741:
540:
590:
409:
295:
1193:
Also note that the second condition guarantees that the reduced basis is length-reduced (adding an integer multiple of one basis vector to another will not decrease its length); the same condition is used in the LLL reduction.
1137:
420:
2000:
160:
156:
826:
1188:
937:
736:{\displaystyle \pi _{i}(\mathbf {x} )=\sum _{j\geq i}{\frac {\langle \mathbf {x} ,\mathbf {b} _{j}^{*}\rangle }{\langle \mathbf {b} _{j}^{*},\mathbf {b} _{j}^{*}\rangle }}\mathbf {b} _{j}^{*}}
1020:
886:
1512:
578:
310:
1052:
84:
766:
153:
1545:
965:
111:
218:
849:
1869:
1505:
1057:
535:{\displaystyle \mu _{i,j}={\frac {\langle \mathbf {b} _{i},\mathbf {b} _{j}^{*}\rangle }{\langle \mathbf {b} _{j}^{*},\mathbf {b} _{j}^{*}\rangle }}}
1737:
1432:
1253:
2065:
1679:
1498:
1608:
1785:
771:
1583:
1694:
1732:
1669:
1613:
1576:
2060:
1874:
1765:
1684:
1674:
1550:
17:
1702:
1955:
1142:
891:
1950:
1879:
1780:
1917:
1831:
970:
1996:
1986:
1945:
1721:
1715:
1689:
1560:
1210:
1981:
1922:
180:
301:
1884:
1757:
1603:
1555:
2070:
1899:
1790:
404:{\displaystyle \mathbf {B} ^{*}=\{\mathbf {b} _{1}^{*},\mathbf {b} _{2}^{*},\dots ,\mathbf {b} _{n}^{*}\},}
2010:
1960:
1940:
857:
210:
164:
1661:
1636:
1565:
1392:
545:
2020:
1028:
60:
2015:
1907:
1889:
1864:
1826:
1570:
1460:
1347:
2025:
1991:
1912:
1816:
1775:
1770:
1747:
1651:
184:
749:
1856:
1803:
1800:
1641:
1540:
1476:
1378:
1363:
1337:
1314:
1259:
1233:
116:
1597:
1590:
1976:
1932:
1646:
1623:
1428:
1249:
48:
1821:
1468:
1420:
1355:
1306:
1241:
944:
176:
1445:
290:{\displaystyle \mathbf {B} =\{\mathbf {b} _{1},\mathbf {b} _{2},\dots ,\mathbf {b} _{n}\},}
89:
1811:
1710:
1405:
1464:
1351:
1234:"A Survey of Solving SVP Algorithms and Recent Strategies for Solving the SVP Challenge"
1841:
1742:
1727:
1631:
1532:
834:
187:. The first algorithm for constructing a KZ-reduced basis was given in 1983 by Kannan.
2054:
1836:
1521:
1318:
1263:
1367:
1846:
1480:
1446:"HKZ and Minkowski Reduction Algorithms for Lattice-Reduction-Aided MIMO Detection"
1245:
1211:
https://sites.math.washington.edu/~rothvoss/lecturenotes/IntOpt-and-Lattices.pdf
1132:{\displaystyle \{\pi _{2}(\mathbf {b} _{2}),\cdots \pi _{2}(\mathbf {b} _{n})\}}
1490:
1424:
1025:
Note that the first condition can be reformulated recursively as stating that
1472:
1359:
1524:
51:
1238:
International
Symposium on Mathematics, Quantum Theory, and Cryptography
159:. KZ has exponential complexity versus the polynomial complexity of the
1310:
1328:
Lyu, Shanxiang; Ling, Cong (2017). "Boosted KZ and LLL Algorithms".
1294:
1383:
1342:
163:
algorithm, however it may still be preferred for solving multiple
821:{\displaystyle \mathbf {b} _{i}^{*},\cdots ,\mathbf {b} _{n}^{*}}
1494:
1377:
Wen, Jinming; Chang, Xiao-Wen (2018). "On the KZ Reduction".
1240:. Mathematics for Industry. Vol. 33. pp. 189–207.
1161:
910:
167:(CVPs) in the same lattice, where it can be more efficient.
