/*********************************************************************************** * Author: Sebastian Jambor, 2011 * * (C) GPL (e-mail from June 6, 2012, 17:00:31 MESZ) * * sebastian@momo.math.rwth-aachen.de * * * * Implementation of an algorithm to compute the minimal polynomial of a * * square matrix A \in \F_p^{n \times n}. * * * * Let V_1, \dotsc, V_k \in \F_p^{1 \times n} be vectors such that * * V_1, V_1*A, V_1*A^2, \dotsc, V_2, V_2*A, V_2*A^2, \dotsc * * generate \F_p^{1 \times n}. * * Let mpV_i be the monic polynomial of smallest degree such that * * V_i*mpV_i(A) = 0. * * Then the minimal polynomial of A is the least common multiple of the mpV_i. * * * * * * The algorithm uses two classes: * * * * 1. LinearDependencyMatrix * * This is used to find a linear dependency between the vectors V, V*A, \ldotsc. * * To to this, it has an internal n \times (2n + 1) matrix. * * Every time a new row VA^i is inserted, it is reduced via Gauss' Algorithm, * * using right hand sides. If VA^i is reduced to zero, then the vectors are * * linearly dependent, and the dependency can be read of at the right hand sides. * * * * Example: Compute the minimal polynomial of A = [[0,1],[1,1]] with V = [1,0] * * over F_5. * * Then LinearDependencyMatrix will be: * * After the first step (i.e., after inserting V = [1,0]): * * ( 1 0 | 1 0 0 ) * * After the second step (i.e., after inserting VA = [0,1]): * * ( 1 0 | 1 0 0 ) * * ( 0 1 | 0 1 0 ) * * In the third step, where VA^2 = [1,1] is inserted, the row * * ( 1 1 | 0 0 1 ) * * is reduced to * * ( 0 0 | 4 4 1) * * Thus VA^2 + 4*VA + 4*V = 0, so mpV = t^2 + 4*t + 4. * * * * * * * * 2. NewVectorMatrix * * If one vector V_1 is not enough to compute the minimal polynomial, i.e. the * * vectors V_1, V_1*A, V_1*A^2, \dotsc don't generate \F_p^{1 \times n}, then * * we have to find a vector V_2 which is not in the span of the V_1*A^i. * * This is done with NewVectorMatrix, which simply holds a reduced n \times n * * matrix, where the rows generate the span of the V_jA^i. * * To find a vector which is not in the span, simply take the k-th standard * * vector, where k is not a pivot element of A. * * * * * * For efficiency reasons, the matrix entries in LinearDependencyMatrix * * and NewVectorMatrix are not initialized to zero. Instead, a variable rows * * is used to indicate the number of rows which are nonzero (all further * * rows are regarded as zero rows). Furthermore, the array pivots stores the * * pivot entries of the rows, i.e., pivots[i] indicates the position of the * * first non-zero entry in the i-th row, which is normalized to 1. * ***********************************************************************************/ #ifndef MINPOLY_H #define MINPOLY_H class NewVectorMatrix; class LinearDependencyMatrix { friend class NewVectorMatrix; private: unsigned p; unsigned long n; unsigned long **matrix; unsigned long *tmprow; unsigned *pivots; unsigned rows; public: LinearDependencyMatrix(unsigned n, unsigned long p); ~LinearDependencyMatrix(); // reset the matrix, so that we can use it to find another linear dependence // Note: there is no need to reinitialize the matrix and vectors! void resetMatrix(); // return the first nonzero entry in row (only the first n entries are checked, // regardless of the size, since we will also apply this for rows with // right hand sides). // If the first n entries are all zero, return -1 (so this gives a check if row is the zero vector) int firstNonzeroEntry(unsigned long *row); void reduceTmpRow(); void normalizeTmp(unsigned i); bool findLinearDependency(unsigned long* newRow, unsigned long* dep); //friend std::ostream& operator<<(std::ostream& out, const LinearDependencyMatrix& mat); }; // This class is used to find a new vector for the next step in the // minimal polynomial algorithm. class