P rt r - - PowerPoint PPT Presentation

❆ P♦❧②♥♦♠✐❛❧✲❚✐♠❡ ❆❧❣♦r✐t❤♠ ❢♦r ❙♦❧✈✐♥❣ t❤❡ ❍✐❞❞❡♥ ❙✉❜s❡t ❙✉♠ Pr♦❜❧❡♠

❏❡❛♥✲❙é❜❛st✐❡♥ ❈♦r♦♥ ❛♥❞ ❆❣♥❡s❡ ●✐♥✐

❯♥✐✈❡rs✐t② ♦❢ ▲✉①❡♠❜♦✉r❣

❈❘❨P❚❖✷✵✷✵

✶ ✴ ✶✽

❍✐❞❞❡♥ ❙✉❜s❡t ❙✉♠ Pr♦❜❧❡♠

✇✐t❤ ❛♥❞ ✳

• ✐✈❡♥

❛♥❞ ✱ r❡❝♦✈❡r ✳

✷ ✴ ✶✽

❍✐❞❞❡♥ ❙✉❜s❡t ❙✉♠ Pr♦❜❧❡♠

h = α1x1 + · · · + αnxn (mod Q) ✇✐t❤ x1, . . . , xn ∈ {0, 1} ❛♥❞ α1, . . . , αn ∈ Z/QZn✳

• ✐✈❡♥ Q, h ❛♥❞ α1, . . . , αn✱ r❡❝♦✈❡r x1, . . . , xn✳

✷ ✴ ✶✽

❍✐❞❞❡♥ ❙✉❜s❡t ❙✉♠ Pr♦❜❧❡♠

h1 = α1x1,1 + · · · + αnxn,1 (mod Q) ✳ ✳ ✳ hm = α1x1,m + · · · + αnxn,m (mod Q) ✇✐t❤ xi,j ∈ {0, 1} ❛♥❞ α1, . . . , αn ∈ Z/QZn✳

• ✐✈❡♥ Q ❛♥❞ h1, . . . , hm✱ r❡❝♦✈❡r α1, . . . , αn ❛♥❞ xi,j ❢♦r

i ∈ [n] ❛♥❞ j ∈ [m]✳ The weights αi’s are hidden!!

✸ ✴ ✶✽

❍✐❞❞❡♥ ❙✉❜s❡t ❙✉♠ Pr♦❜❧❡♠

h1 = α1x1,1 + · · · + αnxn,1 (mod Q) ✳ ✳ ✳ hm = α1x1,m + · · · + αnxn,m (mod Q) ✇✐t❤ xi,j ∈ {0, 1} ❛♥❞ α1, . . . , αn ∈ Z/QZn✳

• ✐✈❡♥ Q ❛♥❞ h1, . . . , hm✱ r❡❝♦✈❡r α1, . . . , αn ❛♥❞ xi,j ❢♦r

i ∈ [n] ❛♥❞ j ∈ [m]✳

h1 · · · · · · hm

• =

α1 · · · αn

• 

   x1,1 · · · · · · x1,m ✳ ✳ ✳ ✳ ✳ ✳ xn,1 · · · · · · xn,m     (mod Q) ✸ ✴ ✶✽

❍✐❞❞❡♥ ❙✉❜s❡t ❙✉♠ Pr♦❜❧❡♠

▲❡t Q ❜❡ ❛♥ ✐♥t❡❣❡r✱ ❛♥❞ ❧❡t α1, . . . , αn ❜❡ r❛♥❞♦♠ ✐♥t❡❣❡rs ✐♥ Z/QZ✳ ▲❡t x1, . . . , xn ∈ Zm ❜❡ r❛♥❞♦♠ ✈❡❝t♦rs ✇✐t❤ ❝♦♠♣♦♥❡♥ts ✐♥ {0, 1}✳ ▲❡t h ∈ Zm s❛t✐s❢②✐♥❣✿ h = α1x1 + · · · + αnxn (mod Q)

• ✐✈❡♥ Q ❛♥❞ h✱ r❡❝♦✈❡r t❤❡ ✐♥t❡❣❡rs αi✬s ❛♥❞ t❤❡ ✈❡❝t♦rs

xi✬s✳ h = α X (mod Q)

