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All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put dup 54 /lessorequalslant put dup 62 /greaterorequalslant put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI9 %!PS-AdobeFont-1.1: CMMI9 1.100 %%CreationDate: 1996 Jul 23 07:53:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueID 5087384 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E3948FFB0B4E70F212EC976D65099D84E0D37A7A771C3101D6AD26A0513378F 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All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 70 /F put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSL10 %!PS-AdobeFont-1.1: CMSL10 1.0 %%CreationDate: 1991 Aug 20 16:40:20 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSL10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-62 -250 1123 750}readonly def /UniqueID 5000798 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9429B9D40924DC059325D9D4CC0344F3F997A99E6CC0676735EBCD685AAC9142 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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Fk(\001)g Fi(7)g Fk(\001)f Fi(37)h Fk(\001)g Fi(18979)h Fk(\001)e Fi(38113)i Fk(\001)f Fi(202109)h Fk(\001)e Fi(3033517343)k Fk(\001)758 1039 y Fi(3004350778635364630447)q(636)q(6422)q(610)q(3312)q(370)q(9704)q(1) p 3507 1063 V 448 1066 3060 4 v 446 1224 4 159 v 502 1122 a(P256)p 707 1224 V 103 w(2)793 1098 y Fg(4)853 1122 y Fk(\001)23 b Fi(3)i Fk(\001)f Fi(71)h Fk(\001)e Fi(131)j Fk(\001)d Fi(373)i Fk(\001)f Fi(3407)i Fk(\001)e Fi(17449)h Fk(\001)f Fi(38189)i Fk(\001)e Fi(187019741)i Fk(\001)e Fi(622491383)j Fk(\001)d Fi(1002328039319)k Fk(\001)758 1201 y Fi(2624747550333869278416)q(773)q(953)p 3507 1224 V 448 1228 3060 4 v 446 1386 4 159 v 498 1283 a(K283)p 707 1386 V 100 w(2)793 1260 y Fg(5)859 1283 y Fk(\001)i Fi(3)h Fk(\001)f Fi(7)1060 1260 y Fg(2)1125 1283 y Fk(\001)g Fi(11)h Fk(\001)g Fi(29)g Fk(\001)f Fi(71)h Fk(\001)f Fi(281)h Fk(\001)g Fi(283)g Fk(\001)f Fi(2671)i Fk(\001)e Fi(338407)i Fk(\001)e Fi(22546501513)k Fk(\001)c Fi(192823925599)k Fk(\001)758 1362 y Fi (1166987951426410268715)q(004)q(0114)q(741)q(1093)q(537)q(981)p 3507 1386 V 448 1389 3060 4 v 446 1547 4 158 v 501 1444 a(B283)p 707 1547 V 102 w(2)24 b Fk(\001)g Fi(3)g Fk(\001)f Fi(5)h Fk(\001)f Fi(1171)i Fk(\001)e Fi(21557)j Fk(\001)d Fi(709278129089)k Fk(\001)c Fi(77278332289855843)29 b Fk(\001)23 b Fi(270232628419903655237)30 b Fk(\001)758 1523 y Fi(6927558641061659265722)q(900)q(93)p 3507 1547 V 448 1550 3060 4 v 446 1629 4 80 v 502 1606 a(P384)p 707 1629 V 103 w(2)16 b Fk(\001)g Fi(3)880 1582 y Fg(2)930 1606 y Fk(\001)g Fi(7)1001 1582 y Fg(2)1051 1606 y Fk(\001)f Fi(13)i Fk(\001)e Fi(112467999998166422996537)q(934)q(7)p Fk(\001)p 3507 1629 V 446 1708 4 79 v 707 1708 V 758 1684 a Fg(30554657881403)q(520)q(02)q(73)q(39)q(469)q(06)q(14)q(45)q (610)q(90)q(64)q(12)q(496)q(06)q(16)q(040)q(78)q(84)q(36)q(539)q(19)q (79)q(70)q(492)q(92)q(68)q(48)q(032)q(63)q(90)q(471)p 3507 1708 V 448 1711 3060 4 v 446 1791 4 80 v 498 1767 a Fi(K409)p 707 1791 V 100 w(2)h Fk(\001)g Fi(3)880 1744 y Fg(3)930 1767 y Fk(\001)g Fi(5)f Fk(\001)h Fi(7)1087 1744 y Fg(3)1137 1767 y Fk(\001)g Fi(137)g Fk(\001)f Fi(193)i Fk(\001)e Fi(409)i Fk(\001)e Fi(2957)i Fk(\001)e Fi(218513)j Fk(\001)d Fi(884595540581)p Fk(\001)p 3507 1791 V 446 1869 4 79 v 707 1869 V 758 1846 a Fg(57739872041429)q(577)q (10)q(68)q(95)q(912)q(23)q(46)q(24)q(093)q(56)q(63)q(61)q(026)q(92)q (38)q(758)q(50)q(15)q(82)q(529)q(98)q(61)q(81)q(411)q(78)q(95)q(37)q (723)q(27)q(25)q(193)q(16)q(50)q(3)p 3507 1869 V 448 1873 3060 4 v 446 2031 4 158 v 501 1928 a Fi(B409)p 707 2031 V 102 w(2)49 b Fk(\001)e Fi(5)h Fk(\001)g Fi(19)g Fk(\001)g Fi(5197)h Fk(\001)e Fi(2967389)j Fk(\001)d Fi(373915204316167)52 b Fk(\001)c Fi(445277563636353902377234)q(1)54 b Fk(\001)758 2007 y Fi(1058692476773986657054)q(662)q(7965)q(598)q (729)22 b Fk(\001)15 b Fi(1279922477462777151320)q(9574)q(192)q(2692)q (460)q(77)p 3507 2031 V 448 2034 3060 4 v 864 2188 a Fr(T)-8 b(able)31 b(1:)42 b(F)-8 b(actorizations)33 b(of)e Fo(n)20 b Fp(\000)g Fr(1)30 b(for)h(some)f(NIST)g(curv)m(es)448 2553 y(group)h(op)s(erations)h(required)f(b)m(y)h(generic)h(discrete)f (logarithm)h(algorithms)f(suc)m(h)g(as)448 2666 y(the)d(Shank's)f(BSGS) g(or)h(P)m(ollard-)p Fo(\032)h Fr(algorithms.)41 b(In)28 b(particular)h(for)g(small)g(v)-5 b(alues)29 b(of)448 2779 y Fo(u)36 b Fr(the)g(algorithm)h(cost)g(will)f(b)s(e)g(close)h(to) f(2)p Fo(C)2074 2713 y Fp(p)p 2150 2713 48 4 v 66 x Fo(v)s Fr(,)i(and)d(so)h(the)g(algorithm)h(ma)m(y)g(b)s(e)448 2892 y(faster)31 b(than)f(generic)i(algorithms)f(ev)m(en)g(for)f Fo(u)g Fr(as)h(small)g(as)f(4)p Fo(C)2676 2859 y Fi(2)2716 2892 y Fr(.)589 3005 y(In)23 b(some)g(groups)f(used)g(for)h (cryptograph)m(y)-8 b(,)25 b(index)d(calculus)i(algorithms)g(for)e (solv-)448 3117 y(ing)k(discrete)h(logarithms)f(ha)m(v)m(e)h(sub)s(exp) s(onen)m(tial)e(cost.)41 b(These)25 b(algorithms)i(are)f(usu-)448 3230 y(ally)33 b(signi\014can)m(tly)g(faster)f(than)g(generic)h (algorithms.)45 b(F)-8 b(or)33 b(example,)g(in)f(the)g(group)448 3343 y Fl(F)504 3310 y Fk(\003)504 3366 y Fn(p)573 3343 y Fr(of)d(units)g(in)g(a)g(\014nite)g(\014eld,)h(where)e Fo(p)d Fp(\031)g Fr(2)2007 3310 y Fi(1024)2153 3343 y Fr(,)30 b(it)f(is)h(exp)s(ected)f(that)h(index)f(calcu-)448 3456 y(lus)35 b(algorithms)g(can)g(\014nd)e(discrete)j(logarithms)f (with)g(a)g(cost)g(of)g(ab)s(out)g(2)3100 3423 y Fi(80)3210 3456 y Fr(group)448 3569 y(op)s(erations.)589 3682 y(The)26 b(algorithm)g(of)g(Theorem)f(1)h(still)h(applies)f(for)f(groups)g (where)g(index)g(calculus)448 3795 y(metho)s(ds)35 b(are)h(a)m(v)-5 b(ailable,)39 b(but)c(it)h(ma)m(y)g(not)f(alw)m(a)m(ys)i(result)f(in)f (a)h(faster)f(metho)s(d)g(to)448 3908 y(\014nd)c(logarithms)h(in)g(the) g(group.)45 b(The)31 b(algorithm)i(in)f(Theorem)f(1)h(has)g(a)g(cost)h (of)f(at)448 4021 y(least)690 3990 y Fg(3)670 3955 y Fp(p)p 746 3955 55 4 v 66 x Fo(n)i Fr(scalar)h(m)m(ultiplications)h (\(including)e(oracle)i(queries\),)g(as)e(noted)h(earlier.)448 4134 y(If)i Fo(n)g Fp(\031)f Fr(2)790 4101 y Fi(1024)936 4134 y Fr(,)j(then)e(this)g(cost)i(is)e(at)h(least)h(2)2074 4101 y Fi(341)2221 4134 y Fr(group)e(op)s(erations,)j(whic)m(h)d(is)g (far)448 4247 y(greater)32 b(than)e(the)h(cost)g(using)f(index)g (calculus)h(metho)s(ds.)589 4359 y(Some)26 b(standards,)g(suc)m(h)f(as) g(DSA)h(in)f(FIPS)g(186-2)i(and)e(ANSI)g(X9.30)i(and)e(Di\016e-)448 4472 y(Hellman)34 b(in)f(ANSI)g(X9.42,)j(use)d(a)h(group)e Fp(h)p Fo(G)p Fp(i)j Fr(that)f(is)f(a)h(m)m(uc)m(h)f(smaller)h (subgroup)448 4585 y(of)28 b(the)h(group)e Fl(F)1015 4552 y Fk(\003)1015 4608 y Fn(p)1054 4585 y Fr(.)40 b(Generally)-8 b(,)31 b(the)d(size)h Fp(h)p Fo(G)p Fp(i)g Fr(is)f(c)m(hosen)g(so)h (that)f(generic)h(DLP)f(algo-)448 4698 y(rithms)23 b(in)g Fp(h)p Fo(G)p Fp(i)h Fr(ha)m(v)m(e)g(appro)m(ximately)h(the)e(same)h (cost)g(as)g(index)e(calculus)i(algorithms)448 4811 y(in)30 b Fl(F)610 4778 y Fk(\003)610 4834 y Fn(p)650 4811 y Fr(.)41 b(In)29 b(cases)j(lik)m(e)g(this)e(Theorem)g(1)h(ma)m(y)g(ma)m (y)g(pro)m(vide)g(a)g(faster)g(attac)m(k)h(on)f(the)448 4924 y(system)g(than)f(is)g(curren)m(tly)h(kno)m(wn.)1920 5225 y(7)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 589 573 a Fr(Some)27 b(systems)g(using)f(hardw)m (are)g(mo)s(dules)g(for)h(protecting)h(priv)-5 b(ate)27 b(k)m(eys)g(migh)m(t)448 686 y(pro)m(vide)37 b(an)g(SDHP)j(oracle)f(to) e(an)g(adv)m(ersary)-8 b(.)60 b(F)-8 b(or)37 b(example,)j(a)d(smart)f (card)h(is)g(a)448 799 y(highly)30 b(constrained)f(en)m(vironmen)m(t,)i (and)d(sometimes)j(all)f(hashing)f(and)g(k)m(ey)h(deriv)-5 b(a-)448 912 y(tion)28 b(is)g(done)g(outside)f(the)h(securit)m(y)h(b)s (oundary)c(of)j(the)g(smart)g(card,)g(usually)f(on)h(the)448 1024 y(smart)h(card)g(reader.)40 b(A)29 b(smart)f(card)h(\(holding)g(a) g(user's)f(priv)-5 b(ate)29 b(k)m(ey)h Fo(q)s Fr(\))f(used)f(in)g(an) 448 1137 y(encryption)j(sc)m(heme)g(ma)m(y)g(b)s(e)f(presen)m(ted)h (with)f(a)h(recipien)m(t)h(public)e(k)m(ey)i Fo(R)f Fr(and)f(the)448 1250 y(card)d(ma)m(y)g(simply)g(return)f(the)h(elemen)m(t)h Fo(q)s(R)f Fr(to)h(the)f(reader,)g(where)g(the)g(subsequen)m(t)448 1363 y(encryption)41 b(pro)s(cessing)f(is)g(p)s(erformed.)69 b(In)40 b(this)g(case,)45 b(a)40 b(malicious)i(smart)e(card)448 1476 y(reader)31 b(could)f(use)g(the)h(smart)f(card)g(as)h(an)f(SDHP)35 b(oracle.)589 1589 y(W)-8 b(e)46 b(no)m(w)e(discuss)f(certain)i(proto)s (cols)g(that)g(pro)m(vide)f(a)h(SDHP)j(oracle)d(to)g(an)448 1702 y(adv)m(ersary)-8 b(.)448 1945 y Fx(3.1)113 b(ElGamal)38 b(Encryption)448 2117 y Fr(ElGamal)e(encryption,)f(in)f(its)h(original) g(form)f(\(see)i([6],)g(or)e([14)q(,)i Fp(x)p Fr(8.4.2]\))h(mak)m(es)e (an)448 2230 y(SDHP)c(oracle)d(a)m(v)-5 b(ailable.)42 b(ElGamal)28 b(encryption)f(of)g(message)g Fo(m)g Fr(to)g(an)g(en)m (tit)m(y)h(with)448 2343 y(public)38 b(k)m(ey)g Fo(Q)g Fr(=)g Fo(g)1165 2310 y Fn(q)1241 2343 y Fr(results)g(in)g(a)g (ciphertext)h Fo(c)f Fr(=)g(\()p Fo(c)2435 2357 y Fi(1)2475 2343 y Fo(;)15 b(c)2554 2357 y Fi(2)2594 2343 y Fr(\))38 b(=)g(\()p Fo(g)2857 2310 y Fn(x)2902 2343 y Fo(;)15 b(mg)3068 2310 y Fn(q)r(x)3147 2343 y Fr(\).)63 b(In)38 b(a)448 2456 y(c)m(hosen)i(ciphertext)h(attac)m(k)g(against)g(an)f (encryption)f(sc)m(heme,)k(an)c(adv)m(ersary)h(can)448 2569 y(select)f(an)m(y)e(ciphertext)h(and)e(obtain)h(its)h(decryption.) 60 b(F)-8 b(or)38 b(ElGamal)g(encryption,)448 2682 y(if)k(this)g(adv)m (ersary)g(c)m(ho)s(oses)h Fo(c)i Fr(=)f(\()p Fo(g)1772 2649 y Fn(x)1817 2682 y Fo(;)15 b(c)1896 2696 y Fi(2)1936 2682 y Fr(\),)45 b(then)d(it)g(obtains)g Fo(m)j Fr(=)f Fo(c)2971 2696 y Fi(2)3011 2682 y Fo(=g)3102 2649 y Fn(q)r(x)3180 2682 y Fr(.)76 b(The)448 2794 y(adv)m(ersary)36 b(can)g(compute)h Fo(g)1459 2761 y Fn(q)r(x)1571 2794 y Fr(=)d Fo(c)1715 2808 y Fi(2)1755 2794 y Fo(=m)p Fr(,)k(whic)m(h)d(solv)m(es)i(SDHP)2722 2808 y Fn(Q)2817 2794 y Fr(for)f(instance)g Fo(g)3367 2761 y Fn(x)3412 2794 y Fr(.)448 2907 y(Therefore)26 b(with)g(just)f(a)i(little)h(arithmetic)f(a)f(c)m(hosen)h(ciphertext)g (adv)m(ersary)f(can)g(use)448 3020 y(the)31 b(ElGamal)h(decryption)e (oracle)i(as)e(an)h(SDHP)j(oracle.)589 3133 y(ElGamal)28 b(encryption)f(is)g(already)g(kno)m(wn)f(to)h(b)s(e)f(vulnerable)h(to)g (c)m(hosen)g(cipher-)448 3246 y(text)41 b(attac)m(ks.)71 b(The)39 b(kno)m(wn)g(attac)m(ks)j(mainly)e(allo)m(w)h(information)f (to)g(b)s(e)f(learned)448 3359 y(ab)s(out)31 b(previously)f(encrypted)h (messages,)h(but)e(they)h(do)f(not)h(leak)h(an)m(ything)f(ab)s(out)448 3472 y(the)h(priv)-5 b(ate)33 b(k)m(ey)-8 b(.)46 b(Finding)32 b(the)g(priv)-5 b(ate)32 b(k)m(ey)g Fo(q)j Fr(is)d(a)g(serious)g(break) g(of)g(the)g(system,)448 3585 y(b)s(ecause)27 b(all)h(encryption)f(is)g (compromised.)39 b(When)27 b(using)f(the)h(victim)h(as)f(an)g(SDHP)448 3698 y(oracle)36 b(in)e(this)g(manner)f(the)h(priv)-5 b(ate)35 b(k)m(ey)g Fo(q)i Fr(can)d(b)s(e)g(found)f(with)g(ab)s(out) 3110 3667 y Fg(3)3090 3632 y Fp(p)p 3166 3632 55 4 v 66 x Fo(n)h Fr(cost,)448 3811 y(assuming)25 b(the)g(adv)m(ersary)g(kno) m(ws)f(a)h(factor)h Fo(u)f Fp(\031)2187 3780 y Fg(3)2167 3745 y Fp(p)p 2242 3745 V 2242 3811 a Fo(n)g Fr(of)g Fo(n)9 b Fp(\000)g Fr(1.)38 b(The)24 b(attac)m(k)j(requires)448 3924 y Fo(u)35 b Fr(decryption)f(queries)h(to)g(the)g(victim,)i(whic)m (h)d(when)g Fo(u)e Fp(\031)2608 3893 y Fg(3)2588 3858 y Fp(p)p 2663 3858 V 2663 3924 a Fo(n)i Fr(ma)m(y)i(b)s(e)e(to)s(o)h (large)448 4036 y(to)f(allo)m(w)g(for)e(a)h(realistic)h(attac)m(k.)50 b(Ho)m(w)m(ev)m(er)35 b(as)d(men)m(tioned)h(b)s(efore)g(smaller)g(v)-5 b(alues)448 4149 y(of)31 b Fo(u)f Fr(can)h(b)s(e)f(used)f(and)h(ma)m(y) h(giv)m(e)h(a)f(practical)h(attac)m(k.)589 4262 y(Tw)m(o)c(coun)m (termeasures)g(to)g(this)f(attac)m(k)j(on)d(ElGamal)i(encryption)e(are) h(to)g(use)f(a)448 4375 y(k)m(ey)34 b(deriv)-5 b(ation)34 b(function)f(or)g(to)h(use)f(symmetric)h(encryption)f(rather)g(than)g (simple)448 4488 y(m)m(ultiplication)26 b(to)f(transform)e(the)h (message.)39 b(Suc)m(h)24 b(coun)m(termeasures)g(are)g(incorp)s(o-)448 4601 y(rated)i(in)g(more)f(mo)s(dern)g(v)-5 b(arian)m(ts)26 b(of)g(the)g(DH-based)g(encryption,)h(suc)m(h)e(as)h(DHAES)448 4714 y(and)k(ECIES)f([11)q(].)589 4827 y(A)g(second,)g(more)f(fundamen) m(tal,)h(kind)e(of)h(coun)m(termeasure)h(is)g(to)f(add)g(in)m(tegrit)m (y)448 4940 y(to)43 b(the)f(ciphertext.)75 b(The)41 b(in)m(tegrit)m(y)j (serv)m(es)e(to)h(ensure)e(that)h(a)g(v)-5 b(alid)42 b(ciphertext)1920 5225 y(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 448 573 a Fr(can)45 b(only)f(b)s(e)g(created)h(b) m(y)f(an)h(en)m(tit)m(y)h(that)e(kno)m(ws)h(the)f(ephemeral)h(priv)-5 b(ate)44 b(k)m(ey)448 686 y