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■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t② ❖❜❢✉s❝❛t✐♦♥ ❲✐t❤♦✉t ▼❛♣s✿ ❆tt❛❝❦s ❛♥❞ ❋✐①❡s ❢♦r ◆♦✐s② ▲✐♥❡❛r ❋❊

❙❤✇❡t❛ ❆❣r❛✇❛❧✶✱ ❆❧✐❝❡ P❡❧❧❡t✲▼❛r②✷

✶ ■■❚ ▼❛❞r❛s✱ ✷ ❑❯ ▲❡✉✈❡♥

❊✉r♦❝r②♣t ✷✵✷✵ ❤tt♣s✿✴✴❡♣r✐♥t✳✐❛❝r✳♦r❣✴✷✵✷✵✴✹✶✺✳♣❞❢

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶ ✴ ✶✾

❲❤❛t ✐s t❤✐s t❛❧❦ ❛❜♦✉t❄

❈r②♣t❛♥❛❧②t✐❝ st✉❞② ♦❢ ❛♥ ✐❖ ❝♦♥str✉❝t✐♦♥ ❬❆❣r✶✾❪✳ ✷ ❛tt❛❝❦s ✶ r❡♣❛✐r❡❞ ❝♦♥str✉❝t✐♦♥

❬❆❣r✶✾❪ ❙✳ ❆❣r❛✇❛❧✳ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t② ♦❜❢✉s❝❛t✐♦♥ ✇✐t❤♦✉t ♠✉❧t✐❧✐♥❡❛r ♠❛♣s✿ ◆❡✇ t❡❝❤♥✐q✉❡s ❢♦r ❜♦♦tstr❛♣♣✐♥❣ ❛♥❞ ✐♥st❛♥t✐❛t✐♦♥✳ ❊✉r♦❝r②♣t✳ ❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✷ ✴ ✶✾

❲❤❛t ✐s t❤✐s t❛❧❦ ❛❜♦✉t❄

❈r②♣t❛♥❛❧②t✐❝ st✉❞② ♦❢ ❛♥ ✐❖ ❝♦♥str✉❝t✐♦♥ ❬❆❣r✶✾❪✳ ⇒ ✷ ❛tt❛❝❦s ⇒ ✶ r❡♣❛✐r❡❞ ❝♦♥str✉❝t✐♦♥

❬❆❣r✶✾❪ ❙✳ ❆❣r❛✇❛❧✳ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t② ♦❜❢✉s❝❛t✐♦♥ ✇✐t❤♦✉t ♠✉❧t✐❧✐♥❡❛r ♠❛♣s✿ ◆❡✇ t❡❝❤♥✐q✉❡s ❢♦r ❜♦♦tstr❛♣♣✐♥❣ ❛♥❞ ✐♥st❛♥t✐❛t✐♦♥✳ ❊✉r♦❝r②♣t✳ ❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✷ ✴ ✶✾

❖❜❢✉s❝❛t✐♦♥

✐❖ ✐s ✏❝r②♣t♦✲❝♦♠♣❧❡t❡✑✿ ✐♠♣❧✐❡s ✇✐t♥❡ss ❡♥❝r②♣t✐♦♥✱ ❢✉♥❝t✐♦♥❛❧ ❡♥❝r②♣t✐♦♥✱ ❞❡♥✐❛❜❧❡ ❡♥❝r②♣t✐♦♥✱ ♦❜❧✐✈✐♦✉s tr❛♥s❢❡r✱ tr❛✐t♦r tr❛❝✐♥❣✱ ♠✉❧t✐❧✐♥❡❛r ♠❛♣s✳✳✳ ❚✇♦ ♠❛✐♥ ❛♣♣r♦❛❝❤❡s t♦ ❜✉✐❧❞ ❝❛♥❞✐❞❛t❡ ✐❖✿ ❉✐r❡❝t ❝♦♥str✉❝t✐♦♥s

◮ ✉s✐♥❣ ♠✉❧t✐❧✐♥❡❛r ♠❛♣s

❇♦♦tstr❛♣♣✐♥❣ ❛♣♣r♦❛❝❤❡s

◮ r❡❞✉❝t✐♦♥ t♦ ✇❡❛❦ ❢♦r♠s ♦❢ ❢✉♥❝t✐♦♥❛❧ ❡♥❝r②♣t✐♦♥ ❘❡❢❡r❡♥❝❡s ❝❛♥ ❜❡ ❢♦✉♥❞ ❛t ❤tt♣s✿✴✴t❡❧✳❛r❝❤✐✈❡s✲♦✉✈❡rt❡s✳❢r✴t❡❧✲✵✷✸✸✼✾✸✵✴❞♦❝✉♠❡♥t✱ ♣❛❣❡ ✶✵✼ ❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✸ ✴ ✶✾

❖❜❢✉s❝❛t✐♦♥

✐❖ ✐s ✏❝r②♣t♦✲❝♦♠♣❧❡t❡✑✿ ✐♠♣❧✐❡s ✇✐t♥❡ss ❡♥❝r②♣t✐♦♥✱ ❢✉♥❝t✐♦♥❛❧ ❡♥❝r②♣t✐♦♥✱ ❞❡♥✐❛❜❧❡ ❡♥❝r②♣t✐♦♥✱ ♦❜❧✐✈✐♦✉s tr❛♥s❢❡r✱ tr❛✐t♦r tr❛❝✐♥❣✱ ♠✉❧t✐❧✐♥❡❛r ♠❛♣s✳✳✳ ❚✇♦ ♠❛✐♥ ❛♣♣r♦❛❝❤❡s t♦ ❜✉✐❧❞ ❝❛♥❞✐❞❛t❡ ✐❖✿ ❉✐r❡❝t ❝♦♥str✉❝t✐♦♥s

◮ ✉s✐♥❣ ♠✉❧t✐❧✐♥❡❛r ♠❛♣s

❇♦♦tstr❛♣♣✐♥❣ ❛♣♣r♦❛❝❤❡s

◮ r❡❞✉❝t✐♦♥ t♦ ✇❡❛❦ ❢♦r♠s ♦❢ ❢✉♥❝t✐♦♥❛❧ ❡♥❝r②♣t✐♦♥ ❘❡❢❡r❡♥❝❡s ❝❛♥ ❜❡ ❢♦✉♥❞ ❛t ❤tt♣s✿✴✴t❡❧✳❛r❝❤✐✈❡s✲♦✉✈❡rt❡s✳❢r✴t❡❧✲✵✷✸✸✼✾✸✵✴❞♦❝✉♠❡♥t✱ ♣❛❣❡ ✶✵✼ ❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✸ ✴ ✶✾

❆❣r❛✇❛❧✬s ❝♦♥str✉❝t✐♦♥ ♦❢ ✐❖

✐❖ ▲❲❊ ◆▲✐♥❋❊

◆▲✐♥❋❊✿ ◆♦✐s② ▲✐♥❡❛r ❋✉♥❝t✐♦♥❛❧ ❊♥❝r②♣t✐♦♥

❛♥❞

❬❆❘✶✼✱❆❣r✶✾✱❏▲❙✶✾❪

P❛✐r✐♥❣s ✰ ✇❡❛❦ ❛♥❞ ❧❡❛❦② ▲❲❊ ♦r

❬❆❣r✶✾❪ ❬❆❏▲▼❙✶✾✱❏▲❙✶✾❪

❬❆❘✶✼❪ ❙✳ ❆❣r❛✇❛❧ ❛♥❞ ❆✳ ❘♦s❡♥✳ ❋✉♥❝t✐♦♥❛❧ ❡♥❝r②♣t✐♦♥ ❢♦r ❜♦✉♥❞❡❞ ❝♦❧❧✉s✐♦♥s✱ r❡✈✐s✐t❡❞✳ ❚❈❈✳ ❬❆❣r✶✾❪ ❙✳ ❆❣r❛✇❛❧✳ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t② ♦❜❢✉s❝❛t✐♦♥ ✇✐t❤♦✉t ♠✉❧t✐❧✐♥❡❛r ♠❛♣s✿ ◆❡✇ t❡❝❤♥✐q✉❡s ❢♦r ❜♦♦tstr❛♣♣✐♥❣ ❛♥❞ ✐♥st❛♥t✐❛t✐♦♥✳ ❊✉r♦❝r②♣t✳ ❬❏▲❙✶✾❪ ❆✳ ❏❛✐♥ ❛♥❞ ❍✳ ▲✐♥ ❛♥❞ ❆✳ ❙❛❤❛✐✳ ❙✐♠♣❧✐❢②✐♥❣ ❈♦♥str✉❝t✐♦♥s ❛♥❞ ❆ss✉♠♣t✐♦♥s ❢♦r ✐❖✳ ❡Pr✐♥t✳ ❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✹ ✴ ✶✾

