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All Rights Reserved) readonly def /FullName (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX9 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{-58 -250 1195 750}readonly def /UniqueID 5000767 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI9 %!PS-AdobeFont-1.1: CMTI9 1.0 %%CreationDate: 1991 Aug 18 21:08:07 % 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 (CMTI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI9 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{-35 -250 1148 750}readonly def /UniqueID 5000827 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E3948FFB3DF7BFF10C9BDA4EFE5F68A8CB1526990D1357AE6D2F7C2D2EF8496 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % 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 (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 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{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % 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 (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 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{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec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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. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 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{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F00F963068B8B731A88D7740B0DDAED1B3F82 7DB9DFB4372D3935C286E39EE7AC9FB6A9B5CE4D2FAE1BC0E55AE02BFC464378 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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. 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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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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 (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 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 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 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{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6CC3F1E9AE32F234EB60FE7D E34995B1ACFF52428EA20C8ED4FD73E3935CEBD40E0EAD70C0887A451E1B1AC8 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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 8 /braceleftbig put dup 9 /bracerightbig put 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: 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. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 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{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % 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 (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 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{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F0364CD5660F74BEE96790DE35AFA90CCF712 B1805DA88AE375A04D99598EADFC625BDC1F9C315B6CF28C9BD427F32C745C99 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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. All Rights Reserved) readonly def /FullName (CMTT9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT9 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{-6 -233 542 698}readonly def /UniqueID 5000831 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F00F963068B8232429ED8B7CF6A3D879A2D1E 2931CE5F5D18C658602059F07BE66E6EFC9239D7AB2FB8A4CBD41675B8ECF279 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % 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 (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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2831 y Fi(23)5003 2904 y Fh(\000)45 b Fp(1)p Fk(;)33 b(H)16 b Fp(\()p Fk(k)5754 2934 y Fo(i)5820 2904 y Fp(\)\).)762 3143 y(All)48 b(2)1169 3071 y Fi(32)1383 3143 y Fp(k)-6 b(ey-searc)g(h)46 b(circuits)f(no)-6 b(w)46 b(run)g(in)g(parallel,)g(pro)6 b(ducing)44 b Fk(Z)14 b Fp(\()p Fk(\014)d(;)33 b Fp(2)6158 3071 y Fi(23)6327 3143 y Fk(;)g(r)6505 3173 y Fo(j)6589 3143 y Fp(\))47 b(for)f(eac)-6 b(h)463 3382 y(random)45 b Fk(r)1264 3412 y Fo(j)1394 3382 y Fp(and)g Fk(Z)14 b Fp(\()p Fk(\014)d(;)33 b Fp(0)p Fk(;)g(H)16 b Fp(\()p Fk(k)2751 3412 y Fo(i)2818 3382 y Fp(\)\))p Fk(;)33 b(:)g(:)g(:)j(;)d(Z)14 b Fp(\()p Fk(\014)d(;)33 b Fp(2)3954 3310 y Fi(23)4127 3382 y Fh(\000)s Fp(1)p Fk(;)g(H)16 b Fp(\()p Fk(k)4836 3412 y Fo(i)4903 3382 y Fp(\)\))47 b(for)e(eac)-6 b(h)45 b(target)h Fk(H)16 b Fp(\()p Fk(k)6746 3412 y Fo(i)6812 3382 y Fp(\).)47 b(This)463 3621 y(tak)-6 b(es)67 b(sligh)-6 b(tly)67 b(more)f(time)h(than)f(2)3190 3549 y Fi(23)3402 3621 y Fh(\001)45 b Fp(2)3602 3549 y Fi(4)3746 3621 y Fp(=)56 b(2)4057 3549 y Fi(27)4291 3621 y Fp(AES)67 b(computations.)762 3860 y(The)89 b(attac)-6 b(k)g(er)91 b(then)f(applies)f(\(for)h (example\))f(Sc)-6 b(himmler's)90 b(sorting)g(algorithm)g(to)463 4099 y(sort)66 b(the)f(2)1305 4027 y Fi(36)1539 4099 y Fk(Z)80 b Fp(v)-11 b(alues.)64 b(This)h(algorithm)h(tak)-6 b(es)65 b(just)g(2)4766 4027 y Fi(21)5000 4099 y Fp(adjacen)-6 b(t)64 b(compare-exc)-6 b(hange)463 4338 y(steps|not)66 b(a)g(b)6 b(ottlenec)-6 b(k)65 b(compared)g(to)h(2)3837 4266 y Fi(27)4072 4338 y Fp(AES)g(computations.)762 4577 y(If)56 b(the)g(sorting)g(encoun)-6 b(ters)55 b(a)h(collision)f(b)6 b(et)-6 b(w)g(een)56 b(t)-6 b(w)g(o)57 b Fk(Z)14 b Fp(\()p Fk(\014)d(;)33 b Fp(2)5498 4505 y Fi(23)5667 4577 y Fk(;)g(r)5845 4607 y Fo(j)5929 4577 y Fp(\)'s,)58 b(it)f(thro)-6 b(ws)56 b(one)463 4817 y(of)44 b(those)g Fk(r)1258 4847 y Fo(j)1342 4817 y Fp('s)g(a)-6 b(w)g(a)g(y)-17 b(.)46 b(If)e(the)g(sorting)g (encoun)-6 b(ters)43 b(a)h(collision)g(b)6 b(et)-6 b(w)g(een)43 b Fk(Z)14 b Fp(\()p Fk(\014)d(;)33 b(n;)g(H)16 b Fp(\()p Fk(k)6792 4847 y Fo(i)6859 4817 y Fp(\)\))46 b(and)463 5056 y Fk(Z)14 b Fp(\()p Fk(\014)d(;)33 b Fp(2)1002 4983 y Fi(23)1171 