86:
it yields a lattice basis with orthogonality defect at most
206:
A KZ-reduced basis for a lattice is defined as follows:
2041:
indicate that algorithm is for numbers of special forms
1183:{\displaystyle \pi _{2}({\mathcal {L}}(\mathbf {B} ))}
932:{\displaystyle \pi _{i}({\mathcal {L}}(\mathbf {B} ))}
1145:
1060:
1031:
973:
947:
894:
860:
837:
774:
752:
593:
548:
423:
313:
221:
119:
92:
63:
1969:
1931:
1898:
1855:
1799:
1756:
1660:
1622:
1531:
1276:Micciancio & Goldwasser, p.133, definition 7.8
1182:
1131:
1046:
1014:
959:
931:
880:
843:
820:
760:
735:
572:
534:
403:
289:
175:The definition of a KZ-reduced basis was given by
147:
105:
78:
1444:Zhang, Wen; Qiao, Sanzheng; Wei, Yimin (2012).
1415:Micciancio, Daniele; Goldwasser, Shafi (2002).
1015:{\displaystyle \left|\mu _{i,j}\right|\leq 1/2}
1506:
8:
1126:
1061:
710:
670:
665:
637:
526:
486:
481:
446:
395:
329:
281:
230:
1513:
1499:
1491:
1382:
1341:
1169:
1160:
1159:
1150:
1144:
1117:
1112:
1102:
1083:
1078:
1068:
1059:
1054:is a shortest vector in the lattice, and
1038:
1033:
1030:
1004:
982:
972:
946:
918:
909:
908:
899:
893:
872:
867:
862:
859:
836:
812:
807:
802:
786:
781:
776:
773:
753:
751:
727:
722:
717:
704:
699:
694:
684:
679:
674:
659:
654:
649:
640:
634:
622:
607:
598:
592:
547:
520:
515:
510:
500:
495:
490:
475:
470:
465:
455:
450:
443:
428:
422:
389:
384:
379:
363:
358:
353:
343:
338:
333:
320:
315:
312:
275:
270:
254:
249:
239:
234:
222:
220:
135:
129:
124:
118:
97:
91:
70:
66:
65:
62:
1295:"Sur les formes quadratiques positives"
1203:
1453:IEEE Transactions on Signal Processing
1401:
1390:
1330:IEEE Transactions on Signal Processing
1139:is a KZ-reduced basis for the lattice
851:is KZ-reduced if the following holds:
7:
1293:Korkine, A.; Zolotareff, G. (1877).
881:{\displaystyle \mathbf {b} _{i}^{*}}
198:) algorithm was introduced in 1987.
183:in 1877, a strengthened version of
888:is the shortest nonzero vector in
573:{\displaystyle 1\leq j<i\leq n}
414:and the Gram-Schmidt coefficients
14:
1722:Special number field sieve (SNFS)
1716:General number field sieve (GNFS)
584:Also define projection functions
33:lattice basis reduction algorithm
1170:
1113:
1079:
1047:{\displaystyle \mathbf {b} _{1}}
1034:
919:
863:
803:
777:
754:
718:
695:
675:
650:
641:
608:
511:
491:
466:
451:
380:
354:
334:
316:
271:
250:
235:
223:
79:{\displaystyle \mathbb {R} ^{n}}
1417:Complexity of Lattice Problems
1177:
1174:
1166:
1156:
1123:
1108:
1089:
1074:
926:
923:
915:
905:
768:orthogonally onto the span of
612:
604:
1:
1680:Lenstra elliptic curve (ECM)
1246:10.1007/978-981-15-5191-8_15
761:{\displaystyle \mathbf {x} }
2066:Computational number theory
148:{\displaystyle 2^{n^{2}/2}}
2087:
1987:Exponentiation by squaring
1670:Continued fraction (CFRAC)
18:Kolmogorov–Zurbenko filter
15:
2034:
1425:10.1007/978-1-4615-0897-7
181:Yegor Ivanovich Zolotarev
37:Hermite–Korkine–Zolotarev
1473:10.1109/TSP.2012.2210708
1360:10.1109/TSP.2017.2708020
16:Not to be confused with
1900:Greatest common divisor
1232:Yasuda, Masaya (2021).