Fo(x)e Fr(asso)s(ciated)i(with)e(the)h (ephemeral)f(public)g(k)m(ey)h Fo(g)2328 653 y Fn(x)2372 686 y Fr(.)76 b(If)42 b(the)h(decryptor)f(rev)m(eals)448 799 y Fo(g)494 766 y Fn(q)r(x)573 799 y Fr(,)37 b(an)e(en)m(tit)m(y)i (who)e(already)h(kno)m(ws)f Fo(x)h Fr(do)s(es)f(not)g(learn)h(an)m (ything)g(new,)g(b)s(ecause)448 912 y Fo(g)494 879 y Fn(q)r(x)606 912 y Fr(is)e(computable)g(from)f(the)h(static)h(public)e (k)m(ey)h Fo(g)2312 879 y Fn(q)2384 912 y Fr(and)f(the)h(ephemeral)g (priv)-5 b(ate)448 1024 y(k)m(ey)40 b Fo(x)p Fr(.)67 b(In)39 b(our)f(attac)m(k)k(the)d(adv)m(ersary)h(c)m(ho)s(oses)g (ephemeral)f(public)g(k)m(eys)h Fo(g)3245 992 y Fn(x)3328 1024 y Fr(for)448 1137 y(whic)m(h)e(it)g(do)s(es)f(not)h(kno)m(w)f(the) h(corresp)s(onding)f(ephemeral)g(priv)-5 b(ate)38 b(k)m(ey)h Fo(x)p Fr(.)62 b(This)448 1250 y(coun)m(termeasure)41 b(is)f(also)h(part)f(of)g(DHAES)g(and)g(ECIES.)f(Cramer)g(and)h (Shoup's)448 1363 y(public-k)m(ey)i(encryption)g(sc)m(heme)g([3)q(])g (also)g(includes)f(this)h(coun)m(termeasure.)75 b(The)448 1476 y(original)23 b(rationale)f(for)f(including)g(in)m(tegrit)m(y)i (in)e(the)g(ciphertext)h(is)f(to)h(prev)m(en)m(t)g(c)m(hosen)448 1589 y(ciphertext)31 b(attac)m(ks)i(b)m(y)d(making)h(them)f(no)h(more)f (e\013ectiv)m(e)j(than)d(passiv)m(e)h(attac)m(ks.)589 1702 y(Inciden)m(tally)-8 b(,)27 b(the)d(original)h(description)f(of)g (the)g(Di\016e-Hellman)i(k)m(ey)e(agreemen)m(t)448 1815 y(proto)s(col)33 b([5)q(])f(did)f(not)h(use)f(k)m(ey)i(deriv)-5 b(ation)32 b(function.)45 b(Therefore,)32 b(if)g(an)g(adv)m(ersary)448 1928 y(has)h(some)g(means)f(of)h(obtaining)g(the)g(DH)g(shared)f (secret)i(k)m(ey)-8 b(,)34 b(our)e(attac)m(k)j(ma)m(y)f(b)s(e)448 2041 y(p)s(ossible.)448 2284 y Fx(3.2)113 b(F)-9 b(ord-Kaliski)39 b(Key)e(Retriev)-6 b(al)448 2456 y Fr(In)33 b(the)g(F)-8 b(ord-Kaliski)35 b(k)m(ey)f(retriev)-5 b(al)34 b(sc)m(heme)g([7)q(],)g (whic)m(h)f(is)h(curren)m(tly)f(b)s(eing)g(stan-)448 2569 y(dardized)25 b(in)g([8)q(])h(and)f([9],)i(an)e(SDHP)30 b(oracle)d(is)e(a)m(v)-5 b(ailable.)42 b(P)m(art)26 b(of)g(this)f(sc)m (heme)h(\(see)448 2682 y(Figure)35 b(1\))g(requires)f(the)g(serv)m(er)h (to)g(compute)f Fo(c)2154 2649 y Fn(q)2227 2682 y Fr(for)g(an)m(y)h(v) -5 b(alue)34 b Fo(c)h Fr(sen)m(t)g(to)g(it)g(b)m(y)f(a)448 2794 y(clien)m(t,)c(where)c Fo(q)j Fr(is)e(a)g(long-term)h(k)m(ey)-8 b(.)40 b(The)26 b(serv)m(er)h(th)m(us)g(pro)m(vides)f(an)h(SDHP)k (oracle)448 2907 y(to)c(an)f(adv)m(ersary)-8 b(,)28 b(so)f(the)f (securit)m(y)h(of)f Fo(q)j Fr(is)d(p)s(oten)m(tially)i(w)m(eak)m(ened)g (b)m(y)e(an)g(adv)m(ersary)448 3020 y(using)k(our)g(attac)m(k.)1371 3217 y(Clien)m(t)656 b(Serv)m(er)p 1321 3254 606 4 v 2266 3254 298 4 v 1371 3330 a Fo(b)25 b Fp(2)1496 3344 y Fn(R)1579 3330 y Fr([1)p Fo(;)15 b(n)21 b Fp(\000)f Fr(1])1371 3443 y Fo(c)26 b Fr(=)f Fo(g)1578 3410 y Fn(b)1575 3465 y(\031)2081 3522 y(c)1972 3574 y Fp(\000)-17 b(\000)c(\000)k(!) 2273 3668 y Fo(s)25 b Fr(=)g Fo(c)2476 3635 y Fn(q)2080 3748 y(s)1972 3799 y Fp( )-17 b(\000)c(\000)k(\000)1371 3904 y Fo(z)30 b Fr(=)25 b Fo(s)1582 3871 y Fn(b)1612 3847 y Fb(\000)p Fg(1)1698 3871 y Fi(mo)r(d)t Fn(n)1210 4092 y Fr(Figure)31 b(1:)41 b(F)-8 b(ord-Kaliski)32 b(Key)e(Retriev)-5 b(al)589 4305 y(The)26 b(generator)h Fo(g)1215 4319 y Fn(\031)1288 4305 y Fr(in)f(Figure)g(1)g(is)g(deriv)m(ed)g(from)g(user) f(passw)m(ord)g Fo(\031)s Fr(.)39 b(The)26 b(v)-5 b(alue)448 4417 y Fo(z)49 b Fr(=)43 b Fo(g)699 4384 y Fn(d)696 4440 y(\031)786 4417 y Fr(dep)s(ends)c(on)j(a)g(user)f(secret)i(and)e(on)g (a)h(serv)m(er)g(secret,)k(and)41 b(is)h(called)h(a)448 4530 y Fj(har)-5 b(dene)g(d)38 b(p)-5 b(asswor)g(d)p Fr(.)51 b(In)32 b(practice,)j(cryptographic)f(k)m(eys)f(will)g(b)s(e)f (deriv)m(ed)h(from)g Fo(z)t Fr(.)448 4643 y(The)21 b(proto)s(col)h (prev)m(en)m(ts)f(dictionary)h(attac)m(ks)h(against)f(the)f(user's)g (passw)m(ord)f(b)s(ecause)448 4756 y Fo(z)35 b Fr(is)c(deriv)m(ed)f (from)h(a)g(strong)f(secret)i Fo(q)h Fr(k)m(ept)f(b)m(y)e(the)h(serv)m (er.)42 b(If)30 b Fo(q)j Fr(is)e(rev)m(ealed,)h(then)448 4869 y(the)f(user)f(b)s(ecomes)g(vulnerable)g(to)h(dictionary)h(searc)m (hes.)1920 5225 y(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 589 573 a Fr(In)32 