The goal of the project is to examine a polynomial-time probabilistic algorithm to compute the minimum distance of an arbitrary, non-binary, linear error-correcting code. Only the binary case was done in A. Foster's honors project [F]. For the non-binary case a new function was needed in the GAP kernel. This function was added by Steve Linton of St. Andrews, Scotland, one of GAP's kernel maintainers, in August 2004. The programming shall be done in the GAP coding-theory package GUAVA. GAP is a computer algebra package whose open source kernel is written in the C programming language [GAP]. However, most packages (such as GUAVA) and the algorithms described here are written in GAP's own interpreted language. Jointly, with my advisor, I extended Foster's work (Algorithm 4) to the non-binary case but with a new feature making it faster (Algorithm 5). The paper ends with applications to the non-binary McEliece public key cryptosystem (see §4) and trace codes (see §3)
Acknowledgement: The GAP examples were performed on an AMD-64 linux PC with 2G of RAM. I am grateful to the Director of Research Professor Reza Malek-Madani for making this available.
Let be a finite field with elements. We call a subset of a (block) code. A linear code is a subspace of the vector space . The length of the code is the integer . A generator matrix of a linear code can be any matrix whose rows form a basis for the subspace. Let be a linear code of length over with generator matrix , where is a power of a prime . In this case if is a row vector in , then is a codeword in C. Moreover, all codewords in C arise from some message vector this way. This defines the encoder from the message space to the code C. If , then the code is called binary. We assume that has been given the standard basis , , ..., . The dimension of is denoted , so the number of elements of is equal to . The quantity is called the rate of the code and measures the amount of information that the code can transmit.
Two codes are equivalent to each other if one can be obtained from the other by a combination of permutations of coordinate positions of the code and multiplication of the symbols appearing in a fixed position by a non-zero scalar. More precisely, two codes are equivalent if the generator matrix of one can be derived from the other using operations of the following types:
Please see [HILL] or MacWilliams and Sloane [MS] for a proof of this fact. A code with generator matrix in the form above is said to be in standard form.
Another important parameter associated to the code is the number of errors which it can, in principle, correct. For this notion, let us introduce the concept of distance between codewords. For any two , let denote the number of coordinates where these two vectors differ:
(1) |
If denotes the smallest distance between distinct codewords in a linear code ,then there exist and such that . If is any matrix with coefficients in , then let denote the minimum distance of the linear code spanned by the rows of .
(2) |
Note, if is equivalent to , then . In general, this parameter is known to be very difficult to efficiently determine [CHA]. In fact, computing it in general is known to be NP-complete [BMT], [VAR].
Let be a linear code. We say is error-correcting provided for any vector in there is exactly one codeword for which .
Proof: Let be a codeword. Let denote a ball of radius centered at . Since is error correcting, any two such balls must be disjoint. Therefore, the distance between their centers cannot be less than or equal to . This proves ``the only if'' part. For the ``if'' part of the proof, note that implies that any two such balls are disjoint. Suppose that is not error-correcting, in other words, the ball of radius centered at contains distinct codewords and . By the triangle inequality for the Hamming metric, contains . This contradicts the fact that they must be disjoint.
Notation: Let denote a field extension of the finite field .
First we must explain how each element of may be represented as an invertible matrix with entries in .
Let denote a generator of the cyclic group . Let denote the minimal polynomial of (the lowest degree monic polynomial which has as a root). Take the matrix associated to , denoted , to be the companion matrix of (namely the characteristic polynomial of is ). If the degree of is , then is an matrix with coefficients in (and the degree of is ). If denotes any other non-zero element, then we can write , for some (because is a cyclic group). Take the matrix associated to to be . The matrix associated to will be the zero matrix.
We define the trace of to be the trace of the corresponding matrix. Denote this trace by . Let denote a linear code over . The trace code of is defined to be
The naive method for calculating the minimum distance is to examine the weight of every possible linear combination of rows in .
Input: A code of length n containing elements over .
Output: The minimum distance of .
In the above algorithm is not required to be linear. For linear codes the following algorithm is much faster.
By ``proper linear combination'', we mean to exclude the case where all coefficients are equal to 0.
Though this is still an exponential time algorithm problem, the vectors being compared are of length rather than length . Additionally, this algorithm works for all Galois fields of order . (We remark that step 2 in the above algorithm was missing from the implementation of MinimumDistance in the last release of GUAVA. This bug is fixed in version 2.0.)
The probabilistic algorithm used to reduce this to polynomial time was initially proposed by J. S. Leon [LEO] and later modified by F. Chabaud [CHA]. One version of the algorithm given in Chabaud's paper is as follows:
(3) |
However, sometimes step (3) is impossible because Gaussian elimination does not always produce a matrix in standard form. For example, take any code with generator matrix in row reduced echelon form which is not of the form .
Here is a faster version of Leon's algorithm.
Notation as in Algorithm 3.
Chabaud [CHA] and Barg [BARG] have discussed finding optimal values for , , and .
