123: − 1 has at least one large prime factor. Most sufficiently large primes are strong; if a prime used for cryptographic purposes turns out to be non-strong, it is much more likely to be through malice than through an accident of
941:
489:
1930:
1058:
240:
1934:
1442:
135:
factorization method is more efficient than
Pollard's algorithm and finds safe prime factors just as quickly as it finds non-safe prime factors of similar size, thus the size of
322:
1475:
854:
416:
1990:
1799:
1435:
56:
279: − 1 by making it a number with very many prime factors; generally, we take the product of all prime powers less than some limit
1667:
1411:
971:
1609:
1428:
1273:
794:
132:
1538:
1715:
1513:
1662:
1599:
1543:
1506:
1804:
1695:
1614:
1604:
1480:
1632:
1286:
1885:
1880:
1809:
1710:
1309:
100:
and thus minimally smooth. These primes are sometimes construed as "safe for cryptographic purposes", but they might be
1847:
116:
48:
1761:
1926:
1916:
1875:
1651:
1645:
1619:
1490:
181:
160:
1911:
1852:
1814:
1687:
1533:
1485:
124:
1829:
1720:
1397:
259:
1940:
1890:
1870:
848:. After completing the first stage, which is the same as the basic algorithm, instead of computing a new
1591:
1566:
1495:
41:
1950:
290:
1945:
1837:
1819:
1794:
1756:
1500:
1375:
1330:
746:
Since the algorithm is incremental, it is able to keep running with the bound constantly increasing.
94:
1955:
1921:
1842:
1746:
1705:
1700:
1677:
1581:
771:
1786:
1733:
1730:
1571:
1470:
1346:
328:
runs through those prime powers. Check at each stage, or once at the end if you prefer, whether
250:
71:
1527:
1520:
112:
1906:
1862:
1576:
1553:
1407:
1197:
1751:
1383:
1338:
1741:
1640:
936:{\displaystyle M'=\prod _{{\text{primes }}q\leq B_{2}}q^{\lfloor \log _{q}B_{2}\rfloor }}
809: = 2 will find a quarter of all 64-bit factors and 1/27 of all 96-bit factors.
1379:
1334:
1771:
1672:
1657:
1561:
1462:
1388:
1359:
1984:
1766:
1451:
1350:
67:
38:
1776:
105:
1276:, use a modified version of the p - 1 algorithm to eliminate potential candidates.
484:{\displaystyle M=\prod _{{\text{primes}}~q\leq B}q^{\lfloor \log _{q}{B}\rfloor }}
51:
in 1974. It is a special-purpose algorithm, meaning that it is only suitable for
1420:
1342:
817:
A variant of the basic algorithm is sometimes used; instead of requiring that
360: − 1 is divisible by small primes, at which point the Pollard
86:
1454:
70:; the essential observation is that, by working in the multiplicative group
44:
801: − 1 method once the factors are at all large; running the
17:
1321:
Pollard, J. M. (1974). "Theorems of factorization and primality testing".
774:, the probability that the largest factor of such a number is less than (
128:
62:
The factors it finds are ones for which the number preceding the factor,
1401:
1265:
1255:
52:
1053:{\displaystyle Q=\prod _{{\text{primes }}q\in (B_{1},B_{2}]}(H^{q}-1)}
1269:
1177:
are even numbers. The distribution of prime numbers is such that the
1406:. Providence, RI: American Mathematical Society. pp. 138–141.
1156:
the difference between consecutive prime numbers. Since typically
825:, we require it to have all but one of its factors less than some
1310:
What are strong primes and are they necessary for the RSA system?
732:
299 / 13 = 23 is prime, thus it is fully factored: 299 = 13 × 23.
78:, we are also working in the multiplicative groups modulo all of
55:
with specific types of factors; it is the simplest example of an
1424:
782:; so there is a probability of about 3 = 1/27 that a
653:
are all the same in some rare cases, this algorithm will fail.
676:
make it run slower, but are more likely to produce a factor.
