

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷ ✴ ✷✷

❈r②♣t♦❣r❛♣❤✐❝ ♠✉❧t✐❧✐♥❡❛r ♠❛♣

❉❡✜♥✐t✐♦♥✿ κ ❛s②♠♠❡tr✐❝ ♠✉❧t✐❧✐♥❡❛r ♠❛♣

❉✐✛❡r❡♥t ❧❡✈❡❧s ♦❢ ❡♥❝♦❞✐♥❣s✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ s✉❜s❡ts ♦❢ {✶, . . . , κ}✳ ❉❡♥♦t❡ ❜② ❊♥❝(a, S) ❛ ❧❡✈❡❧✲S ❡♥❝♦❞✐♥❣ ♦❢ t❤❡ ♠❡ss❛❣❡ a✱ ❢♦r S ⊆ {✶, . . . , κ} =: [κ]✳ ❋✉♥❝t✐♦♥❛❧✐t②✿ ❆❞❞✐t✐♦♥✿ ❆❞❞✭❊♥❝

✶

✱ ❊♥❝

✷

✮ ❂ ❊♥❝

✶ ✷

▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ▼✉❧t✭❊♥❝

✶ ✶ ✱ ❊♥❝ ✷ ✷ ✮ ❂ ❊♥❝ ✶ ✷ ✶ ✷

✐❢

✶ ✷

❩❡r♦✲t❡st✿ ❩❡r♦✲t❡st✭❊♥❝ ✮ ❂ ❚r✉❡ ✐✛ ✵ ❙❡❝✉r✐t②✿ ♠✉❧t✐♣❧❡ s❡❝✉r✐t② ❞❡✜♥✐t✐♦♥s

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✸ ✴ ✷✷

❈r②♣t♦❣r❛♣❤✐❝ ♠✉❧t✐❧✐♥❡❛r ♠❛♣

❉❡✜♥✐t✐♦♥✿ κ ❛s②♠♠❡tr✐❝ ♠✉❧t✐❧✐♥❡❛r ♠❛♣

❉✐✛❡r❡♥t ❧❡✈❡❧s ♦❢ ❡♥❝♦❞✐♥❣s✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ s✉❜s❡ts ♦❢ {✶, . . . , κ}✳ ❉❡♥♦t❡ ❜② ❊♥❝(a, S) ❛ ❧❡✈❡❧✲S ❡♥❝♦❞✐♥❣ ♦❢ t❤❡ ♠❡ss❛❣❡ a✱ ❢♦r S ⊆ {✶, . . . , κ} =: [κ]✳ ❋✉♥❝t✐♦♥❛❧✐t②✿ ❆❞❞✐t✐♦♥✿ ❆❞❞✭❊♥❝(a✶, S)✱ ❊♥❝(a✷, S)✮ ❂ ❊♥❝(a✶ + a✷, S) ▼✉❧t✐♣❧✐❝❛t✐♦♥✿ ▼✉❧t✭❊♥❝(a✶, S✶)✱ ❊♥❝(a✷, S✷)✮ ❂ ❊♥❝(a✶ · a✷, S✶ ∪ S✷) ✐❢ S✶ ∩ S✷ = ∅ ❩❡r♦✲t❡st✿ ❩❡r♦✲t❡st✭❊♥❝(a, [κ])✮ ❂ ❚r✉❡ ✐✛ a = ✵ ❙❡❝✉r✐t②✿ ♠✉❧t✐♣❧❡ s❡❝✉r✐t② ❞❡✜♥✐t✐♦♥s

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✸ ✴ ✷✷

▼♠❛♣✿ ❛♣♣❧✐❝❛t✐♦♥s ❛♥❞ ❝❛♥❞✐❞❛t❡s

❆♣♣❧✐❝❛t✐♦♥s✿ ❖♥❡✲r♦✉♥❞ ❦❡②✲❡①❝❤❛♥❣❡ ❜❡t✇❡❡♥ κ + ✶ ✉s❡rs ✭❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ♣❛✐r✐♥❣s✮ ❆ttr✐❜✉t❡ ❜❛s❡❞ ❡♥❝r②♣t✐♦♥✱ ✇✐t♥❡ss ❡♥❝r②♣t✐♦♥✱ . . . ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t② ♦❜❢✉s❝❛t✐♦♥ ✭✐❖✮

❚❤r❡❡ ♠❛✐♥ ❝❛♥❞✐❞❛t❡s

• ●❍✶✸✱ ❈▲❚✶✸✱ ●●❍✶✺
• ●❍✶✸✿ ●❛r❣✱ ●❡♥tr② ❛♥❞ ❍❛❧❡✈✐ ✭❊✉r♦❝r②♣t ✷✵✶✸✮

❈▲❚✶✸✿ ❈♦r♦♥✱ ▲❡♣♦✐♥t✱ ❚✐❜♦✉❝❤✐ ✭❈r②♣t♦ ✷✵✶✸✮

• ●❍✶✺✿ ●❡♥tr②✱ ●♦r❜✉♥♦✈✱ ❍❛❧❡✈✐ ✭❚❈❈ ✷✵✶✺✮

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✹ ✴ ✷✷

▼♠❛♣✿ ❛♣♣❧✐❝❛t✐♦♥s ❛♥❞ ❝❛♥❞✐❞❛t❡s

❆♣♣❧✐❝❛t✐♦♥s✿ ❖♥❡✲r♦✉♥❞ ❦❡②✲❡①❝❤❛♥❣❡ ❜❡t✇❡❡♥ κ + ✶ ✉s❡rs ✭❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ♣❛✐r✐♥❣s✮ ❆ttr✐❜✉t❡ ❜❛s❡❞ ❡♥❝r②♣t✐♦♥✱ ✇✐t♥❡ss ❡♥❝r②♣t✐♦♥✱ . . . ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t② ♦❜❢✉s❝❛t✐♦♥ ✭✐❖✮

❚❤r❡❡ ♠❛✐♥ ❝❛♥❞✐❞❛t❡s

• ●❍✶✸✱ ❈▲❚✶✸✱ ●●❍✶✺
• ●❍✶✸✿ ●❛r❣✱ ●❡♥tr② ❛♥❞ ❍❛❧❡✈✐ ✭❊✉r♦❝r②♣t ✷✵✶✸✮

❈▲❚✶✸✿ ❈♦r♦♥✱ ▲❡♣♦✐♥t✱ ❚✐❜♦✉❝❤✐ ✭❈r②♣t♦ ✷✵✶✸✮

• ●❍✶✺✿ ●❡♥tr②✱ ●♦r❜✉♥♦✈✱ ❍❛❧❡✈✐ ✭❚❈❈ ✷✵✶✺✮

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✹ ✴ ✷✷

Pr❡✈✐♦✉s ❛tt❛❝❦s ♦♥ ●●❍✶✸ ♠❛♣

❩❡r♦✐③✐♥❣ ❛tt❛❝❦s

❚❤❡♦r❡♠ ❬❍✉ ❛♥❞ ❏✐❛✱ ❊✉r♦❝r②♣t✬✶✻❪

❚❤❡ ●●❍✶✸ ♠❛♣ ✐s ✐♥s❡❝✉r❡ ✐❢ ❡♥❝♦❞✐♥❣s ♦❢ ③❡r♦ ❛r❡ ♣r♦✈✐❞❡❞✳

• ●❍✶✸ ✐s ✐♥s❡❝✉r❡ ❢♦r ❛❧♠♦st ❛❧❧

❛♣♣❧✐❝❛t✐♦♥s✱ ❡①❝❡♣t ♦❜❢✉s❝❛t✐♦♥✳ ❩❡r♦✐③✐♥❣ ❛tt❛❝❦ ♦♥ s♦♠❡ ❝❛♥❞✐❞❛t❡ ♦❜❢✉s❝❛t♦rs✿

▼✐❧❡s✱ ❙❛❤❛✐✱ ❩❤❛♥❞r②✱ ❈r②♣t♦✬✶✻ ❈❤❡♥✱ ●❡♥tr②✱ ❍❛❧❡✈✐✱ ❊❈✬✶✼

❙t❛t✐st✐❝❛❧ ❛tt❛❝❦s ♠❡♥t✐♦♥❡❞ ✐♥ ❬●●❍✶✸❪

✷ s❛♠♣❧✐♥❣ ♠❡t❤♦❞s ♣r♦♣♦s❡❞

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✺ ✴ ✷✷

Pr❡✈✐♦✉s ❛tt❛❝❦s ♦♥ ●●❍✶✸ ♠❛♣

❩❡r♦✐③✐♥❣ ❛tt❛❝❦s

❚❤❡♦r❡♠ ❬❍✉ ❛♥❞ ❏✐❛✱ ❊✉r♦❝r②♣t✬✶✻❪

❚❤❡ ●●❍✶✸ ♠❛♣ ✐s ✐♥s❡❝✉r❡ ✐❢ ❡♥❝♦❞✐♥❣s ♦❢ ③❡r♦ ❛r❡ ♣r♦✈✐❞❡❞✳

• ●❍✶✸ ✐s ✐♥s❡❝✉r❡ ❢♦r ❛❧♠♦st ❛❧❧

❛♣♣❧✐❝❛t✐♦♥s✱ ❡①❝❡♣t ♦❜❢✉s❝❛t✐♦♥✳ ❩❡r♦✐③✐♥❣ ❛tt❛❝❦ ♦♥ s♦♠❡ ❝❛♥❞✐❞❛t❡ ♦❜❢✉s❝❛t♦rs✿

