Locally Decodable Codes

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Locally Decodable Codes
Uri Nadav
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Contents

• What is Locally Decodable Code (LDC) ?
• Constructions
• Lower Bounds
• Reduction from Private Information Retrieval
  (PIR) to LDC

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Minimum Distance

• For every x?y that satisfy d(C(x),C(y)) d
• Error correction problem is solvable for less
  than d/2 errors
• Error Detection problem is solvable for less than
  d errors

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Error-correction
Codeword
Encoding
x
C(x)
Input
Worst case error assumption
Errors
Corrupted codeword
y
Decoding
i
xi
Bit to decode
Decoded bit
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Query Complexity

• Number of indices decoder is allowed to read from
  (corrupted) codeword
• Decoding can be done with query complexity
  O(C(x))
• We are interested in constant query complexity

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Adversarial Model

• We can view the errors model as an adversary that
  chooses positions to destroy, and has access to
  the decoding/encoding scheme (but not to random
  coins)

The adversary is allowed to insert at most ?m
errors
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Why not decode in blocks?

• Adversary is worst case so it can destroy more
  than d fraction of some blocks, and less from
  others.

Nice errors
Worst Case
Many errors in the same block
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Ideal Code C0,1n??m

• Constant information rate n/m gt c
• Resilient against constant fraction of errors
  (linear minimum distance)
• Efficient Decoding (constant query complexity)

No Such Code!
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Definition of LDC
C0,1n??m is a (q,?,?) locally decodable code
if there exists a prob. algorithm A such that
?x ? 0,1n, y ? ?m with distance d(y,C(x))lt?m
and ?i ? 1,..,n, Pr A(y,i)xi gt ½ ?
The Probability is over the coin tosses of A
A reads at most q indices of y (of its choice)
Queries are not allowed to be adaptive
A has oracle access to y
A must be probabilistic if qlt ?m
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Example Hadamard Code

• Hadamard is (2,d, ½ -2d) LDC
• Construction

Relative minimum distance ½
Encoding
x1
x2
xn
ltx,1gt
ltx,2gt
ltx,2n-1gt
source word
codeword
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Example Hadamard Code

• Reconstruction

Pick a?R0,1n
2 queries
reconstruction formula
ltx,agt
ltx,aeigt


Decoding
ltx,1gt
x1
x2
xn
ltx,2gt
ltx,2n-1gt
source word
codeword
If less than d fraction of errors,
then reconstruction probability is at least 1-2d
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Another Construction
Probability of 1-4? for correct decoding
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Generalization
2k queries m2kn1/k
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Smoothly Decodable Code
C0,1n??m is a (q,c,?) smoothly decodable code
if there exists a prob. algorithm A such that
1
?x ? 0,1n and ?i ? 1,..,n, Pr A(C(x),i)xi
gt ½ ?
The Probability is over the coin tosses of A
A has access to a non corrupted codeword
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A reads at most q indices of C(x) (of its choice)
Queries are not allowed to be adaptive
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?i ? 1,..,n and ?j ? 1,..,m, Pr A(,i)
reads j c/m
The event is A reads index j of C(x) to
reconstruct index i
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LDC is also Smooth Code

• Claim Every (q,d,e) LDC is a (q,q/d,e) smooth
  code.
• Intuition If the code is resilient against
  linear number of errors, then no bit of the
  output can be queried too often (or else
  adversary will choose it)

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Proof LDC is Smooth

• A - a reconstruction algorithm for (q,d,e) LDC
• Si j PrA query j gt q/dm
• There are at mostq queries, so sum of prob. over
  j is q , thus Si lt dm

Set of indices read too often
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ProofLDC is Smooth

• A uses A as black box, returns whatever A
  returns as xi
• A gives A oracle access to corrupted codeword
  C(x), return only indices not in S

• A reconstructs xi with probability at least 1/2
  e, because there are at most Si lt dm errors

A is a (q,q/d, e) Smooth decoding algorithm
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Proof LDC is Smooth
indices that A reads too often
C(x)
what A wants
A
what A gets
C(x)
0
0
0
indices that A fixed arbitrarily
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Smooth Code is LDC

• A bit can be reconstructed using q uniformly
  distributed queries, with e advantage , when no
  errors
• With probability (1-qd) all the queries are to
  non-corrupted indices.

