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1 Overview Some basic math Error correcting codes Low degree polynomials Introduction to consistent readers and consistency tests H.W 2 Fields +,·,0, 1, -a and a-1 are only notations! Definition (field): A set F with two binary operations + (addition) and · (multiplication) is called a field if 1 a,bF, a+bF 2 a,b,cF, (a+b)+c=a+(b+c) 3 a,bF, a+b=b+a 4 0F, aF, a+0=a 5 aF, -aF, a+(-a)=0 6 a,bF, a·bF 7 a,b,cF, (a·b)·c=a·(b·c) 8 a,bF, a·b=b·a 9 1F, aF, a·1=a 10 a0F, a-1F, a·a-1=1 11 a,b,cF, a·(b+c)=a·b+a·c 3 Finite Fields Definition (finite field): A finite set F with two binary operations + (addition) and · (multiplication) is called a finite field if it is a field. Example: Zp denotes {0,1,...,p-1}. We define + and · as the addition and multiplication modulo p respectively. One can prove that (Zp,+,·) is a field iff p is prime. Throughout the presentations we’ll usually refer to Zp when we’ll mention finite fields. 4 Strings & Functions (1) Let = 0 2 . . . n-1, where i. We can describe the string as a function : {0…n-1} , such that i (i) = i. Let f be a function f : D R. Then f can be described as a string in R|D|, spelling f’s value on each point of D. 5 Strings & Functions - Example For example, let f be a function f : Z5 Z5, and let = Z5. f(x) = x2 = 0, 1, 4, 4, 1 6 Introduction to Error Correcting Codes Motivation: original message 1001110 “noise” 1 1001110 received message 1101110 communication line We’d like to still be able to reconstruct the original message 7 Error Correcting Codes Note that : R is indeed a distance function, because it satisfies: m m + m (x,y)0 and (x,y)=0 iff x=y (1) x,y Definition (encoding): An encoding E is a m (x,y)=(y,x) function E : n(2)x,y m, where m >> n. (3) x,y,zm (x,z)(x,y)+(y,z) Definition (-code): An encoding E is an -code if n (E(),E()) 1 - , where (x,y) (the Hamming distance), denotes the fraction of entries on which x and y differ. 8 -code: illustration E1- D R 9 Univariate Polynomials Definition (univariate polynomial): a polynomial in x over a field F is a function P:FF, which can be written as r 1 P( x) a j x j j 0 for some series of coefficients a0,...,ar-1F. The natural number r is called the degree-bound of the polynomial. Note: A polynomial whose degree-bound is r is of degree at most r-1 ! 10 If there are two such polynomials: p1 & p2, then p1-p2 is a polynomial with degree-bound r, which has r roots. This contradicts the fundamental theorem of Algebra! Univariate Interpolation Given x0,y0,...,xr-1,yr-1F there is a single univariate polynomial P and degree-bound r, which satisfies 0kr-1 P(xk)=yk (((xxx( x xxxjj )x)j )j ) tt jjjkjkk k P( x)t)) yt yyykkyk k 0 (((xxx(kkxkkxxxjjx)j))j ) kk k00k0 t jjj kkjk k 11 rrr1 (Lagrange’s formula) (x xj) j t yt ( xt x j ) j t t a-b denotes a+(-b) a/b denoted a•(b-1) Since the degree-bound Let’s check the value of of this polynomial is r, we process the coefficients of a thisThe polynomial in xof= finding xt in fact proved the polynomial for some 0 t given r-1: its value in r points is called correctness of the formula interpolation. 11 A Generic -code Set F to be the finite field Zp 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 polynomial of degree-bound n such that f(i) = i for all 0 i n-1. 12 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. 13 A Generic -code - Example p = m = 5, n = 2 = 1, 2 = 3, 1 f(x) = x + 1 f(x) = 3x + 3 E() = 1, 2, 3, 4, 0 E() = 3, 1, 4, 2, 0 14 Strings & Functions (2) We can describe any string as a function f:Hd H (H is a finite field, d is a positive integer). Given a n we’ll achieve that by choosing H=Zq, where q is the smallest prime greater than ||, and d=logqn. 15 Multivariate Polynomials Definition (polynomial): Let F be a field and let d be some positive integer number. A function p:FdF is a polynomial if it can be written as h 1 h 1 i0 0 id 1 0 p( x0 ,..., xd 1 ) ... ai0 ,..., id 1 x0i0 ... xdid11 for some series of coefficients in the field. h is the degree-bound on each one of the variables. The total-degree of the polynomial is max{ i0+…+id-1 : ai …i 0 }. 0 d-1 16 -Codes - Home Assignment We’ve seen that univariate polynomials over a finite field F with degree-bound r are -codes for = (r-1)/|F|. For which multivariate polynomials (over a finite field F, with degree-bound h in each variable and dimension d) are -codes? Next 17 Curves Definition (curve): Let F be a field and let d be some natural number. A (univariate) curve is a function :F Fd of the form ( x) ( p1 ( x),..., pd ( x)) where p1,...,pd are univariate polynomials over F. The degree-bound of is the maximum over the degree-bounds of the polynomials. 