b(early)h(drafts)f(of)h([8])g (and)f([9],)i(elliptic)g(curv)m(e)e(groups)g(are)h(allo)m(w)m(ed)h (with)e(this)448 686 y(proto)s(col,)j(so)d(our)g(attac)m(k)j(ma)m(y)e (b)s(e)f(relev)-5 b(an)m(t)34 b(to)f(the)g(proto)s(col)g(as)g(describ)s (ed)f(there.)448 799 y(Recen)m(t)e(v)m(ersions)e(of)g(these)h(draft)f (standards)f(ha)m(v)m(e)i(incorp)s(orated)f(measures)g(sp)s(ecif-)448 912 y(ically)k(designed)e(to)i(coun)m(ter)f(our)e(attac)m(k.)448 1155 y Fx(3.3)113 b(Chaum)38 b(and)h(v)-6 b(an)38 b(An)m(t)m(w)m(erp)s (en's)f(Undeniable)j(Signatures)448 1327 y Fr(Another)e(sc)m(heme)h(in) f(whic)m(h)f(an)h(SDHP)k(oracle)e(is)e(a)m(v)-5 b(ailable)40 b(is)e(Chaum)f(and)g(v)-5 b(an)448 1440 y(An)m(t)m(w)m(erp)s(en's)26 b(undeniable)f(signature)i(sc)m(heme)f([2)q(].)39 b(In)26 b(this)f(sc)m(heme,)j(the)e(signer)g(has)448 1552 y(priv)-5 b(ate)38 b(k)m(ey)f Fo(q)j Fr(and)c(computes)h(her)f(signature)h(on)f (message)i Fo(m)f Fr(as)g Fo(s)e Fr(=)h Fo(m)3158 1519 y Fn(q)3196 1552 y Fr(.)60 b(The)448 1665 y(groups)30 b(suggested)g(for)g(use)g(in)f([2)q(])h(are)g Fl(F)1885 1632 y Fk(\003)1885 1688 y Fn(p)1955 1665 y Fr(where)f Fo(p)h Fr(is)g(a)g(strong)g(prime.)40 b(Therefore,)448 1778 y(this)32 b(proto)s(col)h(is)f(not)h(a\013ected)g(b)m(y)f(the)g (attac)m(k,)j(b)s(ecause)e(index)e(calculus)i(metho)s(ds)448 1891 y(solv)m(e)22 b(the)f(DLP)g(faster)h(than)e(our)g(attac)m(k.)40 b(Ho)m(w)m(ev)m(er)23 b(if)e(the)g(proto)s(col)h(is)f(implemen)m(ted) 448 2004 y(with)41 b(other)g(groups)f(suc)m(h)h(as)g(elliptic)i(curv)m (e)e(groups)f(or)h(DSA)g(groups,)i(whic)m(h)e(is)448 2117 y(reasonable)j(b)s(ecause)f(they)g(mak)m(e)h(the)f(sc)m(heme)g(m)m (uc)m(h)g(more)g(e\016cien)m(t,)48 b(then)43 b(our)448 2230 y(attac)m(ks)c(ma)m(y)e(b)s(e)f(viable.)59 b(A)37 b(v)m(ery)f(simple)h(coun)m(termeasure)g(to)g(this)f(attac)m(k)j(is)d (to)448 2343 y(hash)30 b(the)h(message)g(b)s(efore)f(exp)s(onen)m (tiating.)448 2629 y Fq(4)135 b(The)45 b(Algorithm)g(as)g(a)h (Reduction)448 2832 y Fr(In)32 b(Di\016e-Hellman)j(\(DH\))f(k)m(ey)g (agreemen)m(t)h([5],)f(Alice)h(and)d(Bob)h(exc)m(hange)h Fo(g)3213 2799 y Fn(x)3291 2832 y Fr(and)448 2945 y Fo(g)494 2912 y Fn(y)569 2945 y Fr(and)e(then)g(compute)h(a)g(shared)f(secret)h Fo(g)2003 2912 y Fn(xy)2085 2945 y Fr(,)h(using)e(their)g(resp)s(ectiv) m(e)i(kno)m(wledge)448 3058 y(of)f Fo(x)g Fr(and)f Fo(y)s Fr(.)47 b(Hardness)32 b(of)g(the)h(DHP)g(is)g(necessary)g(to)g(ensure)f (that)h(an)g(adv)m(ersary)448 3171 y(who)d(sees)h Fo(g)869 3138 y Fn(x)944 3171 y Fr(and)f Fo(g)1167 3138 y Fn(y)1239 3171 y Fr(cannot)h(easily)g(compute)g(the)g(shared)e(secret)j Fo(g)2906 3138 y Fn(xy)2987 3171 y Fr(.)589 3284 y(In)c(the)g(static)h (v)-5 b(arian)m(t)29 b(of)f(DH)g(k)m(ey)h(agreemen)m(t,)h(Alice)g(has)d (a)i(static)g(priv)-5 b(ate)28 b(k)m(ey)448 3397 y Fo(q)s Fr(,)37 b(meaning)e(she)g(alw)m(a)m(ys)h(uses)f Fo(x)e Fr(=)g Fo(q)s Fr(.)54 b(It)35 b(is)g(unclear)g(ho)m(w)g(this)g(reuse)g (a\013ects)i(her)448 3510 y(securit)m(y)-8 b(.)39 b(If)22 b(the)g(instances)g(of)g(the)g(DHP)g(are)g(arranged)g(in)g(a)g(square)f (matrix)i(indexed)448 3623 y(b)m(y)39 b Fp(h)p Fo(G)p Fp(i)28 b(\002)e(h)p Fo(G)p Fp(i)p Fr(,)43 b(as)c(in)g(Figure)h (2\(a\),)j(then)c(Alice)i(is)e(more)g(concerned)h(ab)s(out)f(the)448 3735 y(hardness)33 b(of)h(the)g(Di\016e-Hellman)i(problem)d(instances)h (in)g(the)g(single)g(ro)m(w)g(de\014ned)448 3848 y(b)m(y)h Fo(x)d Fr(=)h Fo(q)k Fr(\(Figure)f(2\(b\).\))54 b(Ev)m(en)35 b(if)g(the)g(DHP)g(is)g(hard)e(for)i(almost)h(all)f(instances,)448 3961 y(it)d(is)f(p)s(ossible)f(that)i(it)f(is)g(easy)h(when)e (restricted)h(to)h(Alice's)g(ro)m(w,)g(and)e(so)h(hardness)448 4074 y(of)37 b(the)g(DHP)g(ma)m(y)h(not)f(b)s(e)f(su\016cien)m(t)h(to)h (ensure)e(securit)m(y)h(for)g(Alice.)61 b(The)36 b Fj(static)448 4187 y(Di\016e-Hel)5 b(lman)38 b(pr)-5 b(oblem)45 b Fr(\(SDHP)5 b(\))37 b(is)f(to)i(solv)m(e)g(this)e(subset)g(of)h(the)g(instances)g (of)448 4300 y(the)31 b(DHP)g(that)g(Alice)g(w)m(ould)g(b)s(e)e (concerned)i(with:)448 4438 y Fs(Problem)36 b(1)f(\(SDHP)5 b(\).)30 b Fm(Giv)m(en)h(\014xed)f Fo(g)j Fm(and)d Fo(g)2178 4405 y Fn(q)2217 4438 y Fm(,)g(and)g(random)g Fo(g)2829 4405 y Fn(y)2871 4438 y Fm(,)g(\014nd)f Fo(g)3154 4405 y Fn(q)r(y)3230 4438 y