1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
|
implement Ida;
#
# M Rabin, ``Efficient Dispersal of Information for Security,
# Load Balancing, and Fault Tolerance'', JACM 36(2), April 1989, pp. 335-348
# the scheme used below is that suggested at the top of page 340
#
include "sys.m";
sys: Sys;
include "rand.m";
rand: Rand;
include "ida.m";
invtab: array of int;
init()
{
sys = load Sys Sys->PATH;
rand = load Rand Rand->PATH;
rand->init(sys->pctl(0, nil)^(sys->millisec()<<8));
# the table is in a separate module so that
# the copy in the module initialisation section is discarded
# after unloading, preventing twice the space being used
idatab := load Idatab Idatab->PATH;
invtab = idatab->init(); # the big fella
idatab = nil;
}
Field: con 65537;
Fmax: con Field-1;
div(a, b: int): int
{
return mul(a, invtab[b]);
}
mul(a, b: int): int
{
if(a == Fmax && b == Fmax) # avoid overflow
return 1;
return int((big(a*b) & 16rFFFFFFFF) % big Field);
}
sub(a, b: int): int
{
return ((a-b)+Field)%Field;
}
add(a, b: int): int
{
return (a + b)%Field;
}
#
# return a fragment representing the encoded version of data
#
fragment(data: array of byte, m: int): ref Frag
{
nb := len data;
nw := (nb+1)/2;
a := array[m] of {* => rand->rand(Fmax)+1}; # no zero elements
f := array[(nw + m - 1)/m] of int;
o := 0;
i := 0;
for(k := 0; k < len f; k++){
c := 0;
for(j := 0; j < m && i < nb; j++){
b := int data[i++] << 8;
if(i < nb)
b |= int data[i++];
c = add(c, mul(b, a[j]));
}
f[o++] = c;
}
return ref Frag(nb, m, a, f, nil);
}
#
# return the data encoded by the given set of fragments
#
reconstruct(frags: array of ref Frag): (array of byte, string)
{
if(len frags < 1 || len frags < (m := frags[0].m))
return (nil, "too few fragments");
fraglen := len frags[0].enc;
a := array[m] of array of int;
for(j := 0; j < len a; j++){
a[j] = frags[j].a;
if(len a[j] != m)
return (nil, "inconsistent encoding matrix");
if(len frags[j].enc != fraglen)
return (nil, "inconsistent fragments");
}
ainv := minvert(a);
out := array[fraglen*2*m] of byte;
o := 0;
for(k := 0; k < fraglen; k++){
for(i := 0; i < m; i++){
row := ainv[i];
b := 0;
for(j = 0; j < m; j++)
b = add(b, mul(frags[j].enc[k], row[j]));
if((b>>16) != 0)
return (nil, "corrupt output");
out[o++] = byte (b>>8);
out[o++] = byte b;
}
}
if(frags[0].dlen < len out)
out = out[0: frags[0].dlen];
return (out, nil);
}
#
# Rabin's paper gives a way of building an encoding matrix that can then
# be inverted in O(m^2) operations, compared to O(m^3) for the following,
# but m is small enough it doesn't seem worth the added complication,
# and it's only done once per set
#
minvert(a: array of array of int): array of array of int
{
m := len a; # it's square
out := array[m] of {* => array[m*2] of {* => 0}};
for(r := 0; r < m; r++){
out[r][0:] = a[r];
out[r][m+r] = 1; # identity matrix
}
for(r = 0; r < m; r++){
x := out[r][r]; # by construction, cannot be zero, unless later corrupted
for(c := 0; c < 2*m; c++)
out[r][c] = div(out[r][c], x);
for(r1 := 0; r1 < m; r1++)
if(r1 != r){
y := div(out[r1][r], out[r][r]);
for(c = 0; c < 2*m; c++)
out[r1][c] = sub(out[r1][c], mul(y, out[r][c]));
}
}
for(r = 0; r < m; r++)
out[r] = out[r][m:];
return out;
}
Val: adt {
v: int;
n: int;
};
addval(vl: list of ref Val, v: int): list of ref Val
{
for(l := vl; l != nil; l = tl l)
if((hd l).v == v){
(hd l).n++;
return vl;
}
return ref Val(v, 1) :: vl;
}
mostly(vl: list of ref Val): ref Val
{
if(len vl == 1)
return hd vl;
v: ref Val;
for(; vl != nil; vl = tl vl)
if(v == nil || (hd vl).n > v.n)
v = hd vl;
return v;
}
#
# return a consistent set of Frags: all parameters agree with the majority,
# and obviously bad fragments have been discarded
#
# in the absence of error, they should all have the same value, so lists are fine;
# could separately return the discarded ones, out of interest
#
consistent(frags: array of ref Frag): array of ref Frag
{
t := array[len frags] of ref Frag;
t[0:] = frags;
frags = t;
ds: list of ref Val; # data size
ms: list of ref Val;
fls: list of ref Val;
for(i := 0; i < len frags; i++){
f := frags[i];
if(f != nil){
ds = addval(ds, f.dlen);
ms = addval(ms, f.m);
fls = addval(fls, len f.enc);
}
}
dv := mostly(ds);
mv := mostly(ms);
flv := mostly(fls);
if(mv == nil || flv == nil || dv == nil)
return nil;
for(i = 0; i < len frags; i++){
f := frags[i];
if(f == nil || f.m != mv.v || f.m != len f.a || len f.enc != flv.v || f.dlen != dv.v || badfrag(f)){ # inconsistent: drop it
if(i+1 < len frags)
frags[i:] = frags[i+1:];
frags = frags[0:len frags-1];
}
}
if(len frags == 0)
return nil;
return frags;
}
badfrag(f: ref Frag): int
{
for(i := 0; i < len f.a; i++){
v := f.a[i];
if(v <= 0 || v >= Field)
return 1;
}
for(i = 0; i < len f.a; i++){
v := f.enc[i];
if(v == 0 || v >= Field)
return 1;
}
return 0;
}
|