The previous two algorithms were discussed in A. Foster's thesis [F], although only implemented in GUAVA for binary codes. The next algorithm is original to this thesis and forms the basis of our application to non-binary McEliece public key cryptosystems. This application is discussed in section 4.
Examples of the above algorithm are given in a later section.
The next algorithm is used in this project as well as Lennon's project [L] for testing purposes.
Input: A code of length n containing elements over , an integer , a word
Output: The list of all codewords within Hamming Distance from
We remark that if then the algorithm still works but may return an empty list.
Input: A linear code of length over where is an extension field of
Output: The Trace Code of
In the example below we compute the minimum distance of several randomly chosen Trace codes where is the field with elements and is the field with elements.
gap> F:=GF(4); GF(2^2) gap> FF:=GF(16); GF(2^4) gap> Read("/home/irons/gapfiles/TraceCode.gap"); gap> C:=RandomLinearCode(15,4,F); a [15,4,?] randomly generated code over GF(4) gap> TC:=TraceCode(C,F);time; a linear [15,4,1..8]6..11 user defined unrestricted code over GF(4) 126 gap> MinimumDistance(C); 7 gap> MinimumDistance(TC); 7 gap> C:=RandomLinearCode(15,4,FF); a [15,4,?] randomly generated code over GF(16) gap> TC:=TraceCode(C,F);time; a linear [15,8,1..4]4..7 user defined unrestricted code over GF(4) 31061 gap> MinimumDistance(C); 10 gap> MinimumDistance(TC); 4 gap> C:=RandomLinearCode(15,4,FF); a [15,4,?] randomly generated code over GF(16) gap> TC:=TraceCode(C,F);time; a linear [15,8,1..6]4..7 user defined unrestricted code over GF(4) 31542 gap> MinimumDistance(C); 10 gap> MinimumDistance(TC); 5 gap> C:=RandomLinearCode(15,4,FF); a [15,4,?] randomly generated code over GF(16) gap> TC:=TraceCode(C,F);time; a linear [15,8,1..4]4..7 user defined unrestricted code over GF(4) 33973 gap> MinimumDistance(C); 10 gap> MinimumDistance(TC); 4
We follow section 1.2 of [CHA].
Here is a brief description of this cryptosystem. Alice is sending a message to Bob. Eve is trying to read Alice's message. Given is a non-binary code with generator matrix and also given is a decoding algorithm that corrects up to errors. Input parameters are an invertible matrix and an permutation matrix . The triplet will be the secret key and the pair will be the public key. Everyone knows the public key, but Alice and Bob are the only two who know the secret key. For Alice to transmit a bit message , she sends the cipher text , where is a random vector of weight at most . To decipher the cipher text Bob decodes using the given algorithm.
To decrypt an encoded message, consider the matrix
The GAP code below illustrates how to use the MinimumDistanceRandom command described in Algorithm 5 to break the McEliece cryptosystem cipher.
gap> F:=GF(16); GF(2^4) gap> C := ReedSolomonCode( 15, 10 ); a cyclic [15,6,10]7..9 Reed-Solomon code over GF(16) gap> G:=GeneratorMat(C);; gap> g:=Random(SymmetricGroup(15));; gap> v:=ShallowCopy(NullWord( 15, F)); [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] gap> v[1]:=One(F);; gap> v[2]:=One(F);; gap> v[3]:=One(F);; gap> v[4]:=One(F);; gap> v[5]:=One(F);; gap> e:=Permuted(v,g);; gap> c0:=Random(C);; gap> w:=VectorCodeword(c0+e);; gap> Gnew:=Concatenation(G,[w]);; gap> Cnew:=GeneratorMatCode(Gnew,F); time; a linear [15,7,1..9]6..8 code defined by generator matrix over GF(16) 49109 gap> MinimumDistanceRandom(Cnew,10,13);time; This is a probabilistic algorithm which may return the wrong answer. [ 5, [ 0 0 0 0 1 0 0 1 0 0 1 1 0 0 1 ] ] 158361 gap> e; [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2), Z(2)^0 ]Notice the error vector matches the minimum weight vector returned by MinimumDistanceRandom, as desired.