5056 y Fk(;)g(r)1349 5086 y Fo(j)1434 5056 y Fp(\),)55 b(the)g(mac)-6 b(hine)54 b(pauses)e(to)j(redo)f(2)4107 4983 y Fi(23)4296 5056 y Fh(\000)21 b Fk(n)54 b Fp(iterations)h(of)f (the)h(computation)e(of)463 5295 y Fk(Z)14 b Fp(\()p Fk(\014)d(;)33 b Fp(2)1002 5222 y Fi(23)1171 5295 y Fk(;)g(r)1349 5325 y Fo(j)1434 5295 y Fp(\).)53 b(If)f(the)g(in)-6 b(termediate)52 b(v)-11 b(alue)52 b(\()p Fk(\014)11 b(;)33 b(n)16 b Fp(+)g(1)p Fk(;)33 b(s)p Fp(\))53 b(satis\014es)e Fk(H)16 b Fp(\()p Fk(s)g Fh(\010)g Fp(\()p Fk(\014)11 b(;)33 b(n)p Fp(\)\))57 b(=)e Fk(H)16 b Fp(\()p Fk(k)7236 5325 y Fo(i)7302 5295 y Fp(\))463 5534 y(then)66 b(the)g(mac)-6 b(hine)66 b(prin)-6 b(ts)66 b Fk(s)45 b Fh(\010)f Fp(\()p Fk(\014)11 b(;)33 b(n)p Fp(\))67 b(as)f(a)g(guess)e(for)i Fk(k)4844 5564 y Fo(i)4910 5534 y Fp(.)463 6132 y Fe(Heuristic)77 b(analysis)463 6530 y Fp(What's)66 b(the)g(c)-6 b(hance)65 b(that)i(a)f(particular)g(k)-6 b(ey)-17 b(,)67 b(sa)-6 b(y)66 b Fk(k)4511 6560 y Fi(1)4601 6530 y Fp(,)g(is)h(found)e(b)-6 b(y)66 b(this)h(mac)-6 b(hine?)762 6769 y(Here)83 b(is)g(a)f(crude)g (estimate.)h(There)e(are)h(2)4120 6697 y Fi(36)4372 6769 y Fp(v)-11 b(alues)82 b Fk(Z)14 b Fp(\()p Fk(\014)d(;)33 b Fp(2)5521 6697 y Fi(23)5690 6769 y Fk(;)g(r)5868 6799 y Fo(j)5952 6769 y Fp(\),)84 b(eac)-6 b(h)82 b(of)g(whic)-6 b(h)463 7008 y(in)g(v)g(olv)g(ed)66 b(2)1342 6936 y Fi(23)1575 7008 y Fp(in)-6 b(termediate)64 b(v)-11 b(alues)64 b(\()p Fk(\014)11 b(;)33 b(n)40 b Fp(+)h(1)p Fk(;)33 b(s)p Fp(\))66 b(and)d(2)4876 6936 y Fi(23)5109 7008 y Fp(inputs)h Fk(s)41 b Fh(\010)g Fp(\()p Fk(\014)11 b(;)33 b(n)p Fp(\))65 b(to)g Fk(H)16 b Fp(,)64 b(for)463 7247 y(a)d(total)h(of)f(2)1417 7175 y Fi(59)1647 7247 y Fp(inputs)g(to)h Fk(H)16 b Fp(.)61 b(If)h(an)-6 b(y)61 b(of)g(those)f(pseudorandom)e(inputs)j(bump)6 b(ed)60 b(in)-6 b(to)62 b Fk(k)7291 7277 y Fi(1)463 7487 y Fp(then)68 b(the)h(mac)-6 b(hine)67 b(will)j(\014nd)e Fk(k)2931 7517 y Fi(1)3020 7487 y Fp(.)h(Sp)6 b(eci\014cally)-17 b(,)67 b(if)i(an)f(in)-6 b(termediate)68 b(v)-11 b(alue)68 b(\()p Fk(\014)11 b(;)33 b(n)46 b Fp(+)g(1)p Fk(;)33 b(s)p Fp(\))463 7726 y(in)66 b(the)f(computation)f(of)h Fk(Z)14 b Fp(\()p Fk(\014)d(;)33 b Fp(2)2963 7653 y Fi(23)3132 7726 y Fk(;)g(r)3310 7756 y Fo(j)3394 7726 y Fp(\))66 b(satis\014es)e Fk(s)43 b Fh(\010)f Fp(\()p Fk(\014)11 b(;)33 b(n)p Fp(\))56 b(=)f Fk(k)5459 7756 y Fi(1)5548 7726 y Fp(,)66 b(then)f Fk(Z)14 b Fp(\()p Fk(\014)d(;)33 b Fp(2)6661 7653 y Fi(23)6830 7726 y Fk(;)g(r)7008 7756 y Fo(j)7092 7726 y Fp(\))56 b(=)463 7965 y Fk(Z)14 b Fp(\()p Fk(\014)d(;)33 b(n;)g(H)16 b Fp(\()p Fk(s)52 b Fh(\010)f Fp(\()p Fk(\014)11 b(;)33 b(n)p Fp(\)\)\))72 b(=)f Fk(Z)14 b Fp(\()p Fk(\014)d(;)33 b(n;)g(H)16 b Fp(\()p Fk(k)3668 7995 y Fi(1)3759 7965 y Fp(\)\);)77 b(the)e(sorting)h(will)h(disco)-6 b(v)g(er)75 b(this)h(collision,)463 8204 y(will)55 b(lo)6 b(cate)53 b(the)g(same)g(in)-6 b(termediate)53 b(v)-11 b(alue)53 b(\()p Fk(\014)11 b(;)33 b(n)19 b Fp(+)g(1)p Fk(;)33 b(s)p Fp(\),)55 b(and)e(will)i(prin)-6 b(t)54 b Fk(s)19 b Fh(\010)g Fp(\()p Fk(\014)11 b(;)33 b(n)p Fp(\))56 b(=)g Fk(k)7291 8234 y Fi(1)463 8443 y Fp(as)66 b(its)h(guess)d(for)i Fk(k)1917 8473 y Fi(1)2006 8443 y Fp(.)h(This)f(o)6 b(ccurs)64 b(with)j(probabilit)-6 b(y)67 b(2)4772 8371 y Fi(59)4940 8443 y Fk(=)p Fp(2)5140 8371 y Fi(128)5443 8443 y Fp(=)56 b(2)5754 8371 y Fd(\000)p Fi(69)6046 8443 y Fp(.)762 8682 y(This)62 b(estimate)g(is)g(sligh)-6 b(tly)63 b(o)-6 b(v)g(eroptimistic,)63 b(for)f(three)f(reasons:)f (\014rst,)i(there)g(are)f(not)463 8921 y(quite)84 b(as)g(man)-6 b(y)83 b Fk(r)1896 8951 y Fo(j)1980 8921 y Fp('s;)i(second,)d(there)h (migh)-6 b(t)84 b(b)6 b(e)83 b(collisions)g(among)g(the)g(inputs)h(to)g Fk(H)16 b Fp(;)463 9160 y(third,)74 b(there)e(migh)-6 b(t)74 b(b)6 b(e)72 b(a)h(collision)g(b)6 b(et)-6 b(w)g(een)72 b(t)-6 b(w)g(o)74 b Fk(Z)14 b Fp(\()p Fk(\014)d(;)33 b Fp(2)5038 9088 y Fi(23)5207 9160 y Fk(;)g(r)5385 9190 y Fo(j)5469 9160 y Fp(\)'s,)74 b(eliminating)f(the)g Fk(r)7296 9190 y Fo(j)463 9399 y Fp(relev)-11 b(an)-6 b(t)59 b(to)h Fk(k)1550 9429 y Fi(1)1639 9399 y Fp(.)g(But)f(the)g (\014rst)g(e\013ect)g(is)g(small)g(and)g(the)g(other)f(e\013ects)h(are) f(conjecturally)463 9638 y(negligible.)66 b(The)g(mac)-6 b(hine)65 b(\014nds)g Fk(k)3175 9668 y Fi(1)3331 9638 y Fp(with)i(probabilit)-6 b(y)67 b(conjecturally)e(close)g(to)h(2)6809 9566 y Fd(\000)p Fi(69)7102 9638 y Fp(.)762 9878 y(Of)94 b(course,)e(the)i(mac)-6 b(hine)93 b(sim)-6 b(ultaneously)94 b(has)f(a)g(c)-6 b(hance)92 b(of)i(\014nding)f Fk(k)6627 9908 y Fi(2)6716 9878 y Fp(,)h(and)f(a)463 10117 y(c)-6 b(hance)87 b(of)h(\014nding)f Fk(k)2172 10147 y Fi(3)2261 10117 y Fp(,)i(and)e(so)h(on.)g(Its)g(c)-6 b(hance)87 b(of)g(\014nding)h Ff(at)j(le)-10 b(ast)92 b(one)102 b Fp(of)88 b(the)g(2)7211 10044 y Fi(10)463 10356 y Fp(target)67 b(k)-6 b(eys)66 b(is,)h(conjecturally)-17 b(,)65 b(close)g(to)i(2)3789 10283 y Fd(\000)p Fi(59)4082 10356 y Fp(.)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 762 -364 a Fp(The)97 b(attac)-6 b(k)g(er)99 b(can)d(increase)g(the)i(c)-6 b(hance)96 b(of)h(success)e(b)-6 b(y)99 b(running)d(the)i(mac)-6 b(hine)463 -125 y(rep)6 b(eatedly)83 b(with)i(new)f(c)-6 b(hoices)82 b(of)i Fk(\014)11 b Fp(.)84 b(F)-17 b(or)85 b(example,)f(after)g(running)f(the)h(mac)-6 b(hine)84 b(2)7291 -197 y Fi(5)463 114 y Fp(times,)77 b(the)f(attac)-6 b(k)g(er)77 b(has)e(c)-6 b(hance)74 b(close)h(to)h(2)4056 42 y Fd(\000)p Fi(64)4426 114 y Fp(of)f(\014nding)g Fk(k)5440 144 y Fi(1)5530 114 y Fp(,)h(and)f(c)-6 b(hance)75 b(close)g(to)463 353 y(2)563 281 y Fd(\000)p Fi(54)949 353 y Fp(of)91 b(\014nding)g(at)i(least)f(one)f(k)-6 b(ey)-17 b(.)93 b(These)e(2)4171 281 y Fi(5)4352 353 y Fp(runs)h(tak)-6 b(e)92 b(only)g(sligh)-6 b(tly)94 b(more)d(time)463 592 y(than)65 b(2)1027 520 y Fi(32)1261 592 y Fp(AES)h(computations.)e(If)h (the)h(mac)-6 b(hine)64 b(do)6 b(es)64 