▼✐❧❡s✱ ❙❛❤❛✐✱ ❩❤❛♥❞r②✱ ❈r②♣t♦✬✶✻ ❈❤❡♥✱ ●❡♥tr②✱ ❍❛❧❡✈✐✱ ❊❈✬✶✼

❙t❛t✐st✐❝❛❧ ❛tt❛❝❦s ♠❡♥t✐♦♥❡❞ ✐♥ ❬●●❍✶✸❪

✷ s❛♠♣❧✐♥❣ ♠❡t❤♦❞s ♣r♦♣♦s❡❞

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✺ ✴ ✷✷

Pr❡✈✐♦✉s ❛tt❛❝❦s ♦♥ ●●❍✶✸ ♠❛♣

❩❡r♦✐③✐♥❣ ❛tt❛❝❦s

❚❤❡♦r❡♠ ❬❍✉ ❛♥❞ ❏✐❛✱ ❊✉r♦❝r②♣t✬✶✻❪

❚❤❡ ●●❍✶✸ ♠❛♣ ✐s ✐♥s❡❝✉r❡ ✐❢ ❡♥❝♦❞✐♥❣s ♦❢ ③❡r♦ ❛r❡ ♣r♦✈✐❞❡❞✳

• ●❍✶✸ ✐s ✐♥s❡❝✉r❡ ❢♦r ❛❧♠♦st ❛❧❧

❛♣♣❧✐❝❛t✐♦♥s✱ ❡①❝❡♣t ♦❜❢✉s❝❛t✐♦♥✳ ❩❡r♦✐③✐♥❣ ❛tt❛❝❦ ♦♥ s♦♠❡ ❝❛♥❞✐❞❛t❡ ♦❜❢✉s❝❛t♦rs✿

◮ ▼✐❧❡s✱ ❙❛❤❛✐✱ ❩❤❛♥❞r②✱ ❈r②♣t♦✬✶✻ ◮ ❈❤❡♥✱ ●❡♥tr②✱ ❍❛❧❡✈✐✱ ❊❈✬✶✼

❙t❛t✐st✐❝❛❧ ❛tt❛❝❦s ♠❡♥t✐♦♥❡❞ ✐♥ ❬●●❍✶✸❪

✷ s❛♠♣❧✐♥❣ ♠❡t❤♦❞s ♣r♦♣♦s❡❞

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✺ ✴ ✷✷

Pr❡✈✐♦✉s ❛tt❛❝❦s ♦♥ ●●❍✶✸ ♠❛♣

❩❡r♦✐③✐♥❣ ❛tt❛❝❦s

❚❤❡♦r❡♠ ❬❍✉ ❛♥❞ ❏✐❛✱ ❊✉r♦❝r②♣t✬✶✻❪

❚❤❡ ●●❍✶✸ ♠❛♣ ✐s ✐♥s❡❝✉r❡ ✐❢ ❡♥❝♦❞✐♥❣s ♦❢ ③❡r♦ ❛r❡ ♣r♦✈✐❞❡❞✳

• ●❍✶✸ ✐s ✐♥s❡❝✉r❡ ❢♦r ❛❧♠♦st ❛❧❧

❛♣♣❧✐❝❛t✐♦♥s✱ ❡①❝❡♣t ♦❜❢✉s❝❛t✐♦♥✳ ❩❡r♦✐③✐♥❣ ❛tt❛❝❦ ♦♥ s♦♠❡ ❝❛♥❞✐❞❛t❡ ♦❜❢✉s❝❛t♦rs✿

◮ ▼✐❧❡s✱ ❙❛❤❛✐✱ ❩❤❛♥❞r②✱ ❈r②♣t♦✬✶✻ ◮ ❈❤❡♥✱ ●❡♥tr②✱ ❍❛❧❡✈✐✱ ❊❈✬✶✼

❙t❛t✐st✐❝❛❧ ❛tt❛❝❦s ♠❡♥t✐♦♥❡❞ ✐♥ ❬●●❍✶✸❪

✷ s❛♠♣❧✐♥❣ ♠❡t❤♦❞s ♣r♦♣♦s❡❞

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✺ ✴ ✷✷

Pr❡✈✐♦✉s ❛tt❛❝❦s ♦♥ ●●❍✶✸ ♠❛♣

❩❡r♦✐③✐♥❣ ❛tt❛❝❦s

❚❤❡♦r❡♠ ❬❍✉ ❛♥❞ ❏✐❛✱ ❊✉r♦❝r②♣t✬✶✻❪

❚❤❡ ●●❍✶✸ ♠❛♣ ✐s ✐♥s❡❝✉r❡ ✐❢ ❡♥❝♦❞✐♥❣s ♦❢ ③❡r♦ ❛r❡ ♣r♦✈✐❞❡❞✳

• ●❍✶✸ ✐s ✐♥s❡❝✉r❡ ❢♦r ❛❧♠♦st ❛❧❧

❛♣♣❧✐❝❛t✐♦♥s✱ ❡①❝❡♣t ♦❜❢✉s❝❛t✐♦♥✳ ❩❡r♦✐③✐♥❣ ❛tt❛❝❦ ♦♥ s♦♠❡ ❝❛♥❞✐❞❛t❡ ♦❜❢✉s❝❛t♦rs✿

◮ ▼✐❧❡s✱ ❙❛❤❛✐✱ ❩❤❛♥❞r②✱ ❈r②♣t♦✬✶✻ ◮ ❈❤❡♥✱ ●❡♥tr②✱ ❍❛❧❡✈✐✱ ❊❈✬✶✼

❙t❛t✐st✐❝❛❧ ❛tt❛❝❦s ♠❡♥t✐♦♥❡❞ ✐♥ ❬●●❍✶✸❪

◮ ✷ s❛♠♣❧✐♥❣ ♠❡t❤♦❞s

♣r♦♣♦s❡❞

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✺ ✴ ✷✷

❲❤❛t ✐s ❛ st❛t✐st✐❝❛❧ ❛tt❛❝❦❄

■♥ t❤❡ ●●❍✶✸ ♠❛♣✿ ❊♥❝♦❞✐♥❣s ❛r❡ r❛♥❞♦♠✐③❡❞ ❜✉t ♠♦❞✉❧♦ q

◮ ❛♥❛❧♦❣♦✉s t♦ ◆❚❘❯ ◮ ❡①♣❡❝t❛t✐♦♥ ❛♥❞ ✈❛r✐❛♥❝❡ r❡✈❡❛❧ ♥♦t❤✐♥❣

❆❢t❡r ③❡r♦✲t❡st✿ ♦❜t❛✐♥ ❛♥ ❡❧❡♠❡♥t ✐♥ ✭♥♦ r❡❞✉❝t✐♦♥ ♠♦❞ ✮

❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❡♥❝♦❞✐♥❣s ❤❡♥❝❡ r❛♥❞♦♠✐③❡❞ ✐ts ✈❛r✐❛♥❝❡ ♠✐❣❤t r❡✈❡❛❧ s❡❝r❡t ✐♥❢♦r♠❛t✐♦♥

■♥ t❤✐s t❛❧❦

❚❤❡ ❧❡❛❦ ✇❡ ❛♥❛❧②s❡ ✐s t❤❡ ✈❛r✐❛♥❝❡ ♦❢ t❤❡ ♣♦st✲③❡r♦✲t❡st❡❞ ❡❧❡♠❡♥ts

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✻ ✴ ✷✷

❲❤❛t ✐s ❛ st❛t✐st✐❝❛❧ ❛tt❛❝❦❄

■♥ t❤❡ ●●❍✶✸ ♠❛♣✿ ❊♥❝♦❞✐♥❣s ❛r❡ r❛♥❞♦♠✐③❡❞ ❜✉t ♠♦❞✉❧♦ q

◮ ❛♥❛❧♦❣♦✉s t♦ ◆❚❘❯ ◮ ❡①♣❡❝t❛t✐♦♥ ❛♥❞ ✈❛r✐❛♥❝❡ r❡✈❡❛❧ ♥♦t❤✐♥❣

❆❢t❡r ③❡r♦✲t❡st✿ ♦❜t❛✐♥ ❛♥ ❡❧❡♠❡♥t ✐♥ Z ✭♥♦ r❡❞✉❝t✐♦♥ ♠♦❞q✮

◮ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❡♥❝♦❞✐♥❣s

❤❡♥❝❡ r❛♥❞♦♠✐③❡❞ ✐ts ✈❛r✐❛♥❝❡ ♠✐❣❤t r❡✈❡❛❧ s❡❝r❡t ✐♥❢♦r♠❛t✐♦♥

■♥ t❤✐s t❛❧❦

❚❤❡ ❧❡❛❦ ✇❡ ❛♥❛❧②s❡ ✐s t❤❡ ✈❛r✐❛♥❝❡ ♦❢ t❤❡ ♣♦st✲③❡r♦✲t❡st❡❞ ❡❧❡♠❡♥ts

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✻ ✴ ✷✷

❲❤❛t ✐s ❛ st❛t✐st✐❝❛❧ ❛tt❛❝❦❄

■♥ t❤❡ ●●❍✶✸ ♠❛♣✿ ❊♥❝♦❞✐♥❣s ❛r❡ r❛♥❞♦♠✐③❡❞ ❜✉t ♠♦❞✉❧♦ q

◮ ❛♥❛❧♦❣♦✉s t♦ ◆❚❘❯ ◮ ❡①♣❡❝t❛t✐♦♥ ❛♥❞ ✈❛r✐❛♥❝❡ r❡✈❡❛❧ ♥♦t❤✐♥❣

❆❢t❡r ③❡r♦✲t❡st✿ ♦❜t❛✐♥ ❛♥ ❡❧❡♠❡♥t ✐♥ Z ✭♥♦ r❡❞✉❝t✐♦♥ ♠♦❞q✮

◮ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❡♥❝♦❞✐♥❣s ◮ ❤❡♥❝❡ r❛♥❞♦♠✐③❡❞ ◮ ✐ts ✈❛r✐❛♥❝❡ ♠✐❣❤t r❡✈❡❛❧ s❡❝r❡t ✐♥❢♦r♠❛t✐♦♥