❆❣r❛✇❛❧✬s ❝♦♥str✉❝t✐♦♥ ♦❢ ✐❖

✐❖ ▲❲❊ ◆▲✐♥❋❊

◆▲✐♥❋❊✿ ◆♦✐s② ▲✐♥❡❛r ❋✉♥❝t✐♦♥❛❧ ❊♥❝r②♣t✐♦♥

❛♥❞

❬❆❘✶✼✱❆❣r✶✾✱❏▲❙✶✾❪

❉✐r❡❝t ❝♦♥str✉❝t✐♦♥ P❛✐r✐♥❣s ✰ ✇❡❛❦ ❛♥❞ ❧❡❛❦② ▲❲❊ ♦r

❬❆❣r✶✾❪ ❬❆❏▲▼❙✶✾✱❏▲❙✶✾❪

❬❆❣r✶✾❪ ❙✳ ❆❣r❛✇❛❧✳ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t② ♦❜❢✉s❝❛t✐♦♥ ✇✐t❤♦✉t ♠✉❧t✐❧✐♥❡❛r ♠❛♣s✿ ◆❡✇ t❡❝❤♥✐q✉❡s ❢♦r ❜♦♦tstr❛♣♣✐♥❣ ❛♥❞ ✐♥st❛♥t✐❛t✐♦♥✳ ❊✉r♦❝r②♣t✳ ❬❆❏▲▼❙✶✾❪ P✳ ❆♥❛♥t❤✱ ❆✳ ❏❛✐♥✱ ❍✳ ▲✐♥✱ ❈✳ ▼❛tt ❛♥❞ ❆✳ ❙❛❤❛✐✳ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t② ❖❜❢✉s❝❛t✐♦♥ ❲✐t❤♦✉t ▼✉❧t✐❧✐♥❡❛r ▼❛♣s✿ ◆❡✇ P❛r❛❞✐❣♠s ✈✐❛ ▲♦✇ ❉❡❣r❡❡ ❲❡❛❦ Ps❡✉❞♦r❛♥❞♦♠♥❡ss ❛♥❞ ❙❡❝✉r✐t② ❆♠♣❧✐✜❝❛t✐♦♥✳ ❈r②♣t♦✳ ❬❏▲❙✶✾❪ ❆✳ ❏❛✐♥✱ ❍✳ ▲✐♥ ❛♥❞ ❆✳ ❙❛❤❛✐✳ ❙✐♠♣❧✐❢②✐♥❣ ❈♦♥str✉❝t✐♦♥s ❛♥❞ ❆ss✉♠♣t✐♦♥s ❢♦r ✐❖✳ ❡Pr✐♥t✳ ❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✹ ✴ ✶✾

❆❣r❛✇❛❧✬s ❝♦♥str✉❝t✐♦♥ ♦❢ ✐❖

✐❖ ▲❲❊ ◆▲✐♥❋❊

◆▲✐♥❋❊✿ ◆♦✐s② ▲✐♥❡❛r ❋✉♥❝t✐♦♥❛❧ ❊♥❝r②♣t✐♦♥

❛♥❞

❬❆❘✶✼✱❆❣r✶✾✱❏▲❙✶✾❪

❉✐r❡❝t ❝♦♥str✉❝t✐♦♥ P❛✐r✐♥❣s ✰ ✇❡❛❦ ❛♥❞ ❧❡❛❦② ▲❲❊ ♦r

❬❆❣r✶✾❪ ❬❆❏▲▼❙✶✾✱❏▲❙✶✾❪

❬❆❣r✶✾❪ ❙✳ ❆❣r❛✇❛❧✳ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t② ♦❜❢✉s❝❛t✐♦♥ ✇✐t❤♦✉t ♠✉❧t✐❧✐♥❡❛r ♠❛♣s✿ ◆❡✇ t❡❝❤♥✐q✉❡s ❢♦r ❜♦♦tstr❛♣♣✐♥❣ ❛♥❞ ✐♥st❛♥t✐❛t✐♦♥✳ ❊✉r♦❝r②♣t✳ ❬❆❏▲▼❙✶✾❪ P✳ ❆♥❛♥t❤✱ ❆✳ ❏❛✐♥✱ ❍✳ ▲✐♥✱ ❈✳ ▼❛tt ❛♥❞ ❆✳ ❙❛❤❛✐✳ ■♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t② ❖❜❢✉s❝❛t✐♦♥ ❲✐t❤♦✉t ▼✉❧t✐❧✐♥❡❛r ▼❛♣s✿ ◆❡✇ P❛r❛❞✐❣♠s ✈✐❛ ▲♦✇ ❉❡❣r❡❡ ❲❡❛❦ Ps❡✉❞♦r❛♥❞♦♠♥❡ss ❛♥❞ ❙❡❝✉r✐t② ❆♠♣❧✐✜❝❛t✐♦♥✳ ❈r②♣t♦✳ ❬❏▲❙✶✾❪ ❆✳ ❏❛✐♥✱ ❍✳ ▲✐♥ ❛♥❞ ❆✳ ❙❛❤❛✐✳ ❙✐♠♣❧✐❢②✐♥❣ ❈♦♥str✉❝t✐♦♥s ❛♥❞ ❆ss✉♠♣t✐♦♥s ❢♦r ✐❖✳ ❡Pr✐♥t✳ ❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✹ ✴ ✶✾

◆▲✐♥❋❊

❆✉t❤♦r✐t② ❯s❡r ❝t

x ← ❊♥❝(▼❙❑,

x) s❦

z ← ❑❡②●❡♥(▼❙❑,

z) ❉❡❝(s❦

z, ❝t x) =

z, x ⇒ ❤✐❞❡s ❡✈❡r②t❤✐♥❣ ❡①❝❡♣t z, x ❉❡❝ s❦ ❝t ♥♦✐s❡ ❤✐❞❡s ❡✈❡r②t❤✐♥❣ ❡①❝❡♣t ❤✐❞❡s t❤❡ ❧❛st ❜✐ts ♦❢ ▲✐♥❋❊ ◆▲✐♥❋❊

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✺ ✴ ✶✾

◆▲✐♥❋❊

❆✉t❤♦r✐t② ❯s❡r ❝t

x ← ❊♥❝(▼❙❑,

x) s❦

z ← ❑❡②●❡♥(▼❙❑,

z) ❉❡❝ s❦ ❝t ❤✐❞❡s ❡✈❡r②t❤✐♥❣ ❡①❝❡♣t ❉❡❝(s❦

z, ❝t x) =

z, x + ♥♦✐s❡ ⇒ ❤✐❞❡s ❡✈❡r②t❤✐♥❣ ❡①❝❡♣t ≈ z, x ⇒ ❤✐❞❡s t❤❡ ❧❛st ❜✐ts ♦❢ z, x ▲✐♥❋❊ ◆▲✐♥❋❊

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✺ ✴ ✶✾

Pr❡✈✐♦✉s ✇♦r❦ ❛♥❞ ❝♦♥tr✐❜✉t✐♦♥s

❬❆❣r✶✾❪✿ ♣r♦✈❡❞ ❤❡r ❝♦♥str✉❝t✐♦♥ s❡❝✉r❡ ✐♥ ❛ ✇❡❛❦ ♠♦❞❡❧ ✭✉♥❞❡r ♥♦♥ st❛♥❞❛r❞ ❛ss✉♠♣t✐♦♥s✮ ✐❢ ♦♥❧② ♦♥❡ ❝✐♣❤❡rt❡①t ❛✈❛✐❧❛❜❧❡ t♦ t❤❡ ❛tt❛❝❦❡r ❖✉r ❝♦♥tr✐❜✉t✐♦♥✿ ♠♦r❡ ❝r②♣t❛♥❛❧②s✐s ❚✇♦ ❛tt❛❝❦s ✭✉s✐♥❣ ♠✉❧t✐♣❧❡ ❝✐♣❤❡rt❡①ts✮

♠✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦ r❛♥❦ ❛tt❛❝❦

❆ ✜①❡❞ ❝♦♥str✉❝t✐♦♥

♣r❡✈❡♥ts t❤❡ t✇♦ ❛tt❛❝❦s ✇❡ ❛❧s♦ st✉❞② ♦t❤❡r ♣♦ss✐❜❧❡ ❛tt❛❝❦s ♣r♦♣♦s❡ ♣❛r❛♠❡t❡rs s❡tt✐♥❣ ✇❤✐❝❤ ✇❡ ❜❡❧✐❡✈❡ ✐s s❡❝✉r❡ ✭❡✈❡♥ q✉❛♥t✉♠❧②✮

