2132:
1849:
1176:
in parallel for a number of vectors equal to the width of a machine word â 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.
2127:{\displaystyle p\geq {\begin{cases}1/64,&{\text{if }}b=k+1{\text{ and }}q=2\\\left(1-{\frac {3}{2^{b-k}}}\right)^{2}\geq 1/16&{\text{if }}b\geq k+2{\text{ and }}q=2\\\left(1-{\frac {2}{q^{b-k}}}\right)^{2}\geq 1/9&{\text{if }}b\geq k+1{\text{ and }}q>2\end{cases}}}
1180:
It turns out, by a generalization of the
BerlekampâMassey 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
1453:
1318:
1063:
1142:
659:
842:
1841:
567:
1371:
955:
1648:
1598:
203:
1229:
136:
753:
2269:
419:
1534:
91:
868:
270:
1741:
1675:
1507:
1480:
706:
476:
1761:
1715:
1695:
1162:
975:
888:
679:
496:
441:
330:
310:
290:
156:
65:
2210:
2390:
2415:
2240: = 4 used to compute a kernel vector of a 484603Ă484603 matrix of entries modulo 2â1, and hence to compute discrete logarithms in the field
1376:
1234:
2410:
2262:
444:
980:
2436:
2369:
2255:
1068:
2400:
572:
2327:
2204:
D. Coppersmith, Solving homogeneous linear equations over GF(2) via block
Wiedemann algorithm, Math. Comp. 62 (1994), 333-350.
758:
712:
2359:
1766:
448:
501:
2199:
Wiedemann, D., "Solving sparse linear equations over finite fields," IEEE Trans. Inf. Theory IT-32, pp. 54-62, 1986.
2313:
1323:
893:
2364:
2226:-based algorithm for computing the vector generating polynomials, and describes a practical implementation with
2278:
1603:
1559:
161:
21:
1184:
1172:
The natural implementation of sparse matrix arithmetic on a computer makes it easy to compute the sequence
2223:
2220:
Subquadratic computation of vector generating polynomials and improvement of the block
Wiedemann algorithm
103:
2323:
1553:
718:
2153:
2318:
28:
1864:
2297:
2165:
335:
2219:
1512:
70:
2183:
847:
208:
2333:
2175:
1720:
1549:
2213:' (the cover material is in French but the content in English) is a reasonable description.
1653:
1485:
1458:
684:
454:
2374:
36:
2292:
1746:
1700:
1680:
1147:
960:
873:
664:
481:
426:
315:
295:
275:
141:
50:
2430:
2338:
93:
97:
32:
24:
1448:{\displaystyle t_{\max }>{\frac {d}{i_{\max }}}+{\frac {d}{j_{\max }}}+O(1)}
2179:
2187:
2354:
2247:
2211:
A study of
Coppersmith's block Wiedemann algorithm using matrix polynomials
1313:{\displaystyle i=0\ldots i_{\max },j=0\ldots j_{\max },t=0\ldots t_{\max }}
2154:"Probabilistic analysis of block Wiedemann for leading invariant factors"
2405:
2395:
844:. Our hope is that this sequence, which by construction annihilates
2170:
2152:
Harrison, Gavin; Johnson, Jeremy; Saunders, B. David (2022-01-01).