192:block Korkine-Zolotarev
165:closest vector problems
2061:Theory of cryptography
2011:Modular exponentiation
1400:Cite journal requires
1184:
1133:
1048:
1016:
961:
960:{\displaystyle j<i}
933:
882:
845:
822:
762:
737:
574:
536:
405:
291:
149:
107:
80:
1738:Shanks's square forms
1662:Integer factorization
1637:Sieve of Eratosthenes
1299:Mathematische Annalen
1222:Zhang et al 2012, p.1
1185:
1134:
1049:
1017:
962:
934:
883:
846:
823:
763:
738:
575:
537:
406:
292:
150:
108:
106:{\displaystyle n^{n}}
81:
2016:Montgomery reduction
1890:Function field sieve
1865:Baby-step giant-step
1711:Quadratic sieve (QS)
1419:. pp. 131–136.
1143:
1058:
1029:
971:
945:
892:
858:
835:
772:
750:
591:
546:
421:
311:
302:Gram–Schmidt process
219:
117:
90:
61:
2026:Trachtenberg system
1992:Integer square root
1933:Modular square root
1652:Wheel factorization
1604:Quadratic Frobenius
1584:Lucas–Lehmer–Riesel
1465:2012ITSP...60.5963Z
1352:2017ITSP...65.4784L
877:
817:
791:
732:
709:
689:
664:
525:
505:
480:
394:
368:
348:
1918:Extended Euclidean
1857:Discrete logarithm
1786:Schönhage–Strassen
1642:Sieve of Pritchard
1311:10.1007/BF01442667
1180:
1129:
1044:
1012:
957:
929:
878:
861:
841:
818:
801:
775:
758:
733:
716:
693:
673:
648:
633:
570:
532:
509:
489:
464:
401:
378:
352:
332:
287:
145:
103:
76:
2048:
2047:
1647:Sieve of Sundaram
1434:978-1-4613-5293-8
1336:(18): 4784–4796.
1255:978-981-15-5190-1
844:{\displaystyle B}
714:
618:
530:
304:orthogonal basis
185:Hermite reduction
49:lattice reduction
25:Korkine–Zolotarev
2078:
1997:Integer relation
1970:Other algorithms
1875:Pollard kangaroo
1766:Ancient Egyptian
1624:Prime-generating
1609:Solovay–Strassen
1522:Number-theoretic
1515:
1508:
1501:
1492:
1484:
1450:
1438:
1409:
1403:
1398:
1396:
1388:
1386:
1371:
1345:
1322:
1277:
1274:
1268:
1267:
1229:
1223:
1220:
1214:
1208:
1189:
1187:
1186:
1181:
1173:
1165:
1164:
1155:
1154:
1138:
1136:
1135:
1130:
1122:
1121:
1116:
1107:
1106:
1088:
1087:
1082:
1073:
1072:
1053:
1051:
1050:
1045:
1043:
1042:
1037:
1021:
1019:
1018:
1013:
1008:
997:
993:
992:
966:
964:
963:
958:
938:
936:
935:
930:
922:
914:
913:
904:
903:
887:
885:
884:
879:
876:
871:
866:
850:
848:
847:
842:
827:
825:
824:
819:
816:
811:
806:
790:
785:
780:
767:
765:
764:
759:
757:
742:
740:
739:
734:
731:
726:
721:
715:
713:
708:
703:
698:
688:
683:
678:
668:
663:
658:
653:
644:
635:
632:
611:
603:
602:
579:
577:
576:
571:
541:
539:
538:
533:
531:
529:
524:
519:
514:
504:
499:
494:
484:
479:
474:
469:
460:
459:
454:
444:
439:
438:
410:
408:
407:
402:
393:
388:
383:
367:
362:
357:
347:
342:
337:
325:
324:
319:
296:
294:
293:
288:
280:
279:
274:
259:
258:
253:
244:
243:
238:
226:
177:Aleksandr Korkin
154:
152:
151:
146:
144:
143:
139:
134:
133:
112:
110:
109:
104:
102:
101:
85:
83:
82:
77:
75:
74:
69:
57:For lattices in
2086:
2085:
2081:
2080:
2079:
2077:
2076:
2075:
2051:
2050:
2049:
2044:
2030:
1965:
1927:
1894:
1851:
1795:
1752:
1656:
1618:
1591:Proth's theorem
1533:Primality tests
1527:
1519:
1488:
1448:
1443:
1435:
1414:
1399:
1389:
1376:
1327:
1292:
1289:
1283:
1281:
1280:
1275:
1271:
1256:
1231:
1230:
1226:
1221:
1217:
1209:
1205:
1200:
1146:
1141:
1140:
1111:
1098:
1077:
1064:
1056:
1055:
1032:
1027:
1026:
978:
974:
969:
968:
943:
942:
895:
890:
889:
856:
855:
833:
832:
831:Then the basis
770:
769:
748:
747:
669:
636:
594:
589:
588:
544:
543:
485:
449:
445:
424:
419:
418:
314:
309:
308:
269:
248:
233:
217:
216:
204:
173:
125:
120:
115:
114:
93:
88:
87:
64:
59:
58:
21:
12:
11:
5:
2084:
2082:
2074:
2073:
2071:Lattice points
2068:
2063:
2053:
2052:
2046:
2045:
2043:
2042:
2035:
2032:
2031:
2029:
2028:
2023:
2018:
2013:
2008:
1994:
1989:
1984:
1979:
1973:
1971:
1967:
1966:
1964:
1963:
1958:
1953:
1951:Tonelli–Shanks
1948:
1943:
1937:
1935:
1929:
1928:
1926:
1925:
1920:
1915:
1910:
1904:
1902:
1896:
1895:
1893:
1892:
1887:
1885:Index calculus
1882:
1880:Pohlig–Hellman
1877:
1872:
1867:
1861:
1859:
1853:
1852:
1850:
1849:
1844:
1839:
1834:
1832:Newton-Raphson
1829:
1824:
1819:
1814:
1808:
1806:
1797:
1796:
1794:
1793:
1788:
1783:
1778:
1773:
1768:
1762:
1760:
1758:Multiplication
1754:
1753:
1751:
1750:
1745:
1743:Trial division
1740:
1735:
1730:
1728:Rational sieve
1725:
1718:
1713:
1708:
1700:
1692:
1687:
1682:
1677:
1672:
1666:
1664:
1658:
1657:
1655:
1654:
1649:
1644:
1639:
1634:
1632:Sieve of Atkin
1628:
1626:
1620:
1619:
1617:
1616:
1611:
1606:
1601:
1594:
1587:
1580:
1573:
1568:
1563:
1558:
1556:Elliptic curve
1553:
1548:
1543:
1537:
1535:
1529:
1528:
1520:
1518:
1517:
1510:
1503:
1495:
1486:
1485:
1440:
1439:
1433:
1411:
1410:
1402:|journal=
1373:
1372:
1324:
1323:
1305:(2): 242–292.