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✻ ✴ ✶✾

Pr❡✈✐♦✉s ✇♦r❦ ❛♥❞ ❝♦♥tr✐❜✉t✐♦♥s

❬❆❣r✶✾❪✿ ♣r♦✈❡❞ ❤❡r ❝♦♥str✉❝t✐♦♥ s❡❝✉r❡ ✐♥ ❛ ✇❡❛❦ ♠♦❞❡❧ ✭✉♥❞❡r ♥♦♥ st❛♥❞❛r❞ ❛ss✉♠♣t✐♦♥s✮ ✐❢ ♦♥❧② ♦♥❡ ❝✐♣❤❡rt❡①t ❛✈❛✐❧❛❜❧❡ t♦ t❤❡ ❛tt❛❝❦❡r ❖✉r ❝♦♥tr✐❜✉t✐♦♥✿ ♠♦r❡ ❝r②♣t❛♥❛❧②s✐s ❚✇♦ ❛tt❛❝❦s ✭✉s✐♥❣ ♠✉❧t✐♣❧❡ ❝✐♣❤❡rt❡①ts✮

◮ ♠✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦ ◮ r❛♥❦ ❛tt❛❝❦

❆ ✜①❡❞ ❝♦♥str✉❝t✐♦♥

♣r❡✈❡♥ts t❤❡ t✇♦ ❛tt❛❝❦s ✇❡ ❛❧s♦ st✉❞② ♦t❤❡r ♣♦ss✐❜❧❡ ❛tt❛❝❦s ♣r♦♣♦s❡ ♣❛r❛♠❡t❡rs s❡tt✐♥❣ ✇❤✐❝❤ ✇❡ ❜❡❧✐❡✈❡ ✐s s❡❝✉r❡ ✭❡✈❡♥ q✉❛♥t✉♠❧②✮

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✻ ✴ ✶✾

Pr❡✈✐♦✉s ✇♦r❦ ❛♥❞ ❝♦♥tr✐❜✉t✐♦♥s

❬❆❣r✶✾❪✿ ♣r♦✈❡❞ ❤❡r ❝♦♥str✉❝t✐♦♥ s❡❝✉r❡ ✐♥ ❛ ✇❡❛❦ ♠♦❞❡❧ ✭✉♥❞❡r ♥♦♥ st❛♥❞❛r❞ ❛ss✉♠♣t✐♦♥s✮ ✐❢ ♦♥❧② ♦♥❡ ❝✐♣❤❡rt❡①t ❛✈❛✐❧❛❜❧❡ t♦ t❤❡ ❛tt❛❝❦❡r ❖✉r ❝♦♥tr✐❜✉t✐♦♥✿ ♠♦r❡ ❝r②♣t❛♥❛❧②s✐s ❚✇♦ ❛tt❛❝❦s ✭✉s✐♥❣ ♠✉❧t✐♣❧❡ ❝✐♣❤❡rt❡①ts✮

◮ ♠✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦ ◮ r❛♥❦ ❛tt❛❝❦

❆ ✜①❡❞ ❝♦♥str✉❝t✐♦♥

◮ ♣r❡✈❡♥ts t❤❡ t✇♦ ❛tt❛❝❦s ◮ ✇❡ ❛❧s♦ st✉❞② ♦t❤❡r ♣♦ss✐❜❧❡ ❛tt❛❝❦s ◮ ♣r♦♣♦s❡ ♣❛r❛♠❡t❡rs s❡tt✐♥❣ ✇❤✐❝❤ ✇❡ ❜❡❧✐❡✈❡ ✐s s❡❝✉r❡ ✭❡✈❡♥ q✉❛♥t✉♠❧②✮ ❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✻ ✴ ✶✾

❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦

✶

▼✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦

✷

❘❛♥❦ ❛tt❛❝❦

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✼ ✴ ✶✾

❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦

✶

▼✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦

✷

❘❛♥❦ ❛tt❛❝❦

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✽ ✴ ✶✾

❘▲❲❊ ✇✐t❤ ❝♦rr❡❧❛t❡❞ ♥♦✐s❡

◆♦t❛t✐♦♥s

❊✈❡r②t❤✐♥❣ ✐♥ Rq = Zq[X]/(X n + ✶) ❜❧✉❡✿ s♠❛❧❧

◆❚❘❯

✉♥✐❢ ▼✉❧t✐♣❧❡✲s♠❛❧❧✲s❡❝r❡ts ❘▲❲❊✿ ❉✐st✐♥❣✉✐s❤ ✉♥✐❢♦r♠ ✐♥ ❢r♦♠ ❬❆❣r✶✾❪✬s ❝♦♥str✉❝t✐♦♥ ♥❡❡❞s ♠✉❧t✐♣❧✐❝❛t✐✈✐t② ♦❢ t❤❡ ❝✐♣❤❡rt❡①ts

t♦♦ ❧❛r❣❡ s♠❛❧❧

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✾ ✴ ✶✾

❘▲❲❊ ✇✐t❤ ❝♦rr❡❧❛t❡❞ ♥♦✐s❡

◆♦t❛t✐♦♥s

❊✈❡r②t❤✐♥❣ ✐♥ Rq = Zq[X]/(X n + ✶) ❜❧✉❡✿ s♠❛❧❧

◆❚❘❯

✉♥✐❢ ▼✉❧t✐♣❧❡✲s♠❛❧❧✲s❡❝r❡ts ❘▲❲❊✿ ❉✐st✐♥❣✉✐s❤ ✉♥✐❢♦r♠ ✐♥ Rq ❢r♦♠ f g

• ai , bij = aisj + eij mod q
• i,j

❬❆❣r✶✾❪✬s ❝♦♥str✉❝t✐♦♥ ♥❡❡❞s ♠✉❧t✐♣❧✐❝❛t✐✈✐t② ♦❢ t❤❡ ❝✐♣❤❡rt❡①ts

t♦♦ ❧❛r❣❡ s♠❛❧❧

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✾ ✴ ✶✾

❘▲❲❊ ✇✐t❤ ❝♦rr❡❧❛t❡❞ ♥♦✐s❡

◆♦t❛t✐♦♥s

❊✈❡r②t❤✐♥❣ ✐♥ Rq = Zq[X]/(X n + ✶) ❜❧✉❡✿ s♠❛❧❧

◆❚❘❯

✉♥✐❢ ▼✉❧t✐♣❧❡✲s♠❛❧❧✲s❡❝r❡ts ❘▲❲❊✿ ❉✐st✐♥❣✉✐s❤ ✉♥✐❢♦r♠ ✐♥ Rq ❢r♦♠ f g

• ai , bij = aisj + eij mod q
• i,j

❬❆❣r✶✾❪✬s ❝♦♥str✉❝t✐♦♥ ♥❡❡❞s ♠✉❧t✐♣❧✐❝❛t✐✈✐t② ♦❢ t❤❡ ❝✐♣❤❡rt❡①ts bijbkℓ = aiak

• a′

· sjsℓ

• s′

+ aisj · ekℓ + aksℓ · eij

• t♦♦ ❧❛r❣❡

+ eijekℓ

s♠❛❧❧

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✾ ✴ ✶✾

❘▲❲❊ ✇✐t❤ ❝♦rr❡❧❛t❡❞ ♥♦✐s❡

◆♦t❛t✐♦♥s

❊✈❡r②t❤✐♥❣ ✐♥ Rq = Zq[X]/(X n + ✶) ❜❧✉❡✿ s♠❛❧❧

◆❚❘❯

fi g mod q ≈c ✉♥✐❢ ❘▲❲❊ ✇✐t❤ ❝♦rr❡❧❛t❡❞ ♥♦✐s❡✿ ❉✐st✐♥❣✉✐s❤ ✉♥✐❢♦r♠ ✐♥ Rq ❢r♦♠ f g

• ai = fi

g , bij = aisj + g · eij mod q

• i,j

❬❆❣r✶✾❪✬s ❝♦♥str✉❝t✐♦♥ ♥❡❡❞s ♠✉❧t✐♣❧✐❝❛t✐✈✐t② ♦❢ t❤❡ ❝✐♣❤❡rt❡①ts bijbkℓ = aiak

• a′

· sjsℓ

• s′

+ aisj · gekℓ + aksℓ · geij

• t♦♦ ❧❛r❣❡

+ g✷eijekℓ

s♠❛❧❧

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✾ ✴ ✶✾

❘▲❲❊ ✇✐t❤ ❝♦rr❡❧❛t❡❞ ♥♦✐s❡

◆♦t❛t✐♦♥s

❊✈❡r②t❤✐♥❣ ✐♥ Rq = Zq[X]/(X n + ✶) ❜❧✉❡✿ s♠❛❧❧

◆❚❘❯

fi g mod q ≈c ✉♥✐❢ ❘▲❲❊ ✇✐t❤ ❝♦rr❡❧❛t❡❞ ♥♦✐s❡✿ ❉✐st✐♥❣✉✐s❤ ✉♥✐❢♦r♠ ✐♥ Rq ❢r♦♠ f g

