❆ P♦❧②♥♦♠✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠ ❢♦r ❙♦❧✈✐♥❣ t❤❡ ❍✐❞❞❡♥ ❙✉❜s❡t ❙✉♠ Pr♦❜❧❡♠ ❏❡❛♥✲❙é❜❛st✐❡♥ ❈♦r♦♥ ❛♥❞ ❆❣♥❡s❡ ●✐♥✐ ❯♥✐✈❡rs✐t② ♦❢ ▲✉①❡♠❜♦✉r❣ ❈❘❨P❚❖✷✵✷✵ ✶ ✴ ✶✽
✇✐t❤ ❛♥❞ ✳ ●✐✈❡♥ ❛♥❞ ✱ r❡❝♦✈❡r ✳ ❍✐❞❞❡♥ ❙✉❜s❡t ❙✉♠ Pr♦❜❧❡♠ ✷ ✴ ✶✽
❍✐❞❞❡♥ ❙✉❜s❡t ❙✉♠ Pr♦❜❧❡♠ h = α 1 x 1 + · · · + α n x n (mod Q ) ✇✐t❤ x 1 , . . . , x n ∈ { 0 , 1 } ❛♥❞ α 1 , . . . , α n ∈ Z /Q Z n ✳ ●✐✈❡♥ Q, h ❛♥❞ α 1 , . . . , α n ✱ r❡❝♦✈❡r x 1 , . . . , x n ✳ ✷ ✴ ✶✽
❍✐❞❞❡♥ ❙✉❜s❡t ❙✉♠ Pr♦❜❧❡♠ h 1 = α 1 x 1 , 1 + · · · + α n x n, 1 (mod Q ) ✳ ✳ ✳ h m = α 1 x 1 ,m + · · · + α n x n,m (mod Q ) ✇✐t❤ x i,j ∈ { 0 , 1 } ❛♥❞ α 1 , . . . , α n ∈ Z /Q Z n ✳ ●✐✈❡♥ Q ❛♥❞ h 1 , . . . , h m ✱ r❡❝♦✈❡r α 1 , . . . , α n ❛♥❞ x i,j ❢♦r i ∈ [ n ] ❛♥❞ j ∈ [ m ] ✳ The weights α i ’s are hidden!! ✸ ✴ ✶✽
❍✐❞❞❡♥ ❙✉❜s❡t ❙✉♠ Pr♦❜❧❡♠ h 1 = α 1 x 1 , 1 + · · · + α n x n, 1 (mod Q ) ✳ ✳ ✳ h m = α 1 x 1 ,m + · · · + α n x n,m (mod Q ) ✇✐t❤ x i,j ∈ { 0 , 1 } ❛♥❞ α 1 , . . . , α n ∈ Z /Q Z n ✳ ●✐✈❡♥ Q ❛♥❞ h 1 , . . . , h m ✱ r❡❝♦✈❡r α 1 , . . . , α n ❛♥❞ x i,j ❢♦r i ∈ [ n ] ❛♥❞ j ∈ [ m ] ✳ x 1 , 1 x 1 ,m · · · · · · � h 1 � α 1 ✳ ✳ h m � α n � = ✳ ✳ (mod Q ) · · · · · · · · · ✳ ✳ x n, 1 x n,m · · · · · · ✸ ✴ ✶✽