b(\014nd)g(a)i(k)-6 b(ey|or)65 b(a)g(collision)h(that)463 831 y(do)6 b(esn't)74 b(pro)6 b(duce)72 b(a)j(k)-6 b(ey|then)75 b(the)g(mac)-6 b(hine)73 b(has)h(to)h(tak)-6 b(e)76 b(extra)f(time;)g(but)g(this)g(is) g(a)463 1071 y(rare)66 b(ev)-6 b(en)g(t,)67 b(so)f(it)h(can)e(b)6 b(e)66 b(ignored)f(in)h(ev)-11 b(aluating)66 b(the)g(mac)-6 b(hine's)65 b(p)6 b(erformance.)762 1310 y(Let)66 b(me)g(summarize.)f (This)h(mac)-6 b(hine)66 b(has,)g(conjecturally)-17 b(,)603 1708 y Fh(\017)99 b Fp(c)-6 b(hance)65 b(close)g(to)h(2)2270 1636 y Fd(\000)p Fi(64)2630 1708 y Fp(of)f(\014nding)h Fk(k)3625 1738 y Fi(1)3780 1708 y Fp(after)g(the)g(time)h(for)f(2)5453 1636 y Fi(32)5688 1708 y Fp(AES)g(computations;)603 1947 y Fh(\017)99 b Fp(c)-6 b(hance)72 b(close)g(to)h(2)2291 1875 y Fd(\000)p Fi(54)2658 1947 y Fp(of)g(\014nding)f(at)i(least)f (one)f(of)h(the)g(2)5343 1875 y Fi(10)5585 1947 y Fp(target)h(k)-6 b(eys)73 b(after)g(the)802 2186 y(time)67 b(for)f(2)1661 2114 y Fi(32)1896 2186 y Fp(AES)g(computations;)603 2426 y Fh(\017)99 b Fp(c)-6 b(hance)65 b(close)g(to)h(2)2270 2353 y Fd(\000)p Fi(32)2630 2426 y Fp(of)f(\014nding)h Fk(k)3625 2456 y Fi(1)3780 2426 y Fp(after)g(the)g(time)h(for)f(2)5453 2353 y Fi(64)5688 2426 y Fp(AES)g(computations;)603 2665 y Fh(\017)99 b Fp(c)-6 b(hance)72 b(close)g(to)h(2)2291 2592 y Fd(\000)p Fi(22)2658 2665 y Fp(of)g(\014nding)f(at)i(least)f (one)f(of)h(the)g(2)5343 2592 y Fi(10)5585 2665 y Fp(target)h(k)-6 b(eys)73 b(after)g(the)802 2904 y(time)67 b(for)f(2)1661 2831 y Fi(64)1896 2904 y Fp(AES)g(computations;)603 3143 y Fh(\017)99 b Fp(c)-6 b(hance)92 b(close)h(to)h(1)g(of)f(\014nding)g (at)h(least)g(one)f(k)-6 b(ey)94 b(after)g(the)g(time)g(for)g(2)6722 3071 y Fi(86)6984 3143 y Fp(AES)802 3382 y(computations;)65 b(and)603 3621 y Fh(\017)99 b Fp(c)-6 b(hance)58 b(close)g(to)i(1)f(of) h(\014nding)e(most)h(of)h(the)f(2)4325 3549 y Fi(10)4553 3621 y Fp(target)h(k)-6 b(eys)59 b(after)h(the)f(time)h(for)f(2)7211 3549 y Fi(96)802 3860 y Fp(AES)66 b(computations.)463 4259 y(The)75 b(mac)-6 b(hine)75 b(has)f(reasonable)f(size:)i(2)3563 4186 y Fi(32)3807 4259 y Fp(AES)h(circuits,)f(plus)g(a)h(comparable)d (amoun)-6 b(t)463 4498 y(of)66 b(memory)-17 b(.)463 5095 y Fe(Asymptotics)463 5494 y Fp(The)68 b(same)f(mac)-6 b(hine)68 b(design)f(can)h(b)6 b(e)67 b(scaled)g(up)h(to)h Fk(p)g Fp(parallel)f(k)-6 b(ey-searc)g(h)68 b(circuits,)g(for)463 5733 y(a)59 b(wide)h(range)e(of)h(v)-11 b(alues)58 b(of)h Fk(p)p Fp(.)h(The)f(size-)p Fk(p)g Fp(mac)-6 b(hine)59 b(has)f(a)i(go)6 b(o)g(d)57 b(c)-6 b(hance)58 b(of)h(disco)-6 b(v)g(ering)463 5972 y(a)66 b(target)h Fk(b)p Fp(-bit)f(k)-6 b(ey)67 b(in)g(the)f(time)h(for)f(2)3473 5900 y Fo(b)3552 5972 y Fk(=p)h Fp(cipher)e(ev)-11 b(aluations.)762 6211 y(Ev)-6 b(en)69 b(b)6 b(etter,)69 b(the)g(mac)-6 b(hine)68 b(can)g(b)6 b(e)68 b(sim)-6 b(ultaneously)69 b(applied)f(to)h Fk(q)77 b Fp(k)-6 b(eys)69 b(for)g(an)-6 b(y)69 b Fk(q)463 6450 y Fp(up)h(to)h(roughly)1733 6326 y Fh(p)p 1899 6326 101 8 v 124 x Fk(p)q Fp(.)g(The)f(mac)-6 b(hine)69 b(then)i(has)e(a)i (go)6 b(o)g(d)68 b(c)-6 b(hance)69 b(of)h(disco)-6 b(v)g(ering)70 b Ff(most)90 b Fp(of)463 6690 y(the)66 b(k)-6 b(eys)67 b(in)f(the)g(time)h(for)f(2)2679 6617 y Fo(b)2759 6690 y Fk(=p)g Fp(cipher)g(ev)-11 b(aluations.)762 6929 y(The)1169 6804 y Fh(p)p 1336 6804 V 1336 6929 a Fk(p)64 b Fp(limit)i(arises)d(as) g(follo)-6 b(ws.)64 b(Eac)-6 b(h)63 b(k)-6 b(ey)65 b(is)f(fed)f(to)i(2) 5181 6856 y Fi(4)5270 6929 y Fk(p=)p Fp(8)p Fk(q)70 b Fp(k)-6 b(ey-searc)g(h)64 b(circuits.)463 7168 y(Eac)-6 b(h)69 b(circuit)i(runs)e(for)h(2)2444 7095 y Fi(4)2533 7168 y Fk(p=)p Fp(8)p Fk(q)76 b Fp(iterations.)71 b(The)e(subsequen)-6 b(t)68 b(sorting)i(of)g(2)6383 7095 y Fi(4)6472 7168 y Fk(p)g Fp(n)-6 b(um)g(b)6 b(ers)463 7428 y(tak)-6 b(es)66 b(8)1073 7253 y Fj(p)p 1271 7253 290 8 v 1271 7428 a Fp(2)1371 7371 y Fi(4)1460 7428 y Fk(p)g Fp(adjacen)-6 b(t)64 b(compare-exc)-6 b(hange)62 b(steps,)j(whic)-6 b(h)65 b(b)6 b(ecome)63 b(a)i(b)6 b(ottlenec)-6 b(k)64 b(as)h Fk(q)463 7667 y Fp(gro)-6 b(ws)66 b(past)1457 7543 y Fh(p)p 1623 7543 101 8 v 124 x Fk(p)p Fp(.)762 7907 y(See)f(Section)h(5)g(for)g(a)g(v)-11 b(arian)-6 b(t)67 b(that)f(e\016cien)-6 b(tly)67 b(handles)e(larger)g Fk(q)7 b Fp(.)762 8146 y(The)66 b(success)f(probabilit)-6 b(y)67 b(of)g(the)f(mac)-6 b(hine)67 b(against)f(eac)-6 b(h)66 b(k)-6 b(ey)67 b(scales)e(linearly)i(with)463 8385 y(time.)88 b(F)-17 b(or)87 b(example,)f(in)h(the)g(time)g(for)g(2) 3744 8312 y Fo(b)3823 8385 y Fk(=pq)94 b Fp(cipher)86 b(ev)-11 b(aluations,)86 b(the)h(attac)-6 b(k)g(er)87 b(has)463 8624 y(a)79 b(go)6 b(o)g(d)77 b(c)-6 b(hance)77 b(of)i(disco)-6 b(v)g(ering)78 b Ff(at)84 b(le)-10 b(ast)82 b(one)94 b Fp(k)-6 b(ey)-17 b(.)80 b(In)f(the)g(time)h(for)f(2)6110 8552 y Fo(b)p Fd(\000)p Fi(20)6473 8624 y Fk(=pq)86 b Fp(cipher)463 8863 y(ev)-11 b(aluations,)74 b(the)g(attac)-6 b(k)g(er)75 b(has)f(roughly)f(a)h(2)4063 8791 y Fd(\000)p Fi(20)4431 8863 y Fp(c)-6 b(hance)72 b(of)j(disco)-6 b(v)g(ering)73 b(at)i(least)f(one)463 9102 y(k)-6 b(ey)-17 b(.)68 b(This)e(pattern)g(con)-6 b(tin)g(ues)65 b(do)-6 b(wn)66 b(to)h(a)f(v)-6 b(ery)67 b(small)f(amoun)-6 b(t)66 b(of)g(time.)762 9341 y(F)-17 b(or)75 b Fk(q)g Fp(=)69 b(1,)74 b(a)g(simpler)g(mac)-6 b(hine)73 b(do)6 b(es)73 b(the)h(same)f(job:)h(distribute)g Fk(H)16 b Fp(\()p Fk(k)6411 9371 y Fi(1)6501 9341 y Fp(\))74 b(to)h(man)-6 b(y)463 9580 y(circuits,)72 b(eac)-6 b(h)70 b(of)h(whic)-6 b(h)71 b(searc)-6 b(hes)69 b(sequen)-6 b(tially)71 b(through)f(a)h (range)f(of)h(p)6 b(ossibilities)70 b(for)463 9819 y Fk(k)567 9849 y Fi(1)656 9819 y Fp(.)64 b(Computations)e(for)h(one)f(k) -6 b(ey)63 b(can)f(b)6 b(e)62 b(merged)g(to)h(some)f(exten)-6 b(t)64 b(with)f(computations)463 10058 y(for)j(the)g(next)h(k)-6 b(ey)-17 b(,)67 b(sa)-6 b(ving)66 b(time.)