■♥ t❤✐s t❛❧❦

❚❤❡ ❧❡❛❦ ✇❡ ❛♥❛❧②s❡ ✐s t❤❡ ✈❛r✐❛♥❝❡ ♦❢ t❤❡ ♣♦st✲③❡r♦✲t❡st❡❞ ❡❧❡♠❡♥ts

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✻ ✴ ✷✷

❲❤❛t ✐s ❛ st❛t✐st✐❝❛❧ ❛tt❛❝❦❄

■♥ t❤❡ ●●❍✶✸ ♠❛♣✿ ❊♥❝♦❞✐♥❣s ❛r❡ r❛♥❞♦♠✐③❡❞ ❜✉t ♠♦❞✉❧♦ q

◮ ❛♥❛❧♦❣♦✉s t♦ ◆❚❘❯ ◮ ❡①♣❡❝t❛t✐♦♥ ❛♥❞ ✈❛r✐❛♥❝❡ r❡✈❡❛❧ ♥♦t❤✐♥❣

❆❢t❡r ③❡r♦✲t❡st✿ ♦❜t❛✐♥ ❛♥ ❡❧❡♠❡♥t ✐♥ Z ✭♥♦ r❡❞✉❝t✐♦♥ ♠♦❞q✮

◮ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❡♥❝♦❞✐♥❣s ◮ ❤❡♥❝❡ r❛♥❞♦♠✐③❡❞ ◮ ✐ts ✈❛r✐❛♥❝❡ ♠✐❣❤t r❡✈❡❛❧ s❡❝r❡t ✐♥❢♦r♠❛t✐♦♥

■♥ t❤✐s t❛❧❦

❚❤❡ ❧❡❛❦ ✇❡ ❛♥❛❧②s❡ ✐s t❤❡ ✈❛r✐❛♥❝❡ ♦❢ t❤❡ ♣♦st✲③❡r♦✲t❡st❡❞ ❡❧❡♠❡♥ts

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✻ ✴ ✷✷

❈♦♥tr✐❜✉t✐♦♥ ✭✶✮

❲❤❛t s❡tt✐♥❣ ♦❢ t❤❡ ●●❍✶✸ ♠❛♣ s❤♦✉❧❞ ✇❡ ❝♦♥s✐❞❡r❄ ❲❡ ❞❡✜♥❡ ♦✉r ♦✇♥ s❡tt✐♥❣ ✐♥s♣✐r❡❞ ❜② ✐❖ ❜✉t s✐♠♣❧❡r s❡❝✉r❡ ✐♥ t❤❡ ✇❡❛❦ ♠✉❧t✐❧✐♥❡❛r ♠❛♣ ♠♦❞❡❧

♥♦ ✏s✐♠♣❧❡✑ ③❡r♦✐③✐♥❣ ❛tt❛❝❦s

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✼ ✴ ✷✷

❈♦♥tr✐❜✉t✐♦♥ ✭✶✮

❲❤❛t s❡tt✐♥❣ ♦❢ t❤❡ ●●❍✶✸ ♠❛♣ s❤♦✉❧❞ ✇❡ ❝♦♥s✐❞❡r❄ ❲❡ ❞❡✜♥❡ ♦✉r ♦✇♥ s❡tt✐♥❣ ✐♥s♣✐r❡❞ ❜② ✐❖ ❜✉t s✐♠♣❧❡r s❡❝✉r❡ ✐♥ t❤❡ ✇❡❛❦ ♠✉❧t✐❧✐♥❡❛r ♠❛♣ ♠♦❞❡❧

♥♦ ✏s✐♠♣❧❡✑ ③❡r♦✐③✐♥❣ ❛tt❛❝❦s

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✼ ✴ ✷✷

❈♦♥tr✐❜✉t✐♦♥ ✭✶✮

❲❤❛t s❡tt✐♥❣ ♦❢ t❤❡ ●●❍✶✸ ♠❛♣ s❤♦✉❧❞ ✇❡ ❝♦♥s✐❞❡r❄ ❲❡ ❞❡✜♥❡ ♦✉r ♦✇♥ s❡tt✐♥❣ ✐♥s♣✐r❡❞ ❜② ✐❖ ❜✉t s✐♠♣❧❡r s❡❝✉r❡ ✐♥ t❤❡ ✇❡❛❦ ♠✉❧t✐❧✐♥❡❛r ♠❛♣ ♠♦❞❡❧

♥♦ ✏s✐♠♣❧❡✑ ③❡r♦✐③✐♥❣ ❛tt❛❝❦s

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✼ ✴ ✷✷

❈♦♥tr✐❜✉t✐♦♥ ✭✶✮

❲❤❛t s❡tt✐♥❣ ♦❢ t❤❡ ●●❍✶✸ ♠❛♣ s❤♦✉❧❞ ✇❡ ❝♦♥s✐❞❡r❄ ❲❡ ❞❡✜♥❡ ♦✉r ♦✇♥ s❡tt✐♥❣ ✐♥s♣✐r❡❞ ❜② ✐❖ ❜✉t s✐♠♣❧❡r s❡❝✉r❡ ✐♥ t❤❡ ✇❡❛❦ ♠✉❧t✐❧✐♥❡❛r ♠❛♣ ♠♦❞❡❧

♥♦ ✏s✐♠♣❧❡✑ ③❡r♦✐③✐♥❣ ❛tt❛❝❦s

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✼ ✴ ✷✷

❈♦♥tr✐❜✉t✐♦♥ ✭✶✮

❲❤❛t s❡tt✐♥❣ ♦❢ t❤❡ ●●❍✶✸ ♠❛♣ s❤♦✉❧❞ ✇❡ ❝♦♥s✐❞❡r❄ ❲❡ ❞❡✜♥❡ ♦✉r ♦✇♥ s❡tt✐♥❣ ✐♥s♣✐r❡❞ ❜② ✐❖ ❜✉t s✐♠♣❧❡r s❡❝✉r❡ ✐♥ t❤❡ ✇❡❛❦ ♠✉❧t✐❧✐♥❡❛r ♠❛♣ ♠♦❞❡❧

◮ ♥♦ ✏s✐♠♣❧❡✑ ③❡r♦✐③✐♥❣ ❛tt❛❝❦s ❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✼ ✴ ✷✷

❈♦♥tr✐❜✉t✐♦♥ ✭✷✮

❲❡ ❝♦♥s✐❞❡r ✹ ❞✐✛❡r❡♥t s❛♠♣❧✐♥❣ ♣r♦❝❡❞✉r❡s ❢♦r t❤❡ ❡♥❝♦❞✐♥❣s✿

◮ ✷ ❢r♦♠ ❬●●❍✶✸❪ ◮ ✷ ❢r♦♠ ❬❉●●+✶✽❪

❙❛♠♣❧✐♥❣ ♠❡t❤♦❞ ❧❡❛❦❛❣❡ r❡❧❛t❡❞ ❢✉❧❧ ❛tt❛❝❦❄ t♦ s❡❝r❡t ❡❧❡♠❡♥ts ❙✐♠♣❧✐st✐❝ ❬●●❍✶✸❪ ②❡s ②❡s ❢♦r s♦♠❡ ♣❛r❛♠s ❊①♣♦♥❡♥t✐❛❧ ❬●●❍✶✸❪ ②❡s ♥♦ ❈♦♥s❡r✈❛t✐✈❡ ❬❉●● ✶✽❪ ②❡s ♥♦ ❆❣❣r❡ss✐✈❡ ❬❉●● ✶✽❪ ②❡s ♥♦ ❈♦♠♣❡♥s❛t✐♦♥ ✭t❤✐s ✇♦r❦✮ ♥♦ ♥♦ ❲❡ ♣r♦♣♦s❡ ❛ ❝♦✉♥t❡r♠❡❛s✉r❡ ❈♦♠♣❡♥s❛t✐♦♥ ♠❡t❤♦❞

■♥ t❤✐s s✐♠♣❧❡ s❡tt✐♥❣ ❆❧♠♦st ❛s ❡✣❝✐❡♥t ❛s t❤❡ s✐♠♣❧✐st✐❝ ♠❡t❤♦❞

❬❉●●+✶✽❪ ❉ött❧✐♥❣✱ ●❛r❣✱ ●✉♣t❛✱ ▼✐❛♦✱ ❛♥❞ ▼✉❦❤❡r❥❡❡✳ ❖❜❢✉s❝❛t✐♦♥ ❢r♦♠ ▲♦✇ ◆♦✐s❡ ▼✉❧t✐❧✐♥❡❛r ▼❛♣s✱ ■♥❞♦❝r②♣t✳