18 Vector Spaces Definition (vector space): Let F be a field and V a set. V is a vector space over F if a binary addition + is defined over V and a scalar multiplication · is defined over V and F s.t 1 u,vV, u+vV 2 u,v,wV, (u+v)+w=u+(v+w) 3 u,vV, u+v=v+u 4 0V, vV, v+0=v 5 vV, -vV, v+(-v)=0 6 vV, aF a·vV 7 u,vV, aF a(u+v)=au+av 8 vV, a,bF (a+b)v=av+bv 9 vV, a,bF (ab)v=a(bv) 10 vV, 1·v=v 19 Vector Spaces - Example Let F be a field and let n be a natural number. Fn = { (a1,...,an) | a1,...,anF } is a vector space over F where for any (a1,...,an),(b1,...,bn)Fn (a1,...,an) + (b1,...,bn) = (a1+b1,...,an+bn) and for any (a1,...,an)Fn and cF c•(a1,...,an) = (c•a1,...,c•an) 20 Subspaces Definition (subspace): A subset W of a vector space V (over a field F) is called a subspace of V if W itself is a vector space over the addition and scalar multiplication operations of V. 21 Affine Subspaces Definition (affine subspace): Let V be a vector space. UV is an affine subspace of V if there exist a subspace W of V and a vV, such that U = { u | wW : u = w + v } 22 Linear Combinations Definition (linear combination): Let V be a vector space over some field F. Let v1,...,vkV and let a1,...,akF. The sum a1v1+...+akvk is called a linear combination of v1,...,vk with the coefficients a1,...,ak. Definition (linear dependent): A set of vectors {v1,...,vk} in some vector space V over a field F is linear dependent if there exist a1,...,akF and an 1ik for which ai0, s.t a1v1+...+akvk=0. Vectors which are not linear dependent are called linear independent. 23 Basis Definition (Span): Let V be a vector space over some field F. Let KV. Span(K) denotes the subspace of all the linear combination of members of K. Definition (Basis): Let B{0} be a subset of a vector space V. B is called a basis for V if (a) B is linear independent. (b) Span(B)=V. 24 Dimensions Definition (dimension): The number of vectors in any basis of a vector space is called its dimension. Similarly, the dimension of an affine subspace is the dimension of its corresponding subspace. 25 Restriction of Polynomials Definition (restriction of a polynomial to an affine subspace): Let U be an affine subspace of Fd (where F is a field and d is a positive integer). Let p:FdF be a polynomial. The restriction of p to U is p’:UF, uU p’(u)=p(u). Definition (restriction of a polynomial to a curve): Let :FFd be a curve (where F is a field and d is a positive integer). Let p:FdF be a polynomial. The restriction of p to is p’(x)=p((x)). 26 Restriction of Polynomials Home Assignment [1] Prove that the restriction of p to U is a polynomial. What are its degree-bound and dimension? [2] The same for . Next 27 Low Degree Extension (LDE) Definition (low degree extension): Let : Hd H be a string (where H is some finite field). Given a finite field F, which is a superset of H, we define a low degree extension of to F as a polynomial LDE : Fd F which satisfies: LDE agrees with on Hd (extension). The degree-bound of LDE is |H| in each variable (low degree). 28 LDE - Home Assignment Let {0,1}n. Write down an expression for LDE. 29 Reading a value Goal: To be able to find the value of an LDE in any point (set of points) of Fd. x LDE LDE(x) 30 Straightforward Approach Represent the LDE by its coefficients. Alas, this will require access to |H|d variables, log|F| bits each, each time! x the coefficients of the dimensiond, degree-bound|H| LDE LDE(x) 31 “Tricky” Approach But now we encounter a new problem: we cannot be sure Represent the LDE by its values in thewe points of Fd.are the values are given Now we only need access to one variable bits)to consistent, i.e. (log|F| correspond each time. a single dimension-d, degreebound-|H| polynomial. x the value of the LDE in every point in Fd LDE(x) 32 Consistent Readers In the upcoming lectures we’ll see how to build readers which: access only a small number of the variables each time. detect inconsistency with high probability. We’ll later weaken this notion 33 Consistency Tests Suppose we have a set of variables which represent the LDE in some manner. A consistency test is a set of local tests. If the values of the variables are consistent, all the local tests accept. Otherwise a random test should reject w.h.p. v v v v v v v v v v v v v v 34 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. 35 Corresponding Game 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(0,1,0) 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) 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) P(6,6,4) 2 P(6,6,5) P(6,6,6) 36