Download when a square matrix is repeatedly applied to a vector

Block Wiedemann Algorithm
Problem Statement
• Solve large sparse systems of homogeneous
linear equations Aw=0, w≠0.
– Matrix A will be a singular N X N matrix over the
Galois field with q elements K=GF(q).
– w is a vector of N unknowns over GF(q)
Linearly Generated Sequence
• Let W be a vector space over an abstract field
K. The infinite sequence {s } with s in W is
said to be linearly generated over K if there
exist scalars c , c , …, c in K with c ≠0 for some
0≦d≦l, such that, for every j≧0,
i i>=0
0
1
l
l
c s
i 0
i
j i
i
d
0
• c( x)  c0  c1 x  ...  cl x l is a generating
polynomial {s }
i i>=0 .
Linearly Generated Sequence (cont’d)
• Suppose V is another vector space and let α:
WV, s  t be a linear map. The sequence
{t } is also linearly generated and its
minimum polynomial divides that of {s }
i
i
i
i≧0
i i≧0
Wiedemann Algorithm
• Wiedemann’s algorithm is based on the fact
that when a square matrix is repeatedly
applied to a vector, the resulting vector
sequence is linear recursive
Wiedemann Algorithm (cont’d)
• Let v be a column N-vector over K.
i
{
B
• Since the space S= v}i 0 is N-dimensional, there is
an integer 1≦l≦N such that v, …, B^(l-1)v are linearly
independent and B^l is a linear combination of these
vectors with coefficients in K.
• The monic polynomial f B,v ( )  l  cl 1l 1  ...  c0
has the least degree, and it’s called the
minimum polynomial of v.
Wiedemann Algorithm (cont’d)
• There exists δ≧0 is the minimum index for
which c is nonzero so that
δ
• Let w-hat denote the vector in parentheses
above and suppose w-hat≠0. Then for some
integer t, with 1≦t≦l,
but
and so Bw=0.
Wiedemann Algorithm (cont’d)
• Suppose u is a column vector over K, then the
tr i
i
{
u
B
v
}
{
B
v}i 0 by
sequence
i  0 is derived from
a linear map.
– The minimum polynomial of {u B v}i 0 will
i
{
B
v}i 0
divides
– Wiedemann algorithm depends on the probability
that the two polynomials are the same.
tr
i
Wiedemann Algorithm (cont’d)
Block Wiedemann Algorithm
• Almost the same but…
– Have to decide two block sizes
– Have to decide a shift parameter, a non-negative
integer
– Use random “blocks” rather than random vectors
• The uB i v sequence will become a sequence of blocks
rather than elements in a particular field.
• Have to design another algorithm to find minimum
generating polynomials (realizations of matrices) rather
than using Berlekamp Massey algorithm.
Block Wiedemann Algorithm (cont’d)
Under Determined Variables
•
•
•
•
“Random” blocks
Constant shift
Generating function
Block size
How to Find the Minimum Realization
• !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Reference
• E. Kaltofen and A. Lobo, Distributed MatrixFree Solution of Large Sparse Linear Systems
over Finite Fields, Algorithmica 1999.
• Gilles Villard, A Study of Coppersmith’s Block
Wiedemann Algorithm Using Matrix
Polynomials, LMC-IMAG, report # 975 IM 1997.
Backup
Block Wiedemann Algorithm
• WHAT is pathological matrix????
– Block Wiedemann algorithm can’t handle it.
• Block版的berlekamp massey出來的minimum
polynomial有確保degree的上限嗎
– 似乎有多重解？