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✽ ✴ ✷✷

❈♦♥tr✐❜✉t✐♦♥ ✭✷✮

❲❡ ❝♦♥s✐❞❡r ✹ ❞✐✛❡r❡♥t s❛♠♣❧✐♥❣ ♣r♦❝❡❞✉r❡s ❢♦r t❤❡ ❡♥❝♦❞✐♥❣s✿

◮ ✷ ❢r♦♠ ❬●●❍✶✸❪ ◮ ✷ ❢r♦♠ ❬❉●●+✶✽❪

❙❛♠♣❧✐♥❣ ♠❡t❤♦❞ ❧❡❛❦❛❣❡ r❡❧❛t❡❞ ❢✉❧❧ ❛tt❛❝❦❄ t♦ s❡❝r❡t ❡❧❡♠❡♥ts ❙✐♠♣❧✐st✐❝ ❬●●❍✶✸❪ ②❡s ②❡s ❢♦r s♦♠❡ ♣❛r❛♠s ❊①♣♦♥❡♥t✐❛❧ ❬●●❍✶✸❪ ②❡s ♥♦ ❈♦♥s❡r✈❛t✐✈❡ ❬❉●●+✶✽❪ ②❡s ♥♦ ❆❣❣r❡ss✐✈❡ ❬❉●●+✶✽❪ ②❡s ♥♦ ❈♦♠♣❡♥s❛t✐♦♥ ✭t❤✐s ✇♦r❦✮ ♥♦ ♥♦ ❲❡ ♣r♦♣♦s❡ ❛ ❝♦✉♥t❡r♠❡❛s✉r❡ ❈♦♠♣❡♥s❛t✐♦♥ ♠❡t❤♦❞

■♥ t❤✐s s✐♠♣❧❡ s❡tt✐♥❣ ❆❧♠♦st ❛s ❡✣❝✐❡♥t ❛s t❤❡ s✐♠♣❧✐st✐❝ ♠❡t❤♦❞

❬❉●●+✶✽❪ ❉ött❧✐♥❣✱ ●❛r❣✱ ●✉♣t❛✱ ▼✐❛♦✱ ❛♥❞ ▼✉❦❤❡r❥❡❡✳ ❖❜❢✉s❝❛t✐♦♥ ❢r♦♠ ▲♦✇ ◆♦✐s❡ ▼✉❧t✐❧✐♥❡❛r ▼❛♣s✱ ■♥❞♦❝r②♣t✳

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✽ ✴ ✷✷

❈♦♥tr✐❜✉t✐♦♥ ✭✷✮

❲❡ ❝♦♥s✐❞❡r ✹ ❞✐✛❡r❡♥t s❛♠♣❧✐♥❣ ♣r♦❝❡❞✉r❡s ❢♦r t❤❡ ❡♥❝♦❞✐♥❣s✿

◮ ✷ ❢r♦♠ ❬●●❍✶✸❪ ◮ ✷ ❢r♦♠ ❬❉●●+✶✽❪

❙❛♠♣❧✐♥❣ ♠❡t❤♦❞ ❧❡❛❦❛❣❡ r❡❧❛t❡❞ ❢✉❧❧ ❛tt❛❝❦❄ t♦ s❡❝r❡t ❡❧❡♠❡♥ts ❙✐♠♣❧✐st✐❝ ❬●●❍✶✸❪ ②❡s ②❡s ❢♦r s♦♠❡ ♣❛r❛♠s ❊①♣♦♥❡♥t✐❛❧ ❬●●❍✶✸❪ ②❡s ♥♦ ❈♦♥s❡r✈❛t✐✈❡ ❬❉●●+✶✽❪ ②❡s ♥♦ ❆❣❣r❡ss✐✈❡ ❬❉●●+✶✽❪ ②❡s ♥♦ ❈♦♠♣❡♥s❛t✐♦♥ ✭t❤✐s ✇♦r❦✮ ♥♦ ♥♦ ❲❡ ♣r♦♣♦s❡ ❛ ❝♦✉♥t❡r♠❡❛s✉r❡ ⇒ ❈♦♠♣❡♥s❛t✐♦♥ ♠❡t❤♦❞

◮ ■♥ t❤✐s s✐♠♣❧❡ s❡tt✐♥❣ ◮ ❆❧♠♦st ❛s ❡✣❝✐❡♥t ❛s t❤❡ s✐♠♣❧✐st✐❝ ♠❡t❤♦❞

❬❉●●+✶✽❪ ❉ött❧✐♥❣✱ ●❛r❣✱ ●✉♣t❛✱ ▼✐❛♦✱ ❛♥❞ ▼✉❦❤❡r❥❡❡✳ ❖❜❢✉s❝❛t✐♦♥ ❢r♦♠ ▲♦✇ ◆♦✐s❡ ▼✉❧t✐❧✐♥❡❛r ▼❛♣s✱ ■♥❞♦❝r②♣t✳

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✽ ✴ ✷✷

❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦

✶

❚❤❡ ●●❍✶✸ ▼❛♣

✷

❙t❛t✐st✐❝❛❧ ▲❡❛❦

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✾ ✴ ✷✷

❚❤❡ ●●❍✶✸ ♠✉❧t✐❧✐♥❡❛r ♠❛♣

❉❡✜♥❡ R = Z[X]/(X n + ✶) ✇✐t❤ n = ✷k ❙❛♠♣❧❡ ❛ ✏s♠❛❧❧✑ ❡❧❡♠❡♥t ✐♥ t❤❡ ♣❧❛✐♥t❡①t s♣❛❝❡ ✐s ❙❛♠♣❧❡ ❛ ✏❧❛r❣❡✑ ✐♥t❡❣❡r t❤❡ ❡♥❝♦❞✐♥❣ s♣❛❝❡ ✐s ✶

◆♦t❛t✐♦♥

❲❡ ✇r✐t❡ ❢♦r t❤❡ ❡❧❡♠❡♥ts ✐♥

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✵ ✴ ✷✷

❚❤❡ ●●❍✶✸ ♠✉❧t✐❧✐♥❡❛r ♠❛♣

❉❡✜♥❡ R = Z[X]/(X n + ✶) ✇✐t❤ n = ✷k ❙❛♠♣❧❡ g ❛ ✏s♠❛❧❧✑ ❡❧❡♠❡♥t ✐♥ R ⇒ t❤❡ ♣❧❛✐♥t❡①t s♣❛❝❡ ✐s P = R/g ❙❛♠♣❧❡ ❛ ✏❧❛r❣❡✑ ✐♥t❡❣❡r t❤❡ ❡♥❝♦❞✐♥❣ s♣❛❝❡ ✐s ✶

◆♦t❛t✐♦♥

❲❡ ✇r✐t❡ ❢♦r t❤❡ ❡❧❡♠❡♥ts ✐♥

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✵ ✴ ✷✷

❚❤❡ ●●❍✶✸ ♠✉❧t✐❧✐♥❡❛r ♠❛♣

❉❡✜♥❡ R = Z[X]/(X n + ✶) ✇✐t❤ n = ✷k ❙❛♠♣❧❡ g ❛ ✏s♠❛❧❧✑ ❡❧❡♠❡♥t ✐♥ R ⇒ t❤❡ ♣❧❛✐♥t❡①t s♣❛❝❡ ✐s P = R/g ❙❛♠♣❧❡ q ❛ ✏❧❛r❣❡✑ ✐♥t❡❣❡r ⇒ t❤❡ ❡♥❝♦❞✐♥❣ s♣❛❝❡ ✐s Rq = R/(qR) = Zq[X]/(X n + ✶)

◆♦t❛t✐♦♥

❲❡ ✇r✐t❡ [r]q ❢♦r t❤❡ ❡❧❡♠❡♥ts ✐♥ Rq

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✵ ✴ ✷✷

❚❤❡ ●●❍✶✸ ♠✉❧t✐❧✐♥❡❛r ♠❛♣✿ ❡♥❝♦❞✐♥❣s

❙❛♠♣❧❡ z✶, . . . , zκ ✉♥✐❢♦r♠❧② ✐♥ Rq ❊♥❝♦❞✐♥❣✿ ❆♥ ❡♥❝♦❞✐♥❣ ♦❢ a ❛t ❧❡✈❡❧ S ⊆ {✶, . . . , κ} ✐s u =

• a
• i∈S zi
• q

✇❤❡r❡ a = a ♠♦❞ g

❆❞❞✐t✐♦♥ ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

❆❞❞✐t✐♦♥✿

✶ ✶ ✷ ✷ ✶ ✷

▼✉❧t✐♣❧✐❝❛t✐♦♥✿

✶ ✶

✶

✷ ✷

✷

✶ ✷

✶ ✷

✭✐❢

✶ ✷

✮

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✶ ✴ ✷✷

❚❤❡ ●●❍✶✸ ♠✉❧t✐❧✐♥❡❛r ♠❛♣✿ ❡♥❝♦❞✐♥❣s

❙❛♠♣❧❡ z✶, . . . , zκ ✉♥✐❢♦r♠❧② ✐♥ Rq ❊♥❝♦❞✐♥❣✿ ❆♥ ❡♥❝♦❞✐♥❣ ♦❢ a ❛t ❧❡✈❡❧ S ⊆ {✶, . . . , κ} ✐s u =