305:
761:, can be modelled as a random number of size less than
629:
in step 7, this usually indicates that all factors were
85:
The existence of this algorithm leads to the concept of
1971:
indicate that algorithm is for numbers of special forms
633:-powersmooth, but in rare cases it could indicate that
1089:. As before, exponentiations can be done modulo
974:
857:
607:
in step 6, this indicates there are no prime factors
419:
293:
275:
The idea is to make the exponent a large multiple of
184:
139:
is the key security parameter, not the smoothness of
104:— in current recommendations for cryptographic
1258:
package includes an efficient implementation of the
1899:
1861:
1828:
1785:
1729:
1686:
1590:
1552:
1461:
1186:will all be relatively small. It is suggested that
1323:Proceedings of the Cambridge Philosophical Society
1052:
935:
641:. Additionally, when the maximum prime factors of
483:
316:
234:
1110:, …} be successive prime numbers in the interval
27:Special-purpose algorithm for factoring integers
376:The basic algorithm can be written as follows:
536:(note: exponentiation can be done modulo
364: − 1 algorithm simply returns
348:It is possible that for all the prime factors
1436:
821: − 1 has all its factors less than
518:= 2, random selection here is not imperative)
235:{\displaystyle a^{K(p-1)}\equiv 1{\pmod {p}}}
8:
1358:Montgomery, P. L.; Silverman, R. D. (1990).
928:
902:
476:
455:
1443:
1429:
1421:
832:, and the remaining factor less than some
1387:
1035:
1017:
1004:
986:
985:
973:
922:
909:
901:
889:
874:
873:
856:
729:Since 1 < 13 < 299, thus return 13.
471:
462:
454:
431:
430:
418:
308:
304:
298:
292:
216:
189:
183:
155:be a composite integer with prime factor
1302:
1245:, saving the need for exponentiations.
57:algebraic-group factorisation algorithm
656:The running time of this algorithm is
1085:produces a nontrivial factor of
7:
1274:Great Internet Mersenne Prime Search
93: − 1 is two times a
224:
805: − 1 method up to
514:is odd, then we can always select
498:randomly pick a positive integer,
272:will be divisible by that factor.
131:by the cryptography industry: the
127:. This terminology is considered
25:
1652:Special number field sieve (SNFS)
1646:General number field sieve (GNFS)
1389:10.1090/S0025-5718-1990-1011444-3
778: − 1) is roughly
1991:Integer factorization algorithms
1229:) can be stored in a table, and
757:is the smallest prime factor of
684:If we want to factor the number
317:{\displaystyle x^{w}{\bmod {n}}}
163:, we know that for all integers
287:, and repeatedly replace it by
217:
1272:, the official clients of the
1047:
1028:
1023:
997:
637:had a small order modulo
228:
218:
205:
193:
171:and for all positive integers
35: − 1 algorithm
1:
491:(note: explicitly evaluating
1610:Lenstra elliptic curve (ECM)
1262: − 1 method.
790:will yield a factorisation.
753: − 1, where
117:necessary but not sufficient
797:is faster than the Pollard
591:and go to step 2 or return
571:and go to step 2 or return
506:(note: we can actually fix
2007:
1917:Exponentiation by squaring
1600:Continued fraction (CFRAC)
1368:Mathematics of Computation
407:select a smoothness bound
372:Algorithm and running time
1964:
1360:"An FFT extension to the
1343:10.1017/S0305004100049252
1312:, RSA Laboratories (2007)
393:: a nontrivial factor of
89:, being primes for which
66: − 1, is
1364:− 1 factoring algorithm"
1225:, … (mod
340:is not equal to 1.
125:random number generation
1830:Greatest common divisor
1398:Samuel S. Wagstaff, Jr.
