216:
91:
377:
797:
868:
Although
Protocol II in the same paper contains a fatal error, Scheme I is feasible; one such group/overgroup pairing is analyzed in
1103:
1098:
211:{\displaystyle R\mapsto {\frac {\operatorname {diam} _{H}(B_{G}(0,R)\cap H)}{\operatorname {diam} _{H}(B_{H}(0,R))}}{\text{,}}}
758:
40:
1071:
Davis, Tara C.; Olshanskii, Alexander Yu. (October 29, 2018). "Relative
Subgroup Growth and Subgroup Distortion".
319:
829:
766:
302:
486:
652:
has a word of much smaller length, which is then transmitted to the receiver along with a number of "decoys" from
925:
912:
1053:
We should note that this notion of distortion differs from Gromov's definition (as defined in
966:
587:
distortion; conversely every superadditive function (up to asymptotic equivalence) can be found this way.
85:
20:
291:
36:
958:
921:
580:
489:
971:
946:
537:
1072:
1022:
984:
900:
882:
859:
841:
733:
715:
541:
419:
870:
825:
793:
770:
706:(2011). "Irreducible Sp-representations and subgroup distortion in the mapping class group".
1014:
976:
892:
851:
821:
725:
595:
The simplification in a word problem induced by subgroup distortion suffices to construct a
478:
873:; Keivan, Mallahi-Karai (2019). "Some applications of arithmetic groups in cryptography".
563:
549:
475:
439:
962:
279:
1092:
1026:
988:
904:
703:
584:
384:
980:
863:
737:
596:
1043:(1994). "The extrinsic geometry of subgroups and the generalized word problem".
1040:
699:
634:
525:
482:
237:
73:
24:
1002:
604:
579:; for this reason, it is often omitted. In that case, a subgroup that is not
774:
600:
545:
32:
1018:
896:
524:. For many non-normal but still abelian subgroups, the distortion of the
264:
612:
599:, algorithms for encoding and decoding secret messages. Formally, the
855:
729:
608:
241:
1077:
887:
846:
720:
949:(1997). "On subgroup distortion in finitely presented groups".
792:. American Mathematical Society, Providence, RI. p. 285.
39:. Like much of geometric group theory, the concept is due to
316:. With respect to the chosen generating sets, the element
322:
94:
1005:(2001). "Subgroup distortions in nilpotent groups".
670:, re-expresses the word in terms of generators of
371:
210:
769:lecture notes 182. Cambridge University Press.
666:. The receiver then picks out the element of
520:is at least exponentially distorted with base
407:is at least exponentially distorted with base
76:on the corresponding group; the distortion of
274:A subgroup with bounded distortion is called
8:
572:The denominator in the definition is always
788:Druţu, Cornelia; Kapovich, Michael (2018).
1076:
1066:
1064:
970:
886:
845:
830:"Cryptosystems Using Subgroup Distortion"
719:
360:
347:
332:
327:
321:
203:
176:
160:
124:
108:
101:
93:
815:
813:
811:
809:
763:Asymptotic Invariants of Infinite Groups
753:
751:
749:
747:
414:On the other hand, any embedded copy of
909:An expository summary of both works is
687:
426:is undistorted, as is any embedding of
301:, embedded as a normal subgroup of the
35:can reduce the complexity of a group's
1054:
693:
691:
372:{\displaystyle b^{2^{n}}=a^{n}ba^{-n}}
72:. Then each generating set defines a
7:
834:Theoretical and Applied Informatics
615:) that can be encoded as a number
544:can be a subgroup distortion, but
14:
708:Commentarii Mathematici Helvetici
662:, to obscure the secret subgroup
552:Lie group always have distortion
481:has distortion determined by the
16:Concept in geometric group theory
914:Group distortion in Cryptography
911:Werner, Nicolas (19 June 2021).
619:. The transmitter then encodes
31:measures the extent to which an
981:10.1070/SM1997v188n11ABEH000276
603:message is any object (such as
463:is at least the distortion of
197:
194:
182:
169:
151:
142:
130:
117:
98:
1:
278:, and is the same thing as a
43:, who introduced it in 1993.