• a
• i∈S zi
• q

✇❤❡r❡ a = a ♠♦❞ g

❆❞❞✐t✐♦♥ ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥

❆❞❞✐t✐♦♥✿ a✶ + r✶g

• i∈S zi
• q

+ a✷ + r✷g

• i∈S zi
• q

= a✶ + a✷ + r′g

• i∈S zi
• q

▼✉❧t✐♣❧✐❝❛t✐♦♥✿

• a✶ + r✶g
• i∈S✶ zi
• q

·

• a✷ + r✷g
• i∈S✷ zi
• q

=

• a✶ · a✷ + r′g
• i∈S✶∪S✷ zi
• q

✭✐❢ S✶ ∩ S✷ = ∅✮

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✶ ✴ ✷✷

❚❤❡ ●●❍✶✸ ♠✉❧t✐❧✐♥❡❛r ♠❛♣✿ ③❡r♦✲t❡st

❙❛♠♣❧❡ h ✐♥ R ♦❢ t❤❡ ♦r❞❡r ♦❢ q✶/✷ ▲❡t z∗ = κ

i=✶ zi

❉❡✜♥❡ pzt = [z∗hg−✶]q

❩❡r♦✲t❡st

❚♦ t❡st ✐❢ ✐s ❛♥ ❡♥❝♦❞✐♥❣ ♦❢ ③❡r♦ ✭✐✳❡✳ ✵ ♠♦❞ ✮✱ ❝♦♠♣✉t❡

✶

❚❤✐s ✐s s♠❛❧❧ ✐✛ ✐s ❛ s♠❛❧❧ ♠✉❧t✐♣❧❡ ♦❢ ✳ ❘❡♠❛r❦✿ ■❢ ✵ ♠♦❞ ✱ t❤❡♥ ♦✈❡r

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✷ ✴ ✷✷

❚❤❡ ●●❍✶✸ ♠✉❧t✐❧✐♥❡❛r ♠❛♣✿ ③❡r♦✲t❡st

❙❛♠♣❧❡ h ✐♥ R ♦❢ t❤❡ ♦r❞❡r ♦❢ q✶/✷ ▲❡t z∗ = κ

i=✶ zi

❉❡✜♥❡ pzt = [z∗hg−✶]q

❩❡r♦✲t❡st

❚♦ t❡st ✐❢ u = [c/z∗]q ✐s ❛♥ ❡♥❝♦❞✐♥❣ ♦❢ ③❡r♦ ✭✐✳❡✳ c = ✵ ♠♦❞ g✮✱ ❝♦♠♣✉t❡ [u · pzt]q = [chg−✶]q ❚❤✐s ✐s s♠❛❧❧ ✐✛ c ✐s ❛ s♠❛❧❧ ♠✉❧t✐♣❧❡ ♦❢ g✳ ❘❡♠❛r❦✿ ■❢ c = ✵ ♠♦❞ g✱ t❤❡♥ [u · pzt]q = ch/g ♦✈❡r R

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✷ ✴ ✷✷

❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦

✶

❚❤❡ ●●❍✶✸ ▼❛♣

✷

❙t❛t✐st✐❝❛❧ ▲❡❛❦

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✸ ✴ ✷✷

❙t❛t✐st✐❝❛❧ ❜❛❝❦❣r♦✉♥❞ ✭✶✮

❉❡✜♥✐t✐♦♥s

❆ ❞✐str✐❜✉t✐♦♥ ✐s s❛✐❞ ❝❡♥t❡r❡❞ ✐❢ ✐ts ♠❡❛♥ ✐s ③❡r♦✳ ❆ ❞✐str✐❜✉t✐♦♥ ✐s s❛✐❞ ✐s♦tr♦♣✐❝ ✐❢ ♥♦ ❞✐r❡❝t✐♦♥ ✐s ♣r✐✈✐❧❡❣❡❞✳

❊①❛♠♣❧❡

◆♦t❛t✐♦♥✿ ❲❡ ✇r✐t❡ ✐♥ r❡❞ t❤❡ ❝❡♥t❡r❡❞ ✐s♦tr♦♣✐❝ ✈❛r✐❛❜❧❡s

• ❛✉ss✐❛♥ ❞✐str✐❜✉t✐♦♥

❲❡ ✇r✐t❡ t❤❡ ❞✐s❝r❡t❡ ●❛✉ss✐❛♥ ❞✐str✐❜✉t✐♦♥ ❝❡♥t❡r❡❞ ✐♥ ✵ ❛♥❞ ♦❢ ✈❛r✐❛♥❝❡ ♣❛r❛♠❡t❡r

✷ ♦✈❡r t❤❡ ❧❛tt✐❝❡

✐s ❛ ❝❡♥t❡r❡❞ ✐s♦tr♦♣✐❝ ❞✐str✐❜✉t✐♦♥

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✹ ✴ ✷✷

❙t❛t✐st✐❝❛❧ ❜❛❝❦❣r♦✉♥❞ ✭✶✮

❉❡✜♥✐t✐♦♥s

❆ ❞✐str✐❜✉t✐♦♥ ✐s s❛✐❞ ❝❡♥t❡r❡❞ ✐❢ ✐ts ♠❡❛♥ ✐s ③❡r♦✳ ❆ ❞✐str✐❜✉t✐♦♥ ✐s s❛✐❞ ✐s♦tr♦♣✐❝ ✐❢ ♥♦ ❞✐r❡❝t✐♦♥ ✐s ♣r✐✈✐❧❡❣❡❞✳

❊①❛♠♣❧❡

◆♦t❛t✐♦♥✿ ❲❡ ✇r✐t❡ ✐♥ r❡❞ t❤❡ ❝❡♥t❡r❡❞ ✐s♦tr♦♣✐❝ ✈❛r✐❛❜❧❡s

• ❛✉ss✐❛♥ ❞✐str✐❜✉t✐♦♥

❲❡ ✇r✐t❡ DL,σ t❤❡ ❞✐s❝r❡t❡ ●❛✉ss✐❛♥ ❞✐str✐❜✉t✐♦♥ ❝❡♥t❡r❡❞ ✐♥ ✵ ❛♥❞ ♦❢ ✈❛r✐❛♥❝❡ ♣❛r❛♠❡t❡r σ✷ ♦✈❡r t❤❡ ❧❛tt✐❝❡ L DL,σ ✐s ❛ ❝❡♥t❡r❡❞ ✐s♦tr♦♣✐❝ ❞✐str✐❜✉t✐♦♥

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✹ ✴ ✷✷

❙t❛t✐st✐❝❛❧ ❜❛❝❦❣r♦✉♥❞ ✭✷✮

❉❡✜♥✐t✐♦♥s ✴ ◆♦t❛t✐♦♥

❋♦r r ∈ R✱ ✇❡ ❞❡♥♦t❡ A(r) = rr t❤❡ ❛✉t♦✲❝♦rr❡❧❛t✐♦♥ ♦❢ r✱ ✇❤❡r❡ r ✐s t❤❡ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ♦❢ r ✇❤❡♥ s❡❡♥ ✐♥ C ❚❤❡ ✈❛r✐❛♥❝❡ ♦❢ ❛ ❝❡♥t❡r❡❞ ✈❛r✐❛❜❧❡ r ✐s ❱❛r(r) := E(rr) Pr♦♣♦s✐t✐♦♥✿ ■❢ ✐s ❝❡♥t❡r❡❞ ❛♥❞ ✐s♦tr♦♣✐❝ t❤❡♥ ✵ ❱❛r ■♥ t❤✐s t❛❧❦✱ ❛ss✉♠❡ ❱❛r ✶

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✺ ✴ ✷✷

❙t❛t✐st✐❝❛❧ ❜❛❝❦❣r♦✉♥❞ ✭✷✮

❉❡✜♥✐t✐♦♥s ✴ ◆♦t❛t✐♦♥

❋♦r r ∈ R✱ ✇❡ ❞❡♥♦t❡ A(r) = rr t❤❡ ❛✉t♦✲❝♦rr❡❧❛t✐♦♥ ♦❢ r✱ ✇❤❡r❡ r ✐s t❤❡ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ♦❢ r ✇❤❡♥ s❡❡♥ ✐♥ C ❚❤❡ ✈❛r✐❛♥❝❡ ♦❢ ❛ ❝❡♥t❡r❡❞ ✈❛r✐❛❜❧❡ r ✐s ❱❛r(r) := E(rr) Pr♦♣♦s✐t✐♦♥✿ ■❢ r ✐s ❝❡♥t❡r❡❞ ❛♥❞ ✐s♦tr♦♣✐❝ t❤❡♥ E(r) = ✵ ❱❛r(r) = µ ∈ R ■♥ t❤✐s t❛❧❦✱ ❛ss✉♠❡ ❱❛r ✶

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✺ ✴ ✷✷

❙t❛t✐st✐❝❛❧ ❜❛❝❦❣r♦✉♥❞ ✭✷✮

❉❡✜♥✐t✐♦♥s ✴ ◆♦t❛t✐♦♥

❋♦r r ∈ R✱ ✇❡ ❞❡♥♦t❡ A(r) = rr t❤❡ ❛✉t♦✲❝♦rr❡❧❛t✐♦♥ ♦❢ r✱ ✇❤❡r❡ r ✐s t❤❡ ❝♦♠♣❧❡① ❝♦♥❥✉❣❛t❡ ♦❢ r ✇❤❡♥ s❡❡♥ ✐♥ C ❚❤❡ ✈❛r✐❛♥❝❡ ♦❢ ❛ ❝❡♥t❡r❡❞ ✈❛r✐❛❜❧❡ r ✐s ❱❛r(r) := E(rr) Pr♦♣♦s✐t✐♦♥✿ ■❢ r ✐s ❝❡♥t❡r❡❞ ❛♥❞ ✐s♦tr♦♣✐❝ t❤❡♥ E(r) = ✵ ❱❛r(r) = µ ∈ R ■♥ t❤✐s t❛❧❦✱ ❛ss✉♠❡ ❱❛r(r) = ✶