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 463 -364 a Fl(3)270 b(The)90 b(serial)e (alternativ)-7 b(e)463 95 y Fp(A)91 b(v)-6 b(ery)90 b(small)g Ff(serial)110 b Fp(computer|a)89 b(single)g(k)-6 b(ey-searc)g(h)89 b(circuit|can)g(compute)f(a)i Fk(b)p Fp(-)463 334 y(bit)f(k)-6 b(ey)88 b Fk(k)94 b Fp(from)88 b Fk(H)16 b Fp(\()p Fk(k)6 b Fp(\))89 b(in)f(at)g(most)g(2)3531 262 y Fo(b)3699 334 y Fp(ev)-11 b(aluations)87 b(of)g Fk(H)16 b Fp(;)89 b(e.g.,)f(at)g(most)g(2)6648 262 y Fi(128)6984 334 y Fp(AES)463 573 y(ev)-11 b(aluations)65 b(for)h(a)g(128-bit)h(AES)f(k)-6 b(ey)-17 b(.)762 812 y(If)57 b(that's)g(to)6 b(o)56 b(m)-6 b(uc)g(h)56 b(time,)i(what)e(do)6 b(es)55 b(the)h(attac)-6 b(k)g(er)58 b(do?)e(The)g(ob)-6 b(vious)55 b(answ)-6 b(er)56 b(is)h(to)463 1052 y(build)69 b(the)f(standard)f(parallel)h (brute-force)f(k)-6 b(ey-searc)g(h)68 b(mac)-6 b(hine)67 b(describ)6 b(ed)66 b(in)j(Section)463 1291 y(2.)e(This)f(is)g(a)g (time-pro)6 b(cessor)64 b(tradeo\013,)i(trading)g(price)g(linearly)g (for)g(p)6 b(erformance.)762 1530 y(A)79 b(m)-6 b(uc)g(h)79 b(w)-6 b(orse)78 b(answ)-6 b(er)77 b(is)i(to)g(use)e(a)h(time-memory)h (tradeo\013:)g(a)f Ff(serial)99 b Fp(computer)463 1769 y(that)74 b(computes)e Fk(k)80 b Fp(from)74 b Fk(H)16 b Fp(\()p Fk(k)6 b Fp(\))74 b(in)g(few)-6 b(er)73 b(than)g(2)4319 1697 y Fo(b)4473 1769 y Fp(ev)-11 b(aluations)72 b(of)i Fk(H)16 b Fp(,)74 b(after)f(a)g(massiv)-6 b(e)463 2008 y(precomputation.)80 b(This)h(time-memory)g(tradeo\013)g(trades)f (price)h(for)f(p)6 b(erformance,)79 b(but)463 2247 y(not)74 b(linearly;)h(it)f(migh)-6 b(t)75 b(set)f(sp)6 b(eed)71 b(records)i(for)g(serial)h(computers,)e(but)i(it)h(is)f(painfully)463 2486 y(slo)-6 b(w)67 b(compared)d(to)i(a)g(prop)6 b(erly)66 b(designed)e(parallel)i(computer.)762 2725 y(Consider,)78 b(for)h(example,)f(the)h(same)f(attac)-6 b(k)80 b(as)e(in)h(Section)f (2,)h(but)g(using)f(memory)463 2964 y(on)73 b(a)g(serial)g(computer,)f (rather)h(than)f(pro)6 b(cessors)70 b(on)j(a)g(massiv)-6 b(ely)73 b(parallel)g(computer.)463 3204 y(W)-17 b(e'll)82 b(see)e(that)h(the)g(serial)g(computer)f(is)h(m)-6 b(uc)g(h,)81 b(m)-6 b(uc)g(h,)82 b(m)-6 b(uc)g(h)81 b(slo)-6 b(w)g(er)81 b(and)f(not)h(m)-6 b(uc)g(h)463 3443 y(smaller.)762 3682 y(The)61 b(attac)-6 b(k)g(er)63 b(selects)d(a)i(12-b)-6 b(yte)62 b(string)g Fk(\014)72 b Fp(and)61 b(2)4646 3609 y Fi(36)4850 3682 y Fh(\000)35 b Fp(2)5140 3609 y Fi(33)5371 3682 y Fp(random)60 b(16-b)-6 b(yte)62 b(strings)463 3921 y Fk(r)553 3951 y Fi(1)643 3921 y Fk(;)33 b(r)821 3951 y Fi(2)910 3921 y Fk(;)g(:)g(:)g(:)r Fp(.)61 b(He)f(computes,)f (serially)-17 b(,)62 b Fk(Z)14 b Fp(\()p Fk(\014)d(;)33 b Fp(2)3836 3849 y Fi(23)4004 3921 y Fk(;)g(r)4182 3951 y Fo(j)4267 3921 y Fp(\))61 b(for)f(eac)-6 b(h)59 b(random)g Fk(r)5961 3951 y Fo(j)6045 3921 y Fp(,)h(and)g(stores)f(the)463 4160 y(results)j(in)h(an)f(asso)6 b(ciativ)-6 b(e)62 b(arra)-6 b(y)-17 b(.)63 b(Then,)f(for)h(eac)-6 b(h)61 b(target)i Fk(H)16 b Fp(\()p Fk(k)5406 4190 y Fo(i)5472 4160 y Fp(\))64 b(in)f(turn,)g(the)f(attac)-6 b(k)g(er)463 4399 y(computes)86 b Fk(Z)14 b Fp(\()p Fk(\014)d(;)33 b Fp(0)p Fk(;)g(H)16 b Fp(\()p Fk(k)2361 4429 y Fo(i)2428 4399 y Fp(\)\))p Fk(;)33 b(:)g(:)g(:)j(;)d(Z)14 b Fp(\()p Fk(\014)d(;)33 b Fp(2)3564 4327 y Fi(23)3792 4399 y Fh(\000)58 b Fp(1)p Fk(;)33 b(H)16 b Fp(\()p Fk(k)4556 4429 y Fo(i)4623 4399 y Fp(\)\),)88 b(and)f(lo)6 b(oks)86 b(up)h(eac)-6 b(h)86 b(result)h(in)463 4638 y(the)79 b(asso)6 b(ciativ)-6 b(e)77 b(arra)-6 b(y)-17 b(.)79 b(Conjecturally)f(he)g(has)g (probabilit)-6 b(y)79 b(close)e(to)i(2)6171 4566 y Fd(\000)p Fi(69)6543 4638 y Fp(of)f(\014nding)463 4877 y Fk(k)567 4907 y Fo(i)633 4877 y Fp(,)67 b(exactly)f(as)g(in)h(Section)e(2.)762 5116 y(This)96 b(serial)g(mac)-6 b(hine)95 b(is)h(billions)g(of)g (times)g(slo)-6 b(w)g(er)96 b(than)g(the)g(parallel)f(mac)-6 b(hine.)463 5355 y(The)95 b(serial)g(mac)-6 b(hine)94 b(p)6 b(erforms)93 b(2)3223 5283 y Fi(23)3392 5355 y Fp(\(2)3569 5283 y Fi(23)3801 5355 y Fh(\000)64 b Fp(1\))p Fk(=)p Fp(2)95 b(AES)g(ev)-11 b(aluations)94 b(for)h(eac)-6 b(h)94 b(target)463 5595 y Fk(k)567 5625 y Fo(i)633 5595 y Fp(,)76 b(totalling)g(ab)6 b(out)75 b(2)2250 5522 y Fi(55)2494 5595 y Fp(AES)g(ev)-11 b(aluations;)75 b(and,)g(ev)-6 b(en)75 b(w)-6 b(orse,)75 b(2)5695 5522 y Fi(23)5939 5595 y Fp(AES)g(ev)-11 b(aluations)463 5834 y(for)79 b(eac)-6 b(h)78 b(random)f Fk(r)2075 5864 y Fo(j)2159 5834 y Fp(,)j(totalling)f(ab)6 b(out)78 b(2)3786 5761 y Fi(59)4033 5834 y Fp(AES)h(ev)-11 b(aluations.)78 b(F)-17 b(or)79 b(comparison,)e(the)463 6073 y(parallel)66 b(mac)-6 b(hine)66 b(\014nishes)e(in)j(the)f(time)h(for)f(just)g(2)4481 6000 y Fi(27)4716 6073 y Fp(AES)g(ev)-11 b(aluations.)762 6312 y(The)81 b(serial)h(mac)-6 b(hine)81 b(is)h(not)g(billions)g(of)g (times)g(less)f(exp)6 b(ensiv)-6 b(e)81 b(than)g(the)h(parallel)463 6551 y(mac)-6 b(hine.)83 b(It)g(isn't)h(ev)-6 b(en)82 b(ten)h(times)g(less)g(exp)6 b(ensiv)-6 b(e.)81 b(It's)j(ab)6 b(out)82 b(half)g(the)h(size)f(of)h(the)463 6790 y(parallel)95 b(mac)-6 b(hine:)95 b(it)h(do)6 b(esn't)94 b(ha)-6 b(v)g(e)95 b(the)g(2)3976 6718 y Fi(32)4240 6790 y Fp(AES)g(circuits,)g(but)g(it)h (do)6 b(es)94 b(ha)-6 b(v)g(e)95 b(the)463 7029 y(same)79 b Fh(\031)f Fp(2)1309 6957 y Fi(42)1558 7029 y Fp(b)-6 b(ytes)80 b(of)f(memory)-17 b(.)81 b(P)-6 b(erhaps)78 b(I'm)i(o)-6 b(v)g(erestimating)81 b(the)e(cost)h(of)f(memory)463 7268 y(compared)53 b(to)i(an)f(AES)g(circuit;)h(if)g(so,)g(simply)g (expand)e(the)h(amoun)-6 b(t)55 b(of)f(memory)g(in)h(b)6 b(oth)463 7507 y(mac)-6 b(hines)60 b(to)h(balance)e(the)i (memory-circuit)g(cost)f(with)h(the)g(AES-circuit)g(cost,)g(and)f(the) 463 7746 y(conclusion)k(will)k(b)6 b(e)65 b(the)h(same.)762 7986 y(T)-17 b(o)54 b(summarize:)g(The)f(time-memory)i(tradeo\013)e (pro)6 b(duces)52 b(a)i(ludicrously)f(un)-6 b(balanced)463 8225 y(mac)g(hine)75 b(with)h(tons)g(of)f(memory)g(w)-6 b(aiting)76 b(for)g(one)e(serial)i(CPU.)g(The)f(attac)-6 