Remember Adversary does not know decoding
procedures random coins
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Lower Bounds

• Non existence for q 1 KT
• Non linear rate for q 2 KT
• Exponential rate for linear code, q2 Goldreich
  et al
• Exponential rate for every code, q2
  Kerenidis,de Wolf (using quantum arguments)

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Information Theory basics

• Entropy

H(x) -?Prxi log(Prxi)

• Mutual Information

I(x,y) H(x)-H(xy)
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Information Theory cont

• Entropy of multiple variable is less than the sum
  of entropies! (equal in case of all variables
  mutually independent
• H(x1x2xn) ? H(xi)

• Highest entropy is of a uniformly distributed
  random variable.

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IT result from KT
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Proof

• Combined

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Single query (q1)
Claim If C0,1n??m, is (1,d,e) locally
decodable then
No such family of codes!
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Good Index

• Index j is said to be good for i, if
• PrA(C(x),i)xi A reads j gt ½ e

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Single query (q1)
By definition of LDC
Conditional prob. summing over disjoint events

• There exist at least a single j1 which is good
  for i.

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Perturbation Vector

• Def Perturbation vector ?j1,j2, takes random
  values uniformly distributed from ?, in position
  j1,j2, and 0 otherwise.

0
0
j1 ?
0
0
j2 ?
0
Destroys specified indices in most unpredicted
way
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Adding perturbation
A resilient Against at least 1 error
So, there exists at least one index, j2 good
for i.
j2 ? j1 , because j1 can not be good!
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Single query (q1)
A resilientAgainst dm errors
So, There are at least dm indices of The
codeword good for every i. By pigeonhole
principle , there exists an index j in 1..m,
good for dn indices.
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Single query (q1)

• Think of C(x1..dn) projected on j as a
  function from the dn indices of the input. The
  range is ?, and each bit of the input can be
  reconstructed w.p. ½ e. Thus by IT result

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Case q2

• m O(n)q/(q-1)
• Constant time reconstruction procedures are
  impossible for codes having constant rate!

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Case q2 Proof Sketch

• A LDC C is also smooth
• A q smooth codeword has a small enough subset of
  indices, that still encodes linear amount of
  information
• So, by IT result, m(q-1)/q O(n)

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Applications?

• Better locally decodable codes have applications
  to PIR
• Applications to the practice of fault-tolerant
  data storage/transmission?

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What about Locally Encodable

• A Respectable Code is resilient against O(m)
  fraction of errors.
• We expect a bit of the encoding to depend on many
  bits of the encoding

Otherwise, there exists a bit which influence
less than 1/n fraction of the encoding.
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Open Issues

• Adaptive vs Non-Adaptive Queries

guess first q-1 answers with succeess probability
?q-1

• Closing the gap

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Logarithmic number of queries

• View message as polynomial pFk-gtF
• of degree d (F is a field, F gtgt d)
• Encode message by evaluating p at all Fk points
• To encode n-bits message, can have F polynomial
  in n, and d,k around
• polylog(n)

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To reconstruct p(x)

• Pick a random line in Fk passing through x
• evaluate p on d1 points of the line
• by interpolation, find degree-d univariate
  polynomial that agrees with p on the line
• Use interpolated polynomial to estimate p(x)
• Algorithm reads p in d1 points, each uniformly
  distributed

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x(d1)y
x2y
xy
x
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Private Information Retrieval (PIR)

• Query a public database, without revealing the
  queried record.
• Example A broker needs to query NASDAQ database
  about a stock, but dont wont anyone to know he
  is interested.

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PIR

• A k server PIR scheme of one round, for database
  length n consists of

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PIR definition

• These function should satisfy

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Simple Construction of PIR

• 2 servers, one round
• Each server holds bits x1,, xn.
• To request bit i, choose uniformly A subset of
  n
• Send first server A.
• Send second server Ai (add i to A if it is not
  there, remove if is there)
• Server returns Xor of bits in indices of request
  S in n.
• Xor the answers.

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Lower Bounds On Communication Complexity

• To achieve privacy in case of single server, we
  need n bits message.
• (not too far from the one round 2 server scheme
  we suggested).

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Reduction from PIR to LDC

• A codeword is a Concatenation of all possible
  answers from both servers
• A query procedure is made of 2 queries to the
  database