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✺ ✴ ✷✷

❙t❛t✐st✐❝❛❧ ❧❡❛❦

❘❡❝❛❧❧

■❢ u = [c/z∗]q ✇✐t❤ c = ✵ ♠♦❞ g✱ t❤❡♥ [u · pzt]q = c · h/g ∈ R ■❞❡❛✿ ✐s ✜①❡❞ ❜✉t ✐s ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❱❛r ❱❛r ❲❡ ❝❛♥ ❛♣♣r♦①✐♠❛t❡ ✐t ✇✐t❤ ♠❛♥② s❛♠♣❧❡s

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✻ ✴ ✷✷

❙t❛t✐st✐❝❛❧ ❧❡❛❦

❘❡❝❛❧❧

■❢ u = [c/z∗]q ✇✐t❤ c = ✵ ♠♦❞ g✱ t❤❡♥ [u · pzt]q = c · h/g ∈ R ■❞❡❛✿ h/g ✐s ✜①❡❞ ❜✉t c ✐s ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❱❛r(c · h/g) = ❱❛r(c) · A(h/g) ❲❡ ❝❛♥ ❛♣♣r♦①✐♠❛t❡ ✐t ✇✐t❤ ♠❛♥② s❛♠♣❧❡s

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✻ ✴ ✷✷

❙✐♠♣❧❡ s❡tt✐♥❣ ✭s✐♠♣❧✐✜❡❞✮

❋♦r ❛❧❧ ✶ ≤ i ≤ κ✱ ✇❡ ❣❡t

◮ [

ai zi ]q ✇✐t❤

ai = ai ♠♦❞ g

◮ [

• bi
• j=i zj ]q ✇✐t❤

bi = bi ♠♦❞ g

s✉❝❤ t❤❛t aibi = ✵

✶ ✷ ✸ · · · i · · · κ

ai bi aibi = ✵

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✼ ✴ ✷✷

▲❡❛❦ ✐♥ t❤❡ s✐♠♣❧❡ s❡tt✐♥❣

❲❡ ❣❡t ❡♥❝♦❞✐♥❣s ♦❢ ③❡r♦✿ ui = ai zi

• q

·

• bi
• j=i zj
• q

=

• ai

bi z∗

• q

❆❢t❡r ③❡r♦✲t❡st✿ ❱❛r✐❛♥❝❡✿ ❱❛r ❱❛r ❱❛r

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✽ ✴ ✷✷

▲❡❛❦ ✐♥ t❤❡ s✐♠♣❧❡ s❡tt✐♥❣

❲❡ ❣❡t ❡♥❝♦❞✐♥❣s ♦❢ ③❡r♦✿ ui = ai zi

• q

·

• bi
• j=i zj
• q

=

• ai

bi z∗

• q

❆❢t❡r ③❡r♦✲t❡st✿ ( ai · bi) · h/g ∈ R ❱❛r✐❛♥❝❡✿ ❱❛r ❱❛r ❱❛r

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✽ ✴ ✷✷

▲❡❛❦ ✐♥ t❤❡ s✐♠♣❧❡ s❡tt✐♥❣

❲❡ ❣❡t ❡♥❝♦❞✐♥❣s ♦❢ ③❡r♦✿ ui = ai zi

• q

·

• bi
• j=i zj
• q

=

• ai

bi z∗

• q

❆❢t❡r ③❡r♦✲t❡st✿ ( ai · bi) · h/g ∈ R ❱❛r✐❛♥❝❡✿ ❱❛r( ai · bi) · A(h/g) = ❱❛r( ai) · ❱❛r( bi) · A(h/g)

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✽ ✴ ✷✷

▲❡❛❦❛❣❡ ❢♦r t❤❡ t✇♦ ♠❡t❤♦❞s ♦❢ ❬●●❍✶✸❪

❘❡❝❛❧❧

❚❤❡ ❧❡❛❦❛❣❡ ✐s ❱❛r( ai) · ❱❛r( bi) · A(h/g)

❚❤❡ s✐♠♣❧✐st✐❝ ♠❡t❤♦❞✿

▲❡❛❦❛❣❡✿

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ♠❡t❤♦❞✿

❢♦r ▲❡❛❦❛❣❡✿

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✾ ✴ ✷✷

▲❡❛❦❛❣❡ ❢♦r t❤❡ t✇♦ ♠❡t❤♦❞s ♦❢ ❬●●❍✶✸❪

❘❡❝❛❧❧

❚❤❡ ❧❡❛❦❛❣❡ ✐s ❱❛r( ai) · ❱❛r( bi) · A(h/g)

❚❤❡ s✐♠♣❧✐st✐❝ ♠❡t❤♦❞✿

• ai ← Dai+gR,σ
• bi ← Dbi+gR,σ

▲❡❛❦❛❣❡✿

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ♠❡t❤♦❞✿

❢♦r ▲❡❛❦❛❣❡✿

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✾ ✴ ✷✷

▲❡❛❦❛❣❡ ❢♦r t❤❡ t✇♦ ♠❡t❤♦❞s ♦❢ ❬●●❍✶✸❪

❘❡❝❛❧❧

❚❤❡ ❧❡❛❦❛❣❡ ✐s ❱❛r( ai) · ❱❛r( bi) · A(h/g)

❚❤❡ s✐♠♣❧✐st✐❝ ♠❡t❤♦❞✿

• ai ← Dai+gR,σ
• bi ← Dbi+gR,σ

▲❡❛❦❛❣❡✿ A(h/g)

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ♠❡t❤♦❞✿

❢♦r ▲❡❛❦❛❣❡✿

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✾ ✴ ✷✷

▲❡❛❦❛❣❡ ❢♦r t❤❡ t✇♦ ♠❡t❤♦❞s ♦❢ ❬●●❍✶✸❪

❘❡❝❛❧❧

❚❤❡ ❧❡❛❦❛❣❡ ✐s ❱❛r( ai) · ❱❛r( bi) · A(h/g)

❚❤❡ s✐♠♣❧✐st✐❝ ♠❡t❤♦❞✿

• ai ← Dai+gR,σ
• bi ← Dbi+gR,σ

▲❡❛❦❛❣❡✿ A(h/g)

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ♠❡t❤♦❞✿

• ai =

ai · zi

• bi =

bi ·

j=i zj

❢♦r ai ← D(ai+gR)/zi, σ

• bi ← D(bi+gR)/(

j=i zj), σ

▲❡❛❦❛❣❡✿

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✾ ✴ ✷✷

▲❡❛❦❛❣❡ ❢♦r t❤❡ t✇♦ ♠❡t❤♦❞s ♦❢ ❬●●❍✶✸❪

❘❡❝❛❧❧

❚❤❡ ❧❡❛❦❛❣❡ ✐s ❱❛r( ai) · ❱❛r( bi) · A(h/g)

❚❤❡ s✐♠♣❧✐st✐❝ ♠❡t❤♦❞✿

• ai ← Dai+gR,σ
• bi ← Dbi+gR,σ

▲❡❛❦❛❣❡✿ A(h/g)

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ♠❡t❤♦❞✿

• ai =

ai · zi

• bi =

bi ·

j=i zj

❢♦r ai ← D(ai+gR)/zi, σ

• bi ← D(bi+gR)/(

j=i zj), σ

▲❡❛❦❛❣❡✿ A(zi) · A(

• j=i

zj) · A(h/g) = A(z∗h/g)

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✶✾ ✴ ✷✷

❈♦✉♥t❡r♠❡❛s✉r❡

❘❡❝❛❧❧

❚❤❡ ❧❡❛❦❛❣❡ ✐s ❱❛r( ai) · ❱❛r( bi) · A(h/g) ❲❛♥t❡❞✿ ❱❛r( ai) · ❱❛r( bi) · A(h/g) = ✶

❚❤❡ ❝♦♠♣❡♥s❛t✐♦♥ ♠❡t❤♦❞✿

❢♦r ▲❡❛❦❛❣❡✿ ✶ ❘❡♠❛r❦✿ ♠♦r❡ ❡✣❝✐❡♥t t❤❛♥ ♦t❤❡r ♠❡t❤♦❞s ✭❡①❝❡♣t s✐♠♣❧✐st✐❝✮

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷✵ ✴ ✷✷

❈♦✉♥t❡r♠❡❛s✉r❡

❘❡❝❛❧❧

❚❤❡ ❧❡❛❦❛❣❡ ✐s ❱❛r( ai) · ❱❛r( bi) · A(h/g) ❲❛♥t❡❞✿ ❱❛r( ai) · ❱❛r( bi) · A(h/g) = ✶