b(k)g(er)76 b(w)-6 b(ould)463 8464 y(ha)g(v)g(e)67 b(to)f(b)6 b(e)65 b(completely)h(insane)f(to)i(use)e(this)h(serial)g(mac)-6 b(hine.)463 9043 y Fe(Asymptotics)463 9422 y Fp(The)103 b(disadv)-11 b(an)-6 b(tage)102 b(of)h(a)g(time-memory)h(tradeo\013,)g (compared)d(to)j(a)f(time-pro)6 b(cessor)463 9661 y(tradeo\013,)68 b(gro)-6 b(ws)68 b(linearly)h(with)g(the)f(size)g(of)g(the)g(mac)-6 b(hine.)68 b(F)-17 b(or)69 b(example,)f(a)g(serial)g(k)-6 b(ey-)463 9900 y(searc)g(h)55 b(mac)-6 b(hine)55 b(with)h(ab)6 b(out)54 b(2)2930 9828 y Fi(74)3155 9900 y Fp(b)-6 b(ytes)56 b(of)f(memory)g(is)h(ab)6 b(out)55 b(2)5491 9828 y Fi(63)5715 9900 y Fp(times)h(slo)-6 b(w)g(er)56 b(than)f(a)463 10139 y(comparably)h(priced)g(parallel)g(k)-6 b(ey-searc)g(h)57 b(mac)-6 b(hine,)56 b(and)g(ab)6 b(out)56 b(2)5643 10067 y Fi(33)5868 10139 y Fp(times)h(slo)-6 b(w)g(er)57 b(than)463 10378 y(a)66 b(parallel)g(k)-6 b(ey-searc)g(h)66 b(mac)-6 b(hine)65 b(costing)h(2)3860 10306 y Fi(30)4095 10378 y Fp(times)g(less.)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 762 -364 a Fp(Sometimes)70 b(I)h(see)f(one)g (cryptanalyst)g(arguing)g(that)h(a)g(system)f(has)g(b)6 b(een)69 b(\\brok)-6 b(en")463 -125 y(b)g(y)99 b(a)f(time-memory)h (tradeo\013)e(on)h(a)g(serial)g(computer)f(with)i(2)5559 -197 y Fi(90)5825 -125 y Fp(b)-6 b(ytes)99 b(of)f(memory)-17 b(,)463 114 y(and)69 b(another)g(cryptanalyst)h(arguing)e(that)j (building)e(2)4783 42 y Fi(90)5021 114 y Fp(b)-6 b(ytes)70 b(of)g(memory)f(is)h(a)-6 b(wfully)463 353 y(di\016cult)52 b(so)f(the)h(system)f(has)g(not)h(b)6 b(een)50 b(\\brok)-6 b(en.")50 b(This)i(is)g(a)f(p)6 b(oin)-6 b(tless)51 b(argumen)-6 b(t)51 b(ab)6 b(out)463 592 y(an)90 b(incomp)6 b(eten)-6 b(t)89 b(mac)-6 b(hine)89 b(design.)g(The)h(standard)f(parallel)h (brute-force)f(k)-6 b(ey-searc)g(h)463 831 y(mac)g(hine)57 b(is)h(billions)f(of)h(times)f(less)g(exp)6 b(ensiv)-6 b(e)56 b(than)h(that)h(serial)f(computer)g Ff(and)78 b Fp(billions)463 1071 y(of)g(times)h(faster.)f(An)-6 b(y)g(one)79 b(who)f(thinks)g(that)h(time-memory)g(tradeo\013s)f(are)g (w)-6 b(orrisome)463 1310 y(should)89 b(b)6 b(e)90 b(utterly)h (terri\014ed)f(b)-6 b(y)91 b(the)f(v)-11 b(astly)91 b(sup)6 b(erior)89 b(price)g(and)h(p)6 b(erformance)87 b(of)j(a)463 1549 y(prop)6 b(erly)65 b(designed)g(parallel)h(mac)-6 b(hine.)463 2147 y Fe(F)-19 b(ancier)77 b(serial)g(attac)-6 b(ks)463 2545 y Fp(A)67 b(recen)-6 b(t)65 b(pap)6 b(er)64 b(adv)-6 b(ertises)66 b(a)f(serial)h(computer)f(with)h(2)4878 2473 y Fi(380)5192 2545 y Fp(bits)g(of)g(fast)g(memory)f(that,)463 2784 y(using)73 b(a)g(complicated)f(algorithm,)h(tak)-6 b(es)73 b(only)h(2)4311 2712 y Fi(534)4632 2784 y Fp(cycles)e(to)h (iden)-6 b(tify)75 b(a)e(544-bit)g(k)-6 b(ey)-17 b(.)463 3023 y(This)68 b(attac)-6 b(k)68 b(is)g(p)6 b(ortra)-6 b(y)g(ed)67 b(as)h(b)6 b(eing)66 b(successful)f(b)6 b(ecause)65 b(it)k(is)f(\(sligh)-6 b(tly\))70 b(\\faster)c(than)463 3262 y(exhaustiv)-6 b(e)66 b(searc)-6 b(h.")762 3501 y(Let's)69 b(compare)f(this)h(serial)g(computer)f(to)h(the)h(standard)d Ff(p)-10 b(ar)g(al)10 b(lel)89 b Fp(brute-force)68 b(k)-6 b(ey-)463 3741 y(searc)g(h)53 b(mac)-6 b(hine)54 b(with,)h(sa)-6 b(y)-17 b(,)55 b(2)2793 3668 y Fi(200)3096 3741 y Fp(k)-6 b(ey-searc)g(h)53 b(circuits.)i(The)e(parallel)h(mac)-6 b(hine)54 b(is)g(nearly)463 3980 y(2)563 3907 y Fi(200)878 3980 y Fp(times)66 b(faster)g(than)g(this)g(serial)g(computer,)f(and)h (v)-11 b(astly)67 b(less)e(exp)6 b(ensiv)-6 b(e.)762 4219 y(Wh)g(y)61 b(did)f(the)g(author)f(of)h(this)g(pap)6 b(er)59 b(c)-6 b(haracterize)58 b(this)j(serial)f(attac)-6 b(k)60 b(as)g(successful)463 4458 y(cryptanalysis?)87 b(It's)i(simply)f(not)g(true)g(that)g(the)g(attac)-6 b(k)88 b(is)g(\\faster)f(than)g(exhaustiv)-6 b(e)463 4697 y(searc)g(h"|unless)74 b(y)-6 b(ou)78 b(assume)d(that)i(the)g (attac)-6 b(k)g(er)78 b(is)f(forcing)f(himself)h(to)g(use)f(a)h(serial) 463 4936 y(computer,)93 b(i.e.,)i(that)f(the)f(attac)-6 b(k)g(er)95 b(is)e(an)h(idiot.)g(P)-6 b(erhaps)92 b(a)i (parallelization)f(of)g(the)463 5175 y(complicated)80 b(algorithm)i(could)e(b)6 b(eat)81 b(the)g(parallel)g(brute-force)g(k) -6 b(ey-searc)g(h)80 b(mac)-6 b(hine,)463 5414 y(but)60 b(I)g(doubt)f(it:)i(at)f(\014rst)f(glance,)g(parallelization)g(will)i (impro)-6 b(v)g(e)60 b(the)f(price-p)6 b(erformance)463 5653 y(ratio)67 b(of)f(the)g(complicated)f(algorithm)h(b)-6 b(y)67 b(a)f(factor)g(of)f(only)i(ab)6 b(out)65 b(2)5873 5581 y Fi(180)6121 5653 y Fp(.)762 5893 y(This)86 b(is)h(certainly)f (not)g(an)g(isolated)g(mistak)-6 b(e.)87 b(Most)e(of)h(the)h(\\breaks") d(that)j(I)f(see)463 6132 y(in)73 b(the)g(literature)g(are)g(sligh)-6 b(tly)74 b(faster)e(than)h Ff(serial)93 b Fp(brute-force)72 b(searc)-6 b(h)72 b(but)h(are)f(m)-6 b(uc)g(h)463 6371 y(slo)g(w)g(er)57 b(than)g(a)g(m)-6 b(uc)g(h)57 b(less)f(exp)6 b(ensiv)-6 b(e)56 b Ff(p)-10 b(ar)g(al)10 b(lel)76 b Fp(brute-force)56 b(searc)-6 b(h.)56 b(Man)-6 b(y)57 b(p)6 b(eople)55 b(seem)463 6610 y(to)65 b(think)f(that)h(the)f(n)-6 b(um)g(b)6 b(er)63 b(of)h(cipher)f(rounds)g(\\brok)-6 b(en")62 b(b)-6 b(y)65 b(a)f(di\013eren)-6 b(tial)64 b(attac)-6 b(k,)65 b(for)463 6849 y(example,)f(is)h(the)g(n)-6 b(um)g(b)6 b(er)64 b(of)g(rounds)f(for)i(whic)-6 b(h)64 b(the)h(di\013eren)-6 b(tial)65 b(attac)-6 b(k)65 b(is)g(marginally)463 7088 y(faster)91 b(than)f(a)h Ff(serial)112 b Fp(brute-force)89 b(searc)-6 b(h|ignoring)89 b(the)i(question)f(of)h(whether)f(the)463 7327 y(di\013eren)-6 b(tial)67 b(attac)-6 b(k)67 b(is)f(faster)g(than)g (a)g Ff(p)-10 b(ar)g(al)10 b(lel)86 b Fp(brute-force)65 b(searc)-6 b(h.)762 7566 y(This)63 b(mistak)-6 b(e)64 b(should)e(not)h(b)6 b(e)62 b(tolerated.)i(A)g(cryptanalytic)f(mac)-6 b(hine)62 b(is)i(a)f(failure)g(if)463 7805 y(it's)54 b(slo)-6 b(w)g(er)54 b(than)f(the)g(standard)f(parallel)h(brute-force)e (k)-6 b(ey-searc)g(h)53 b(mac)-6 b(hine)52 b(at)i(the)f(same)463 8044 y(price.)73 