Let me add that the situations I care about are n,m <= 20, the entries are 
<=5 and the matrices are sparsely filled. An random and typical example is

sage: M = matrix([(0, -1, 0, 0, 0, 0, 0, 1),
....:  (1, 0, 1, 0, 0, 0, 0, 0),
....:  (0, -1, 0, 0, 1, 0, 0, 0),
....:  (0, 0, 0, 0, 0, 1, 0, 0),
....:  (0, 0, -1, 0, 0, 0, 1, 0),
....:  (0, 0, 0, -1, 0, 0, -1, 0),
....:  (0, 0, 0, 0, -1, 1, 0, 0),
....:  (-2, 0, 0, 0, 0, 0, 0, 0),
....:  (-1, 1, 0, 0, 0, 0, 0, 0),
....:  (0, 1, 0, 0, 0, 0, 0, -1),
....:  (0, 1, 0, 1, 0, -1, 0, -1),
....:  (0, 2, -1, 1, 0, -1, 0, -1),
....:  (0, 2, -1, 0, 0, -1, 0, -1),
....:  (0, 2, 0, 0, -1, -1, 0, -1),
....:  (0, 2, 0, 0, -1, 0, -1, -1),
....:  (0, 2, 0, 0, 0, 0, -2, -1)]
....: )
sage: M
[ 0 -1  0  0  0  0  0  1]
[ 1  0  1  0  0  0  0  0]
[ 0 -1  0  0  1  0  0  0]
[ 0  0  0  0  0  1  0  0]
[ 0  0 -1  0  0  0  1  0]
[ 0  0  0 -1  0  0 -1  0]
[ 0  0  0  0 -1  1  0  0]
[-2  0  0  0  0  0  0  0]
[-1  1  0  0  0  0  0  0]
[ 0  1  0  0  0  0  0 -1]
[ 0  1  0  1  0 -1  0 -1]
[ 0  2 -1  1  0 -1  0 -1]
[ 0  2 -1  0  0 -1  0 -1]
[ 0  2  0  0 -1 -1  0 -1]
[ 0  2  0  0 -1  0 -1 -1]
[ 0  2  0  0  0  0 -2 -1]
sage: matrix_canonical_hash(M, 8, 8)
2718289463783950847

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