❚❤❡ ❝♦♠♣❡♥s❛t✐♦♥ ♠❡t❤♦❞✿

• ai =

ai ·

• g/h
• bi =

bi ·

• g/h

❢♦r ai ← D(ai+gR)/√

g/h, σ

• bi ← D(bi+gR)/√

g/h, σ

▲❡❛❦❛❣❡✿ ✶ ❘❡♠❛r❦✿ ♠♦r❡ ❡✣❝✐❡♥t t❤❛♥ ♦t❤❡r ♠❡t❤♦❞s ✭❡①❝❡♣t s✐♠♣❧✐st✐❝✮

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷✵ ✴ ✷✷

❈♦✉♥t❡r♠❡❛s✉r❡

❘❡❝❛❧❧

❚❤❡ ❧❡❛❦❛❣❡ ✐s ❱❛r( ai) · ❱❛r( bi) · A(h/g) ❲❛♥t❡❞✿ ❱❛r( ai) · ❱❛r( bi) · A(h/g) = ✶

❚❤❡ ❝♦♠♣❡♥s❛t✐♦♥ ♠❡t❤♦❞✿

• ai =

ai ·

• g/h
• bi =

bi ·

• g/h

❢♦r ai ← D(ai+gR)/√

g/h, σ

• bi ← D(bi+gR)/√

g/h, σ

▲❡❛❦❛❣❡✿ A(

• g/h) · A(
• g/h) · A(h/g)

= ✶ ❘❡♠❛r❦✿ ♠♦r❡ ❡✣❝✐❡♥t t❤❛♥ ♦t❤❡r ♠❡t❤♦❞s ✭❡①❝❡♣t s✐♠♣❧✐st✐❝✮

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷✵ ✴ ✷✷

❲❤❛t t♦ ❞♦ ✇✐t❤ t❤❡ ❧❡❛❦❛❣❡

❙✐♠♣❧✐st✐❝ ♠❡t❤♦❞ ❊①♣♦♥❡♥t✐❛❧ ♠❡t❤♦❞ ▲❡❛❦❛❣❡ ≈ A(h/g) ≈ A(z∗h/g) Pr♦❜❧❡♠✿ ❚❤❡ ❧❡❛❦❡❞ ✈❛❧✉❡s ❛r❡ ❢r❛❝t✐♦♥s ❙♦❧✉t✐♦♥✿ ❢♦r t❤❡ s✐♠♣❧✐st✐❝ ♠❡t❤♦❞ ❩❡r♦✲t❡st r❡❝♦✈❡r ♠✉❧t✐♣❧❡ ♦❢ ✿ ❈♦♠❜✐♥❡ ✐t ✇✐t❤ t❤❡ ❧❡❛❦❛❣❡ t♦ ❣❡t✿

■♥t❡❣❡r ❝♦❡✣❝✐❡♥ts

■❢ ✐s ♣♦❧②✭ ✮

✐s ♣♦❧②✭♥✮ ✐t ❝❛♥ ❜❡ r❡❝♦✈❡r❡❞ ❡①❛❝t❧② ✇✐t❤ ♣♦❧②♥♦♠✐❛❧❧② ♠❛♥② s❛♠♣❧❡s ♦❜t❛✐♥ ❛ ♠✉❧t✐♣❧❡ ♦❢

❘❡♠❛r❦✿ ❞♦❡s ♥♦t ✇♦r❦ ❢♦r

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷✶ ✴ ✷✷

❲❤❛t t♦ ❞♦ ✇✐t❤ t❤❡ ❧❡❛❦❛❣❡

❙✐♠♣❧✐st✐❝ ♠❡t❤♦❞ ❊①♣♦♥❡♥t✐❛❧ ♠❡t❤♦❞ ▲❡❛❦❛❣❡ ≈ A(h/g) ≈ A(z∗h/g) Pr♦❜❧❡♠✿ ❚❤❡ ❧❡❛❦❡❞ ✈❛❧✉❡s ❛r❡ ❢r❛❝t✐♦♥s ❙♦❧✉t✐♦♥✿ ❢♦r t❤❡ s✐♠♣❧✐st✐❝ ♠❡t❤♦❞ ❩❡r♦✲t❡st ⇒ r❡❝♦✈❡r ♠✉❧t✐♣❧❡ ♦❢ h✿ r · h ❈♦♠❜✐♥❡ ✐t ✇✐t❤ t❤❡ ❧❡❛❦❛❣❡ t♦ ❣❡t✿

■♥t❡❣❡r ❝♦❡✣❝✐❡♥ts

■❢ ✐s ♣♦❧②✭ ✮

✐s ♣♦❧②✭♥✮ ✐t ❝❛♥ ❜❡ r❡❝♦✈❡r❡❞ ❡①❛❝t❧② ✇✐t❤ ♣♦❧②♥♦♠✐❛❧❧② ♠❛♥② s❛♠♣❧❡s ♦❜t❛✐♥ ❛ ♠✉❧t✐♣❧❡ ♦❢

❘❡♠❛r❦✿ ❞♦❡s ♥♦t ✇♦r❦ ❢♦r

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷✶ ✴ ✷✷

❲❤❛t t♦ ❞♦ ✇✐t❤ t❤❡ ❧❡❛❦❛❣❡

❙✐♠♣❧✐st✐❝ ♠❡t❤♦❞ ❊①♣♦♥❡♥t✐❛❧ ♠❡t❤♦❞ ▲❡❛❦❛❣❡ ≈ A(h/g) ≈ A(z∗h/g) Pr♦❜❧❡♠✿ ❚❤❡ ❧❡❛❦❡❞ ✈❛❧✉❡s ❛r❡ ❢r❛❝t✐♦♥s ❙♦❧✉t✐♦♥✿ ❢♦r t❤❡ s✐♠♣❧✐st✐❝ ♠❡t❤♦❞ ❩❡r♦✲t❡st ⇒ r❡❝♦✈❡r ♠✉❧t✐♣❧❡ ♦❢ h✿ r · h ❈♦♠❜✐♥❡ ✐t ✇✐t❤ t❤❡ ❧❡❛❦❛❣❡ t♦ ❣❡t✿ ≈ A(rg)

◮ ■♥t❡❣❡r ❝♦❡✣❝✐❡♥ts

■❢ ✐s ♣♦❧②✭ ✮

✐s ♣♦❧②✭♥✮ ✐t ❝❛♥ ❜❡ r❡❝♦✈❡r❡❞ ❡①❛❝t❧② ✇✐t❤ ♣♦❧②♥♦♠✐❛❧❧② ♠❛♥② s❛♠♣❧❡s ♦❜t❛✐♥ ❛ ♠✉❧t✐♣❧❡ ♦❢

❘❡♠❛r❦✿ ❞♦❡s ♥♦t ✇♦r❦ ❢♦r

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷✶ ✴ ✷✷

❲❤❛t t♦ ❞♦ ✇✐t❤ t❤❡ ❧❡❛❦❛❣❡

❙✐♠♣❧✐st✐❝ ♠❡t❤♦❞ ❊①♣♦♥❡♥t✐❛❧ ♠❡t❤♦❞ ▲❡❛❦❛❣❡ ≈ A(h/g) ≈ A(z∗h/g) Pr♦❜❧❡♠✿ ❚❤❡ ❧❡❛❦❡❞ ✈❛❧✉❡s ❛r❡ ❢r❛❝t✐♦♥s ❙♦❧✉t✐♦♥✿ ❢♦r t❤❡ s✐♠♣❧✐st✐❝ ♠❡t❤♦❞ ❩❡r♦✲t❡st ⇒ r❡❝♦✈❡r ♠✉❧t✐♣❧❡ ♦❢ h✿ r · h ❈♦♠❜✐♥❡ ✐t ✇✐t❤ t❤❡ ❧❡❛❦❛❣❡ t♦ ❣❡t✿ ≈ A(rg)

◮ ■♥t❡❣❡r ❝♦❡✣❝✐❡♥ts

■❢ q ✐s ♣♦❧②✭n✮

◮ A(rg) ✐s ♣♦❧②✭♥✮ ◮ ✐t ❝❛♥ ❜❡ r❡❝♦✈❡r❡❞ ❡①❛❝t❧② ✇✐t❤ ♣♦❧②♥♦♠✐❛❧❧② ♠❛♥② s❛♠♣❧❡s ◮ ♦❜t❛✐♥ ❛ ♠✉❧t✐♣❧❡ ♦❢ g

❘❡♠❛r❦✿ ❞♦❡s ♥♦t ✇♦r❦ ❢♦r

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷✶ ✴ ✷✷

❲❤❛t t♦ ❞♦ ✇✐t❤ t❤❡ ❧❡❛❦❛❣❡

❙✐♠♣❧✐st✐❝ ♠❡t❤♦❞ ❊①♣♦♥❡♥t✐❛❧ ♠❡t❤♦❞ ▲❡❛❦❛❣❡ ≈ A(h/g) ≈ A(z∗h/g) Pr♦❜❧❡♠✿ ❚❤❡ ❧❡❛❦❡❞ ✈❛❧✉❡s ❛r❡ ❢r❛❝t✐♦♥s ❙♦❧✉t✐♦♥✿ ❢♦r t❤❡ s✐♠♣❧✐st✐❝ ♠❡t❤♦❞ ❩❡r♦✲t❡st ⇒ r❡❝♦✈❡r ♠✉❧t✐♣❧❡ ♦❢ h✿ r · h ❈♦♠❜✐♥❡ ✐t ✇✐t❤ t❤❡ ❧❡❛❦❛❣❡ t♦ ❣❡t✿ ≈ A(rg)