❍✐❞❞❡♥ ❙✉❜s❡t ❙✉♠ Pr♦❜❧❡♠ ▲❡t Q ❜❡ ❛♥ ✐♥t❡❣❡r✱ ❛♥❞ ❧❡t α 1 , . . . , α n ❜❡ r❛♥❞♦♠ ✐♥t❡❣❡rs ✐♥ Z /Q Z ✳ ▲❡t x 1 , . . . , x n ∈ Z m ❜❡ r❛♥❞♦♠ ✈❡❝t♦rs ✇✐t❤ ❝♦♠♣♦♥❡♥ts ✐♥ { 0 , 1 } ✳ ▲❡t h ∈ Z m s❛t✐s❢②✐♥❣✿ α 1 x 1 + · · · + α n x n h = (mod Q ) ●✐✈❡♥ Q ❛♥❞ h ✱ r❡❝♦✈❡r t❤❡ ✐♥t❡❣❡rs α i ✬s ❛♥❞ t❤❡ ✈❡❝t♦rs x i ✬s✳ = (mod Q ) h X α ✸ ✴ ✶✽
❚✐♠❡❧✐♥❡ ✶✾✾✽ ❇♦②❦♦✱ P❡✐♥❛❞♦ ❛♥❞ ❱❡♥❦❛t❡s❛♥ ♣r❡s❡♥t❡❞ ❛ ❢❛st ❣❡♥❡r❛t♦r ♦❢ r❛♥❞♦♠ ♣❛✐rs ( x, g x (mod p )) ✐♥tr♦❞✉❝✐♥❣ t❤❡ ❍❙❙P✳ ✶✾✾✾ ◆❣✉②❡♥ ❛♥❞ ❙t❡r♥ ❞❡s❝r✐❜❡❞ ❛ ❧❛tt✐❝❡ ❜❛s❡❞ ❛❧❣♦r✐t❤♠ ❢♦r s♦❧✈✐♥❣ t❤❡ ❍❙❙P✳ ✷✵✷✵ ❖✉r ♠❛✐♥ ❝♦♥tr✐❜✉t✐♦♥s✿ • ❞❡t❛✐❧❡❞ ❛♥❛❧②s✐s ♦❢ t❤❡ ◆❣✉②❡♥✲❙t❡r♥ ❛❧❣♦r✐t❤♠✱ • ✈❛r✐❛♥t ✇♦r❦✐♥❣ ✐♥ ♣♦❧②♥♦♠✐❛❧✲t✐♠❡✳ ✹ ✴ ✶✽
❖✈❡r✈✐❡✇ • ❚❤❡ ◆❣✉②❡♥✲❙t❡r♥ ❛tt❛❝❦✳ • ❖✉r ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ ❛tt❛❝❦✳ • ❚❤❡ ❛✣♥❡ ❤✐❞❞❡♥ s✉❜s❡t s✉♠✳ • ❋✐♥❛❧ r❡♠❛r❦s ❛♥❞ ♦♣❡♥ q✉❡st✐♦♥s✳ ✺ ✴ ✶✽
❚❤❡ ◆❣✉②❡♥✲❙t❡r♥ ❆tt❛❝❦ h = α 1 x 1 + · · · + α n x n (mod Q ) ❚❤❡ ✐❞❡❛✿ • ■❢ ❛ ✈❡❝t♦r u ✐s ♦rt❤♦❣♦♥❛❧ t♦ h ♠♦❞✉❧♦ Q ✿ � u , h � ≡ α 1 � u , x 1 � + · · · + α n � u , x n � ≡ 0 (mod Q ) ⇒ p u = ( � u , x 1 � , . . . , � u , x n � ) ✐s ♦rt❤♦❣♦♥❛❧ t♦ α ♠♦❞✉❧♦ Q ✳ • ■❢ � p u � < λ 1 (Λ ⊥ Q ( α )) ✱ ✇❡ ♠✉st ❤❛✈❡ p u = 0 ✱ ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ ✈❡❝t♦r u ✐s ♦rt❤♦❣♦♥❛❧ ✐♥ Z t♦ ❛❧❧ ✈❡❝t♦rs x i ✳ ✻ ✴ ✶✽
❚❤❡ ◆❣✉②❡♥✲❙t❡r♥ ❆tt❛❝❦ = (mod Q ) h X α ❚❤❡ ❛❧❣♦r✐t❤♠✿ ❙t❡♣ ✶ = γ (mod Q ) h C ❙t❡♣ ✷ X α ✼ ✴ ✶✽