1548:
The block
Wiedemann algorithm can be used to calculate the leading
661:; so the minimal polynomial of the matrix annihilates the sequence
1482:
are a series of vectors of length n; but in practice you can take
2251:
1058:{\displaystyle M\sum _{i=0}^{L}q_{i}M^{i}x_{\mathrm {base} }=0}
451:
we know that this polynomial is of degree (which we will call
1509:
as a sequence of unit vectors and simply write out the first
1137:{\displaystyle \sum _{i=0}^{L}q_{i}M^{i}x_{\mathrm {base} }}
715:
allows us to calculate relatively efficiently some sequence
272:
obtained by repeatedly multiplying the vector by the matrix
2120:
654:{\displaystyle \sum _{r=0}^{n_{0}}y\cdot (p_{r}(M^{r}x))=0}
1168:
The block
Wiedemann (or Coppersmith-Wiedemann) algorithm
837:{\displaystyle \sum _{i=0}^{L}q_{i}S_{y}=0\;\forall \;r}
957:. We then take advantage of the initial definition of
1852:
1769:
1749:
1723:
1703:
1683:
1656:
1606:
1562:
1515:
1488:
1461:
1379:
1326:
1237:
1187:
1150:
1071:
983:
963:
896:
876:
850:
761:
721:
687:
667:
575:
504:
484:
457:
429:
338:
332:, and consider the sequence of finite-field elements
318:
298:
278:
211:
164:
144:
106:
73:
53:
2383:
2347:
2306:
2285:
2126:
1836:{\displaystyle \sum _{i=0}^{2n-1}UM^{i}V^{T}x^{i}}
1835:
1755:
1735:
1709:
1689:
1669:
1642:
1592:
1528:
1501:
1474:
1447:
1365:
1312:
1223:
1156:
1136:
1057:
969:
949:
882:
862:
836:
747:
700:
673:
653:
561:
490:
470:
435:
413:
324:
304:
284:
264:
197:
150:
130:
85:
59:
1521:
1423:
1403:
1385:
1358:
1345:
1332:
1305:
1280:
1255:
562:{\displaystyle \sum _{r=0}^{n_{0}}p_{r}M^{r}=0}
2263:
1552:of the matrix, ie, the largest blocks of the
1366:{\displaystyle i_{\max },j_{\max },t_{\max }}
8:
950:{\displaystyle \sum _{i=0}^{L}q_{i}M^{i}x=0}
2270:
2256:
2248:
830:
826:
2169:
2103:
2083:
2073:
2061:
2042:
2033:
2003:
1983:
1973:
1961:
1942:
1933:
1903:
1883:
1870:
1859:
1851:
1827:
1817:
1807:
1785:
1774:
1768:
1748:
1722:
1702:
1682:
1661:
1655:
1628:
1623:
1605:
1578:
1573:
1561:
1520:
1514:
1493:
1487:
1466:
1460:
1422:
1413:
1402:
1393:
1384:
1378:
1357:
1344:
1331:
1325:
1304:
1279:
1254:
1236:
1215:
1205:
1192:
1186:
1149:
1144:is a hopefully non-zero kernel vector of
1118:
1117:
1107:
1097:
1087:
1076:
1070:
1033:
1032:
1022:
1012:
1002:
991:
982:
962:
932:
922:
912:
901:
895:
875:
849:
806:
797:
787:
777:
766:
760:
739:
726:
720:
692:
686:
666:
630:
617:
596:
591:
580:
574:
547:
537:
525:
520:
509:
503:
483:
462:
456:
428:
394:
343:
337:
317:
297:
277:
242:
210:
179:
178:
163:
143:
112:
111:
105:
72:
52:
1643:{\displaystyle U,V\in F_{q}^{b\times n}}
2401:Basic Linear Algebra Subprograms (BLAS)
2144:
39:of an algorithm due to Doug Wiedemann.