◮ ■♥t❡❣❡r ❝♦❡✣❝✐❡♥ts

■❢ q ✐s ♣♦❧②✭n✮

◮ A(rg) ✐s ♣♦❧②✭♥✮ ◮ ✐t ❝❛♥ ❜❡ r❡❝♦✈❡r❡❞ ❡①❛❝t❧② ✇✐t❤ ♣♦❧②♥♦♠✐❛❧❧② ♠❛♥② s❛♠♣❧❡s ◮ ♦❜t❛✐♥ ❛ ♠✉❧t✐♣❧❡ ♦❢ g

❘❡♠❛r❦✿ ❞♦❡s ♥♦t ✇♦r❦ ❢♦r A(z∗h/g)

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷✶ ✴ ✷✷

❈♦♥❝❧✉s✐♦♥

❙❛♠♣❧✐♥❣ ♠❡t❤♦❞ ❧❡❛❦❛❣❡ ❢✉❧❧ ❛tt❛❝❦❄ ❙✐♠♣❧✐st✐❝ ❬●●❍✶✸❪ A(h/g) ②❡s ✐❢ q ✐s ♣♦❧② ❊①♣♦♥❡♥t✐❛❧ ❬●●❍✶✸❪ A(z∗h/g) ♥♦ ❈♦♥s❡r✈❛t✐✈❡ ❬❉●●+✶✽❪ A(h/g) ♥♦ ❆❣❣r❡ss✐✈❡ ❬❉●●+✶✽❪ A(z∗h/g) ♥♦ ❈♦♠♣❡♥s❛t✐♦♥ ✭t❤✐s ✇♦r❦✮ ✶ ♥♦ ❖♣❡♥ ♣r♦❜❧❡♠s✿ ▼❛❦❡ t❤❡ ❢✉❧❧ ❛tt❛❝❦ ✇♦r❦ ❢♦r t❤❡ ❝♦♥s❡r✈❛t✐✈❡ ♠❡t❤♦❞❄ ■s t❤❡ ❧❡❛❦ ❝r✐t✐❝❛❧❄ ❊①t❡♥❞ t❤❡ s✐♠♣❧❡ s❡tt✐♥❣

■s t❤❡ ❝♦♠♣❡♥s❛t✐♦♥ ♠❡t❤♦❞ st✐❧❧ s❛❢❡ ✐♥ ♦t❤❡r s❡tt✐♥❣s❄

◗✉❡st✐♦♥s❄

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷✷ ✴ ✷✷

❈♦♥❝❧✉s✐♦♥

❙❛♠♣❧✐♥❣ ♠❡t❤♦❞ ❧❡❛❦❛❣❡ ❢✉❧❧ ❛tt❛❝❦❄ ❙✐♠♣❧✐st✐❝ ❬●●❍✶✸❪ A(h/g) ②❡s ✐❢ q ✐s ♣♦❧② ❊①♣♦♥❡♥t✐❛❧ ❬●●❍✶✸❪ A(z∗h/g) ♥♦ ❈♦♥s❡r✈❛t✐✈❡ ❬❉●●+✶✽❪ A(h/g) ♥♦ ❆❣❣r❡ss✐✈❡ ❬❉●●+✶✽❪ A(z∗h/g) ♥♦ ❈♦♠♣❡♥s❛t✐♦♥ ✭t❤✐s ✇♦r❦✮ ✶ ♥♦ ❖♣❡♥ ♣r♦❜❧❡♠s✿ ▼❛❦❡ t❤❡ ❢✉❧❧ ❛tt❛❝❦ ✇♦r❦ ❢♦r t❤❡ ❝♦♥s❡r✈❛t✐✈❡ ♠❡t❤♦❞❄ ■s t❤❡ ❧❡❛❦ ❝r✐t✐❝❛❧❄ ❊①t❡♥❞ t❤❡ s✐♠♣❧❡ s❡tt✐♥❣

■s t❤❡ ❝♦♠♣❡♥s❛t✐♦♥ ♠❡t❤♦❞ st✐❧❧ s❛❢❡ ✐♥ ♦t❤❡r s❡tt✐♥❣s❄

◗✉❡st✐♦♥s❄

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷✷ ✴ ✷✷

❈♦♥❝❧✉s✐♦♥

❙❛♠♣❧✐♥❣ ♠❡t❤♦❞ ❧❡❛❦❛❣❡ ❢✉❧❧ ❛tt❛❝❦❄ ❙✐♠♣❧✐st✐❝ ❬●●❍✶✸❪ A(h/g) ②❡s ✐❢ q ✐s ♣♦❧② ❊①♣♦♥❡♥t✐❛❧ ❬●●❍✶✸❪ A(z∗h/g) ♥♦ ❈♦♥s❡r✈❛t✐✈❡ ❬❉●●+✶✽❪ A(h/g) ♥♦ ❆❣❣r❡ss✐✈❡ ❬❉●●+✶✽❪ A(z∗h/g) ♥♦ ❈♦♠♣❡♥s❛t✐♦♥ ✭t❤✐s ✇♦r❦✮ ✶ ♥♦ ❖♣❡♥ ♣r♦❜❧❡♠s✿ ▼❛❦❡ t❤❡ ❢✉❧❧ ❛tt❛❝❦ ✇♦r❦ ❢♦r t❤❡ ❝♦♥s❡r✈❛t✐✈❡ ♠❡t❤♦❞❄ ■s t❤❡ ❧❡❛❦ A(z∗h/g) ❝r✐t✐❝❛❧❄ ❊①t❡♥❞ t❤❡ s✐♠♣❧❡ s❡tt✐♥❣

■s t❤❡ ❝♦♠♣❡♥s❛t✐♦♥ ♠❡t❤♦❞ st✐❧❧ s❛❢❡ ✐♥ ♦t❤❡r s❡tt✐♥❣s❄

◗✉❡st✐♦♥s❄

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷✷ ✴ ✷✷

❈♦♥❝❧✉s✐♦♥

❙❛♠♣❧✐♥❣ ♠❡t❤♦❞ ❧❡❛❦❛❣❡ ❢✉❧❧ ❛tt❛❝❦❄ ❙✐♠♣❧✐st✐❝ ❬●●❍✶✸❪ A(h/g) ②❡s ✐❢ q ✐s ♣♦❧② ❊①♣♦♥❡♥t✐❛❧ ❬●●❍✶✸❪ A(z∗h/g) ♥♦ ❈♦♥s❡r✈❛t✐✈❡ ❬❉●●+✶✽❪ A(h/g) ♥♦ ❆❣❣r❡ss✐✈❡ ❬❉●●+✶✽❪ A(z∗h/g) ♥♦ ❈♦♠♣❡♥s❛t✐♦♥ ✭t❤✐s ✇♦r❦✮ ✶ ♥♦ ❖♣❡♥ ♣r♦❜❧❡♠s✿ ▼❛❦❡ t❤❡ ❢✉❧❧ ❛tt❛❝❦ ✇♦r❦ ❢♦r t❤❡ ❝♦♥s❡r✈❛t✐✈❡ ♠❡t❤♦❞❄ ■s t❤❡ ❧❡❛❦ A(z∗h/g) ❝r✐t✐❝❛❧❄ ❊①t❡♥❞ t❤❡ s✐♠♣❧❡ s❡tt✐♥❣

◮ ■s t❤❡ ❝♦♠♣❡♥s❛t✐♦♥ ♠❡t❤♦❞ st✐❧❧ s❛❢❡ ✐♥ ♦t❤❡r s❡tt✐♥❣s❄

◗✉❡st✐♦♥s❄

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷✷ ✴ ✷✷

❈♦♥❝❧✉s✐♦♥

❙❛♠♣❧✐♥❣ ♠❡t❤♦❞ ❧❡❛❦❛❣❡ ❢✉❧❧ ❛tt❛❝❦❄ ❙✐♠♣❧✐st✐❝ ❬●●❍✶✸❪ A(h/g) ②❡s ✐❢ q ✐s ♣♦❧② ❊①♣♦♥❡♥t✐❛❧ ❬●●❍✶✸❪ A(z∗h/g) ♥♦ ❈♦♥s❡r✈❛t✐✈❡ ❬❉●●+✶✽❪ A(h/g) ♥♦ ❆❣❣r❡ss✐✈❡ ❬❉●●+✶✽❪ A(z∗h/g) ♥♦ ❈♦♠♣❡♥s❛t✐♦♥ ✭t❤✐s ✇♦r❦✮ ✶ ♥♦ ❖♣❡♥ ♣r♦❜❧❡♠s✿ ▼❛❦❡ t❤❡ ❢✉❧❧ ❛tt❛❝❦ ✇♦r❦ ❢♦r t❤❡ ❝♦♥s❡r✈❛t✐✈❡ ♠❡t❤♦❞❄ ■s t❤❡ ❧❡❛❦ A(z∗h/g) ❝r✐t✐❝❛❧❄ ❊①t❡♥❞ t❤❡ s✐♠♣❧❡ s❡tt✐♥❣

◮ ■s t❤❡ ❝♦♠♣❡♥s❛t✐♦♥ ♠❡t❤♦❞ st✐❧❧ s❛❢❡ ✐♥ ♦t❤❡r s❡tt✐♥❣s❄

◗✉❡st✐♦♥s❄

❆✳ P❡❧❧❡t✲▼❛r② ❙t❛t✐st✐❝❛❧ ▲❡❛❦ ♦❢ ●●❍✶✸ ♠❛♣ ❆s✐❛❝r②♣t ✷✵✶✽ ✷✷ ✴ ✷✷