Block Wiedemann algorithm

The Block Wiedemann algorithm for computing kernel vectors of a matrix over a finite field is a generalisation of an algorithm due to Don Coppersmith.

Coppersmith's algorithm

Let M be an n imes n square matrix over some finite field F, let x_{mathrm {base be a random vector of length n, and let x = M x_{mathrm {base. Consider the sequence of vectors S = left [x, Mx, M^2x, ldots ight] obtained by repeatedly multiplying the vector by the matrix M; let y be any other vector of length n, and consider the sequence of finite-field elements S_y = left [y cdot x, y cdot Mx, y cdot M^2x ldots ight]

We know that the matrix M has a minimal polynomial; by the Cayley-Hamilton Theorem we know that this polynomial is of degree (which we will call n_0) no more than n. Say sum_{r=0}^{n_0} p_rM^r = 0. Then sum_{r=0}^{n_0} y cdot (p_r (M^r x)) = 0; so the minimal polynomial of the matrix annihilates the sequence S and hence S_y.

But the Berlekamp-Massey algorithm allows us to calculate relatively efficiently some sequence q_0 ldots q_L with sum_{i=0}^L q_i S_y [{i+r}] =0 forall r. Our hope is that this sequence, which by construction annihilates y cdot S, actually annihilates S; so we have sum_{i=0}^L q_i M^i x = 0. We then take advantage of the initial definition of x to say M sum_{i=0}^L q_i M^i x_{mathrm {base = 0 and so sum_{i=0}^L q_i M^i x_{mathrm {base is a hopefully non-zero kernel vector of M.

The Block Wiedemann algorithm

The natural implementation of sparse matrix arithmetic on a computer makes it easy to compute the sequence S in parallel for a number of vectors equal to the width of a machine word - indeed, it will normally take no longer to compute for that many vectors than for one. If you have several processors, you can compute the sequence S for a different set of random vectors in parallel on all the computers.

It turns out, by a generalisation of the Berlekamp-Massey algorithm to provide a sequence of small matrices, that you can take the sequence produced for a large number of vectors and generate a kernel vector of the original large matrix. You need to compute y_i cdot M^t x_j for some i = 0 ldots i_max, j=0 ldots j_max, t = 0 ldots t_max where i_max, j_max, t_max need to satisfy t_max > frac{d}{i_max} + frac{d}{j_max} + O(1) and y_i are a series of vectors of length n; but in practice you can take y_i as a sequence of unit vectors and simply write out the first i_max entries in your vectors at each time t.

References

Villard's 1997 research report 'A study of Coppersmith's block Wiedemann algorithm using matrix polynomials' (available at [http://citeseer.ist.psu.edu/cache/papers/cs/4204/ftp:zSzzSzftp.imag.frzSzpubzSzCALCUL_FORMELzSzRAPPORTzSz1997zSzRR975.pdf/villard97study.pdf] - the cover material is in French but the content in English) is a reasonable description.

Thomé's paper 'Subquadratic computation of vector generating polynomials and improvement of the block Wiedemann algorithm' (available at [http://citeseer.ist.psu.edu/rd/13850609%2C564537%2C1%2C0.25%2CDownload/http://citeseer.ist.psu.edu/cache/papers/cs/27081/http:zSzzSzwww.lix.polytechnique.frzSzLabozSzEmmanuel.ThomezSzpubliszSzjsc.pdf/subquadratic-computation-of-vector.pdf] ) uses a more sophisticated FFT-based algorithm for computing the vector generating polynomials, and describes a practical implementation with i_max = j_max = 4 used to compute a kernel vector of a 484603x484603 matrix of entries modulo 2607-1, and hence to compute discrete logarithms in the field GF(2^{607}).


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