1593:{\displaystyle M\in F_{q}^{n\times n}}
198:{\displaystyle x=Mx_{\mathrm {base} }}
1536:entries in your vectors at each time
1224:{\displaystyle y_{i}\cdot M^{t}x_{j}}
7:
205:. Consider the sequence of vectors
131:{\displaystyle x_{\mathrm {base} }}
1128:
1125:
1122:
1119:
1043:
1040:
1037:
1034:
827:
189:
186:
183:
180:
122:
119:
116:
113:
14:
748:{\displaystyle q_{0}\ldots q_{L}}
2209:Villard's 1997 research report '
2158:Journal of Symbolic Computation
1442:
1436:
817:
803:
642:
639:
623:
610:
312:be any other vector of length
1:
138:be a random vector of length
2222:' uses a more sophisticated
1544:Invariant Factor Calculation
414:{\displaystyle S_{y}=\left}
2453:
2314:System of linear equations
1677:is a finite field of size
713:BerlekampâMassey algorithm
2365:Cache-oblivious algorithm
2180:10.1016/j.jsc.2021.06.005
1529:{\displaystyle i_{\max }}
86:{\displaystyle n\times n}
42:
18:block Wiedemann algorithm
2437:Numerical linear algebra
2416:General purpose software
2279:Numerical linear algebra
863:{\displaystyle y\cdot S}
423:We know that the matrix
870:, actually annihilates
449:CayleyâHamilton theorem
265:{\displaystyle S=\left}
35:is a generalization by
2128:
1837:
1799:
1757:
1737:
1736:{\displaystyle k<b}
1711:
1691:
1671:
1644:
1594:
1530:
1503:
1476:
1449:
1367:
1314:
1225:
1158:
1138:
1092:
1059:
1007:
971:
951:
917:
884:
864:
838:
782:
749:
702:
675:
655:
603:
563:
532:
492:
472:
437:
415:
326:
306:
286:
266:
199:
152:
132:
87:
61:
2411:Specialized libraries
2324:Matrix multiplication
2319:Matrix decompositions
2129:
1838:
1770:
1758:
1743:invariant factors of
1738:
1712:
1692:
1672:
1670:{\displaystyle F_{q}}
1645:
1595:
1554:Frobenius normal form
1531:
1504:
1502:{\displaystyle y_{i}}
1477:
1475:{\displaystyle y_{i}}
1450:
1368:
1315:
1226:
1159:
1139:
1072:
1060:
987:
972:
952:
897:
885:
865:
839:
762:
750:
703:
701:{\displaystyle S_{y}}
676:
656:
576:
564:
505:
493:
473:
471:{\displaystyle n_{0}}
438:
416:
327:
307:
287:
267:
200:
153:
133:
88:
62:
43:Wiedemann's algorithm
1850:
1767:
1747:
1721:
1701:
1681:
1654:
1604:
1560:
1513:
1486:
1459:
1377:
1324:
1235:
1185:
1148:
1069:
981:
961:
894:
874:
848:
759:
719:
685:
665:
573:
502:
482:
455:
427:
336:
316:
296:
276:
209:
162:
142:
104:
71:
51:
2298:Numerical stability
1639:
1589:
2124:
2119:
1833:
1753:
1733:
1707:
1697:, the probability
1687:
1667:
1640:
1619:
1590:
1569:
1526:
1499:
1472:
1445:
1363:
1310:
1221:
1154:
1134:
1055:
967:
947:
880:
860:
834:
745:
698:
671:
651:
559:
488:
468:
445:minimal polynomial
433:
411:
322:
302:
282:
262:
195:
148:
128:
83:
57:
2424:
2423:
2106:
2086:
2054:
2006:
1986:
1954:
1906:
1886:
1763:are preserved in
1756:{\displaystyle M}
1717:that the leading
1710:{\displaystyle p}
1690:{\displaystyle q}
1550:invariant factors
1428:
1408:
1157:{\displaystyle M}
970:{\displaystyle x}
883:{\displaystyle S}
674:{\displaystyle S}