Fm(.)589 4576 y Fr(It)c(is)f(not)g(apparen)m(t)h (from)e(existing)j(results)d(that)i(computing)g(the)f(SDHP)3154 4590 y Fn(Q)3238 4576 y Fr(func-)448 4689 y(tion)h(is)f(hard)f(if)h Fo(q)j Fr(is)d(unkno)m(wn.)37 b(In)24 b(this)g(section)h(w)m(e)g (discuss)e(a)i(v)m(ersion)f(of)g(this)g(state-)448 4801 y(men)m(t)k(that)g(is)f(implied)g(b)m(y)g(Theorem)g(1.)40 b(Algorithms)28 b(used)f(in)g(previous)f(reductions)448 4914 y(of)37 b(this)e(sort)i(ha)m(v)m(e)g(required)e(an)h(input)f (function)h(computing)g(the)g(Di\016e-Hellman)1897 5225 y(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 1109 644 a Fo(g)108 b(:)15 b(:)g(:)120 b(g)1531 611 y Fn(y)1675 644 y Fo(:)15 b(:)g(:)1120 737 y Fr(.)1120 770 y(.)1120 803 y(.)1273 745 y(.)1308 770 y(.)1344 795 y(.)1516 737 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y(inputs)41 b(is)h(\014xed)f(to)i(b)s(e)e(a)i(group)e(elemen)m(t)i Fo(Q)p Fr(,)i(but)d(as)g(a)g(consequence)h(can)f(only)448 2235 y(compute)31 b(the)g(discrete)g(logarithm)g(corresp)s(onding)e(to) j Fo(Q)p Fr(.)589 2348 y(den)h(Bo)s(er)h([4])g(sho)m(ws)f(that)h(if)f Fo(n)22 b Fp(\000)g Fr(1)33 b(is)h(smo)s(oth,)g(then)f(DLP)d Fh(6)2857 2362 y Fn(P)2946 2348 y Fr(DHP,)35 b(where)448 2461 y Fh(6)519 2475 y Fn(P)620 2461 y Fr(indicates)44 b(a)f(p)s(olynomial-cost)h(reduction.)78 b(The)42 b(in)m(tuition)i(is)f (that)g(giv)m(en)h(a)448 2574 y(DHP-oracle,)31 b(one)d(can)g(basically) h(p)s(erform)d(the)i(P)m(ohlig-Hellman)i(algorithm)f(in)e(the)448 2687 y(exp)s(onen)m(t)j(space,)g(to)h(\014nd)d(a)h(discrete)i (logarithm)f(in)f(the)h(m)m(ultiplicativ)m(e)i(group)d Fl(F)3365 2654 y Fk(\003)3365 2709 y Fn(n)3412 2687 y Fr(,)448 2800 y(whic)m(h)h(in)g(turn)g(pro)m(vides)g(a)h(logarithm)g (in)f Fp(h)p Fo(G)p Fp(i)p Fr(.)589 2913 y(Maurer)g(and)f(W)-8 b(olf)30 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b(Then)448 2230 y(use)33 b Fp(A)g Fr(to)i(solv)m(e)f(the)g(DHP)g(with)f (probabilit)m(y)h Fo(p)f Fr(of)h(\014nding)e Fo(g)2663 2197 y Fn(xy)2745 2230 y Fr(.)49 b(The)33 b(solution)h(to)448 2343 y(the)d(SDHP)k(is)c(\()p Fo(g)1063 2310 y Fn(xy)1145 2343 y Fr(\))1180 2310 y Fn(z)1216 2286 y Fb(\000)p Fg(1)1303 2310 y Fi(mo)r(d)t Fn(n)1489 2343 y Fr(.)42 b(W)-8 b(e)32 b(write)f(SDHP)f Fh(6)c Fr(DHP,)32 b(or)e(SDHP)h Fh(6)3042 2357 y Fn(P)3126 2343 y Fr(DHP)g(to)448 2456 y(emphasize)g(that)g(this) f(is)h(a)g(p)s(olynomial-time)g(reduction.)589 2569 y(This)24 b(relationship)h(b)s(et)m(w)m(een)h(the)f(SDHP)j(and)d(the)f(DHP)i(is)e (akin)h(to)h(the)f Fj(r)-5 b(andom)448 2682 y(self-r)g(e)g(ducibility) 45 b Fr(\(RSR\))36 b(prop)s(ert)m(y)g(of)g(the)g(DHP)-8 b(.)37 b(An)f(algorithm)h(for)f(solving)h(the)448 2794 y(DHP)42 b(with)g(probabilit)m(y)g Fo(p)f Fr(can)h(b)s(e)f(con)m(v)m (erted)i(in)m(to)f(algorithm)h(for)e(solving)i(an)m(y)448 2907 y(instance)c(of)f(the)g(SDHP)t(,)i(except)f(that)g(is)f(slo)m(w)m (er)h(b)m(y)f(factor)h(of)f(1)p Fo(=p)p Fr(.)64 b(T)-8 b(o)38 b(solv)m(e)i(a)448 3020 y(SDHP)30 b(instance)c(with)f (probabilit)m(y)h(close)h(to)f(1,)h(ev)m(en)f(if)f(the)h(probabilit)m (y)g Fo(p)f Fr(is)g(small,)448 3133 y(w)m(e)k(can)g(k)m(eep)g(c)m(ho)s (osing)h Fo(z)j Fr(un)m(til)28 b(w)m(e)h(can)g(solv)m(e)h(the)f(DHP)g (instance.)41 b(This)28 b(will)g(tak)m(e)448 3246 y(1)p Fo(=p)j Fr(attempts)h(on)e(a)m(v)m(erage.)589 3359 y(Con)m(v)m(ersely) -8 b(,)30 b(an)c(algorithm)i(for)e(solving)i(the)e(SDHP)31 b(can)c(b)s(e)f(regarded)h(an)f(algo-)448 3472 y(rithm)j(for)g(solving) h(the)f(DHP)h(with)f(a)g(1)p Fo(=n)h Fr(c)m(hance)g(of)f(success.)41 b(Therefore)29 b(random)448 3585 y(self-reducibilit)m(y)j(can)g(b)s(e)e (applied.)42 b(Unfortunately)-8 b(,)32 b(it)f(is)g(not)g(useful,)g(b)s (ecause)g(the)448 3698 y(success)36 b(probabilit)m(y)h(is)f(so)g(lo)m (w.)58 b(W)-8 b(e)37 b(w)m(ould)f(get)h(a)f(DHP)h(solv)m(er)g(that)f (is)g Fo(n)f Fr(times)448 3811 y(slo)m(w)m(er)c(than)f(the)g(SDHP)k (solv)m(er)d(and)e(this)h(is)g(m)m(uc)m(h)g(slo)m(w)m(er)g(than)g(the)g Fo(O)s Fr(\()3064 3745 y Fp(p)p 3140 3745 55 4 v 66 x Fo(n)p Fr(\)-time)448 3924 y(solution)h(to)h(the)e(DHP)h(that)g (\014rst)f(solv)m(es)h(a)g(DLP)g(instance)g(\(using)f(P)m(ollard-)p Fo(\032)p Fr(,)i(sa)m(y\))448 4036 y(and)e(then)g(exp)s(onen)m (tiating.)589 4149 