1207:. Hence, the values of
645:for each prime factors
283:. Start with a random
161:Fermat's little theorem
1941:Modular exponentiation
1054:
937:
587:then select a smaller
502:, which is coprime to
485:
318:
236:
1668:Shanks's square forms
1592:Integer factorization
1567:Sieve of Eratosthenes
1055:
938:
795:elliptic curve method
567:then select a larger
495:may not be necessary)
486:
319:
237:
42:integer factorization
1946:Montgomery reduction
1820:Function field sieve
1795:Baby-step giant-step
1641:Quadratic sieve (QS)
1403:The Joy of Factoring
972:
855:
739:Methods of choosing
417:
387:: a composite number
291:
182:
95:Sophie Germain prime
1956:Trachtenberg system
1922:Integer square root
1863:Modular square root
1582:Wheel factorization
1534:Quadratic Frobenius
1514:Lucas–Lehmer–Riesel
1380:1990MaCom..54..839M
1335:1974PCPS...76..521P
672:; larger values of
74:a composite number
1848:Extended Euclidean
1787:Discrete logarithm
1716:Schönhage–Strassen
1572:Sieve of Pritchard
1050:
1027:
933:
896:
619:-powersmooth. If
481:
449:
314:
249:is congruent to 1
232:
1978:
1977:
1577:Sieve of Sundaram
1413:978-1-4704-1048-3
1235:be computed from
989:
981:
877:
869:
813:Two-stage variant
793:In practice, the
438:
434:
426:
16:(Redirected from
1998:
1927:Integer relation
1900:Other algorithms
1805:Pollard kangaroo
1696:Ancient Egyptian
1554:Prime-generating
1539:Solovay–Strassen
1452:Number-theoretic
1445:
1438:
1431:
1422:
1417:
1393:
1391:
1374:(190): 839–854.
1354:
1313:
1307:
1244:
1234:
1224:
1218:
1212:
1206:
1176:
1165:
1127:
1084:
1072:
1059:
1057:
1056:
1051:
1040:
1039:
1026:
1022:
1021:
1009:
1008:
990:
987:
964:
942:
940:
939:
934:
932:
931:
927:
926:
914:
913:
895:
894:
893:
878:
875:
865:
847:
769:
768:
671:
628:
606:
586:
566:
553:
535:
490:
488:
487:
482:
480:
479:
475:
467:
466:
448:
436:
435:
432:
344:Multiple factors
339:
323:
321:
320:
315:
313:
312:
303:
302:
271:
241:
239:
238:
233:
231:
209:
208:
39:number theoretic
21:
2006:
2005:
2001:
2000:
1999:
1997:
1996:
1995:
1981:
1980:
1979:
1974:
1960:
1895:
1857:
1824:
1781:
1725:
1682:
1586:
1548:
1521:Proth's theorem
1463:Primality tests
1457:
1449:
1414:
1396:
1357:
1320:
1317:
1316:
1308:
1304:
1299:
1283:
1251:
1249:Implementations
1236:
1230:
1220:
1214:
1208:
1205:
1195:
1187:
1185:
1175:
1167:
1163:
1157:
1155:
1145:
1136:
1125:
1118:
1111:
1109:
1102:
1074:
1064:
1031:
1013:
1000:
970:
969:
954:
952:
918:
905:
897:
885:
858:
853:
852:
846:
839:
833:
831:
815:
772:Dixon's theorem
764:
762:
744:
682:
657:
620:
601:
578:
561:
544:
522:
458:
450:
415:
414:
374:
346:
329:
294:
289:
288:
258:
185:
180:
179:
149:
28:
23:
22:
15:
12:
11:
5:
2004:
2002:
1994:
1993:
1983:
1982:
1976:
1975:
1973:
1972:
1965:
1962:
1961:
1959:
1958:
1953:
1948:
1943:
1938:
1924:
1919:
1914:
1909:
1903:
1901:
1897:
1896:
1894:
1893:
1888:
1883:
1881:Tonelli–Shanks
1878:
1873:
1867:
1865:
1859:
1858:
1856:
1855:
1850:
1845:
1840:
1834:
1832:
1826:
1825:
1823:
1822:
1817:
1815:Index calculus
1812:
1810:Pohlig–Hellman
1807:
1802:
1797:
1791:
1789:
1783:
1782:
1780:
1779:
1774:
1769:
1764:
1762:Newton-Raphson
1759:
1754:
1749:
1744:
1738:
1736:
1727:
1726:
1724:
1723:
1718:
1713:
1708:
1703:
1698:
1692:
1690:
1688:Multiplication
1684:
1683:
1681:
1680:
1675:
1673:Trial division
1670:
1665:
1660:
1658:Rational sieve
1655:
1648:
1643:
1638:
1630:
1622:
1617:
1612:
1607:
1602:
1596:
1594:
1588:
1587:
1585:
1584:
1579:
1574:
1569:
1564:
1562:Sieve of Atkin
1558:
1556:
1550:
1549:
1547:
1546:
1541:
1536:
1531:
1524:
1517:
1510:
1503:
1498:
1493:
1488:
1486:Elliptic curve
1483:
1478:
1473:
1467:
1465:
1459:
1458:
1450:
1448:
1447:
1440:
1433:
1425:
1419:
1418:
1412:
1394:
1355:
1329:(3): 521–528.
1315:
1314:
1301:
1300:
1298:
1295:
1294:
1293:
1282:
1279:
1278:
1277:
1263:
1250:
1247:
1203:
1191:
1181:
1171:
1161:
1150:
1141:
1132:
1123:
1116:
1107:
1100:
1061:
1060:
1049:
1046:
1043:
1038:
1034:
1030:
1025:
1020:
1016:
1012:
1007:
1003:
999:
996:
993:
984:
980:
977:
950:
944:
943:
930:
925:
921:
917:
912:
908:
904:
900:
892:
888:
884:
881:
872:
868:
864:
861:
844:
837:
829:
814:
811:
743:
737:
736:
735:
734:
733:
730:
727:
713:
706:
699:
681:
678:
598:
597:
596:
595:
575:
558:
541:
519:
496:
478:
474:
470:
465:
461:
457:
453:
447:
444:
441:
429:
425:
422:
411:
402:
401:
388:
373:
370:
345:
342:
311:
307:
301:
297:
243:
242:
230:
227:
223:
220:
215:
212:
207:
204:
201:
198:
195:
192:
188:
148:
145:
47:, invented by
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2003:
1992:
1989:
1988:
1986:
1970:
1967:
1966:
1963:
1957:
1954:
1952:
1949:
1947:
1944:
1942:
1939:
1936:
1932:
1928:
1925:
1923:
1920:
1918:
1915:
1913:
1910:
1908:
1905:
1904:
1902:
1898:
1892:
1889:
1887:
1884:
1882:
1879:
1877:
1876:Pocklington's
1874:
1872:
1869:
1868:
1866:
1864:
1860:
1854:
1851:
1849:
1846:
1844:
1841:
1839:
1836:
1835:
1833:
1831:
1827:
1821:
1818:
1816:
1813:
1811:
1808:
1806:
1803:
1801:
1798:
1796:
1793:
1792:
1790:
1788:
1784:
1778:
1775:
1773:
1770:
1768:
1765:
1763:
1760:
1758:
1755:
1753:
1750:
1748:
1745:
1743:
1740:
1739:
1737:
1735:
1732:
1728:
1722:
1719:
1717:
1714:
1712:
1709:
1707:
1704:
1702:
1699:
1697:
1694:
1693:
1691:
1689:
1685:
1679:
1676:
1674:
1671:
1669:
1666:
1664:
1661:
1659:
1656:
1654:
1653:
1649:
1647:
1644:
1642:
1639:
1637:
1635:
1631:
1629:
1627:
1623:
1621:
1620:Pollard's rho
1618:
1616:
1613:
1611:
1608:
1606:
1603:
1601:
1598:
1597:
1595:
1593:
1589:
1583:
1580:
1578:
1575:
1573:
1570:
1568:
1565:
1563:
1560:
1559:
1557:
1555:
1551:
1545:
1542:
1540:
1537:
1535:
1532:
1530:
1529:
1525:
1523:
1522:
1518:
1516:
1515:
1511:
1509:
1508:
1504:
1502:
1499:
1497:
1494:
1492:
1489:
1487:
1484:
1482:
1479:
1477:
1474:
1472:
1469:
1468:
1466:
1464:
1460:
1456:
1453:
1446:
1441:
1439:
1434:
1432:
1427:
1426:
1423:
1415:
1409:
1405:
1404:
1399:
1395:
1390:
1385:
1381:
1377:
1373:
1369:
1365:
1363:
1356:
1352:
1348:
1344:
1340:
1336:
1332:
1328:
1324:
1319:
1318:
1311:
1306:
1303:
1296:
1292:
1291:+ 1 algorithm
1290:
1285:
1284:
1280:
1275:
1271:
1267:
1264:
1261:
1257:
1253:
1252:
1248:
1246:
1243:
1239:
1233:
1228:
1223:
1217:
1211:
1202:
1199:
1194:
1190:
1184:
1180:
1174:
1170:
1160:
1153:
1149:
1146: −
1144:
1140:
1137: =
1135:
1131:
1122:
1115:
1106:
1099:
1094:
1092:
1088:
1082:
1078:
1073:and check if
1071:
1067:
1044:
1041:
1036:
1032:
1018:
1014:
1010:
1005:
1001:
994:
991:
982:
978:
975:
968:
967:
966:
965:, we compute
962:
958:
953:and checking