NewVectorMatrix { private: unsigned p; unsigned long n; unsigned long **matrix; unsigned *pivots; unsigned *nonPivots; unsigned rows; public: NewVectorMatrix(unsigned n, unsigned long p); ~NewVectorMatrix(); // return the first nonzero entry in row (only the first n entries are checked, // regardless of the size, since we will also apply this for rows with // right hand sides). // If the first n entries are all zero, return -1 (so this gives a check if row is the zero vector) int firstNonzeroEntry(unsigned long *row); // // let piv be the pivot position of row i. then this method eliminates entry piv of row // void subtractIthRow(unsigned long *row, unsigned i); void normalizeRow(unsigned long *row, unsigned i); void insertRow(unsigned long* row); // insert each row of the matrix void insertMatrix(LinearDependencyMatrix& mat); // Finds the smallest integer between 0 and n-1, which is not a pivot position. // If no such number exists, return -1. int findSmallestNonpivot(); int findLargestNonpivot(); }; // compute the minimal polynomial of matrix \in \F_p^{n \times n}. // The result is an array of length n + 1, where the i-th entry represents the i-th coefficient // of the minimal polynomial. // // result should be deleted with delete[] unsigned long* computeMinimalPolynomial(unsigned long** matrix, unsigned n, unsigned long p); ///////////////////////////////// // auxiliary methods ///////////////////////////////// // compute x^(-1) mod p // // NOTE: this uses long long instead of unsigned long, for the XEA to work. // This shouldn't be a problem, since p has to be < 2^31 for the multiplication to work anyway. // // There is no need to distinguish between 32bit and 64bit architectures: On 64bit, long long // is the same as long, and on 32bit, we need long long so that the variables can hold negative values. unsigned long modularInverse(long long x, long long p); void vectorMatrixMult(unsigned long* vec, unsigned long **mat, unsigned **nonzeroIndices, unsigned *nonzeroCounts, unsigned long* result, unsigned n, unsigned long p); // a is a vector of length at least dega + 1, and q is a vector of length at least degq + 1, // representing polynomials \sum_i a[i]t^i \in \F_p[t]. // After this method, a will be a mod q. // Method will change dega accordingly. void rem(unsigned long* a, unsigned long* q, unsigned long p, int & dega, int degq); // a is a vector of length at least dega + 1, and q is a vector of length at least degq + 1, // representing polynomials \sum_i a[i]t^i \in \F_p[t]. // After this method, a will be a / q. // Method will change dega accordingly. void quo(unsigned long* a, unsigned long* q, unsigned long p, int & dega, int degq); // NOTE: since we don't know the size of result (the list can be longer than the degree of the polynomial), // every entry has to be preinitialized to zero! void mult(unsigned long* result, unsigned long* a, unsigned long* b, unsigned long p, int dega, int degb); // g = gcd(a,b). // returns deg(g) // // NOTE: since we don't know the size of g, every entry has to be preinitialized to zero! int gcd(unsigned long* g, unsigned long* a, unsigned long* b, unsigned long p, int dega, int degb); // l = lcm(a,b). // returns deg(l) // // has side effects for a // // NOTE: since we don't know the size of l, every entry has to be preinitialized to zero! int lcm(unsigned long* l, unsigned long* a, unsigned long* b, unsigned long p, int dega, int degb); // method suggested by Hans Schoenemann to multiply elements in finite fields // on 32bit and 64bit machines static inline unsigned long multMod(unsigned long a, unsigned long b, unsigned long p) { #if SIZEOF_LONG == 4 #define ULONG64 (unsigned long long) #else #define ULONG64 (unsigned long) #endif return (unsigned long)((ULONG64 a)*(ULONG64 b) % (ULONG64 p)); } #endif // MINPOLY_H