❚❤❡ ◆❣✉②❡♥✲❙t❡r♥ ❆tt❛❝❦ = (mod Q ) h X α ❚❤❡ ❛❧❣♦r✐t❤♠✿ ❙t❡♣ ✶ ❋r♦♠ t❤❡ s❛♠♣❧❡s h ❛♥❞ Q ✱ ❞❡t❡r♠✐♥❡ t❤❡ ❧❛tt✐❝❡ ¯ L x = ( L ⊥ x ) ⊥ ✱ ✇❤❡r❡ L x ✐s t❤❡ ❧❛tt✐❝❡ ❣❡♥❡r❛t❡❞ ❜② t❤❡ x i ✬s✳ h C ❙t❡♣ ✷ ❋r♦♠ ¯ L x ⊇ L x ✱ r❡❝♦✈❡r t❤❡ ❤✐❞❞❡♥ ✈❡❝t♦rs x i ✬s✳ ❋r♦♠ h ✱ t❤❡ x i ✬s ❛♥❞ Q ✱ r❡❝♦✈❡r t❤❡ ✇❡✐❣❤ts α i ✬s✳ C X α ✼ ✴ ✶✽
❖✉r ❛♥❛❧②s✐s✿ ❙t❡♣ ✶✿ • ❲✐t❤ ♣r♦❜❛❜✐❧✐t② ❛t ❧❡❛st 1 / 2 ♦✈❡r t❤❡ ❝❤♦✐❝❡ ♦❢ α ✱ t❤❡ ❛❧❣♦r✐t❤♠ r❡❝♦✈❡rs ❛ ❜❛s✐s ♦❢ ¯ L x ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✱ ❛ss✉♠✐♥❣ t❤❛t Q ✐s ❛ ♣r✐♠❡ ✐♥t❡❣❡r ♦❢ ❜✐ts✐③❡ ❛t ❧❡❛st 2 mn log m ✳ • ❋♦r m = 2 n ✱ ✐❢ t❤❡ ❞❡♥s✐t② ✐s d = n/ log Q = O (1 / ( n log n )) ✇❡ r❡❝♦✈❡r ¯ L x ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳ • ❍❡✉r✐st✐❝❛❧❧②✱ d = O (1 /n ) ✐s s✉✣❝✐❡♥t✳ ❙t❡♣ ✷✿ • ❚❤❡ x i ✬s ❛r❡ s❤♦rt ✈❡❝t♦rs ♦❢ ¯ L x ✳ • ❯s✐♥❣ ❇❑❩ t❤❡ ❛s②♠♣t♦t✐❝ ❝♦♠♣❧❡①✐t② ✐s 2 Ω( n/ log n ) ✳ ✽ ✴ ✶✽
❖✉r ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ ❛❧❣♦r✐t❤♠ ❙t❡♣ ✶ = γ (mod Q ) h C ❙t❡♣ ✷ X α • ❲❡ r❡q✉✐r❡ m ≈ ( n 2 + n ) / 2 ✐♥st❡❛❞ ♦❢ m = 2 n ✳ • ■♠♣r♦✈❡❞ st❡♣ ✶✿ ❢❛st ❣❡♥❡r❛t✐♦♥ ♦❢ ♦rt❤♦❣♦♥❛❧ ✈❡❝t♦rs✳ • ◆❡✇ st❡♣ ✷✿ r❡❝♦✈❡r ❜✐♥❛r② ✈❡❝t♦rs✳ ✾ ✴ ✶✽
◆❡✇ st❡♣ ✷✿ ♠✉❧t✐✈❛r✐❛t❡ ❛♣♣r♦❛❝❤ ■♥❣r❡❞✐❡♥ts✿ L x ✿ t❤❡r❡ ❡①✐sts W ∈ Z n × n ∩ GL( Q , n ) • L x ✐s ❛ s✉❜❧❛tt✐❝❡ ♦❢ ¯ = X W C • ❇❡✐♥❣ ❜✐♥❛r② ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ❝♦♥❞✐t✐♦♥✿ ⇒ y 2 − y = 0 y ∈ { 0 , 1 } ⇐ ✶✵ ✴ ✶✽
▼✐①✐♥❣✳✳✳ = x i w i · · · ˜ ˜ c 1 c m ❋♦r ❡❛❝❤ i = 1 , . . . , n ❛♥❞ j = 1 , . . . , m ✇❡ ❤❛✈❡ • x i,j ∈ { 0 , 1 } • w i · ˜ c j = x i,j c j ) 2 − w i · ˜ ⇒ ( w i · ˜ = c j = 0 ✶✶ ✴ ✶✽