949:
923:
919:
915:
910:
906:
898:
890:
886:
882:
879:
870:
866:
862:
859:
851:
850:
849:
843:
836:
828:
824:
820:
812:
810:
808:
804:
800:
796:
791:
789:
785:
781:
777:
773:
767:
760:
756:
752:
747:
742:
738:
731:
728:
725:
721:
717:
714:
711:
707:
704:
700:
697:
693:
692:
691:
690:
689:
687:
679:
677:
675:
669:
665:
661:
654:
652:
648:
644:
640:
636:
632:
627:
623:
618:
614:
610:
604:
594:
590:
585:
581:
576:
574:
570:
564:
559:
557:
552:
548:
542:
539:
533:
529:
525:
520:
517:
513:
509:
505:
501:
497:
494:
472:
468:
463:
459:
451:
445:
442:
439:
427:
423:
420:
412:
410:
406:
405:
404:
403:
400:
396:
392:
389:
386:
382:
379:
378:
377:
371:
369:
367:
363:
359:
355:
351:
343:
341:
337:
333:
327:
309:
299:
295:
286:
282:
278:
273:
269:
265:
261:
256:
252:
248:
225:
221:
213:
210:
202:
199:
196:
190:
186:
178:
177:
176:
174:
170:
166:
162:
158:
154:
147:Base concepts
146:
144:
142:
138:
134:
130:
126:
122:
118:
114:
111:
107:
106:strong primes
103:
99:
96:
92:
88:
83:
81:
77:
73:
69:
65:
60:
58:
54:
50:
46:
43:
40:
36:
34:
19:
1968:
1650:
1633:
1625:
1624:
1544:Miller–Rabin
1526:
1519:
1512:
1507:Lucas–Lehmer
1505:
1402:
1371:
1367:
1361:
1326:
1322:
1305:
1288:
1259:
1241:
1237:
1231:
1226:
1221:
1215:
1209:
1200:
1192:
1188:
1182:
1178:
1172:
1168:
1158:
1151:
1147:
1142:
1138:
1133:
1129:
1120:
1113:
1104:
1097:
1095:
1090:
1086:
1080:
1076:
1069:
1065:
1062:
988:primes
960:
956:
947:
945:
876:primes
841:
834:
826:
822:
818:
816:
806:
802:
798:
792:
787:
783:
779:
775:
765:
758:
754:
750:
749:Assume that
748:
745:
740:
723:
719:
715:
709:
705:= 2 × 3 × 5.
702:
695:
685:
683:
673:
667:
663:
659:
655:
650:
646:
642:
638:
634:
630:
625:
621:
616:
612:
608:
602:
599:
592:
588:
583:
579:
572:
568:
562:
555:
554:then return
550:
546:
537:
531:
527:
523:
515:
511:
507:
503:
499:
492:
408:
398:
394:
390:
384:
380:
375:
365:
361:
357:
353:
349:
347:
335:
331:
325:
284:
280:
276:
274:
267:
263:
254:
253:a factor of
246:
245:If a number
244:
172:
168:
164:
156:
152:
150:
140:
136:
120:
109:
101:
97:
90:
84:
79:
75:
63:
61:
49:John Pollard
32:
30:
29:
1800:Pollard rho
1757:Goldschmidt
1491:Pocklington
1481:Baillie–PSW
1287:Williams's
334:− 1,
266:− 1,
257:, then the
167:coprime to
87:safe primes
82:s factors.
68:powersmooth
18:Pollard P-1
1912:Cornacchia
1907:Chakravala
1455:algorithms
1297:References
708:We select
694:We select
611:for which
510:, e.g. if
113:ANSI X9.31
31:Pollard's
1886:Berlekamp
1843:Euclidean
1731:Euclidean
1711:Toom–Cook
1706:Karatsuba
1351:122817056
1042:−
995:∈
983:∏
929:⌋
916:
903:⌊
883:≤
871:∏
786:value of
477:⌋
469:
456:⌊
443:≤
428:∏
211:≡
200:−
115:), it is
45:algorithm
1985:Category
1853:Lehmer's
1747:Chunking
1734:division
1663:Fermat's
1400:(2013).