• ai = fi

g , bij = aisj + g · eij mod q

• i,j

❬❆❣r✶✾❪✬s ❝♦♥str✉❝t✐♦♥ ♥❡❡❞s ♠✉❧t✐♣❧✐❝❛t✐✈✐t② ♦❢ t❤❡ ❝✐♣❤❡rt❡①ts bijbkℓ = aiak

• a′

· sjsℓ

• s′

+ fisj · ekℓ + fksℓ · eij

• s♠❛❧❧

+ g✷eijekℓ

s♠❛❧❧

✓

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✾ ✴ ✶✾

❙✐♠♣❧❡ ❞✐st✐♥❣✉✐s❤✐♥❣ ❛tt❛❝❦

■♥♣✉t✿ (a✶ = f✶

g , b✶ = a✶s + ge✶)

✭✷ ❧❛❜❡❧s✱ ✶ s❡❝r❡t✮

(a✷ = f✷

g , b✷ = a✷s + ge✷)

❆tt❛❝❦✿

❧ ✶ ✷ ✷ ✶ ✶ ✷ ✶ ✷ ✷ ✶ ✷ ✶

❝❛♥ ❜❡ ❞✐st✐♥❣✉✐s❤❡❞ ❢r♦♠ ✉♥✐❢♦r♠ ❋✐①✿ t❤❡ ✬s ♥❡❡❞ ♥♦t ❜❡ ♣✉❜❧✐❝

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✵ ✴ ✶✾

❙✐♠♣❧❡ ❞✐st✐♥❣✉✐s❤✐♥❣ ❛tt❛❝❦

■♥♣✉t✿ (a✶ = f✶

g , b✶ = a✶s + ge✶)

✭✷ ❧❛❜❡❧s✱ ✶ s❡❝r❡t✮

(a✷ = f✷

g , b✷ = a✷s + ge✷)

❆tt❛❝❦✿ f

• ❧

a✶b✷ − a✷b✶ = a✶a✷s + a✶ge✷ − a✷a✶s − a✷ge✶ ❝❛♥ ❜❡ ❞✐st✐♥❣✉✐s❤❡❞ ❢r♦♠ ✉♥✐❢♦r♠ ❋✐①✿ t❤❡ ✬s ♥❡❡❞ ♥♦t ❜❡ ♣✉❜❧✐❝

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✵ ✴ ✶✾

❙✐♠♣❧❡ ❞✐st✐♥❣✉✐s❤✐♥❣ ❛tt❛❝❦

■♥♣✉t✿ (a✶ = f✶

g , b✶ = a✶s + ge✶)

✭✷ ❧❛❜❡❧s✱ ✶ s❡❝r❡t✮

(a✷ = f✷

g , b✷ = a✷s + ge✷)

❆tt❛❝❦✿ f

• ❧

a✶b✷ − a✷b✶ = a✶a✷s + a✶ge✷ − a✷a✶s − a✷ge✶ ❝❛♥ ❜❡ ❞✐st✐♥❣✉✐s❤❡❞ ❢r♦♠ ✉♥✐❢♦r♠ ❋✐①✿ t❤❡ ✬s ♥❡❡❞ ♥♦t ❜❡ ♣✉❜❧✐❝

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✵ ✴ ✶✾

❙✐♠♣❧❡ ❞✐st✐♥❣✉✐s❤✐♥❣ ❛tt❛❝❦

■♥♣✉t✿ (a✶ = f✶

g , b✶ = a✶s + ge✶)

✭✷ ❧❛❜❡❧s✱ ✶ s❡❝r❡t✮

(a✷ = f✷

g , b✷ = a✷s + ge✷)

❆tt❛❝❦✿ f

• ❧

a✶b✷ − a✷b✶ = f✶e✷ − f✷e✶

• s♠❛❧❧

❝❛♥ ❜❡ ❞✐st✐♥❣✉✐s❤❡❞ ❢r♦♠ ✉♥✐❢♦r♠ ❋✐①✿ t❤❡ ✬s ♥❡❡❞ ♥♦t ❜❡ ♣✉❜❧✐❝

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✵ ✴ ✶✾

❙✐♠♣❧❡ ❞✐st✐♥❣✉✐s❤✐♥❣ ❛tt❛❝❦

■♥♣✉t✿ (a✶ = f✶

g , b✶ = a✶s + ge✶)

✭✷ ❧❛❜❡❧s✱ ✶ s❡❝r❡t✮

(a✷ = f✷

g , b✷ = a✷s + ge✷)

❆tt❛❝❦✿ f

• ❧

a✶b✷ − a✷b✶ = f✶e✷ − f✷e✶

• s♠❛❧❧

⇒ ❝❛♥ ❜❡ ❞✐st✐♥❣✉✐s❤❡❞ ❢r♦♠ ✉♥✐❢♦r♠ ❋✐①✿ t❤❡ ✬s ♥❡❡❞ ♥♦t ❜❡ ♣✉❜❧✐❝

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✵ ✴ ✶✾

❙✐♠♣❧❡ ❞✐st✐♥❣✉✐s❤✐♥❣ ❛tt❛❝❦

■♥♣✉t✿ (a✶ = f✶

g , b✶ = a✶s + ge✶)

✭✷ ❧❛❜❡❧s✱ ✶ s❡❝r❡t✮

(a✷ = f✷

g , b✷ = a✷s + ge✷)

❆tt❛❝❦✿ f

• ❧

a✶b✷ − a✷b✶ = f✶e✷ − f✷e✶

• s♠❛❧❧

⇒ ❝❛♥ ❜❡ ❞✐st✐♥❣✉✐s❤❡❞ ❢r♦♠ ✉♥✐❢♦r♠ ❋✐①✿ t❤❡ ai✬s ♥❡❡❞ ♥♦t ❜❡ ♣✉❜❧✐❝

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✵ ✴ ✶✾

▼✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦

■♥♣✉t✿ b✶✶ = a✶s✶ + ge✶✶

✶✷ ✶ ✷ ✶✷

✭a✶ = f✶

g ✮

✭✷ ❧❛❜❡❧s✱ ✶ s❡❝r❡t s ✮

b✷✶ = a✷s✶ + ge✷✶

✷✷ ✷ ✷ ✷✷

✭a✷ = f✷

g ✮

❆tt❛❝❦✿

✶✶ ✶✷ ✷✶ ✷✷ ✶ ✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷ ✶ ✶ ✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷ ✶

r❛♥❦ ✶ r❛♥❦ ✷

❞❡t

✶

✷ ❞❡t

❞❡t

✶ ✷

❞❡t

✶ ✷ ✷ ❞❡t ✵ s♠❛❧❧

❝❛♥ ❜❡ ❞✐st✐♥❣✉✐s❤❡❞ ❢r♦♠ ✉♥✐❢♦r♠

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✶ ✴ ✶✾

▼✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦

■♥♣✉t✿ b✶✶ = a✶s✶ + ge✶✶ b✶✷ = a✶s✷ + ge✶✷

✭a✶ = f✶

g ✮

✭✷ ❧❛❜❡❧s✱ ✷ s❡❝r❡ts✮

b✷✶ = a✷s✶ + ge✷✶ b✷✷ = a✷s✷ + ge✷✷

✭a✷ = f✷

g ✮

❆tt❛❝❦✿

✶✶ ✶✷ ✷✶ ✷✷ ✶ ✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷ ✶ ✶ ✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷ ✶

r❛♥❦ ✶ r❛♥❦ ✷

❞❡t

✶

✷ ❞❡t

❞❡t

✶ ✷

❞❡t

✶ ✷ ✷ ❞❡t ✵ s♠❛❧❧

❝❛♥ ❜❡ ❞✐st✐♥❣✉✐s❤❡❞ ❢r♦♠ ✉♥✐❢♦r♠

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✶ ✴ ✶✾

▼✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦

■♥♣✉t✿ b✶✶ = a✶s✶ + ge✶✶ b✶✷ = a✶s✷ + ge✶✷

✭a✶ = f✶

g ✮

✭✷ ❧❛❜❡❧s✱ ✷ s❡❝r❡ts✮

b✷✶ = a✷s✶ + ge✷✶ b✷✷ = a✷s✷ + ge✷✷

✭a✷ = f✷

g ✮

❆tt❛❝❦✿ B = b✶✶ b✶✷ b✷✶ b✷✷

• ✶

✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷ ✶ ✶ ✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷ ✶