✸ ✴ ✶✽

❚✐♠❡❧✐♥❡

✶✾✾✽ ❇♦②❦♦✱ P❡✐♥❛❞♦ ❛♥❞ ❱❡♥❦❛t❡s❛♥ ♣r❡s❡♥t❡❞ ❛ ❢❛st ❣❡♥❡r❛t♦r ♦❢ r❛♥❞♦♠ ♣❛✐rs (x, gx (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 = α1x1 + · · · + αnxn (mod Q) ❚❤❡ ✐❞❡❛✿

• ■❢ ❛ ✈❡❝t♦r u ✐s ♦rt❤♦❣♦♥❛❧ t♦ h ♠♦❞✉❧♦ Q✿

u, h ≡ α1u, x1 + · · · + αnu, xn ≡ 0 (mod Q) ⇒ pu = (u, x1, . . . , u, xn) ✐s ♦rt❤♦❣♦♥❛❧ t♦ α ♠♦❞✉❧♦ Q✳

• ■❢ pu < λ1(Λ⊥

Q(α))✱ ✇❡ ♠✉st ❤❛✈❡ pu = 0✱ ❛♥❞ t❤❡r❡❢♦r❡

t❤❡ ✈❡❝t♦r u ✐s ♦rt❤♦❣♦♥❛❧ ✐♥ Z t♦ ❛❧❧ ✈❡❝t♦rs xi✳

✻ ✴ ✶✽

❚❤❡ ◆❣✉②❡♥✲❙t❡r♥ ❆tt❛❝❦

h

=

α X

(mod Q) ❚❤❡ ❛❧❣♦r✐t❤♠✿

h

=

γ C X (mod Q) ❙t❡♣ ✶ ❙t❡♣ ✷ α

✼ ✴ ✶✽

❚❤❡ ◆❣✉②❡♥✲❙t❡r♥ ❆tt❛❝❦

h

=

α X

(mod Q) ❚❤❡ ❛❧❣♦r✐t❤♠✿

❙t❡♣ ✶ ❋r♦♠ t❤❡ s❛♠♣❧❡s h ❛♥❞ Q✱ ❞❡t❡r♠✐♥❡ t❤❡ ❧❛tt✐❝❡

¯ Lx = (L⊥

x )⊥✱ ✇❤❡r❡ Lx ✐s t❤❡ ❧❛tt✐❝❡ ❣❡♥❡r❛t❡❞ ❜② t❤❡

xi✬s✳ h C

❙t❡♣ ✷ ❋r♦♠ ¯

Lx ⊇ Lx✱ r❡❝♦✈❡r t❤❡ ❤✐❞❞❡♥ ✈❡❝t♦rs xi✬s✳ ❋r♦♠ h✱ t❤❡ xi✬s ❛♥❞ Q✱ r❡❝♦✈❡r t❤❡ ✇❡✐❣❤ts αi✬s✳ X C

α

✼ ✴ ✶✽

❖✉r ❛♥❛❧②s✐s✿ ❙t❡♣ ✶✿

• ❲✐t❤ ♣r♦❜❛❜✐❧✐t② ❛t ❧❡❛st 1/2 ♦✈❡r t❤❡ ❝❤♦✐❝❡ ♦❢ α✱ t❤❡

❛❧❣♦r✐t❤♠ r❡❝♦✈❡rs ❛ ❜❛s✐s ♦❢ ¯ Lx ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✱ ❛ss✉♠✐♥❣ t❤❛t Q ✐s ❛ ♣r✐♠❡ ✐♥t❡❣❡r ♦❢ ❜✐ts✐③❡ ❛t ❧❡❛st 2mn log m✳

• ❋♦r m = 2n✱ ✐❢ t❤❡ ❞❡♥s✐t② ✐s d = n/ log Q = O(1/(n log n))