1281:See also
863:′
521:compute
129:obsolete
53:integers
1969:Italics
1891:Kunerth
1871:Cipolla
1752:Fourier
1721:Fürer's
1615:Euler's
1605:Dixon's
1528:Pépin's
1376:Bibcode
1331:Bibcode
1266:Prime95
1256:GMP-ECM
763:√
726:) = 13.
688:= 299.
680:Example
593:failure
573:failure
545:1 <
413:define
399:failure
1951:Schoof
1838:Binary
1742:Binary
1678:Shor's
1496:Fermat
1410:
1349:
1270:MPrime
1164:> 2
1063:where
780:ε
718:= gcd(
666:× log
662:× log
526:= gcd(
437:
433:primes
391:Output
381:Inputs
251:modulo
159:. By
102:unsafe
72:modulo
1772:Short
1501:Lucas
1347:S2CID
1096:Let {
959:− 1,
770:. By
722:− 1,
701:Thus
549:<
530:− 1,
119:that
37:is a
1767:Long
1701:Long
1408:ISBN
1268:and
1254:The
1128:and
1075:gcd(
955:gcd(
946:for
712:= 2.
698:= 5.
330:gcd(
151:Let
110:e.g.
1931:LLL
1777:SRT
1636:+ 1
1628:− 1
1476:APR
1471:AKS
1384:doi
1339:doi
907:log
649:of
643:p-1
615:is
613:p-1
605:= 1
600:If
577:if
565:= 1
560:if
543:if
460:log
397:or
352:of
324:as
306:mod
260:gcd
222:mod
141:p-1
133:ECM
1987::
1935:KZ
1933:;
1382:.
1372:54
1370:.
1366:.
1345:.
1337:.
1327:76
1325:.
1219:,
1213:,
1198:ln
1196:≤
1166:,
1154:−1
1119:,
1103:,
1093:.
1079:,
1068:=
840:≫
658:O(
624:=
582:=
383::
368:.
356:,
175::
143:.
80:N'
59:.
1937:)
1929:(
1634:p
1626:p
1444:e
1437:t
1430:v
1416:.
1392:.
1386::
1378::
1362:P
1353:.
1341::
1333::
1289:p
1260:p
1242:H
1240:⋅
1238:H
1232:H
1227:n
1222:H
1216:H
1210:H
1204:2
1201:B
1193:n
1189:d
1183:n
1179:d
1173:n
1169:d
1162:1
1159:B
1152:n
1148:q
1143:n
1139:q
1134:n
1130:d
1126:]
1124:2
1121:B
1117:1
1114:B
1112:(
1108:2
1105:q
1101:1
1098:q
1091:n
1087:n
1083:)
1081:n
1077:Q
1070:a
1066:H
1048:)
1045:1
1037:q
1033:H
1029:(
1024:]
1019:2
1015:B
1011:,
1006:1
1002:B
998:(
992:q
979:=
976:Q
963:)
961:n
957:a
951:2
948:B
924:2
920:B
911:q
899:q
891:2
887:B
880:q
867:=
860:M
845:1
842:B
838:2
835:B
830:1
827:B
823:B
819:p
807:B
803:p
799:p
788:n
784:B
776:p
766:n
759:n
755:p
751:p
741:B
724:n
720:a
716:g
710:a
703:M
696:B
686:n
674:B
670:)
668:n
664:B
660:B
651:n
647:p
639:n
635:a
631:B
626:n
622:g
617:B
609:p
603:g
589:B
584:n
580:g
569:B
563:g
556:g
551:n
547:g
540:)
538:n
534:)
532:n
528:a
524:g
516:a
512:n
508:a
504:n
500:a
493:M
473:B
464:q
452:q
446:B
440:q
424:=
421:M
409:B
395:n
385:n
366:n
362:p
358:p
354:n
350:p
338:)
336:n
332:x
326:w
310:n
300:w
296:x
285:x
281:B
277:p
270:)
268:n
264:x
262:(
255:n
247:x
229:)
226:p
219:(
214:1
206:)
203:1
197:p
194:(
191:K
187:a
173:K
169:p
165:a
157:p
153:n
137:p
121:p
108:(
98:q
91:p
76:N
64:p
33:p
20:)
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