BruteForceMindist:=function(C) local n,c,w, mindist, minword, Cwords; Cwords:=Elements(C); n:=WordLength(C); mindist:=n; #initialize minword:=Cwords[1]; #initialize for c in Cwords do w:=WeightCodeword(c); if w >0 and w < mindist then mindist:=w; minword:=c; fi; od; return [mindist,minword]; end; NearestNeighborDecodeword:=function(r,C,dist) local Cwords,NearbyWords,c; Cwords:=Elements(C); NearbyWords:=[]; for c in Cwords do if WeightCodeword(r-c) < dist then NearbyWords:=Concatenation(NearbyWords,[c]); fi; od; return NearbyWords; end; C:=NordstromRobinsonCode( ); a (16,256,6)4 Nordstrom-Robinson code over GF(2) r:=Codeword("1100000000000000"); [ 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] r in C; false NearestNeighborDecodeword(r,C,3); NearestNeighborDecodeword(r,C,4); NearestNeighborDecodeword(r,C,5); CodeLength:= function (C) return WordLength(C); end; TraceCode:= function (C,F) #C is defined over an extension of F #F is a base field local FF, TC, TrC, i, n, c; n:= CodeLength(C); FF:= LeftActingDomain(C); TC:= List(C,c->Codeword(List([1..n],i->Trace(FF,F,c[i])))); TrC:=ElementsCode(TC, F); IsLinearCode(TrC); return TrC; end; ################################################### ## MinimumDistanceRandom( <C>, <num>, <s> ) ## ## This is a simpler version than Leon's method, which does not put G in st form. ## (this works welland is in some cases faster than the st form one) ## Input: C is a linear code ## num is an integer >0 which represents the number of iterations ## s is an integer between 1 and n which represents the columns considered ## in the algorithm. ## Output: an integer >= min dist(C), and hopefully equal to it! ## a codework of that weight ## ## Algorithm: randomly permute the columns of the gen mat G of C ## by a permutation rho - call new mat Gp ## break Gp into (A,B), where A is kxs and B is kx(n-s) ## compute code C_A generated by rows of A ## find min weight codeword c_A of C_A and w_A=wt(c_A) ## using AClosestVectorCombinationsMatFFEVecFFECoords ## extend c_A to a corresponding codeword c_p in C_Gp ## return c=rho^(-1)(c_p) and wt=wt(c_p)=wt(c) ## MinimumDistanceRandom:= function(C,num,s) local A,HasZeroRow,majority,G0, Gp, Gpt, Gt, L, k, n, i, j, m, dimMat, J, d1,M, arrayd1, Combo, rows, row, rowSum, G, F, zero, AClosestVec, B, ZZ, p, numrow0, bigwtrow,bigwtvec, x, d, v, ds, vecs,rho,perms,g,newv,pos,rowcombos,v1,v2; # returns the estimated distance, and corresponding vector of that weight Print("\n This is a probabilistic algorithm which may return the wrong answer.\n"); G0 := GeneratorMat(C); p:=5; #this seems to be an optimal value # it's the max number of rows used in Z to find a small # codewd in C_Z G := List(G0,ShallowCopy); F:=LeftActingDomain(C); dimMat := DimensionsMat(G); n:=dimMat[2]; k:=dimMat[1]; if n=k then C!.lowerBoundMinimumDistance := 1; C!.upperBoundMinimumDistance := 1; return 1; fi; # added 11-2004 if s > n-1 then Print("Resetting s to ",n-2," ... \n"); s:=n-k; fi; arrayd1:=[n]; numrow0:=0; # initialize perms:=[]; # initialize for m in [1..num] do ##Permute the columns of C randomly Gt := TransposedMat(G); Gp := NullMat(n,k); L := SymmetricGroup(n); rho := Random(L); L:=List([1..n],i->OnPoints(i,rho)); for i in [1..n] do Gp[i] := Gt[L[i]]; od; Gp := TransposedMat(Gp); Gp := List(Gp,ShallowCopy); ##generate the matrix A from Gp=(A|B) Gpt := TransposedMat(Gp); A := NullMat(s,k); for i in [1..s] do A[i] := Gpt[i]; od; A := TransposedMat(A); ##generate the matrix B from Gp=(A|B) Gpt := TransposedMat(Gp); B := NullMat(n-s,k); for i in [s+1..n] do B[i-s] := Gpt[i]; od; B := TransposedMat(B); zero := Zero(F)*A[1]; if (s<n-k and A=Zero(F)*A) then Error("This method fails for these parameters. Try increasing s.\n"); fi; if (s=n and A=Zero(F)*A) then return 1; fi; ## search for all rows of weight p ## J is the list of all triples representing the codeword in C which ## corresponds to AClosestVec[1]. J := []; #col number of codewords to compute the length of for i in [1..p] do AClosestVec:=AClosestVectorCombinationsMatFFEVecFFECoords(A, F, zero, i, 1); ### AClosestVec[1]=ZZ*AClosestVec[2]... v1:=AClosestVec[2]*Gp; v2:=Permuted(v1,(rho)^(-1)); Add(J,[WeightVecFFE(v2),Codeword(v2,n,F),AClosestVec[2]]); od; ds:=List(J,x->x[1]); vecs:=List(J,x->x[2]); rowcombos:=List(J,x->x[3]); d:=Minimum(ds); i:=Position(ds,d); arrayd1[m]:=[d,vecs[i],rowcombos[i]]; perms[m]:=rho; od; ## m ds:=List(arrayd1,x->x[1]); vecs:=List(arrayd1,x->x[2]); rowcombos:=List(arrayd1,x->x[3]); d:=MostCommonInList(ds); pos:=Position(ds,d); v:=vecs[pos]; L:=List([1..n],i->OnPoints(i,perms[pos]^(-1))); newv:=Codeword(List(L,i->v[L[i]])); return([d,newv]); end;
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