✇❡ r❡❝♦✈❡r ¯ Lx ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳

• ❍❡✉r✐st✐❝❛❧❧②✱ d = O(1/n) ✐s s✉✣❝✐❡♥t✳

❙t❡♣ ✷✿

• ❚❤❡ xi✬s ❛r❡ s❤♦rt ✈❡❝t♦rs ♦❢ ¯

Lx✳

• ❯s✐♥❣ ❇❑❩ t❤❡ ❛s②♠♣t♦t✐❝ ❝♦♠♣❧❡①✐t② ✐s 2Ω(n/ log n)✳

✽ ✴ ✶✽

❖✉r ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ ❛❧❣♦r✐t❤♠

h

=

γ C X (mod Q) ❙t❡♣ ✶ ❙t❡♣ ✷ α

• ❲❡ r❡q✉✐r❡ m ≈ (n2 + n)/2 ✐♥st❡❛❞ ♦❢ m = 2n✳
• ■♠♣r♦✈❡❞ st❡♣ ✶✿ ❢❛st ❣❡♥❡r❛t✐♦♥ ♦❢ ♦rt❤♦❣♦♥❛❧ ✈❡❝t♦rs✳
• ◆❡✇ st❡♣ ✷✿ r❡❝♦✈❡r ❜✐♥❛r② ✈❡❝t♦rs✳

✾ ✴ ✶✽

◆❡✇ st❡♣ ✷✿ ♠✉❧t✐✈❛r✐❛t❡ ❛♣♣r♦❛❝❤

■♥❣r❡❞✐❡♥ts✿

• Lx ✐s ❛ s✉❜❧❛tt✐❝❡ ♦❢ ¯

Lx✿ t❤❡r❡ ❡①✐sts W ∈ Zn×n ∩ GL(Q, n) X = W C

• ❇❡✐♥❣ ❜✐♥❛r② ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ❝♦♥❞✐t✐♦♥✿

y ∈ {0, 1} ⇐ ⇒ y2 − y = 0

✶✵ ✴ ✶✽

▼✐①✐♥❣✳✳✳ xi = wi ˜ c1 · · · ˜ cm ❋♦r ❡❛❝❤ i = 1, . . . , n ❛♥❞ j = 1, . . . , m ✇❡ ❤❛✈❡

• xi,j ∈ {0, 1}
• wi · ˜

cj = xi,j = ⇒ (wi · ˜ cj)2 − wi · ˜ cj = 0

✶✶ ✴ ✶✽

❚❤❡ r♦✇s ♦❢ W ❛r❡ s♦❧✉t✐♦♥s ♦❢ ♠✉❧t✐✈❛r✐❛t❡ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ s②st❡♠        w · ˜ c1˜ c⊺

1 · w⊺ − w · ˜

c1 = 0 ✳ ✳ ✳ w · ˜ cm˜ c⊺

m · w⊺ − w · ˜

cm = 0

• ❋♦r m ≈ (n2 + n)/2 ✇❡ ❡①♣❡❝t t♦ s♦❧✈❡ t❤✐s s②st❡♠ ❛♥❞

r❡❝♦✈❡r t❤❡ xi✬s ❜② O(n6) ❜✐t ♦♣❡r❛t✐♦♥s ❛♥❞ O(n4) s♣❛❝❡ ❝♦♠♣❧❡①✐t②✱ ✈✐❛ ❧✐♥❡❛r ❛❧❣❡❜r❛✳

✶✷ ✴ ✶✽

❚❤❡ ❝♦❡✣❝✐❡♥t ♦❢ wiwk ✐♥ t❤❡ jt❤ ❡q✉❛t✐♦♥ ✐s (2 − δi,k)CijCkj✳ C R

• −C

E

• 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❛♥❦ n2+n

2

✱ t❤❡♥ t❤❡ ✈❡❝t♦rs xi✬s ❝❛♥ ❜❡ r❡❝♦✈❡r❡❞ ✐♥ O(n6) ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s✳ ❘❡❞✉❝✐♥❣ t❤❡ ♠❛tr✐① r❡❧❛t✐♦♥ X = WC mod p ✇❡ ❝❛♥ ♦❜t❛✐♥ ❛ s②st❡♠ ❞❡✜♥❡❞ ♦✈❡r Fp✳ ⇒ ❋♦r m ≈ (n2 + n)/2 ✇❡ ❡①♣❡❝t t♦ s♦❧✈❡ t❤✐s s②st❡♠ ❛♥❞ r❡❝♦✈❡r t❤❡ xi✬s ❜② O(n6) ❜✐t ♦♣❡r❛t✐♦♥s ❛♥❞ O(n4) s♣❛❝❡ ❝♦♠♣❧❡①✐t②✱ ✈✐❛ ❧✐♥❡❛r ❛❧❣❡❜r❛✳

