Encodings

1
Encodings

• Low-Degree Polynomials
• and
• Consistency

2
Strings Functions

• One can describe an n-bit string s ? ?ns s1,
  s2, , sn
• as a function from an index to the letter
  appearing on the indexs entry of the string
• s 1..n a ?
• On the other hand, a function D a Rcan be
  thought of as a string ? RD spelling s
  value on each point of D

3
Error Correcting Codes

• An encoding E maps strings of length n to strings
  of length m (gtgt n)
• E ?n ?m
• It is an ?-code if for any s1 ¹ s2 ? ?n D(
  Es1, Es2 ) ³ 1 - ?
• (where D( x, y ) denotes the fraction of entries
  x and y differ on)

4
A Generic ?-code

• Set ? to be the finite field Zp (for prime p) and
  assume, for simplicity, that ? ? and m p
• Given s ? ?n , let Es be the (string-ization of
  the) function ? ? that satisfies is the
  (unique) degree n-1 polynomial so that (t)
  st for all 1 t n
• Es can be interpolated from any n points (being
  a univariate polynomial of degree n-1) hence,
  Es1 Es2 may agree on at most n-1 points
• Therefore, E is an n-1 / ?-code

5
Polynomials over Finite-Fields

• Let now ? ? d be a geometric-space
• (this is to say that no complex properties of ?
  would be used, only simple, geometric ones)
• denote by d the dimension of ?.
• Consider now a polynomial
• ? a ?
• of degree h in each variable
• hd is s total degree (or simply degree).

6
Useful Properties of Low-Degree Polynomials

• Interpolation the value of on any point of ?
  can be interpolated from many (almost all) sets
  of (h1)d points of ?.
• In fact, s value is some linear combination
  of the values of on those (h1)d points
• If, however, ? a ? is not of degree d(h1),
  interpolation of a single point p ? ? from
  different sets of (h1)d points may give
  different valuesnamely be inconsistent

7
More Useful Properties of Low-Degree Polynomials

• ?-codeTwo distinct polynomials, 1 and 2, can
  agree on at most a very small (?) fraction of ?,
  i.e.
• D( 1 , 2 ) ³ 1 - ?
• Figure out what ? is.
• That is, what is the largest fractionof points
  in ? two distinctdegree-r polynomials may agree
  on
• (Enough to analyze the case were is of degree
  r in each variable)

8
Geometrical Properties of Low-Degree Polynomials

• Show the restriction of a degree-r polynomial,
  to any dimension-d affine-subspace (i.e., cube
  space shifted by an arbitrary vector), is a
  dimension-d degree-r polynomial
• Show the restriction of a degree-r polynomial to
  a curve ? ? a ? of degree k ( ?(t) )is a
  univariate polynomial of degree rk

9
Def Low-Degree-Extension (LDE)

• Let us now extend the generic encoding above of
  univariate polynomials to higher dimension
• Assume H ? ? ( Hltlt? ) and d logHn (thus n
  ? Hd) and an isomorphism Hd ? 1..n say by
  writing each index in baseH A string s is
  describable as a d-ary function s Hd a ?
• Def LDE(s) ?d a ? such that
• extension LDE(s) agrees with s on Hd
• low-degree LDE(s) is of degree H-1 in each
  variable (i.e., of degree (H-1)d )

10

• Write down the low-degree-extension of a string s
  -- LDE(s) -- as an explicit function of its
  entries s1, .., sn

11
Consistent Reading

• We need to be able to read a value of an LDE in a
  globally consistent manner
• That is, have a representation scheme (a set of
  variables) for LDEs, and a reading procedure,
  that for any arbitrary point p??, accesses a very
  small number of representation variables and
• rejects if inconsistency detected
• otherwise, returns w.h.p. a value for LDEs(p),
  consistent among all points p

12
Consistency

• Simplistic procedure interpolate value from a
  random set of hd points
• This, however, requires hd accesses, while we
  would need a consistent-reader that accesses only
  a very small ( preferably poly-log(n), ultimately
  constant) number of variables, and be guaranteed
  global consistency

13
Consistency Test for Low-degree Polynomials

• A consistency test requires only that value for a
  random p?? (not an arbitrary one) corresponds to
  a global degree-r polynomial
• Fix a Representation a set of variables
• A test is a family of Boolean-functions over
  representation variables,each depending on a
  small set of variables - hence referred to as
  local-test - which may
• reject if detects an inconsistency
• accept w.h.p. values conform to global
  consistency

14
Global Consistency

• Pure global consistency would be for all entries
  to be assigned values consistent with a single
  low-degree polynomial
• Accessing only a small number of variables,
  however, small deviations (say changing one
  value) can be detected only with very small
  probability
• Part of the definition of a test would be a
  weaker notion of global consistency

15
Corresponding Game

• Prover sets values to all variables in the
  representation
• Verifier picks randomly a single local-test and
  accepts or rejects according to its outcome
• The error-probability of a test, is the fraction
  of local-tests that can be made True despite the
  assigned values not conforming to global
  consistency.
• I.e. the probability the verifier erroneously
  accepts