875:Groups Complexity Cryptology
528:gives a strong lower bound.
280:quasi-isometrically embedded
767:London Mathematical Society
306:BS(1, 2) = ⟨
1120:
290:For example, consider the
1007:Communications in Algebra
640:. In a public overgroup
1104:Low-dimensional topology
926:Universitat de Barcelona
490:overgroup representation
951:Matematicheskii Sbornik
947:Olshanskii, A. Yu.
1099:Geometric group theory
1056:) by a linear factor.
1045:Proc. London Math. Soc
790:Geometric Group Theory
373:
303:Baumslag–Solitar group
212:
88:class of the function
86:asymptotic equivalence
21:geometric group theory
1019:10.1081/AGB-100107938
897:10.1515/gcc-2019-2002
434:Elementary properties
374:
292:infinite cyclic group
213:
828:; Ni Yen Lu (2017).
455:, the distortion of
320:
92:
58:be an overgroup for
963:1997SbMat.188.1617O
644:with that distorts
538:computable function
399:from the origin in
67: ∪
29:subgroup distortion
871:Kahrobaei, Delaram
826:Kahrobaei, Delaram
698:Broaddus, Nathan;
542:exponential growth
422:on two generators
420:free abelian group
403:. In particular,
369:
208:
23:, a discipline of
1013:(12): 5439–5463.
856:10.20904/291-2014
822:Chatterji, Indira
799:978-1-4704-1104-6
206:
201:
1111:
1083:
1082:
1080:
1068:
1059:
1058:
1037:
1031:
1030:
1003:Osin, D. V.
999:
993:
992:
974:
943:
937:
936:
934:
932:
919:
908:
890:
867:
849:
817:
804:
803:
785:
779:
778:
755:
742:
741:
723:
695:
677:
673:
669:
665:
661:
651:
647:
643:
639:
632:
622:
618:
578:
568:
561:
523:
519:
515:
512:with eigenvalue
511:
501:
479:abelian subgroup
470:
466:
462:
458:
454:
429:
425:
417:
410:
406:
402:
398:
390:
382:
378:
376:
375:
370:
368:
367:
352:
351:
339:
338:
337:
336:
315:
300:
270:
262:
254:
250:
246:
235:
217:
215:
214:
209:
207:
204:
202:
200:
181:
180:
165:
164:
154:
129:
128:
113:
112:
102:
83:
79:
71:
61:
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53:
49:
1119:
1118:
1114:
1113:
1112:
1110:
1109:
1108:
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1088:
1087:
1086:
1070:
1069:
1062:
1039:
1038:
1034:
1001:
1000:
996:
972:10.1.1.115.1717
945:
944:
940:
930:
928:
917:
910:
869:
820:
818:
807:
800:
787:
786:
782:
757:
756:
745:
730:10.4171/CMH/233
697:
696:
689:
684:
675:
674:, and recovers
671:
667:
663:
653:
649:
645:
641:
637:
624:
620:
616:
593:
591:In cryptography
573:
566:
553:
534:
521:
517:
513:
503:
493:
492:; formally, if
468:
464:
460:
456:
442:
440:tower of groups
436:
427:
423:
415:
408:
404:
400:
392:
391:, but distance
388:
380:
356:
343:
328:
323:
318:
317:
305:
295:ℤ = ⟨
294:
288:
268:
256:
252:
248:
244:
224:
219:
172:
156:
155:
120:
104:
103:
90:
89:
81:
77:
63:
59:
55:
51:
50:generate group
47:
17:
12:
11:
5:
1117:
1115:
1107:
1106:
1101:
1091:
1090:
1085:
1084:
1060:
1032:
994:
938:
924:). Barcelona:
819:Protocol I in
805:
798:
780:
743:
704:Putman, Andrew
686:
685:
683:
680:
648:, the element
623:as an element
592:
589:
581:locally finite
533:
530:
435:
432:
397: + 1
366:
363:
359:
355:
350:
346:
342:
335:
331:
326:
287:
284:
222:
199:
196:
193:
190:
187:
184:
179:
175:
171:
168:
163:
159:
153:
150:
147:
144:
141:
138:
135:
132:
127:
123:
119:
116:
111:
107:
100:
97:
46:Formally, let
15:
13:
10:
9:
6:
4:
3:
2:
1116:
1105:
1102:
1100:
1097:
1096:
1094:
1079:
1074:
1067:
1065:
1061:
1057:
1055:
1050:
1046:
1042:
1036:
1033:
1028:
1024:
1020:
1016:
1012:
1008:
1004:
998:
995:
990:
986:
982:
978:
973:
968:
964:
960:
957:(11): 51–98.