r❛♥❦ ✶ r❛♥❦ ✷

❞❡t

✶

✷ ❞❡t

❞❡t

✶ ✷

❞❡t

✶ ✷ ✷ ❞❡t ✵ s♠❛❧❧

❝❛♥ ❜❡ ❞✐st✐♥❣✉✐s❤❡❞ ❢r♦♠ ✉♥✐❢♦r♠

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✶ ✴ ✶✾

▼✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦

■♥♣✉t✿ b✶✶ = a✶s✶ + ge✶✶ b✶✷ = a✶s✷ + ge✶✷

✭a✶ = f✶

g ✮

✭✷ ❧❛❜❡❧s✱ ✷ s❡❝r❡ts✮

b✷✶ = a✷s✶ + ge✷✶ b✷✷ = a✷s✷ + ge✷✷

✭a✷ = f✷

g ✮

❆tt❛❝❦✿ B

✶✶ ✶✷ ✷✶ ✷✷

= a✶ a✷ s✶ s✷

• + g ·

e✶✶ e✶✷ e✷✶ e✷✷

• ✶

✶ ✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷ ✶

r❛♥❦ ✶ r❛♥❦ ✷

❞❡t

✶

✷ ❞❡t

❞❡t

✶ ✷

❞❡t

✶ ✷ ✷ ❞❡t ✵ s♠❛❧❧

❝❛♥ ❜❡ ❞✐st✐♥❣✉✐s❤❡❞ ❢r♦♠ ✉♥✐❢♦r♠

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✶ ✴ ✶✾

▼✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦

■♥♣✉t✿ b✶✶ = a✶s✶ + ge✶✶ b✶✷ = a✶s✷ + ge✶✷

✭a✶ = f✶

g ✮

✭✷ ❧❛❜❡❧s✱ ✷ s❡❝r❡ts✮

b✷✶ = a✷s✶ + ge✷✶ b✷✷ = a✷s✷ + ge✷✷

✭a✷ = f✷

g ✮

❆tt❛❝❦✿ B

✶✶ ✶✷ ✷✶ ✷✷ ✶ ✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷

= ✶

g ·

f✶ f✷ s✶ s✷

• + g ·

e✶✶ e✶✷ e✷✶ e✷✷

• ✶

r❛♥❦ ✶ r❛♥❦ ✷

❞❡t

✶

✷ ❞❡t

❞❡t

✶ ✷

❞❡t

✶ ✷ ✷ ❞❡t ✵ s♠❛❧❧

❝❛♥ ❜❡ ❞✐st✐♥❣✉✐s❤❡❞ ❢r♦♠ ✉♥✐❢♦r♠

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✶ ✴ ✶✾

▼✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦

■♥♣✉t✿ b✶✶ = a✶s✶ + ge✶✶ b✶✷ = a✶s✷ + ge✶✷

✭a✶ = f✶

g ✮

✭✷ ❧❛❜❡❧s✱ ✷ s❡❝r❡ts✮

b✷✶ = a✷s✶ + ge✷✶ b✷✷ = a✷s✷ + ge✷✷

✭a✷ = f✷

g ✮

❆tt❛❝❦✿ B

✶✶ ✶✷ ✷✶ ✷✷ ✶ ✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷ ✶ ✶ ✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷

= ✶

g ·

C + g· E r❛♥❦

• C
• = ✶

r❛♥❦

• E
• = ✷

❞❡t

✶

✷ ❞❡t

❞❡t

✶ ✷

❞❡t

✶ ✷ ✷ ❞❡t ✵ s♠❛❧❧

❝❛♥ ❜❡ ❞✐st✐♥❣✉✐s❤❡❞ ❢r♦♠ ✉♥✐❢♦r♠

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✶ ✴ ✶✾

▼✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦

■♥♣✉t✿ b✶✶ = a✶s✶ + ge✶✶ b✶✷ = a✶s✷ + ge✶✷

✭a✶ = f✶

g ✮

✭✷ ❧❛❜❡❧s✱ ✷ s❡❝r❡ts✮

b✷✶ = a✷s✶ + ge✷✶ b✷✷ = a✷s✷ + ge✷✷

✭a✷ = f✷

g ✮

❆tt❛❝❦✿ B

✶✶ ✶✷ ✷✶ ✷✷ ✶ ✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷ ✶ ✶ ✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷

= ✶

g ·

C + g· E r❛♥❦

• C
• = ✶

r❛♥❦

• E
• = ✷

❞❡t

• B
• =

✶ g✷ ·❞❡t

• C
• +❞❡t
• C✶ E✷
• +❞❡t
• E✶ C✷
• + g✷·❞❡t
• E
• ✵

s♠❛❧❧

❝❛♥ ❜❡ ❞✐st✐♥❣✉✐s❤❡❞ ❢r♦♠ ✉♥✐❢♦r♠

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✶ ✴ ✶✾

▼✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦

■♥♣✉t✿ b✶✶ = a✶s✶ + ge✶✶ b✶✷ = a✶s✷ + ge✶✷

✭a✶ = f✶

g ✮

✭✷ ❧❛❜❡❧s✱ ✷ s❡❝r❡ts✮

b✷✶ = a✷s✶ + ge✷✶ b✷✷ = a✷s✷ + ge✷✷

✭a✷ = f✷

g ✮

❆tt❛❝❦✿ B

✶✶ ✶✷ ✷✶ ✷✷ ✶ ✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷ ✶ ✶ ✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷

= ✶

g ·

C + g· E r❛♥❦

• C
• = ✶

r❛♥❦

• E
• = ✷

❞❡t

• B
• =

✶ g✷ ·❞❡t

• C
• +❞❡t
• C✶ E✷
• +❞❡t
• E✶ C✷
• + g✷·❞❡t
• E
• =✵
• s♠❛❧❧

❝❛♥ ❜❡ ❞✐st✐♥❣✉✐s❤❡❞ ❢r♦♠ ✉♥✐❢♦r♠

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✶ ✴ ✶✾

▼✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦

■♥♣✉t✿ b✶✶ = a✶s✶ + ge✶✶ b✶✷ = a✶s✷ + ge✶✷

✭a✶ = f✶

g ✮

✭✷ ❧❛❜❡❧s✱ ✷ s❡❝r❡ts✮

b✷✶ = a✷s✶ + ge✷✶ b✷✷ = a✷s✷ + ge✷✷

✭a✷ = f✷

g ✮

❆tt❛❝❦✿ B

✶✶ ✶✷ ✷✶ ✷✷ ✶ ✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷ ✶ ✶ ✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷

= ✶

g ·

C + g· E r❛♥❦

• C
• = ✶

r❛♥❦

• E
• = ✷

❞❡t

• B
• =

✶ g✷ ·❞❡t

• C
• +❞❡t
• C✶ E✷
• +❞❡t
• E✶ C✷
• + g✷·❞❡t
• E
• =✵
• s♠❛❧❧

⇒ ❝❛♥ ❜❡ ❞✐st✐♥❣✉✐s❤❡❞ ❢r♦♠ ✉♥✐❢♦r♠

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✶ ✴ ✶✾

❋✐①✐♥❣ t❤❡ ♠✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦

❋✐①✿ ❡♥s✉r❡ t❤❛t r❛♥❦(C) = ✷

■♥♣✉t✿

✶✶ ✶ ✶ ✶✶ ✶✷ ✶ ✷ ✶✷

✭ ✶

✶ ✮

✷✶ ✷ ✶ ✷✶ ✷✷ ✷ ✷ ✷✷

✭ ✷

✷ ✮

❆tt❛❝❦✿

✶ ✶ ✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷ ✶

✖

✶

✖ ✖

✷

✖

✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷ ✶

r❛♥❦ ✶ ✷ r❛♥❦ ✷

❞❡t

✶

✷ ❞❡t

❞❡t

✶ ✷

❞❡t

✶ ✷ ✷ ❞❡t ✵ ❧❛r❣❡ s♠❛❧❧

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✷ ✴ ✶✾

❋✐①✐♥❣ t❤❡ ♠✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦

❋✐①✿ ❡♥s✉r❡ t❤❛t r❛♥❦(C) = ✷

■♥♣✉t✿ b✶✶ = a✶s✶ + ge✶✶ b✶✷ = a✶s✷ + ge✶✷

✭a✶ = f✶

g ✮

b✷✶ = a✷s✶ + ge✷✶ b✷✷ = a✷s✷ + ge✷✷

✭a✷ = f✷

g ✮

❆tt❛❝❦✿ B

✶ ✶ ✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷ ✶

✖

✶

✖ ✖

✷

✖

✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷

= ✶

g ·

C + g· E r❛♥❦

• C
• = ✶

✷ r❛♥❦

• E
• = ✷

❞❡t

• B
• =

✶ g✷ ·❞❡t

• C
• +❞❡t
• C✶ E✷
• +❞❡t
• E✶ C✷
• + g✷·❞❡t
• E
• =✵

❧❛r❣❡

• s♠❛❧❧

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✷ ✴ ✶✾

❋✐①✐♥❣ t❤❡ ♠✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦

❋✐①✿ ❡♥s✉r❡ t❤❛t r❛♥❦(C) = ✷

■♥♣✉t✿ b✶✶ = a✶s✶ + ge✶✶ b✶✷ = a✶s✷ + ge✶✷

✭a✶ = f✶

g ✮

b✷✶ = a✷s✶ + ge✷✶ b✷✷ = a✷s✷ + ge✷✷

✭a✷ = f✷

g ✮

❆tt❛❝❦✿ B

✶ ✶ ✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷ ✶

✖

✶

✖ ✖

✷

✖

✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷

= ✶

g ·

C + g· E r❛♥❦

• C

✶

• = ✷

r❛♥❦

• E
• = ✷

❞❡t

• B
• =

✶ g✷ ·❞❡t

• C
• +❞❡t
• C✶ E✷
• +❞❡t
• E✶ C✷
• + g✷·❞❡t
• E
• ✵
• ❧❛r❣❡
• s♠❛❧❧

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✷ ✴ ✶✾

❋✐①✐♥❣ t❤❡ ♠✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦

❋✐①✿ ❡♥s✉r❡ t❤❛t r❛♥❦(C) = ✷

■♥♣✉t✿ b✶✶ = a✶s✶ + ge✶✶ b✶✷ = a✶s✷ + ge✶✷

✭a✶ = f✶

g ✮

b✷✶ = a✷s✶ + ge✷✶ b✷✷ = a✷s✷ + ge✷✷

✭a✷ = f✷

g ✮

❆tt❛❝❦✿ B = ✶

g ·

f✶ f✷ s✶ s✷

• + g ·

e✶✶ e✶✷ e✷✶ e✷✷

• ✶

✖

✶

✖ ✖

✷

✖

✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷ ✶

r❛♥❦ ✶ ✷ r❛♥❦ ✷

❞❡t

✶

✷ ❞❡t

❞❡t

✶ ✷

❞❡t

✶ ✷ ✷ ❞❡t ✵ ❧❛r❣❡ s♠❛❧❧

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✷ ✴ ✶✾

❋✐①✐♥❣ t❤❡ ♠✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦

❋✐①✿ ❡♥s✉r❡ t❤❛t r❛♥❦(C) = ✷

■♥♣✉t✿ b✶✶ = a✶,s✶ + ge✶✶ b✶✷ = a✶,s✷ + ge✶✷

✭a✶ = f✶

g ✮

b✷✶ = a✷,s✶ + ge✷✶ b✷✷ = a✷,s✷ + ge✷✷

✭a✷ = f✷

g ✮

❆tt❛❝❦✿ B

✶ ✶ ✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷

= ✶

g ·

✖ f✶ ✖ ✖ f✷ ✖   | | s✶ s✷ | |   + g · e✶✶ e✶✷ e✷✶ e✷✷

• ✶

r❛♥❦ ✶ ✷ r❛♥❦ ✷

❞❡t

✶

✷ ❞❡t

❞❡t

✶ ✷

❞❡t

✶ ✷ ✷ ❞❡t ✵ ❧❛r❣❡ s♠❛❧❧

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✷ ✴ ✶✾

❋✐①✐♥❣ t❤❡ ♠✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦

❋✐①✿ ❡♥s✉r❡ t❤❛t r❛♥❦(C) = ✷

■♥♣✉t✿ b✶✶ = a✶,s✶ + ge✶✶ b✶✷ = a✶,s✷ + ge✶✷

✭a✶ = f✶

g ✮

b✷✶ = a✷,s✶ + ge✷✶ b✷✷ = a✷,s✷ + ge✷✷

✭a✷ = f✷

g ✮

❆tt❛❝❦✿ B

✶ ✶ ✷ ✶ ✷ ✶✶ ✶✷ ✷✶ ✷✷

= ✶

g ·

✖ f✶ ✖ ✖ f✷ ✖   | | s✶ s✷ | |   + g · e✶✶ e✶✷ e✷✶ e✷✷

• ✶

r❛♥❦ ✶ ✷ r❛♥❦ ✷

❞❡t

✶

✷ ❞❡t

❞❡t

✶ ✷

❞❡t

✶ ✷ ✷ ❞❡t ✵ ❧❛r❣❡ s♠❛❧❧

⇒ ✏▼♦❞✉❧❡✲▲❲❊ ✇✐t❤ ❝♦rr❡❧❛t❡❞ ♥♦✐s❡✑ s❡❡♠s t♦ ♣r❡✈❡♥t t❤❡ ❛tt❛❝❦

✭✐❢ ❞✐♠ ♦❢ ✈❡❝t♦rs ✐s ❧❛r❣❡ ❡♥♦✉❣❤✮

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✷ ✴ ✶✾

❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦

✶

▼✉❧t✐✲❝✐♣❤❡rt❡①ts ❛tt❛❝❦

✷

❘❛♥❦ ❛tt❛❝❦

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✸ ✴ ✶✾

❈♦♥t❡①t

❚❤❡ ❛❞✈❡rs❛r② ❝❛♥ ❤♦♥❡st❧② ♣❧❛② t❤❡ ❢♦❧❧♦✇✐♥❣ ❣❛♠❡ A C s❛♠♣❧❡ µ s♠❛❧❧ s❛♠♣❧❡ b ← {✵, ✶} µ ❝♦♠♣✉t❡ ❧❛r❣❡ ♥♦✐s❡ N ✐❢ b = ✵ x ← N ✐❢ b = ✶ x ← N + µ x ❣✉❡ss b r❡♣❡❛t

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✹ ✴ ✶✾

❚❤❡ ♥♦✐s❡ t❡r♠ N

N =

• ℓ,i,j

v×

ij

• p✷

✶·(gℓ ✷ · ˜

ξℓ

✶i · gℓ ✶ · ˜

ξℓ

✷j)

+ p✶p✵·(gℓ

✷ · ˜

ξℓ

✶i · gℓ ✶ · ξℓ ✷j + gℓ ✷ · ξℓ ✶i · gℓ ✶ · ˜

ξℓ

✷j)

+ p✶(f ℓ

✶i · t✶ · ˜

ξℓ

✷j + f ℓ ✷j · t✷ · ˜

ξℓ

✶i)

+ p✷

✵ · (gℓ ✷ · ξℓ ✶i · gℓ ✶ · ξℓ ✷j)

+ p✵(f ℓ

✶i · t✶ · ξℓ ✷j + f ℓ ✷j · t✷ · ξℓ ✶i)

• ✵

✶ ❛r❡ ❦♥♦✇♥ ❛♥❞ ✶ ✵

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

✷ ✶✱ ✶ ✵✱ ✶✱ ✷ ✵ ❛♥❞ ✵✳

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✺ ✴ ✶✾

❚❤❡ ♥♦✐s❡ t❡r♠ N

N =

• ℓ,i,j

v×

ij

• p✷

✶·(gℓ ✷ · ˜

ξℓ

✶i · gℓ ✶ · ˜

ξℓ

✷j)

+ p✶p✵·(gℓ

✷ · ˜

ξℓ

✶i · gℓ ✶ · ξℓ ✷j + gℓ ✷ · ξℓ ✶i · gℓ ✶ · ˜

ξℓ

✷j)

+ p✶(f ℓ

✶i · t✶ · ˜

ξℓ

✷j + f ℓ ✷j · t✷ · ˜

ξℓ

✶i)

+ p✷

✵ · (gℓ ✷ · ξℓ ✶i · gℓ ✶ · ξℓ ✷j)

+ p✵(f ℓ

✶i · t✶ · ξℓ ✷j + f ℓ ✷j · t✷ · ξℓ ✶i)

• p✵, p✶ ❛r❡ ❦♥♦✇♥ ❛♥❞ p✶ ≫ p✵ ≫ ❛❧❧ t❤❡ r❡st

⇒ ❝❛♥ s♣❧✐t t❤❡ ♥♦✐s❡ t❡r♠s ❛❝❝♦r❞✐♥❣ t♦ p✷

✶✱ p✶p✵✱ p✶✱ p✷ ✵ ❛♥❞ p✵✳

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✺ ✴ ✶✾

❚❤❡ ♥♦✐s❡ t❡r♠ N

N mod p✷

✶ =

• ℓ,i,j

v×

ij

• p✷

✶·(gℓ ✷ · ˜

ξℓ

✶i · gℓ ✶ · ˜

ξℓ

✷j)

+ p✶p✵·(gℓ

✷ · ˜

ξℓ

✶i · gℓ ✶ · ξℓ ✷j + gℓ ✷ · ξℓ ✶i · gℓ ✶ · ˜

ξℓ

✷j)

+ p✶(f ℓ

✶i · t✶ · ˜

ξℓ

✷j + f ℓ ✷j · t✷ · ˜

ξℓ

✶i)

+ p✷

✵ · (gℓ ✷ · ξℓ ✶i · gℓ ✶ · ξℓ ✷j)

+ p✵(f ℓ

✶i · t✶ · ξℓ ✷j + f ℓ ✷j · t✷ · ξℓ ✶i)

• p✵, p✶ ❛r❡ ❦♥♦✇♥ ❛♥❞ p✶ ≫ p✵ ≫ ❛❧❧ t❤❡ r❡st

⇒ ❝❛♥ s♣❧✐t t❤❡ ♥♦✐s❡ t❡r♠s ❛❝❝♦r❞✐♥❣ t♦ p✷

✶✱ p✶p✵✱ p✶✱ p✷ ✵ ❛♥❞ p✵✳

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✺ ✴ ✶✾

❚❤❡ ♥♦✐s❡ t❡r♠ N

(N mod p✷

✶) mod p✶p✵ =

• ℓ,i,j

v×

ij

• p✷

✶·(gℓ ✷ · ˜

ξℓ

✶i · gℓ ✶ · ˜

ξℓ

✷j)

+ p✶p✵·(gℓ

✷ · ˜

ξℓ

✶i · gℓ ✶ · ξℓ ✷j + gℓ ✷ · ξℓ ✶i · gℓ ✶ · ˜

ξℓ

✷j)

+ p✶(f ℓ

✶i · t✶ · ˜

ξℓ

✷j + f ℓ ✷j · t✷ · ˜

ξℓ

✶i)

+ p✷

✵ · (gℓ ✷ · ξℓ ✶i · gℓ ✶ · ξℓ ✷j)

+ p✵(f ℓ

✶i · t✶ · ξℓ ✷j + f ℓ ✷j · t✷ · ξℓ ✶i)

• p✵, p✶ ❛r❡ ❦♥♦✇♥ ❛♥❞ p✶ ≫ p✵ ≫ ❛❧❧ t❤❡ r❡st

⇒ ❝❛♥ s♣❧✐t t❤❡ ♥♦✐s❡ t❡r♠s ❛❝❝♦r❞✐♥❣ t♦ p✷

✶✱ p✶p✵✱ p✶✱ p✷ ✵ ❛♥❞ p✵✳

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✺ ✴ ✶✾

❚❤❡ r❛♥❦ ❛tt❛❝❦

p✷

✶·

• ℓ,i,j

v×

ij

(gℓ

✷ · ˜

ξℓ

✶i · gℓ ✶ · ˜

ξℓ

✷j)

p✶p✵·

• ℓ,i,j

v×

ij

(gℓ

✷ · ˜

ξℓ

✶i · gℓ ✶ · ξℓ ✷j + gℓ ✷ · ξℓ ✶i · gℓ ✶ · ˜

ξℓ

✷j)

p✶·

• ℓ,i,j

v×

ij

(f ℓ

✶i · t✶ · ˜

ξℓ

✷j + f ℓ ✷j · t✷ · ˜

ξℓ

✶i)

p✷

✵·

• ℓ,i,j

v×

ij

(gℓ

✷ · ξℓ ✶i · gℓ ✶ · ξℓ ✷j)

p✵·

• ℓ,i,j

v×

ij

(f ℓ

✶i · t✶ · ξℓ ✷j + f ℓ ✷j · t✷ · ξℓ ✶i)

✵ ♦r ❣r❡❡♥✿ ❣♦♦❞ ♥♦✐s❡ t❡r♠s ✭❤✐❞❡ t❤❡ ❝❤❛❧❧❡♥❣❡✮ r❡❞✿ ❜❛❞ ♥♦✐s❡ t❡r♠s ✭❞♦ ♥♦t ❤✐❞❡ t❤❡ ❝❤❛❧❧❡♥❣❡✮

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✻ ✴ ✶✾

❚❤❡ r❛♥❦ ❛tt❛❝❦

• ℓ,i,j

v×

ij

(gℓ

✷ · ˜

ξℓ

✶i · gℓ ✶ · ˜

ξℓ

✷j)

• ℓ,i,j

v×

ij

(gℓ

✷ · ˜

ξℓ

✶i · gℓ ✶ · ξℓ ✷j + gℓ ✷ · ξℓ ✶i · gℓ ✶ · ˜

ξℓ

✷j)

• ℓ,i,j

v×

ij

(f ℓ

✶i · t✶ · ˜

ξℓ

✷j + f ℓ ✷j · t✷ · ˜

ξℓ

✶i)

• ℓ,i,j

v×

ij

(gℓ

✷ · ξℓ ✶i · gℓ ✶ · ξℓ ✷j)

• ℓ,i,j

v×

ij

(f ℓ

✶i · t✶ · ξℓ ✷j + f ℓ ✷j · t✷ · ξℓ ✶i)

✵ ♦r ❣r❡❡♥✿ ❣♦♦❞ ♥♦✐s❡ t❡r♠s ✭❤✐❞❡ t❤❡ ❝❤❛❧❧❡♥❣❡✮ r❡❞✿ ❜❛❞ ♥♦✐s❡ t❡r♠s ✭❞♦ ♥♦t ❤✐❞❡ t❤❡ ❝❤❛❧❧❡♥❣❡✮

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✻ ✴ ✶✾

❚❤❡ r❛♥❦ ❛tt❛❝❦

• ℓ,i,j

v×

ij

(gℓ

✷ · ˜

ξℓ

✶i · gℓ ✶ · ˜

ξℓ

✷j)

• ℓ,i,j

v×

ij

(gℓ

✷ · ˜

ξℓ

✶i · gℓ ✶ · ξℓ ✷j + gℓ ✷ · ξℓ ✶i · gℓ ✶ · ˜

ξℓ

✷j)

• ℓ,i,j

v×

ij

(f ℓ

✶i · t✶ · ˜

ξℓ

✷j + f ℓ ✷j · t✷ · ˜

ξℓ

✶i)

• ℓ,i,j

v×

ij

(gℓ

✷ · ξℓ ✶i · gℓ ✶ · ξℓ ✷j)

• ℓ,i,j

v×

ij

(f ℓ

✶i · t✶ · ξℓ ✷j + f ℓ ✷j · t✷ · ξℓ ✶i)

✵ ♦r ❣r❡❡♥✿ ❣♦♦❞ ♥♦✐s❡ t❡r♠s ✭❤✐❞❡ t❤❡ ❝❤❛❧❧❡♥❣❡✮ r❡❞✿ ❜❛❞ ♥♦✐s❡ t❡r♠s ✭❞♦ ♥♦t ❤✐❞❡ t❤❡ ❝❤❛❧❧❡♥❣❡✮

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✻ ✴ ✶✾

❚❤❡ r❛♥❦ ❛tt❛❝❦

• ℓ,i,j

v×

ij

(gℓ

✷ · ˜

ξℓ

✶i · gℓ ✶ · ˜

ξℓ

✷j)

• ℓ,i,j

v×

ij

(gℓ

✷ · ˜

ξℓ

✶i · gℓ ✶ · ξℓ ✷j + gℓ ✷ · ξℓ ✶i · gℓ ✶ · ˜

ξℓ

✷j)

• ℓ,i,j

v×

ij

(f ℓ

✶i · t✶ · ˜

ξℓ

✷j + f ℓ ✷j · t✷ · ˜

ξℓ

✶i)

• ℓ,i,j

v×

ij

(gℓ

✷ · ξℓ ✶i · gℓ ✶ · ξℓ ✷j)

• ℓ,i,j

v×

ij

(f ℓ

✶i · t✶ · ξℓ ✷j + f ℓ ✷j · t✷ · ξℓ ✶i) + (✵ ♦r µ)