491:{\displaystyle n}
436:{\displaystyle M}
325:{\displaystyle n}
305:{\displaystyle y}
285:{\displaystyle M}
151:{\displaystyle n}
60:{\displaystyle M}
2444:
2334:Matrix splitting
2272:
2265:
2258:
2249:
2192:
2191:
2173:
2149:
2133:
2131:
2130:
2125:
2123:
2122:
2107:
2104:
2087:
2084:
2077:
2066:
2065:
2060:
2056:
2055:
2053:
2052:
2034:
2007:
2004:
1987:
1984:
1977:
1966:
1965:
1960:
1956:
1955:
1953:
1952:
1934:
1907:
1904:
1887:
1884:
1874:
1842:
1840:
1839:
1834:
1832:
1831:
1822:
1821:
1812:
1811:
1798:
1784:
1762:
1760:
1759:
1754:
1742:
1740:
1739:
1734:
1716:
1714:
1713:
1708:
1696:
1694:
1693:
1688:
1676:
1674:
1673:
1668:
1666:
1665:
1649:
1647:
1646:
1641:
1638:
1627:
1599:
1597:
1596:
1591:
1588:
1577:
1535:
1533:
1532:
1527:
1525:
1524:
1508:
1506:
1505:
1500:
1498:
1497:
1481:
1479:
1478:
1473:
1471:
1470:
1454:
1452:
1451:
1446:
1429:
1427:
1426:
1414:
1409:
1407:
1406:
1394:
1389:
1388:
1373:need to satisfy
1372:
1370:
1369:
1364:
1362:
1361:
1349:
1348:
1336:
1335:
1319:
1317:
1316:
1311:
1309:
1308:
1284:
1283:
1259:
1258:
1230:
1228:
1227:
1222:
1220:
1219:
1210:
1209:
1197:
1196:
1163:
1161:
1160:
1155:
1143:
1141:
1140:
1135:
1133:
1132:
1131:
1112:
1111:
1102:
1101:
1091:
1086:
1064:
1062:
1061:
1056:
1048:
1047:
1046:
1027:
1026:
1017:
1016:
1006:
1001:
976:
974:
973:
968:
956:
954:
953:
948:
937:
936:
927:
926:
916:
911:
889:
887:
886:
881:
869:
867:
866:
861:
843:
841:
840:
835:
816:
802:
801:
792:
791:
781:
776:
754:
752:
751:
746:
744:
743:
731:
730:
707:
705:
704:
699:
697:
696:
680:
678:
677:
672:
660:
658:
657:
652:
635:
634:
622:
621:
602:
601:
600:
590:
568:
566:
565:
560:
552:
551:
542:
541:
531:
530:
529:
519:
497:
495:
494:
489:
477:
475:
474:
469:
467:
466:
442:
440:
439:
434:
420:
418:
417:
412:
410:
406:
399:
398:
348:
347:
331:
329:
328:
323:
311:
309:
308:
303:
291:
289:
288:
283:
271:
269:
268:
263:
261:
257:
247:
246:
204:
202:
201:
196:
194:
193:
192:
157:
155:
154:
149:
137:
135:
134:
129:
127:
126:
125:
92:
90:
89:
84:
66:
64:
63:
58:
2452:
2451:
2447:
2446:
2445:
2443:
2442:
2441:
2427:
2426:
2425:
2420:
2379:
2375:Multiprocessing
2343:
2339:Sparse problems
2302:
2281:
2276:
2239:
2232:
2218:Thomé's paper '
2196:
2195:
2151:
2150:
2146:
2141:
2136:
2118:
2117:
2105: and
2081:
2038:
2026:
2022:
2021:
2018:
2017:
2005: and
1981:
1938:
1926:
1922:
1921:
1918:
1917:
1905: and
1881:
1860:
1848:
1847:
1823:
1813:
1803:
1765:
1764:
1745:
1744:
1719:
1718:
1699:
1698:
1679:
1678:
1657:
1652:
1651:
1602:
1601:
1558:
1557:
1546:
1516:
1511:
1510:
1489:
1484:
1483:
1462:
1457:
1456:
1418:
1398:
1380:
1375:
1374:
1353:
1340:
1327:
1322:
1321:
1300:
1275:
1250:
1233:
1232:
1211:
1201:
1188:
1183:
1182:
1170:
1146:
1145:
1113:
1103:
1093:
1067:
1066:
1028:
1018:
1008:
979:
978:
959:
958:
928:
918:
892:
891:
872:
871:
846:
845:
793:
783:
757:
756:
735:
722:
717:
716:
688:
683:
682:
663:
662:
626:
613:
592:
571:
570:
543:
533:
521:
500:
499:
480:
479:
478:) no more than
458:
453:
452:
425:
424:
390:
356:
352:
339:
334:
333:
314:
313:
294:
293:
274:
273:
238:
222:
218:
207:
206:
174:
160:
159:
140:
139:
107:
102:
101:
69:
68:
49:
48:
45:
37:Don Coppersmith
12:
11:
5:
2450:
2448:
2440:
2439:
2429:
2428:
2422:
2421:
2419:
2418:
2413:
2408:
2403:
2398:
2393:
2387:
2385:
2381:
2380:
2378:
2377:
2372:
2367:
2362:
2357:
2351:
2349:
2345:
2344:
2342:
2341:
2336:
2331:
2321:
2316:
2310:
2308:
2304:
2303:
2301:
2300:
2295:
2293:Floating point
2289:
2287:
2283:
2282:
2277:
2275:
2274:
2267:
2260:
2252:
2246:
2245:
2237:
2230:
2215:
2214:
2206:
2205:
2201:
2200:
2194:
2193:
2143:
2142:
2140:
2137:
2121:
2116:
2113:
2110:
2102:
2099:
2096:
2093:
2090:
2082:
2080:
2076:
2072:
2069:
2064:
2059:
2051:
2048:
2045:
2041:
2037:
2032:
2029:
2025:
2020:
2019:
2016:
2013:
2010:
2002:
1999:
1996:
1993:
1990:
1982:
1980:
1976:
1972:
1969:
1964:
1959:
1951:
1948:
1945:
1941:
1937:
1932:
1929:
1925:
1920:
1919:
1916:
1913:
1910:
1902:
1899:
1896:
1893:
1890:
1882:
1880:
1877:
1873:
1869:
1866:
1865:
1863:
1858:
1855:
1845:
1830:
1826:
1820:
1816:
1810:
1806:
1802:
1797:
1794:
1791:
1788:
1783:
1780:
1777:
1773:
1752:
1732:
1729:
1726:
1706:
1686:
1664:
1660:
1637:
1634:
1631:
1626:
1622:
1618:
1615:
1612:
1609:
1587:
1584:
1581:
1576:
1572:
1568:
1565:
1545:
1542:
1523:
1519:
1496:
1492:
1469:
1465:
1444:
1441:
1438:
1435:
1432:
1425:
1421:
1417:
1412:
1405:
1401:
1397:
1392:
1387:
1383:
1360:
1356:
1352:
1347:
1343:
1339:
1334:
1330:
1307:
1303:
1299:
1296:
1293:
1290:
1287:
1282:
1278:
1274:
1271:
1268:
1265:
1262:
1257:
1253:
1249:
1246:
1243:
1240:
1218:
1214:
1208:
1204:
1200:
1195:
1191:
1169:
1166:
1153:
1130:
1127:
1124:
1121:
1116:
1110:
1106:
1100:
1096:
1090:
1085:
1082:
1079:
1075:
1054:
1051:
1045:
1042:
1039:
1036:
1031:
1025:
1021:
1015:
1011:
1005:
1000:
997:
994:
990:
986:
966:
946:
943:
940:
935:
931:
925:
921:
915:
910:
907:
904:
900:
879:
859:
856:
853:
833:
829:
825:
822:
819:
815:
812:
809:
805:
800:
796:
790:
786:
780:
775:
772:
769:
765:
742:
738:
734:
729:
725:
695:
691:
670:
650:
647:
644:
641:
638:
633:
629:
625:
620:
616:
612:
609:
606:
599:
595:
589:
586:
583:
579:
558:
555:
550:
546:
540:
536:
528:
524:
518:
515:
512:
508:
487:
465:
461:
432:
409:
405:
402:
397:
393:
389:
386:
383:
380:
377:
374:
371:
368:
365:
362:
359:
355:
351:
346:
342:
321:
301:
281:
260:
256:
253:
250:
245:
241:
237:
234:
231:
228:
225:
221:
217:
214:
191:
188:
185:
182:
177:
173:
170:
167:
147:
124:
121:
118:
115:
110:
82:
79:
76:
56:
44:
41:
20:for computing
13:
10:
9:
6:
4:
3:
2:
2449:
2438:
2435:
2434:
2432:
2417:
2414:
2412:
2409:
2407:
2404:
2402:
2399:
2397:
2394:
2392:
2389:
2388:
2386:
2382:
2376:
2373:
2371:
2368:
2366:
2363:
2361:
2358:
2356:
2353:
2352:
2350:
2346:
2340:
2337:
2335:
2332:
2329:
2325:
2322:
2320:
2317:
2315:
2312:
2311:
2309:
2305:
2299:
2296:
2294:
2291:
2290:
2288:
2284:
2280:
2273:
2268:
2266:
2261:
2259:
2254:
2253:
2250:
2243:
2236:
2233: =
2229:
2225:
2221:
2217:
2216:
2212:
2208:
2207:
2203:
2202:
2198:
2197:
2189:
2185:
2181:
2177:
2172:
2167:
2163:
2159:
2155:
2148:
2145:
2138:
2135:
2114:
2111:
2108:
2100:
2097:
2094:
2091:
2088:
2078:
2074:
2070:
2067:
2062:
2057:
2049:
2046:
2043:
2039:
2035:
2030:
2027:
2023:
2014:
2011:
2008:
2000:
1997:
1994:
1991:
1988:
1978:
1974:
1970:
1967:
1962:
1957:
1949:
1946:
1943:
1939:
1935:
1930:
1927:
1923:
1914:
1911:
1908:
1900:
1897:
1894:
1891:
1888:
1878:
1875:
1871:
1867:
1861:
1856:
1853:
1844:
1828:
1824:
1818:
1814:
1808:
1804:
1800:
1795:
1792:
1789:
1786:
1781:
1778:
1775:
1771:
1750:
1730:
1727:
1724:
1704:
1684:
1662:
1658:
1635:
1632:
1629:
1624:
1620:
1616:
1613:
1610:
1607:
1585:
1582:
1579:
1574:
1570:
1566:
1563:
1555:
1551:
1543:
1541:
1539:
1517:
1494:
1490:
1467:
1463:
1439:
1433:
1430:
1419:
1415:
1410:
1399:
1395:
1390:
1381:
1354:
1350:
1341:
1337:
1328:
1301:
1297:
1294:
1291:
1288:
1285:
1276:
1272:
1269:
1266:
1263:
1260:
1251:
1247:
1244:
1241:
1238:
1216:
1212:
1206:
1202:
1198:
1193:
1189:
1178:
1175:
1167:
1165:
1151:
1114:
1108:
1104:
1098:
1094:
1088:
1083:
1080:
1077:
1073:
1052:
1049:
1029:
1023:
1019:
1013:
1009:
1003:
998:
995:
992:
988:
984:
964:
944:
941:
938:
933:
929:
923:
919:
913:
908:
905:
902:
898:
890:; so we have
877:
857:
854:
851:
831:
823:
820:
813:
810:
807:
798:
794:
788:
784:
778:
773:
770:
767:
763:
740:
736:
732:
727:
723:
714:
709:
693:
689:
668:
648:
645:
636:
631:
627:
618:
614:
607:
604:
597:
593:
587:
584:
581:
577:
556:
553:
548:
544:
538:
534:
526:
522:
516:
513:
510:
506:
485:
463:
459:
450:
446:
430:
421:
407:
403:
400:
395:
391:
387:
384:
381:
378:
375:
372:
369:
366:
363:
360:
357:
353:
349:
344:
340:
319:
299:
279:
258:
254:
251:
248:
243:
239:
235:
232:
229:
226:
223:
219:
215:
212:
175:
171:
168:
165:
145:
108:
99:
95:
94:square matrix
80:
77:
74:
54:
40:
38:
34:
30:
26:
23:
19:
2286:Key concepts
2241:
2234:
2227:
2161:
2157:
2147:
1846:
1547:
1537:
1179:
1173:
1171:
710:
422:
98:finite field
46:
33:finite field
17:
15:
2328:algorithms
2171:1803.03864
2164:: 98â116.
2139:References
681:and hence
158:, and let
96:over some
2355:CPU cache
2188:0747-7171
2092:≥
2068:≥
2047:−
2031:−
1992:≥
1968:≥
1947:−
1931:−
1857:≥
1793:−
1772:∑
1633:×
1617:∈
1583:×
1567:∈
1556:. Given
1298:…
1273:…
1248:…
1231:for some
1199:⋅
1074:∑
989:∑
899:∑
855:⋅
828:∀
764:∑
733:…
608:⋅
578:∑
507:∑
447:; by the
404:…
388:⋅
373:⋅
361:⋅
255:…
78:×
2431:Category
2384:Software
2348:Hardware
2307:Problems
2085:if
1985:if
1885:if
711:But the
569:. Then
1065:and so
977:to say
100:F, let
31:over a
25:vectors
2406:LAPACK
2396:MATLAB
2186:
1650:where
1320:where
498:. Say
443:has a
292:; let
67:be an
29:matrix
22:kernel
2391:ATLAS
2166:arXiv
755:with
27:of a
2370:SIMD
2244:(2).
2184:ISSN
2112:>
1728:<
1600:and
1455:and
1391:>
47:Let
16:The
2360:TLB
2238:max
2231:max
2224:FFT
2176:doi
2162:108
1843:is
1522:max
1424:max
1404:max
1386:max
1359:max
1346:max
1333:max
1306:max
1281:max
1256:max
2433::
2242:GF
2182:.
2174:.
2160:.
2156:.
2134:.
1979:16
1876:64
1540:.
1164:.
708:.
2330:)
2326:(
2271:e
2264:t
2257:v
2235:j
2228:i
2190:.
2178::
2168::
2115:2
2109:q
2101:1
2098:+
2095:k
2089:b
2079:9
2075:/
2071:1
2063:2
2058:)
2050:k
2044:b
2040:q
2036:2
2028:1
2024:(
2015:2
2012:=
2009:q
2001:2
1998:+
1995:k
1989:b
1975:/
1971:1
1963:2
1958:)
1950:k
1944:b
1940:2
1936:3
1928:1
1924:(
1915:2
1912:=
1909:q
1901:1
1898:+
1895:k
1892:=
1889:b
1879:,
1872:/
1868:1
1862:{
1854:p
1829:i
1825:x
1819:T
1815:V
1809:i
1805:M
1801:U
1796:1
1790:n
1787:2
1782:0
1779:=
1776:i
1751:M
1731:b
1725:k
1705:p
1685:q
1663:q
1659:F
1636:n
1630:b
1625:q
1621:F
1614:V
1611:,
1608:U
1586:n
1580:n
1575:q
1571:F
1564:M
1538:t
1518:i
1495:i
1491:y
1468:i
1464:y
1443:)
1440:1
1437:(
1434:O
1431:+
1420:j
1416:d
1411:+
1400:i
1396:d
1382:t
1355:t
1351:,
1342:j
1338:,
1329:i
1302:t
1295:0
1292:=
1289:t
1286:,
1277:j
1270:0
1267:=
1264:j
1261:,
1252:i
1245:0
1242:=
1239:i
1217:j
1213:x
1207:t
1203:M
1194:i
1190:y
1174:S
1152:M
1129:e
1126:s
1123:a
1120:b
1115:x
1109:i
1105:M
1099:i
1095:q
1089:L
1084:0
1081:=
1078:i
1053:0
1050:=
1044:e
1041:s
1038:a
1035:b
1030:x
1024:i
1020:M
1014:i
1010:q
1004:L
999:0
996:=
993:i
985:M
965:x
945:0
942:=
939:x
934:i
930:M
924:i
920:q
914:L
909:0
906:=
903:i
878:S
858:S
852:y
832:r
824:0
821:=
818:]
814:r
811:+
808:i
804:[
799:y
795:S
789:i
785:q
779:L
774:0
771:=
768:i
741:L
737:q
728:0
724:q
694:y
690:S
669:S
649:0
646:=
643:)
640:)
637:x
632:r
628:M
624:(
619:r
615:p
611:(
605:y
598:0
594:n
588:0
585:=
582:r
557:0
554:=
549:r
545:M
539:r
535:p
527:0
523:n
517:0
514:=
511:r
486:n
464:0
460:n
431:M
408:]
401:x
396:2
392:M
385:y
382:,
379:x
376:M
370:y
367:,
364:x
358:y
354:[
350:=
345:y
341:S
320:n
300:y
280:M
259:]
252:,
249:x
244:2
240:M
236:,
233:x
230:M
227:,
224:x
220:[
216:=
213:S
190:e
187:s
184:a
181:b
176:x
172:M
169:=
166:x
146:n
123:e
120:s
117:a
114:b
109:x
81:n
75:n
55:M
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