b(New)g(attac)-6 b(ks)73 b(m)-6 b(ust)73 b(b)6 b(e)72 b(compared)f(to)i(the)f(b)6 b(est)72 b(previous)g(attac)-6 b(ks,)73 b(not)g(merely)463 8284 y(the)66 b(b)6 b(est)66 b(previous)f Ff(serial)87 b Fp(attac)-6 b(ks.)463 8921 y Fl(4)270 b(Defending)91 b(against)e(the)h(standard)g(attac)-7 b(k)463 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b(e)463 114 y(the)66 b(attac)-6 b(k)g(er's)67 b(job)f(more)g(di\016cult.)463 683 y Fe(Input-space)79 b(separation)463 1053 y Fp(The)62 b(standard)g(parallel)g(mac)-6 b(hine)62 b(attac)-6 b(ks)63 b(a)f(large)g(batc)-6 b(h)63 b(of)f(target)h(k)-6 b(eys)63 b(at)g(ab)6 b(out)61 b(the)463 1292 y(same)66 b(cost)f(as)h(attac)-6 b(king)66 b(a)g(single)g(k)-6 b(ey)-17 b(.)762 1531 y(Often)94 b(the)g(attac)-6 b(k)g(er's)95 b(b)6 b(ene\014t)93 b(is)h(prop)6 b(ortional)93 b(to)h(the)g(n)-6 b(um)g(b)6 b(er)94 b(of)g(target)g(k)-6 b(eys)463 1770 y(successfully)91 b(found.)g(P)-6 b(erhaps)91 b(the)i(attac)-6 b(k)g(er)93 b(is)g(trying)g(to)g(steal)f(the)h(computational)463 2009 y(p)6 b(o)-6 b(w)g(er)90 b(of)g(as)f(man)-6 b(y)90 b(target)h(computers)d(as)i(p)6 b(ossible,)88 b(and)i(eac)-6 b(h)89 b(extra)h(k)-6 b(ey)90 b(lets)h(him)463 2248 y(steal)86 b(p)6 b(o)-6 b(w)g(er)86 b(from)f(an)h(extra)g(computer.)f(The)g(attac) -6 b(k)g(er's)87 b(cost-b)6 b(ene\014t)84 b(ratio)i(is)g(then)463 2487 y(divided)66 b(b)-6 b(y)67 b(the)f(n)-6 b(um)g(b)6 b(er)66 b(of)g(target)g(k)-6 b(eys.)762 2727 y(This)95 b(amortization)f(relies)g(on)h(the)f(attac)-6 b(k)g(er)96 b(b)6 b(eing)93 b(giv)-6 b(en)95 b Fk(H)16 b Fp(\()p Fk(k)5940 2757 y Fi(1)6029 2727 y Fp(\))p Fk(;)33 b(H)16 b Fp(\()p Fk(k)6557 2757 y Fi(2)6648 2727 y Fp(\))p Fk(;)33 b(:)g(:)g(:)97 b Fp(for)463 2966 y(the)84 b(same)g(function)f Fk(H)16 b Fp(:)85 b(e.g.,)f(AES)3296 2996 y Fo(k)3381 3016 y Fc(1)3468 2966 y Fp(\(8675309\))p Fk(;)33 b Fp(AES)4805 2996 y Fo(k)4890 3016 y Fc(2)4977 2966 y Fp(\(8675309\))p Fk(;)g(:)g(:)g(:)q Fp(.)85 b(If)f(there's)f(no)463 3205 y(o)-6 b(v)g(erlap)59 b(b)6 b(et)-6 b(w)g(een)58 b(the)h(inputs)g(to)g (AES)3481 3235 y Fo(k)3566 3255 y Fc(1)3712 3205 y Fp(and)f(the)h (inputs)g(to)g(AES)5663 3235 y Fo(k)5748 3255 y Fc(2)5894 3205 y Fp(then)g(the)f(attac)-6 b(k)g(er)463 3444 y(can't)66 b(sim)-6 b(ultaneously)66 b(attac)-6 b(k)67 b Fk(k)3027 3474 y Fi(1)3182 3444 y Fp(and)f Fk(k)3674 3474 y Fi(2)3763 3444 y Fp(.)762 3683 y(\\Aha!")51 b(one)g(migh)-6 b(t)53 b(sa)-6 b(y)-17 b(.)52 b(\\W)-17 b(e)52 b(should)e(design)h(our)g (cryptographic)f(proto)6 b(cols)50 b(so)i(that)463 3922 y(di\013eren)-6 b(t)67 b(k)-6 b(eys)66 b(are)g(applied)f(to)i(disjoin) -6 b(t)66 b(input)h(sets!")762 4161 y(Example:)60 b(De)h(Canni)-6 b(\023)-94 b(ere,)60 b(Lano,)f(and)h(Preneel)f(in)i([2,)g(Section)f(5]) h(suggest)e(designing)463 4400 y(stream)88 b(ciphers)g(so)g(that)h (nonces)d(are)i(as)g(long)g(as)g(k)-6 b(eys,)90 b(and)d(then)i(c)-6 b(ho)6 b(osing)86 b(nonces)463 4639 y(pseudorandomly)-17 b(.)67 b(This)i(suggestion)d(fails)j(to)g(ac)-6 b(hiev)g(e)68 b(the)h(goal|a)f(t)-6 b(ypical)70 b(nonce)d(will)463 4879 y(still)e(b)6 b(e)63 b(used)g(for)g(man)-6 b(y)65 b(k)-6 b(eys)64 b(if)g(the)g(n)-6 b(um)g(b)6 b(er)63 b(of)g(nonces)f(is)i(b)6 b(elo)-6 b(w)63 b(the)h(n)-6 b(um)g(b)6 b(er)63 b(of)h(k)-6 b(eys)463 5118 y Ff(times)67 b(the)e(numb)-10 b(er)65 b(of)h(messages)g(p)-10 b(er)65 b(key)18 b Fp(|but)60 b(one)f(can)h(mak)-6 b(e)60 b(nonces)e(ev)-6 b(en)60 b(longer)g(to)463 5357 y(prev)-6 b(en)g(t)67 b(rep)6 b(etition.)762 5596 y(Unfortunately)-17 b(,)65 b([2])f(fo)6 b(cuses)61 b(en)-6 b(tirely)65 b(on)e(the)h(costs)f(and)g (b)6 b(ene\014ts)62 b(for)h(the)h(attac)-6 b(k)g(er,)463 5835 y(and)66 b(neglects)f(to)h(consider)f(the)h(costs)f(and)h(b)6 b(ene\014ts)64 b(for)i(the)g(cryptographic)e(users:)603 6197 y Fh(\017)99 b Fp(Man)-6 b(y)83 b(stream)h(ciphers|coun)-6 b(ter-mo)6 b(de)80 b(AES,)k(for)f(example|ha)-6 b(v)g(e)83 b(small)h(input)802 6437 y(sizes)69 b(and)h(w)-6 b(ould)70 b(ha)-6 b(v)g(e)70 b(to)h(b)6 b(e)69 b(radically)h(rew)-6 b(ork)g(ed,)71 b(presumably)e(losing)g(sp)6 b(eed,)69 b(to)802 6676 y(handle)77 b(long)h(nonces.)f(The)h(authors)g(of)g([2])h (assert)f(that)h(\\the)f(state)g(is)h(already)f(at)802 6915 y(least)g(t)-6 b(wice)78 b(the)g(k)-6 b(ey)78 b(size";)f(this)h (is)g(true)g(for)g(some)e(stream)i(ciphers)e(but)i(false)f(for)802 7154 y(coun)-6 b(ter)66 b(mo)6 b(de.)603 7386 y Fh(\017)99 b Fp(Ev)-6 b(en)80 b(when)f(a)h(stream)f(cipher)h(easily)g(accepts)e(a) i(long)f(nonce,)g(generating)g(a)h(long)802 7625 y(pseudorandom)j (nonce)i(is)i(m)-6 b(uc)g(h)87 b(more)f(exp)6 b(ensiv)-6 b(e)85 b(than)i(generating)e(a)h(standard)802 7864 y(sequen)-6 b(tial)54 b(nonce.)f(Long)g(pseudorandom)d(nonces)i(cost)i(CPU)g (cycles)f(to)i(compute|)802 8103 y(the)85 b(pseudorandom)d(n)-6 b(um)g(b)6 b(er)84 b(generator)g(is)h(another)f(attac)-6 b(k)86 b(target,)f(so)g(it)h(needs)802 8342 y(to)h(b)6 b(e)85 b(at)i(least)f(as)g(strong)f(as)h(the)g(main)g(cipher|and)f (they)i(cost)e(bandwidth)h(to)802 8581 y(transmit.)603 8813 y Fh(\017)99 b Fp(As)73 b(for)f(b)6 b(ene\014ts:)70 b(Input-space)h(separation)g(do)6 b(esn't)71 b(mak)-6 b(e)72 b Ff(my)90 b Fp(k)-6 b(ey)73 b(more)f(di\016cult)802 9052 y(to)67 b(\014nd.)f(It)h(stops)f(the)h(attac)-6 b(k)g(er)67 b(from)g(\014nding)e(other)h(p)6 b(eople's)65 b(k)-6 b(eys)67 b(as)f(part)h(of)f(the)802 9291 y(same)74 b(computation,)h(and)g(p)6 b(erhaps)73 b(this)i(di\013erence)f(will)i (deter)f(the)h(attac)-6 b(k)g(er,)76 b(but)802 9531 y(p)6 b(erhaps)64 b(it)j(w)-6 b(on't.)463 9900 y(T)-17 b(o)68 b(summarize,)e(input-space)f(separation)h(has)g(limited)i(b)6 b(ene\014ts)65 b(and)i(quite)g(noticeable)463 10139 y(costs.)55 b(See)g(b)6 b(elo)-6 b(w)55 b(for)h(a)f(di\013eren)-6 b(t)56 b(approac)-6 b(h)54 b(that)i(has)f(m)-6 b(uc)g(h)56 b(larger)f(b)6 b(ene\014ts)54 b(and)h(m)-6 b(uc)g(h)463 10378 y(smaller)66 b(costs.)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 463 -364 a Fe(Larger)77 b(k)-6 b(eys)463 34 y Fp(A)72 b(more)e(ob)-6 b(vious)71 b(w)-6 b(a)g(y)72 b(to)f(mak)-6 b(e)71 b(the)g(attac)-6 b(k)g(er's)71 b(job)g(m)-6 b(uc)g(h)71 b(more)f(di\016cult)i(is)f(to)g Ff(use)76 b(a)463 274 y(lar)-10 b(ger)75 b(key)p Fp(.)c(Wh)-6 b(y)72 b(fo)6 b(ol)71 b(around)f(with)i(128-bit)f(k)-6 b(eys)72 b(when)f(w)-6 b(e)71 b(can)g(simply)g(use)g(256-bit)463 513 y(k)-6 b(eys?)762 752 y(The)68 b(b)6 b(ene\014ts)66 b(to)i(users)f(are)h(clear,)g(and)f(far)h(out)-6 b(w)g(eigh)68 b(the)g(b)6 b(ene\014ts)67 b(of)g(input-space)463 991 y(separation.)62 b(Brute-force)f(k)-6 b(ey-searc)g(h)62 b(attac)-6 b(ks)63 b(suddenly)e(b)6 b(ecome)60 b(2)5808 919 y Fi(128)6119 991 y Fp(times)j(slo)-6 b(w)g(er|)463 1230 y(i.e.,)68 b(completely)d(impractical)i(for)f(the)g(foreseeable)d (future.)762 1469 y(\\But)70 b(256-bit)f(AES)h(tak)-6 b(es)70 b(14)f(rounds,)f(while)i(128-bit)f(AES)h(tak)-6 b(es)69 b(only)h(10!")f(some)463 1708 y(p)6 b(eople)82 b(will)i(argue.)e(\\In)h(general,)f(256-bit)h(ciphers)f(ha)-6 b(v)g(e)84 b(more)e(rounds)g(than)h(128-bit)463 1947 y(ciphers.)67 b(That's)g(a)h(quite)g(serious)e(cost)i(in)f(sp)6 b(eed;)66 b(w)-6 b(e)68 b(need)e(to)i(consider)e(other)i(w)-6 b(a)g(ys)68 b(to)463 2186 y(use)e(the)g(same)f(CPU)h(cycles.")762 2426 y(But)j(this)g(argumen)-6 b(t)69 b(confuses)d(t)-6 b(w)g(o)70 b(di\013eren)-6 b(t)69 b(c)-6 b(hanges.)67 b(W)-17 b(e)68 b(can)g(switc)-6 b(h)69 b(to)g(256-bit)463 2665 y(k)-6 b(eys,)72 b(e\013ectiv)-6 b(ely)72 b(eliminating)f (brute-force)f(attac)-6 b(ks,)72 b Ff(without)91 b Fp(increasing)69 b(the)j(n)-6 b(um)g(b)6 b(er)463 2904 y(of)63 b(rounds.)e(The)h(only)h (cost)f(is)h(the)f(cost)g(of)h(generating)e(and)h(storing)g(larger)g(k) -6 b(eys,)63 b(whic)-6 b(h)463 3143 y(is)92 b(normally)g(m)-6 b(uc)g(h)91 b(smaller)h(than)f(the)g(cost)g(of)h(generating)e(and)g (transmitting)j(large)463 3382 y(random)65 b(nonces)g(for)h(ev)-6 b(ery)66 b(message.)e(The)i(b)6 b(ene\014t)64 b(is)j(m)-6 b(uc)g(h)66 b(larger.)762 3621 y(\\But)61 b(256-bit)f(AES)g(delib)6 b(erately)60 b(uses)f(14)g(rounds)g(to)i(k)-6 b(eep)60 b(us)f(secure)g(against)g(non-)463 3860 y(brute-force)k(attac)-6 b(ks!")64 b(some)f(p)6 b(eople)61 b(will)k(argue.)e(\\There's)f(an)i (8-round)f(attac)-6 b(k)64 b(taking)463 4099 y(time)78 b(only)g(2)1477 4027 y Fi(204)1725 4099 y Fp(,)g(for)f(example.)g(Ok) -11 b(a)-6 b(y)-17 b(,)79 b(ok)-11 b(a)-6 b(y)-17 b(,)78 b(that's)g(on)f(a)g(mac)-6 b(hine)77 b(with)h(2)6494 4027 y Fi(104)6820 4099 y Fp(bits)f(of)463 4338 y(memory)-17 b(,)65 b(but)g(ma)-6 b(yb)6 b(e)64 b(some)g(additional)g(ideas)g(will)h (pro)6 b(duce)63 b(a)h(10-round)f(attac)-6 b(k)66 b(more)463 4577 y(e\016cien)-6 b(t)66 b(than)g(a)g(256-bit)h(parallel)f (brute-force)f(k)-6 b(ey-searc)g(h)65 b(attac)-6 b(k.")762 4817 y(But)58 b(this)f(argumen)-6 b(t)57 b(confuses)d(securit)-6 b(y)57 b(lev)-6 b(el)58 b(with)g(k)-6 b(ey)57 b(size.)g(I)g(nev)-6 b(er)57 b(said)f(that)i(this)463 5056 y(c)-6 b(hange)85 b(w)-6 b(ould)87 b(pro)6 b(duce)84 b(a)j(256-bit)f(securit)-6 b(y)87 b(lev)-6 b(el.)88 b(I)f(said)f(that)h(it)h(w)-6 b(ould)86 b(mak)-6 b(e)87 b(the)463 5295 y(attac)-6 b(k)g(er's)77 b(job)f(m)-6 b(uc)g(h)76 b(more)g(di\016cult|pro)6 b(ducing)74 b(larger)i(b)6 b(ene\014ts)74 b(than)i(input-space)463 5534 y(separation)64 b(at)h(lo)-6 b(w)g(er)66 b(cost.)e(If)h(the)g (resulting)g(securit)-6 b(y)65 b(lev)-6 b(el)66 b(is,)f(sa)-6 b(y)-17 b(,)66 b(192)e(bits,)h(then)g(the)463 5773 y(mission)h(has)f(b) 6 b(een)65 b(accomplished.)762 6012 y(\\But)82 b(y)-6 b(ou're)82 b(not)g(allo)-6 b(w)g(ed)82 b(to)h(use)e(k)-6 b(eys)82 b(larger)f(than)h(the)f(target)h(securit)-6 b(y)82 b(lev)-6 b(el!")463 6251 y(some)81 b(p)6 b(eople)79 b(will)j(argue.)f(\\It's,)h(um,)f(against)g(the)g(la)-6 b(w!)82 b(The)f(standard)f(de\014nition)g(of)463 6490 y(securit)-6 b(y)67 b(v)-11 b(aries)65 b(with)i(y)-6 b(our)67 b(k)-6 b(ey)66 b(size!")762 6729 y(But)89 b(that's)g(a)g (silly)g(de\014nition)f(of)g(securit)-6 b(y)-17 b(.)90 b(P)-6 b(erhaps)87 b(the)h(most)h(e\016cien)-6 b(t)88 b(w)-6 b(a)g(y)90 b(to)463 6968 y(ac)-6 b(hiev)g(e)61 b(a)g(192-bit)g(securit)-6 b(y)62 b(lev)-6 b(el)62 b(is)f(with)h(a)f (system)g(ha)-6 b(ving)61 b(a)g(256-bit)g(k)-6 b(ey)-17 b(.)63 b(If)e(the)g(user)463 7208 y(w)-6 b(an)g(ts)73 b(a)f(192-bit)g(securit)-6 b(y)72 b(lev)-6 b(el,)73 b(then)f(the)g (user)f(should)g(select)g(that)i(system.)f(There's)463 7447 y(no)66 b(justi\014cation)f(for)h(demanding)f(a)h(reduced)e(k)-6 b(ey)67 b(size.)762 7686 y(\(Helix,)51 b(in)-6 b(tro)6 b(duced)47 b(b)-6 b(y)50 b(F)-17 b(erguson,)47 b(Whiting,)j(Sc)-6 b(hneier,)48 b(Kelsey)-17 b(,)49 b(Luc)-6 b(ks,)48 b(and)g(Kohno)463 7925 y(in)63 b([3],)h(is)f(an)g(example)f(of)h(a)f(fast)h(stream)g (cipher)f(that)h(uses)f(a)g(k)-6 b(ey)64 b(size)e(ab)6 b(o)-6 b(v)g(e)62 b(its)i(target)463 8164 y(securit)-6 b(y)83 b(lev)-6 b(el.)84 b(I)e(don't)h(kno)-6 b(w)83 b(whether)f(this)g(is)h(the)g(most)f(e\016cien)-6 b(t)83 b(approac)-6 b(h,)81 b(but)i(I)463 8403 y(certainly)67 b(w)-6 b(ould)66 b(not)g(w)-6 b(an)g(t)67 b(it)g(to)g(b)6 b(e)65 b(excluded)g(from)h(consideration.\))762 8642 y(I'm)77 b(not)f(sa)-6 b(ying)76 b(that)h(increasing)e(the)h(n)-6 b(um)g(b)6 b(er)76 b(of)g(rounds)f(is)h(a)g(bad)g(idea.)g(On)g(the)463 8881 y(con)-6 b(trary:)80 b(extra)f(rounds)f(ha)-6 b(v)g(e,)80 b(historically)-17 b(,)80 b(b)6 b(een)78 b(quite)h(e\013ectiv)-6 b(e)80 b(at)f(stopping)f(non-)463 9120 y(brute-force)87 b(attac)-6 b(ks.)88 b(But)f(this)h(has)f(nothing)g(to)g(do)g(with)i (the)e(question)g(of)g(whether)463 9360 y(randomness)64 b(should)h(b)6 b(e)65 b(added)g(to)h(nonces)f(or)h(to)g(k)-6 b(eys.)