956:
952:
948:
942:
939:
927:
923:
916:
915:
906:
902:
898:
894:
889:
884:
880:
876:
872:
865:
861:
857:
853:
848:
843:
839:
835:
831:
827:
823:
816:
814:
812:
810:
806:
801:
795:
791:
784:
781:
776:
772:
768:
764:
760:
754:
752:
750:
748:
744:
739:
735:
731:
727:
722:
717:
713:
709:
705:
701:
694:
692:
688:
681:
679:
660:
657: \
656:
636:
631:
628: ∈
627:
614:
610:
606:
602:
598:
590:
588:
586:
585:superadditive
582:
577:
570:
565:
560:
557: ↦
556:
551:
547:
546:Lie subgroups
543:
540:with at most
539:
531:
529:
527:
510:
507: ≤
506:
500:
497: ∈
496:
491:
488:
484:
480:
477:
472:
453:
450: ≤
449:
446: ≤
445:
441:
433:
431:
430:into itself.
421:
412:
401:BS(1, 2)
396:
386:
364:
361:
357:
353:
348:
344:
340:
333:
329:
324:
313:
309:
304:
298:
293:
285:
283:
281:
277:
272:
266:
260:
247:about center
243:
239:
233:
229:
225:
191:
188:
185:
177:
173:
166:
161:
157:
148:
145:
139:
136:
133:
125:
121:
114:
109:
105:
95:
87:
75:
70:
66:
62:generated by
44:
42:
38:
34:
30:
26:
22:
1052:
1048:
1044:
1041:Farb, Benson
1035:
1010:
1006:
997:
954:
950:
941:
931:13 September
929:. Retrieved
913:
881:(1): 25–33.
878:
874:
840:(2): 14–24.
837:
833:
789:
783:
762:
711:
707:
700:Farb, Benson
658:
654:
629:
625:
597:cryptosystem
594:
575:
571:
558:
554:
535:
532:Known values
508:
504:
498:
494:
473:
451:
447:
443:
437:
413:
394:
379:is distance
311:
307:
296:
289:
275:
273:
258:
231:
227:
220:
68:
64:
45:
41:Misha Gromov
37:word problem
28:
18:
1078:1212.5208v1
714:: 537–556.
635:word length
526:normal core
487:conjugation
483:eigenvalues
276:undistorted
74:word metric
25:mathematics
1093:Categories
1051:(3): 578.
888:1803.11528
847:1610.07515
759:Gromov, M.
682:References
282:subgroup.
54:, and let
1027:122842195
989:250919942
967:CiteSeerX
905:119676551
775:842851469
721:0707.2262
601:plaintext
562:for some
550:nilpotent
383:from the
362:−
167:
146:∩
115:
99:↦
33:overgroup
864:16899700
761:(1993).
564:rational
502:acts on
286:Examples
265:diameter
959:Bibcode
738:7665268
613:numbers
516:, then
485:of the
418:in the
310:,
263:is the
236:is the
230:,
84:is the
1025:
987:
969:
903:
862:
796:
773:
736:
609:images
536:Every
476:normal
385:origin
242:radius
218:where
1073:arXiv
1023:S2CID
985:S2CID
922:grado
918:(PDF)
901:S2CID
883:arXiv
860:S2CID
842:arXiv
836:. 1.
734:S2CID
716:arXiv
633:with
611:, or
548:of a
438:In a
257:diam(
933:2022
794:ISBN
771:OCLC
605:text
583:has
255:and
238:ball
158:diam
106:diam
1015:doi
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955:188
893:doi
852:doi
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467:in
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267:of
251:in
240:of
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424:ℤ
416:ℤ
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