y(A)25 b(subtle)g(p)s(oin)m(t)g(when)e(comparing)i (the)g(SDHP)k(and)24 b(the)h(DLP)g(is)g(that)g(in)f(Corol-)448 4262 y(lary)h(2)h(only)f(one)g(instance)h(of)f(the)g(discrete)h (logarithm)g(can)f(b)s(e)f(solv)m(ed,)j(the)e(instance)448 4375 y Fo(g)494 4342 y Fn(q)533 4375 y Fr(.)41 b(This)29 b(leads)i(one)f(to)i(consider)e(the)g Fj(single-instanc)-5 b(e)33 b(Discr)-5 b(ete)33 b(L)-5 b(o)g(garithm)35 b(Pr)-5 b(ob-)448 4488 y(lem)49 b Fr(\(SDLP\),)c(in)c(whic)m(h)g(one)g(is)h (giv)m(en)g(\014xed)f Fo(g)2208 4455 y Fn(q)2247 4488 y Fr(,)j(with)d(the)h(task)g(b)s(eing)e(to)j(\014nd)448 4601 y Fo(q)s Fr(.)55 b(Previous)35 b(w)m(ork)g([4,)g(12)q(,)h(1])f (compared)g(the)g(DHP)h(and)e(DLP)-8 b(,)36 b(while)f(ours)f(com-)448 4714 y(pares)27 b(the)h(SDHP)j(and)c(SDLP)o(.)40 b(Th)m(us)26 b(our)h(w)m(ork)g(can)h(b)s(e)f(considered)g(as)g Fj(p)-5 b(ar)g(al)5 b(lel)39 b Fr(to)p 448 4775 1196 4 v 552 4829 a Fg(2)587 4861 y Ff(Inciden)n(tally)-6 b(,)22 b(the)f(lo)n(w)n (er)j(b)r(ounds)d(in)h([15)q(])g(for)h(hardness)f(of)h(the)f(DHP)f(are) i(similar)h(to)e(our)g(lo)n(w)n(er)448 4952 y(b)r(ound.)1897 5225 y Fr(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 448 573 a Fr(previous)30 b(w)m(ork,)h(in)f(the) h(sense)f(that)h(it)g(w)m(orks)f(in)g(the)h Fj(single-instanc)-5 b(e)33 b(setting)p Fr(.)448 816 y Fx(4.3)113 b(Analogy)38 b(with)f(the)g(Rabin)h(Cryptosystem)448 988 y Fr(Although)29 b(the)f(RSA)g(cryptosystem)h(is)f(v)m(ery)h(similar)g(to)g(the)f(Rabin) g(cryptosystem,)448 1101 y(nob)s(o)s(dy)33 b(has)i(pro)m(v)m(en)g(that) h(duplicating)f(the)g(ra)m(w)g(RSA)f(priv)-5 b(ate)35 b(k)m(ey)h(op)s(eration)f(is)448 1214 y(as)30 b(hard)e(as)h(\014nding)f (the)h(RSA)g(priv)-5 b(ate)30 b(k)m(ey)-8 b(.)41 b(The)29 b(RSA)f(in)m(v)m(ersion)i(problem,)f(whic)m(h)448 1327 y(m)m(ust)45 b(b)s(e)f(hard)f(for)h(RSA)g(to)i(b)s(e)e(secure,)k(is)d (therefore)g(p)s(oten)m(tially)h(easier)f(than)448 1440 y(in)m(teger)32 b(factorization.)589 1552 y(The)f(original)h(Rabin)e (cryptosystem)h([17)q(])g(has)f(the)h(prop)s(ert)m(y)f(that)i (duplicating)448 1665 y(the)d(priv)-5 b(ate)28 b(k)m(ey)h(op)s(eration) f(is)g(almost)h(as)f(hard)f(as)i(\014nding)d(the)i(priv)-5 b(ate)29 b(k)m(ey)-8 b(.)41 b(Ho)m(w-)448 1778 y(ev)m(er)d(the)e (original)i(Rabin)e(cryptosystem)h(is)f(vulnerable)g(to)i(activ)m(e)g (attac)m(ks.)62 b(The)448 1891 y(mo)s(dern)26 b(Rabin)h(cryptosystem)g (uses)g(hash-based)f(padding)h(to)g(protect)i(against)f(ac-)448 2004 y(tiv)m(e)41 b(attac)m(ks.)69 b(F)-8 b(urthermore,)41 b(this)e(padding)g(also)h(allo)m(ws)g(for)f(securit)m(y)h(pro)s(ofs)e (of)448 2117 y(the)44 b(sc)m(heme)h(in)f(the)g(random)f(oracle)j(mo)s (del.)81 b(Activ)m(e)46 b(attac)m(ks)g(on)e(the)g(original)448 2230 y(Rabin)33 b(cryptosystem)i(are)e(analogous)i(to)g(our)e(attac)m (k,)k(and)c(the)h(mo)s(dern)e(padding)448 2343 y(mec)m(hanism)38 b(for)f(Rabin)g(cryptosystems)g(is)h(analogous)g(the)g(k)m(ey)g(deriv) -5 b(ation)38 b(func-)448 2456 y(tion)29 b(mec)m(hanism)g(for)f(DH)h (cryptosystems.)41 b(Both)29 b(these)g(mec)m(hanisms)g(prev)m(en)m(t)g (the)448 2569 y(reduction)24 b(algorithms)h(that)f(pro)m(v)m(e)h (securit)m(y)g(from)e(b)s(eing)h(turned)e(in)m(to)j(attac)m(ks)h(that) 448 2682 y(h)m(urt)k(the)h(securit)m(y)-8 b(.)589 2794 y(With)30 b(DH)g(based)f(cryptosystems)h(it)g(w)m(as)f(p)s(oten)m (tially)i(the)f(case)g(that)g(breaking)448 2907 y(DH)42 b(cryptosystems)f(w)m(as)h(p)s(oten)m(tially)g(m)m(uc)m(h)f(easier)h (than)f(solving)h(the)f(DLP)-8 b(.)42 b(In)448 3020 y(other)35 b(w)m(ords,)f(the)h(situation)g(w)m(as)f(lik)m(e)i(the)e(situation)h (for)f(RSA)f(to)s(da)m(y)-8 b(.)53 b(With)35 b(the)448 3133 y(pioneering)30 b(w)m(ork)f(of)h(den)e(Bo)s(er's)i(and)f(others,)h (DH)g(cryptosystems)f(w)m(ere)h(put)e(on)i(a)448 3246 y(b)s(etter)25 b(fo)s(oting:)38 b(they)25 b(w)m(ere)g(based)f(on)g(a)h (problem)e(pro)m(v)-5 b(ably)25 b(almost)g(as)g(hard)e(as)i(the)448 3359 y(DLP)-8 b(.)30 b(Our)d(reduction)i(can)g(b)s(e)f(though)m(t)h(of) g(as)g(extending)g(this)g(e\013ort.)41 b(Sp)s(eci\014cally)-8 b(,)448 3472 y(w)m(e)31 b(establish)f(that)h(duplicating)f(an)g (individual)f(en)m(tit)m(y's)j(priv)-5 b(ate)31 b(k)m(ey)g(op)s (eration)f(is)448 3585 y(almost)i(as)e(hard)g(as)g(\014nding)f(the)i (individual's)f(priv)-5 b(ate)31 b(k)m(ey)-8 b(.)448 3871 y Fq(5)135 b(Ac)l(kno)l(wledgmen)l(ts)448 4074 y Fr(Thanks)24 b(to)i(Alfred)e(Menezes)j(for)d(man)m(y)h(helpful)g (discussions.)38 b(Scott)26 b(V)-8 b(anstone)26 b(and)448 4187 y(Certicom)32 b(generously)e(supp)s(orted)f(our)h(pursuit)f(of)h (this)g(researc)m(h)h(topic.)448 4474 y Fq(References)494 4676 y Fr([1])46 b(D.)d(Boneh)f(and)g(R.J.)g(Lipton,)j Fj(A)n(lgorithms)g(for)f(black-b)-5 b(ox)44 b(\014elds)g(and)g(their) 635 4789 y(applic)-5 b(ation)39 b(to)d(crypto)-5 b(gr)g(aphy)p Fr(,)39 b(Adv)-5 b(ances)34 b(in)g(Cryptology)g(|)g(CR)-8 b(YPTO)33 b('96)635 4902 y(\(N.)e(Koblitz,)h(ed.\),)f(LNCS,)f(v)m(ol.)h (1070,)i(Springer,)c(1996,)j(pp.)e(283{297.)1897 5225 y(15)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 494 573 a Fr([2])46 b(D.)e(Chaum)d(and)h(H.)h (v)-5 b(an)43 b(An)m(t)m(w)m(erp)s(en,)j Fj(Undeniable)e(signatur)-5 b(es)p Fr(,)47 b(Adv)-5 b(ances)635 686 y(in)42 b(Cryptology)i(|)e(CR) -8 b(YPTO)42 b('89)h(\(G.)h(Brassard,)h(ed.\),)i(LNCS,)42 b(v)m(ol.)h(435,)635 799 y(Springer,)30 b(1989,)i(pp.)e(212{217.)494 986 y([3])46 b(R.)d(Cramer)f(and)h(V.)g(Shoup,)i Fj(A)e(pr)-5 b(actic)g(al)47 b(public)d(key)f(encryption)j(scheme)635 1099 y(pr)-5 b(ovably)45 b(se)-5 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y([9])46 b(ISO)20 b(11770-4,)26 b Fj(Information)h(te)-5 b(chnolo)g(gy)26 b(|)e(se)-5 b(curity)25 b(te)-5 b(chniques)24 b(|)g(key)g(man-)635 3693 y(agement)32 b(|)f(p)-5 b(art)33 b(4:)42 b(Me)-5 b(chanisms)32 b(b)-5 b(ase)g(d)33 b(on)f(we)-5 b(ak)32 b(se)-5 b(cr)g(ets)p Fr(,)31 b(ISO,)d(No)m(v)m(em)m(b)s(er)635 3806 y(2004.)448 3993 y([10])47 b(D.)28 b(E.)f(Kn)m(uth,)h Fj(Seminumeric)-5 b(al)31 b(algorithms)p Fr(,)f(3rd)d(ed.,)h(The)f(Art) g(of)h(Computer)635 4106 y(Programming,)j(v)m(ol.)h(2,)f(Addison-W)-8 b(esley)g(,)32 b(1998.)448 4294 y([11])47 b(P)-8 b(.)29 b(Roga)m(w)m(a)m(y)i(M.)d(Ab)s(dalla,)h(M.)g(Bellare,)h Fj(The)h(or)-5 b(acle)32 b(di\016e-hel)5 b(lman)31 b(assump-)635 4407 y(tions)48 b(and)f(an)g(analysis)i(of)d(dhies)p Fr(,)51 b(T)-8 b(opics)46 b(in)f(Cryptology)i(-)e(CT-RSA)g(01)635 4520 y(\(D.)32 b(Naccac)m(he,)h(ed.\),)e(LNCS,)f(v)m(ol.)h(2020,)i (Springer-V)-8 b(erlag,)31 b(2001.)448 4707 y([12])47 b(U.)32 b(M.)g(Maurer)g(and)f(S.)g(W)-8 b(olf,)33 b Fj(Di\016e-Hel)5 b(lman)33 b(or)-5 b(acles)p Fr(,)34 b(Adv)-5 b(ances)32 b(in)f(Cryp-)635 4820 y(tology)38 b(|)d(CR)-8 b(YPTO)35 b('96)i(\(N.)g(Koblitz,)h(ed.\),)g(LNCS,)d(v)m(ol.)i(1070,)i(Springer,) 635 4933 y(1996,)33 b(pp.)c(268{282.)1897 5225 y(16)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 448 573 a Fr([13])p 635 573 273 4 v 335 w(,)39 b Fj(The)g(Di\016e-Hel)5 b(lman)39 b(pr)-5 b(oto)g(c)g(ol)p Fr(,)42 b(Designs,)e(Co)s(des)d(and)g(Cryptogra-)635 686 y(ph)m(y)31 b Fs(19)g Fr(\(2000\),)i(141{171.)448 873 y([14])47 b(A.)86 b(J.)g(Menezes,)101 b(P)-8 b(.)86 b(C.)f(v)-5 b(an)86 b(Oorsc)m(hot,)100 b(and)85 b(S.)h(A.)g(V)-8 b(anstone,)635 986 y Fj(Handb)j(o)g(ok)103 b(of)f(applie)-5 b(d)103 b(crypto)-5 b(gr)g(aphy)p Fr(,)127 b(CR)m(C)105 b(Press,)123 b(1996,)j(See)635 1099 y(www.cacr.math.u)m(w)m(aterlo)s (o.ca/hac.)448 1287 y([15])47 b(A.)23 b(Muzereau,)h(N.)f(P)-8 b(.)23 b(Smart,)g(and)f(F.)h(V)-8 b(ercauteren,)25 b Fj(The)g(e)-5 b(quivalenc)g(e)25 b(b)-5 b(etwe)g(en)635 1400 y(the)24 b(DHP)f(and)i(DLP)e(for)h(el)5 b(liptic)24 b(curves)f(use)-5 b(d)25 b(in)e(pr)-5 b(actic)g(al)27 b(applic)-5 b(ations)p Fr(,)25 b(LMS)635 1513 y(J.)30 b(Comput.)g(Math.)h Fs(7)g Fr(\(2004\),)i(50{72,)g(See)e (www.lms.ac.uk.)448 1700 y([16])47 b(J.)32 b(P)m(ollard,)h Fj(Monte-c)-5 b(arlo)35 b(metho)-5 b(ds)36 b(for)e(index)g(c)-5 b(omputation)37 b(mo)-5 b(d)35 b Fo(p)p Fr(,)d(Mathe-)635 1813 y(matics)g(of)e(Computation)h Fs(32)g Fr(\(1978\),)i(918{924.)448 2001 y([17])47 b(M.)h(O.)e(Rabin,)51 b Fj(Digitalize)-5 b(d)49 b(signatur)-5 b(es)48 b(and)h(public-key)e(functions)h(as)h(in-) 635 2114 y(tr)-5 b(actable)45 b(as)f(factorization)p Fr(,)j(T)-8 b(ec)m(h.)43 b(Rep)s(ort)f(212,)k(Massac)m(h)m(ussets)e (Institute)635 2227 y(of)31 b(T)-8 b(ec)m(hnology)32 b(Lab)s(oratory)f(for)f(Computer)f(Science,)j(1979.)1897 5225 y(17)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF