2. 1

3. 2 Introduction In this lecture we’ll cover: Definition of strings as functions and vice versa Error correcting codes Low degree polynomials Low degree extension Consistent readers Consistency tests

4. 3 Strings & Functions Let  =  1  2...  3, where  i . We can describe the string  as a function  : [1…n]  , such that  i  (i) =  i. Let f be a function f : D  R. Then f can be described as a string over the alphabet  = R |D|, spelling f’s value on each point of D.

5. 4 Strings & Functions - Example For example, let f be a function f : Z 5  Z 5, and let  = Z 5. f(x) = x 2   = 0, 1, 4, 4, 1

6. 5 Error Correcting Codes Definition (encoding): An encoding E is a function E :  n   m, where m >> n. Definition (  -code): An encoding E is an  -code if  n  (E(  ),E(  ))  1 - , where  (x,y), denotes the fraction of entries on which x and y differ.

7. 6 A Generic  -code Set F to be the finite field Z p for some prime p, and assume for simplicity that  = F and m = p. Given  n, let E(  ) be the string of the function f  : F  F that satisfies: f  is the unique n – 1 degree polynomial such that f  (i - 1) =  i for all 1  i  n.

8. 7 A Generic  -code (2) E(  ) can be interpolated from any n points. Hence, for any , E(  ) and E(  ) may agree on at most n – 1 points. Therefore, E is an (n – 1) / m - code.

9. 8 A Generic  -code - Example p = m = 5, n = 2  = 1, 2  = 3, 1 f  (x) = x + 1f  (x) = 3x + 3 E(  ) = 1, 2, 3, 4, 0E(  ) = 3, 1, 4, 2, 0

10. 9 Polynomials Over Finite Fields Let V = F d be a geometric space of dimension d. Let p be a polynomial p : V  F of degree h in each variable. The total degree of p is h  d.

11. 10 Properties of Low Degree Polynomials The value of p on any point of V can be interpolated from many (almost all) sets of (h+1) d points of V. (This is true when the points are linearly independent). If p is of a degree greater than h  d, then interpolation of a single point x  V from different sets of (h+1) d points may give different values (be inconsistent).

12. 11 Properties of Low Degree Polynomials (2) Two distinct polynomials, p 1 and p 2, can agree on at most a very small (  ) fraction of V, i.e.  (p 1,p 2 )  1 - .

13. 12 Properties of Low Degree Polynomials (3) The restriction of a polynomial of degree h in each variable to an affine sub-space of dimension d’ is a polynomial of dimension d’ and degree h in each variable. The restriction of a polynomial p of degree h in each variable to a curve  : F  V of degree k, p(  (t)) is a univariate polynomial of degree h  k.

14. 13 Low Degree Extension (LDE) Definition (low degree extension): Assume H  F, such that |H| << |F|, and let d = log |H| n. A string  n is describable as a function  : H d  . LDE(  ) is a function LDE(  ) : F d  F such that:  LDE(  ) agrees with  on H d (extension).  LDE(  ) is of degree |H| - 1 in each variable (low degree).

15. 14 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 LDE(  ), and a reading procedure, that for any x  V accesses a very small number of representation variables and:  Rejects if inconsistency detected.  Otherwise, returns with high probability a value for LDE(  )(x), consistent among all points x.

16. 15 Consistency A simple procedure: interpolate value from a set of (h+1) d points. This, however, requires (h+1) d points, while we would need a consistent-reader which accesses only a very small (preferably poly-log(n), ultimately constant) number of variables.

17. 16 Consistency Test A consistency test requires only that value for a random x  V corresponds to a global degree-h polynomial. Notice that this is a weaker requirement than the one stated in the consistent reader definition, since it allows some x’s to be accepted with high probability although they do not agree with the global consistency, as long as the probability over all x’s remains low.

18. 17 Consistency Test (2) 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 a local test – which may:  Reject if inconsistency detected.  Accept with high probability values conform to global consistency.

19. 18 Consistency Test (3) P(0,0,0)P(0,0,1)P(0,0,2)P(0,0,3)P(0,0,4)P(0,0,5)P(0,0,6) P(0,1,0)P(0,1,1)P(0,1,2)P(0,1,3)P(0,1,4)P(0,1,5)P(0,1,6) P(0,2,0)P(0,2,1)P(0,2,2)P(0,2,3)P(0,2,4)P(0,2,5)P(0,2,6) P(0,3,0)P(0,3,1)P(0,3,2)P(0,3,3)P(0,3,4)P(0,3,5)P(0,3,6) P(6,6,0)P(6,6,1)P(6,6,2)P(6,6,3)P(6,6,4)P(6,6,5)P(6,6,6) P(0,0,0)P(0,0,1)P(0,0,2)P(0,0,3)P(0,0,4)P(0,0,5)P(0,0,6) 3 P(0,1,1)P(0,1,2)P(0,1,3)P(0,1,4)P(0,1,5)P(0,1,6) P(0,2,0)P(0,2,1) 5 P(0,2,3)P(0,2,4)P(0,2,5)P(0,2,6) P(0,3,0)P(0,3,1)P(0,3,2)P(0,3,3)P(0,3,4)P(0,3,5)P(0,3,6) P(6,6,0)P(6,6,1)P(6,6,2)P(6,6,3) 2 P(6,6,5)P(6,6,6)

20. 19 Global Consistency Definition (pure global consistency): Pure global consistency would be for all x  V to be assigned values consistent with a single low degree polynomial. This cannot be detected by a local test, since changing the values in a small fraction of points will be detected only with low probability.

21. 20 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 output. The error-probability of a test is the fraction of local tests that may accept although the assigned values do not conform to global consistency.