Chinese Remaindering with Errors

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Chinese Remaindering with Errors

• Alex Bulach

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Chinese Remaindering with Errors

• What about?
• General Information
• Prerequisites
• Small Errors
• Large Errors
• Small Errors vs. Large Errors

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Chinese Remaindering with Errors

• What about? Secret Sharing

• Aim Sharing information so
• Any large enough collection of parties can
  efficiently recover the secrets
• No small coalition of parties can obtain any
  information about the secret

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Chinese Remaindering with Errors

• General Information

• Codewords
• Distance d between to codewords
• Can detect up to d
• errors
• Can fix only d/2 errors

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Chinese Remaindering with Errors

• General Information

• Dist. d n-k (redundant information)
• e (n-k) /2 max. number of error.
• Proof
• d(u,v) 0 d(u,v) 0 iff u v
• d(u,v) d(v,u) for all u,v
• d(u,v) d(u,w) d(w,v) for all u,v,w

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Chinese Remaindering with Errors

• General Information

• Proof (contd.)
• d(C) mind(u,v)u,v ? C, u ? v
• d(C) 2e1
• d(c,r) e gt d(c1,r) e1
• Assume d(c1,r) e
• 2e1 d(C) d(c,c1) d(c,r) d(c1,r) e e

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Chinese Remaindering with Errors

• Prerequisites

• n remainder p1 pn
• p1 lt p2 lt pn
• k with k lt n
• Message m with m lt ?i0, i k pi
• pn p1O(1)

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Chinese Remaindering with Errors

• Small Errors

• Error e lt (n-k)/2
• N ?i0, im pi
• K ?i0, ik pi
• Given (p1, , pn, k, r1, , rn)
• Assumption either
• mpi ri or
• ypi 0

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Chinese Remaindering with Errors

• Small Errors

• Find r with CRT so that r ? ZN and r rpi
• Two Steps
• find integers y and z so that.
• 1 y E
• 0 z lt N/E
• y r z (mod N)
• Output z/y

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Chinese Remaindering with Errors

• Small Errors

• m 17
• p1 2
• p2 3
• p3 5
• p4 7
• p5 13

mp1 1 mp2 2 mp3 2 mp4 3 mp5 4
Bad Example! Why?
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Chinese Remaindering with Errors

• Small Errors

• n 5
• k 3
• N 2730
• K 30
• E lt v(N/(K-1))
• E 9
• N/E 303

mp1 1 mp2 2 mp3 2 mp4 3 mp5 4
r1 1 r2 1 r3 2 r4 3 r5 4
Amplitude of Distance 3
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Chinese Remaindering with Errors

• Small Errors

• CRT gt r 1837
• r ? m

• y Amplitude of Distance

? ?
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Chinese Remaindering with Errors

• Small Errors

• CRT gt r 1837
• r ? m
• z 51, y 3
• z/y 51/3 17 m

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Chinese Remaindering with Errors

• Small Errors

• Why? How?

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Chinese Remaindering with Errors

• Small Errors

• Set y ?ri ?mp1 pi (Amplitude of Dist.)
• z y m
• Notice
• y ? 0 y E
• m K-1

z y m (K-1)E
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Chinese Remaindering with Errors

• Small Errors

• z y m (K-1)E
• E lt N / ((K-1)E) gt (K-1)Elt N/E
• z 0
• y r z (mod N)
• gt z ym yr (mod pi)

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Chinese Remaindering with Errors

• Small Errors

• For every mpi ri
• ym ympi yri yr (mod pi)
• Set T ?ri mp1 pi
• (Amplitude of Agreement)
• T N/E
• CRT ym z (mod T)
• But z lt N/E and my (K-1)E lt N/E

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Chinese Remaindering with Errors

• Large Errors

• Error e gt (n-k)/2
• N ?i0, im pi
• K ?i0, ik pi
• F 2(l2)/2 v(l2) N1/(l1) K(l1)/2
• l ? v(2nlog pn)/k log p1 ? (optimal)
• Given (p1, , pn, k, r1, , rn)
• Find r with CRT so that r ? ZN and r rpi

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Chinese Remaindering with Errors

• Large Errors

• CRT gt r
• Two Steps
• find integers c so that.
• For 0 i l ci lt (F/Ki)
• And ?i0, il ciri 0 mod N
• ltc0, , cngt ? lt0, , 0gt
• Output All roots for polynomial
• C(x) ?i0, il cixi

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Chinese Remaindering with Errors

• Large Errors

• n 5
• k 3
• N 2730
• K 30
• F 18372,13
• l 2

mp1 1 mp2 2 mp3 2 mp4 3 mp5 4
er1 0 er2 1 er3 2 er4 3 er5 4
Amplitude of Distance 6
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Chinese Remaindering with Errors

• Large Errors

• Note that
• ?cjmj (l1)maxcjmj
• (l1)maxcjKj
• (l1)maxF

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Chinese Remaindering with Errors

• Large Errors

• For every mpi ri (Slide 16)
• ?cjmj ?cj mjpi ?cjrjj ?cjrj
• 0(mod pi)
• Set P ?ri mp1 pi
• CRT gt ?j0 j l cjmj 0(mod P)
• Pgt2(l1)F gt ?j0 j l cjmj 0

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Chinese Remaindering with Errors

• Small vs. large error algorithm

• Small error
• geb r(Nachricht), N(?pi), E(Fehlertoleranz)
• ges y, z mit yr z mod N
• Large errors
• geb r(Nachricht), N(?pi), E(Fehlertoleranz)
• ges ci mit ?ciri 0 mod N

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Chinese Remaindering with Errors

• Small vs. large error algorithm

• Small
• yr z mod N
• z-yr 0 mod N

• Large (mit l 1)
• ?ciri 0 mod N
• c0 c1r 0 mod N

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Chinese Remaindering with Errors

• Small vs. large error algorithm

• Small Errors
• yr z mod N gt yr z dN gt yr - dN z

(
)
1 r 0 N
B
(y -k)B (y, yr-dN) (y,z)
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Chinese Remaindering with Errors

• Small vs. large error algorithm

• Large Errors (l 2)
• yrxr² z mod N gt yrxr² z dN
• gt yrxr² -dN z mod N

(
)
1 0 r 0 1 r² 0 0 N
B
(y x -k)B (y,x,yrxr2-dN) (y,x,z)