✶✹ ✴ ✶✽

❈♦♠♣❛r✐s♦♥

◆❙✾✾ ❈●✷✵ m 2n (n2 + n)/2 log Q O(n2) O(n2) t✐♠❡ 2Ω(n/ log n) O(n9)

Nguyen-Stern Coron-Gini

18 26820

sec

100 150 200 250 number of variables

Experimental timing comparison

✶✺ ✴ ✶✽

❆✣♥❡ ❍✐❞❞❡♥ ❙✉❜s❡t ❙✉♠ Pr♦❜❧❡♠

h + se = α1x1 + · · · + αnxn (mod Q)

• ✐✈❡♥ Q✱ h ❛♥❞ e✱ r❡❝♦✈❡r t❤❡ αi✬s ❛♥❞ s ❛♥❞ t❤❡ ✈❡❝t♦rs xi✬s✳

❙t❡♣ ✶ ❋r♦♠ (h, e)✱ ❞❡t❡r♠✐♥❡ t❤❡ ❧❛tt✐❝❡ ¯

Lx✱ ✇❤❡r❡ Lx ✐s t❤❡ ❧❛tt✐❝❡ ❣❡♥❡r❛t❡❞ ❜② t❤❡ xi✬s✳

❙t❡♣ ✷ ❋r♦♠ ¯

Lx ⊇ Lx✱ r❡❝♦✈❡r t❤❡ ❤✐❞❞❡♥ ✈❡❝t♦rs xi✬s✳ ❋r♦♠ h✱ t❤❡ xi✬s ❛♥❞ Q✱ r❡❝♦✈❡r t❤❡ ✇❡✐❣❤ts αi✬s ❛♥❞ s✳

✶✻ ✴ ✶✽

❈♦♥❝❧✉s✐♦♥s

• ❲❡ ❛r❣✉❡ t❤❛t t❤❡ ❛s②♠♣t♦t✐❝ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ❢✉❧❧

◆❣✉②❡♥✲❙t❡r♥ ❛❧❣♦r✐t❤♠ ✐s 2Ω(n/ log n)✳

• ❲❡ ♣r♦♣♦s❡ ❛ ♥❡✇ s❡❝♦♥❞ st❡♣ t♦ r❡❝♦✈❡r s❤♦rt ✭❜✐♥❛r②✮

✈❡❝t♦rs ✉s✐♥❣ m ≃ n2/2 s❛♠♣❧❡s✱ ✈✐❛ ❛ ♠✉❧t✐✈❛r✐❛t❡ t❡❝❤♥✐q✉❡✳

• ❆s②♠♣t♦t✐❝❛❧❧② t❤❡ ❤❡✉r✐st✐❝ ❝♦♠♣❧❡①✐t② ♦❢ ♦✉r ❢✉❧❧

❛❧❣♦r✐t❤♠ ✐s O(n9)✳ ❈❛♥ ✇❡ ❢✉rt❤❡r r❡❞✉❝❡ m❄ ❛♥❞ log Q❄ ■♥ ❛❞❞✐t✐♦♥✱ ✇❡ s❤♦✇ ❤♦✇ t♦ s❧✐❣❤t❧② r❡❞✉❝❡ t❤❡ ♥✉♠❜❡r ♦❢ s❛♠♣❧❡s m r❡q✉✐r❡❞ ❢♦r ♦✉r ❛tt❛❝❦✱ ✇✐t❤ t✇♦ ❞✐✛❡r❡♥t ♠❡t❤♦❞s❀ ✐♥ ❜♦t❤ ❝❛s❡s t❤❡ ❛tt❛❝❦ r❡♠❛✐♥s ❤❡✉r✐st✐❝❛❧❧② ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✉♥❞❡r t❤❡ ❝♦♥❞✐t✐♦♥ m = n2/2 − O(n log n)✳

✶✼ ✴ ✶✽

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

❋✉❧❧ ♣❛♣❡r ❛t ❤tt♣s✿✴✴✐❛✳❝r✴✷✵✷✵✴✹✻✶ ❈♦❞❡ ❛t ❤tt♣s✿✴✴♣❛st❡❜✐♥✳❝♦♠✴❩❋❦✶q❥❢P

✶✽ ✴ ✶✽