❣r❡❡♥✿ ❣♦♦❞ ♥♦✐s❡ t❡r♠s ✭❤✐❞❡ t❤❡ ❝❤❛❧❧❡♥❣❡✮ r❡❞✿ ❜❛❞ ♥♦✐s❡ t❡r♠s ✭❞♦ ♥♦t ❤✐❞❡ t❤❡ ❝❤❛❧❧❡♥❣❡✮

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✻ ✴ ✶✾

❋✐①✐♥❣ t❤❡ r❛♥❦ ❛tt❛❝❦

■❞❡❛✿ r❡♠♦✈❡ t❤❡ ♠♦❞✉❧✐ p✵ ❛♥❞ p✶ ⇒ ❝❛♥♥♦t s♣❧✐t t❤❡ ♥♦✐s❡ t❡r♠ ❛♥②♠♦r❡ N =

• ℓ,i,j

v×

ij

• p✷

✶(gℓ ✷ · ˜

ξℓ

✶i · gℓ ✶ · ˜

ξℓ

✷j)

+ p✶p✵(gℓ

✷ · ˜

ξℓ

✶i · gℓ ✶ · ξℓ ✷j + gℓ ✷ · ξℓ ✶i · gℓ ✶ · ˜

ξℓ

✷j)

+ p✶(f ℓ

✶i · t✶ · ˜

ξℓ

✷j + f ℓ ✷j · t✷ · ˜

ξℓ

✶i)

+ p✷

✵(gℓ ✷ · ξℓ ✶i · gℓ ✶ · ξℓ ✷j)

+ p✵(f ℓ

✶i · t✶ · ξℓ ✷j + f ℓ ✷j · t✷ · ξℓ ✶i)

• ❆✳ P❡❧❧❡t✲▼❛r②

◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✼ ✴ ✶✾

❋✉rt❤❡r ❝r②♣t❛♥❛❧②s✐s

❉❡s❝r✐❜❡ ♦t❤❡r ♣♦t❡♥t✐❛❧ ❛tt❛❝❦s

◮ ✇❤❛t ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡s❡ ❛tt❛❝❦s ◮ ✇❤② t❤✐s ❞♦❡s ♥♦t ❜r❡❛❦ t❤❡ s❝❤❡♠❡

♦r ✇❤❛t ❝♦♥str❛✐♥t ♦♥ ♣❛r❛♠❡t❡rs ♣r❡✈❡♥ts t❤❡ ❛tt❛❝❦

◗✉❛♥t✉♠ ❝♦♠♣✉t❡r ❞♦❡s ♥♦t s❡❡♠ t♦ ❤❡❧♣ t❤❡ ❛tt❛❝❦❡r Pr♦♣♦s❡ ❛ ❝♦♥❝r❡t❡ s❡t ♦❢ ♣❛r❛♠❡t❡rs ✭❛s②♠♣t♦t✐❝✮

s❡❡ ❙❡❝t✐♦♥ ✼✳✼

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✽ ✴ ✶✾

❋✉rt❤❡r ❝r②♣t❛♥❛❧②s✐s

❉❡s❝r✐❜❡ ♦t❤❡r ♣♦t❡♥t✐❛❧ ❛tt❛❝❦s

◮ ✇❤❛t ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡s❡ ❛tt❛❝❦s ◮ ✇❤② t❤✐s ❞♦❡s ♥♦t ❜r❡❛❦ t❤❡ s❝❤❡♠❡

♦r ✇❤❛t ❝♦♥str❛✐♥t ♦♥ ♣❛r❛♠❡t❡rs ♣r❡✈❡♥ts t❤❡ ❛tt❛❝❦

◗✉❛♥t✉♠ ❝♦♠♣✉t❡r ❞♦❡s ♥♦t s❡❡♠ t♦ ❤❡❧♣ t❤❡ ❛tt❛❝❦❡r Pr♦♣♦s❡ ❛ ❝♦♥❝r❡t❡ s❡t ♦❢ ♣❛r❛♠❡t❡rs ✭❛s②♠♣t♦t✐❝✮

s❡❡ ❙❡❝t✐♦♥ ✼✳✼

❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✽ ✴ ✶✾

❋✉rt❤❡r ❝r②♣t❛♥❛❧②s✐s

❉❡s❝r✐❜❡ ♦t❤❡r ♣♦t❡♥t✐❛❧ ❛tt❛❝❦s

◮ ✇❤❛t ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡s❡ ❛tt❛❝❦s ◮ ✇❤② t❤✐s ❞♦❡s ♥♦t ❜r❡❛❦ t❤❡ s❝❤❡♠❡

♦r ✇❤❛t ❝♦♥str❛✐♥t ♦♥ ♣❛r❛♠❡t❡rs ♣r❡✈❡♥ts t❤❡ ❛tt❛❝❦

◗✉❛♥t✉♠ ❝♦♠♣✉t❡r ❞♦❡s ♥♦t s❡❡♠ t♦ ❤❡❧♣ t❤❡ ❛tt❛❝❦❡r Pr♦♣♦s❡ ❛ ❝♦♥❝r❡t❡ s❡t ♦❢ ♣❛r❛♠❡t❡rs ✭❛s②♠♣t♦t✐❝✮

◮ s❡❡ ❙❡❝t✐♦♥ ✼✳✼ ❤tt♣s✿✴✴❡♣r✐♥t✳✐❛❝r✳♦r❣✴✷✵✷✵✴✹✶✺✳♣❞❢ ❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✽ ✴ ✶✾

❖♣❡♥ ♣r♦❜❧❡♠s

Pr♦✈❡ t❤❡ s❝❤❡♠❡ ❢r♦♠ s✐♠♣❧❡r ❛ss✉♠♣t✐♦♥s ✭❝❢ ❬❏▲❙✶✾❪✮❄

◮ ❡✳❣✳✱ ♠♦❞✉❧❡✲▲❲❊ ✇✐t❤ ❝♦rr❡❧❛t❡❞ ♥♦✐s❡ ✰ · · · ❄

❋✐♥❞ ❞✐✛❡r❡♥t ❛tt❛❝❦s❄

◮ ❚❤❡ ✷ ❛tt❛❝❦s s❤❛r❡ s✐♠✐❧❛r✐t✐❡s ✇✐t❤ ❛tt❛❝❦s ❛❣❛✐♥st ♠✉❧t✐❧✐♥❡❛r ♠❛♣

❜❛s❡❞ ♦❜❢✉s❝❛t♦rs✱ ✇❤②❄

❚❤❛♥❦ ②♦✉

❬❏▲❙✶✾❪ ❆✳ ❏❛✐♥ ❛♥❞ ❍✳ ▲✐♥ ❛♥❞ ❆✳ ❙❛❤❛✐✳ ❙✐♠♣❧✐❢②✐♥❣ ❈♦♥str✉❝t✐♦♥s ❛♥❞ ❆ss✉♠♣t✐♦♥s ❢♦r ✐❖✳ ❡Pr✐♥t✳ ❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✾ ✴ ✶✾

❖♣❡♥ ♣r♦❜❧❡♠s

Pr♦✈❡ t❤❡ s❝❤❡♠❡ ❢r♦♠ s✐♠♣❧❡r ❛ss✉♠♣t✐♦♥s ✭❝❢ ❬❏▲❙✶✾❪✮❄

◮ ❡✳❣✳✱ ♠♦❞✉❧❡✲▲❲❊ ✇✐t❤ ❝♦rr❡❧❛t❡❞ ♥♦✐s❡ ✰ · · · ❄

❋✐♥❞ ❞✐✛❡r❡♥t ❛tt❛❝❦s❄

◮ ❚❤❡ ✷ ❛tt❛❝❦s s❤❛r❡ s✐♠✐❧❛r✐t✐❡s ✇✐t❤ ❛tt❛❝❦s ❛❣❛✐♥st ♠✉❧t✐❧✐♥❡❛r ♠❛♣

❜❛s❡❞ ♦❜❢✉s❝❛t♦rs✱ ✇❤②❄

❚❤❛♥❦ ②♦✉

❬❏▲❙✶✾❪ ❆✳ ❏❛✐♥ ❛♥❞ ❍✳ ▲✐♥ ❛♥❞ ❆✳ ❙❛❤❛✐✳ ❙✐♠♣❧✐❢②✐♥❣ ❈♦♥str✉❝t✐♦♥s ❛♥❞ ❆ss✉♠♣t✐♦♥s ❢♦r ✐❖✳ ❡Pr✐♥t✳ ❆✳ P❡❧❧❡t✲▼❛r② ◆▲✐♥❋❊✿ ❛tt❛❝❦s ❛♥❞ ✜①❡s ❊✉r♦❝r②♣t ✷✵✷✵ ✶✾ ✴ ✶✾