1288:
1285:
1279:
1278:
1269:
1254:
1224:
1215:
1202:
1201:
1199:
1196:
1179:
1176:
1172:
1168:
1163:
1158:
1153:
1149:
1128:
1125:
1120:
1115:
1110:
1105:
1101:
1097:
1094:
1091:
1086:
1081:
1076:
1071:
1067:
1063:
1041:
1036:
1023:
1022:
1011:
1007:
1003:
1000:
996:
991:
988:
985:
981:
977:
956:
953:
950:
939:
928:
925:
921:
917:
912:
907:
902:
898:
875:
870:
865:
840:
815:
810:
805:
800:
797:
794:
789:
784:
779:
756:
746:which project
744:
743:
730:
725:
720:
712:
707:
702:
697:
692:
687:
682:
677:
672:
667:
662:
657:
652:
647:
643:
639:
631:
628:
625:
621:
617:
614:
610:
606:
601:
597:
582:
581:
569:
566:
563:
560:
557:
554:
551:
528:
523:
518:
513:
508:
503:
498:
493:
488:
483:
478:
473:
468:
463:
458:
453:
448:
442:
437:
434:
431:
427:
412:
411:
400:
397:
392:
387:
382:
377:
374:
371:
366:
361:
356:
351:
346:
341:
336:
331:
328:
323:
318:
298:
297:
286:
283:
278:
273:
268:
265:
262:
257:
252:
247:
242:
237:
232:
229:
225:
203:
200:
172:
169:
142:
138:
132:
128:
123:
100:
96:
73:
68:
13:
10:
9:
6:
4:
3:
2:
2083:
2072:
2069:
2067:
2064:
2062:
2059:
2058:
2056:
2040:
2037:
2036:
2033:
2027:
2024:
2022:
2019:
2017:
2014:
2012:
2009:
2006:
2002:
1998:
1995:
1993:
1990:
1988:
1985:
1983:
1980:
1978:
1975:
1974:
1972:
1968:
1962:
1959:
1957:
1954:
1952:
1949:
1947:
1946:Pocklington's
1944:
1942:
1939:
1938:
1936:
1934:
1930:
1924:
1921:
1919:
1916:
1914:
1911:
1909:
1906:
1905:
1903:
1901:
1897:
1891:
1888:
1886:
1883:
1881:
1878:
1876:
1873:
1871:
1868:
1866:
1863:
1862:
1860:
1858:
1854:
1848:
1845:
1843:
1840:
1838:
1835:
1833:
1830:
1828:
1825:
1823:
1820:
1818:
1815:
1813:
1810:
1809:
1807:
1805:
1802:
1798:
1792:
1789:
1787:
1784:
1782:
1779:
1777:
1774:
1772:
1769:
1767:
1764:
1763:
1761:
1759:
1755:
1749:
1746:
1744:
1741:
1739:
1736:
1734:
1731:
1729:
1726:
1724:
1723:
1719:
1717:
1714:
1712:
1709:
1707:
1705:
1701:
1699:
1697:
1693:
1691:
1690:Pollard's rho
1688:
1686:
1683:
1681:
1678:
1676:
1673:
1671:
1668:
1667:
1665:
1663:
1659:
1653:
1650:
1648:
1645:
1643:
1640:
1638:
1635:
1633:
1630:
1629:
1627:
1625:
1621:
1615:
1612:
1610:
1607:
1605:
1602:
1600:
1599:
1595:
1593:
1592:
1588:
1586:
1585:
1581:
1579:
1578:
1574:
1572:
1569:
1567:
1564:
1562:
1559:
1557:
1554:
1552:
1549:
1547:
1544:
1542:
1539:
1538:
1536:
1534:
1530:
1526:
1523:
1516:
1511:
1509:
1504:
1502:
1497:
1496:
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155:bound of the
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1577:Lucas–Lehmer
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1459:(11): 5963.
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1870:Pollard rho
1827:Goldschmidt
1561:Pocklington
1551:Baillie–PSW
1213:, pp. 18-19
300:define its
2055:Categories
1982:Cornacchia
1977:Chakravala
1525:algorithms
1384:1702.08152
1343:1703.03303
1287:References
542:, for any
202:Definition
1956:Berlekamp
1913:Euclidean
1801:Euclidean
1781:Toom–Cook
1776:Karatsuba
1319:121803621
1264:226333525
1148:π
1100:π
1096:⋯
1066:π
999:≤
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264:…
52:algorithm
45:algorithm
1923:Lehmer's
1817:Chunking
1804:division
1733:Fermat's
1368:16832357
941:For all
209:Given a
2039:Italics
1961:Kunerth
1941:Cipolla
1822:Fourier
1791:Fürer's
1685:Euler's
1675:Dixon's
1598:Pépin's
1481:5962834
1461:Bibcode
1348:Bibcode
171:History
2021:Schoof
1908:Binary
1812:Binary
1748:Shor's
1566:Fermat
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1842:Short
1571:Lucas
1477:S2CID
1449:(PDF)
1379:arXiv
1364:S2CID
1338:arXiv
1315:S2CID
1260:S2CID
1198:Notes
211:basis
47:is a
1837:Long
1771:Long
1429:ISBN
1406:help
1250:ISBN
952:<
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190:The
179:and
23:The
2001:LLL
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