❚❤❡ r♦✇s ♦❢ W ❛r❡ s♦❧✉t✐♦♥s ♦❢ ♠✉❧t✐✈❛r✐❛t❡ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ s②st❡♠ 1 · w ⊺ − w · ˜ w · ˜ c ⊺ c 1 = 0 c 1 ˜ ✳ ✳ ✳ m · w ⊺ − w · ˜ w · ˜ c ⊺ c m = 0 c m ˜ • ❋♦r m ≈ ( n 2 + n ) / 2 ✇❡ ❡①♣❡❝t t♦ s♦❧✈❡ t❤✐s s②st❡♠ ❛♥❞ r❡❝♦✈❡r t❤❡ x i ✬s ❜② O ( n 6 ) ❜✐t ♦♣❡r❛t✐♦♥s ❛♥❞ O ( n 4 ) s♣❛❝❡ ❝♦♠♣❧❡①✐t②✱ ✈✐❛ ❧✐♥❡❛r ❛❧❣❡❜r❛✳ ✶✷ ✴ ✶✽
❚❤❡ ❝♦❡✣❝✐❡♥t ♦❢ w i w k ✐♥ t❤❡ j t❤ ❡q✉❛t✐♦♥ ✐s (2 − δ i,k ) C ij C kj ✳ R � C E − C • E ✐s t❤❡ ♠❛tr✐① ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ s②st❡♠✳ • ❚❤❡ r♦✇s ♦❢ W ❛r❡ ❡✐❣❡♥✈❡❝t♦rs ♦❢ ❝❡rt❛✐♥ s✉❜♠❛tr✐❝❡s ♦❢ ❛ ❜❛s✐s ♦❢ ker E ✳ ✶✸ ✴ ✶✽
▲❡♠♠❛ ■❢ R ❤❛s r❛♥❦ n 2 + n ✱ t❤❡♥ t❤❡ ✈❡❝t♦rs x i ✬s ❝❛♥ ❜❡ r❡❝♦✈❡r❡❞ ✐♥ 2 O ( n 6 ) ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s✳ ❘❡❞✉❝✐♥❣ t❤❡ ♠❛tr✐① r❡❧❛t✐♦♥ X = WC mod p ✇❡ ❝❛♥ ♦❜t❛✐♥ ❛ s②st❡♠ ❞❡✜♥❡❞ ♦✈❡r F p ✳ ⇒ ❋♦r m ≈ ( n 2 + n ) / 2 ✇❡ ❡①♣❡❝t t♦ s♦❧✈❡ t❤✐s s②st❡♠ ❛♥❞ r❡❝♦✈❡r t❤❡ x i ✬s ❜② O ( n 6 ) ❜✐t ♦♣❡r❛t✐♦♥s ❛♥❞ O ( n 4 ) s♣❛❝❡ ❝♦♠♣❧❡①✐t②✱ ✈✐❛ ❧✐♥❡❛r ❛❧❣❡❜r❛✳ ✶✹ ✴ ✶✽
❈♦♠♣❛r✐s♦♥ ◆❙✾✾ ❈●✷✵ ( n 2 + n ) / 2 m 2 n O ( n 2 ) O ( n 2 ) log Q 2 Ω( n/ log n ) O ( n 9 ) t✐♠❡ Experimental timing comparison 26820 Nguyen-Stern sec Coron-Gini 18 100 150 200 250 number of variables ✶✺ ✴ ✶✽
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