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 463 -364 a Fl(5)270 b(The)90 b(v)-15 b(arian)-7 b(t)88 b(parallel)g(brute-force)i(k)-7 b(ey-searc)g(h)87 b(attac)-7 b(k)463 114 y Fp(This)74 b(section)g(describ)6 b(es)72 b(a)i(v)-11 b(arian)-6 b(t)75 b(of)f(the)g(standard)f(parallel) h(brute-force)g(k)-6 b(ey-searc)g(h)463 353 y(mac)g(hine.)61 b(This)g(v)-11 b(arian)-6 b(t)62 b(is)g(a)f(little)i(slo)-6 b(w)g(er)61 b(but)h(has)e(the)i(adv)-11 b(an)-6 b(tage)60 b(that)h(it)i(can)d(handle)463 592 y(man)-6 b(y)67 b(more)e(k)-6 b(eys)67 b(sim)-6 b(ultaneously)-17 b(.)463 1190 y Fe(The)77 b(problem)463 1589 y Fp(The)c(problem)f(is)h(the)g(same)f(as)h(in)g (Section)f(2:)i(the)f(attac)-6 b(k)g(er)73 b(is)g(trying)h(to)f(\014nd) g(16-b)-6 b(yte)463 1828 y(AES)46 b(k)-6 b(eys)46 b Fk(k)1427 1858 y Fi(1)1516 1828 y Fk(;)33 b(k)1708 1858 y Fi(2)1798 1828 y Fk(;)g(:)g(:)g(:)r Fp(,)47 b(giv)-6 b(en)46 b Fk(H)16 b Fp(\()p Fk(k)3084 1858 y Fi(1)3173 1828 y Fp(\))p Fk(;)33 b(H)16 b Fp(\()p Fk(k)3701 1858 y Fi(2)3792 1828 y Fp(\))p Fk(;)33 b(:)g(:)g(:)r Fp(.)47 b(Let's)e(again)h(assume)e (that)i(the)g(attac)-6 b(k)g(er)463 2067 y(is)67 b(facing)e(2)1345 1995 y Fi(10)1580 2067 y Fp(k)-6 b(eys)66 b(o)-6 b(v)g(erall.)463 2665 y Fe(The)77 b(v)-13 b(arian)-6 b(t)78 b(k)-6 b(ey-searc)g(h)77 b(circuit)463 3063 y Fp(The)e(attac)-6 b(k)g(er)76 b(builds)f(a)g(v)-6 b(ery)76 b(small)f(k)-6 b(ey-searc)g(h)75 b(circuit.)h(The)f(k)-6 b(ey-searc)g(h)74 b(circuit)i(has)463 3302 y(three)d(inputs:)h(a)g (23-bit)g(string)f Fk(\013)q Fp(,)i(a)e(16-b)-6 b(yte)74 b(string)g Fk(\014)11 b Fp(,)73 b(and)g(a)h(16-b)-6 b(yte)73 b(string)h Fk(s)p Fp(.)h(The)463 3541 y(k)-6 b(ey-searc)g(h)59 b(circuit)h(has)f(one)g(output:)h(a)g(16-b)-6 b(yte)59 b(string)h Fk(D)6 b Fp(\()p Fk(\013)q(;)33 b(\014)11 b(;)33 b(s)p Fp(\))61 b(de\014ned)d(recursiv)-6 b(ely)463 3780 y(as)66 b Fk(\014)54 b Fh(\010)45 b Fk(s)66 b Fp(if)h Fk(\014)55 b Fh(\010)44 b Fk(s)67 b Fp(b)6 b(egins)64 b(with)j Fk(\013)q Fp(,)g(and)e(as)h Fk(D)6 b Fp(\()p Fk(\013)q(;)33 b(\014)11 b(;)33 b(H)16 b Fp(\()p Fk(\014)55 b Fh(\010)45 b Fk(s)p Fp(\)\))67 b(otherwise.)762 4020 y(This)62 b(k)-6 b(ey-searc)g(h)61 b(circuit)i(is)f(attac)-6 b(hed)61 b(to)h(a)g(comparable-size)e(memory)h(circuit)i(that)463 4259 y(bu\013ers)j(2)1213 4186 y Fi(4)1368 4259 y Fp(inputs)g(and)f(2) 2465 4186 y Fi(4)2621 4259 y Fp(outputs,)h(as)f(in)i(Section)e(2.)762 4498 y(Ho)-6 b(w)85 b(long)f(do)6 b(es)83 b(the)h(k)-6 b(ey-searc)g(h)84 b(circuit)h(tak)-6 b(e)84 b(to)h(pro)6 b(duce)82 b(its)j(2)6019 4425 y Fi(4)6193 4498 y Fp(outputs?)f(The)463 4737 y(16-b)-6 b(yte)69 b(strings)e Fk(\014)56 b Fh(\010)45 b Fk(s)69 b Fp(b)6 b(ounce)65 b(around)i(pseudorandomly)-17 b(,)66 b(and)h(one)g(out)i(of)e(ev)-6 b(ery)68 b(2)7211 4665 y Fi(23)463 4976 y Fp(strings)63 b(b)6 b(egins)61 b(with)j Fk(\013)q Fp(,)g(so)e(the)h(n)-6 b(um)g(b)6 b(er)62 b(of)h(iterations)g(p)6 b(er)62 b(output)h(is)g(t)-6 b(ypically)65 b(on)d(the)463 5215 y(scale)75 b(of)h(2)1287 5143 y Fi(23)1455 5215 y Fp(.)h(Ma)-6 b(yb)6 b(e)75 b(less;)g(ma)-6 b(yb)6 b(e)76 b(more;)f(o)6 b(ccasionally)74 b(the)i(circuit)g(falls)g (in)-6 b(to)77 b(a)f(lo)6 b(op)463 5454 y(and)90 b(nev)-6 b(er)91 b(pro)6 b(duces)88 b(an)j(output;)g(but)g(the)g(k)-6 b(ey-searc)g(h)90 b(circuit)h(t)-6 b(ypically)93 b(computes)463 5693 y(outputs)66 b(for)g(most)g(of)g(its)h(inputs)f(within)h(2)3823 5621 y Fi(27)4058 5693 y Fp(iterations.)463 6291 y Fe(The)77 b(v)-13 b(arian)-6 b(t)78 b(k)-6 b(ey-searc)g(h)77 b(mac)-6 b(hine)463 6690 y Fp(The)78 b(attac)-6 b(k)g(er)78 b(no)-6 b(w)78 b(builds)g(a)f(mac)-6 b(hine)77 b(with)i(2)4228 6617 y Fi(32)4475 6690 y Fp(k)-6 b(ey-searc)g(h)77 b(circuits)h(in)g(a) g(2)6684 6617 y Fi(16)6904 6690 y Fh(\002)52 b Fp(2)7211 6617 y Fi(16)463 6929 y Fp(mesh.)67 b(Eac)-6 b(h)67 b(k)-6 b(ey-searc)g(h)67 b(circuit)h(has)f(its)h(o)-6 b(wn)68 b(comparable-size)d(memory)j(circuit,)g(and)463 7168 y(is)f(connected)c(to)k(its)g(immediate)f(neigh)-6 b(b)6 b(ors,)65 b(as)g(in)i(Section)e(2.)762 7407 y(The)k(attac)-6 b(k)g(er)70 b(feeds)e(2)2545 7335 y Fi(36)2783 7407 y Fp(inputs)h(to)h(his)f(2)4057 7335 y Fi(32)4295 7407 y Fp(k)-6 b(ey-searc)g(h)69 b(circuits.)h(Sp)6 b(eci\014cally)-17 b(,)68 b(the)463 7646 y(attac)-6 b(k)g(er)71 b(selects)d(a)i(23-bit)g (string)g Fk(\013)q Fp(,)g(a)g(16-b)-6 b(yte)69 b(string)h Fk(\014)11 b Fp(,)70 b(and)f(2)5625 7574 y Fi(36)5840 7646 y Fh(\000)46 b Fp(2)6141 7574 y Fi(10)6380 7646 y Fp(random)68 b(16-)463 7885 y(b)-6 b(yte)89 b(strings)f Fk(r)1685 7915 y Fi(1)1774 7885 y Fk(;)33 b(r)1952 7915 y Fi(2)2042 7885 y Fk(;)g(:)g(:)g(:)r Fp(;)89 b(generates)d(one)h (input)h(\()p Fk(\013)q(;)33 b(\014)11 b(;)33 b(r)4940 7915 y Fo(j)5025 7885 y Fp(\))89 b(for)f(eac)-6 b(h)87 b(random)g Fk(r)6831 7915 y Fo(j)6915 7885 y Fp(;)i(and)463 8124 y(generates)64 b(one)i(input)g(\()p Fk(\013)q(;)33 b(\014)11 b(;)33 b(H)16 b Fp(\()p Fk(k)3104 8154 y Fo(i)3171 8124 y Fp(\)\))68 b(for)e(eac)-6 b(h)65 b(target)h Fk(H)16 b Fp(\()p Fk(k)5096 8154 y Fo(i)5163 8124 y Fp(\).)762 8363 y(All)78 b(2)1199 8291 y Fi(32)1445 8363 y Fp(k)-6 b(ey-searc)g(h)77 b(circuits)f(no)-6 b(w)77 b(run)g(in)g(parallel)g (for)f(2)5311 8291 y Fi(27)5557 8363 y Fp(iterations,)h(pro)6 b(ducing)463 8602 y Fk(D)g Fp(\()p Fk(\013)q(;)33 b(\014)11 b(;)33 b(r)1229 8632 y Fo(j)1314 8602 y Fp(\))92 b(for)f (\(conjecturally\))g(most)h Fk(j)11 b Fp('s)92 b(and)e Fk(D)6 b Fp(\()p Fk(\013)q(;)33 b(\014)11 b(;)33 b(H)16 b Fp(\()p Fk(k)5462 8632 y Fo(i)5529 8602 y Fp(\)\))93 b(for)e(\(conjecturally\))463 8841 y(most)66 b Fk(i)p Fp('s.)h(This)f(tak)-6 b(es)66 b(sligh)-6 b(tly)68 b(more)d(time)i (than)f(2)4457 8769 y Fi(27)4692 8841 y Fp(AES)g(computations.)762 9081 y(The)83 b(attac)-6 b(k)g(er)85 b(then)e(sorts)h(the)g Fk(D)89 b Fp(v)-11 b(alues.)83 b(If)h(the)g(sorting)g(encoun)-6 b(ters)82 b(a)h(collision)463 9320 y(b)6 b(et)-6 b(w)g(een)83 b(t)-6 b(w)g(o)84 b Fk(D)6 b Fp(\()p Fk(\013)q(;)33 b(\014)11 b(;)33 b(r)2409 9350 y Fo(j)2494 9320 y Fp(\)'s,)84 b(it)h(thro)-6 b(ws)83 b(one)g(of)g(those)f Fk(r)4982 9350 y Fo(j)5066 9320 y Fp('s)i(a)-6 b(w)g(a)g(y)-17 b(.)85 b(If)e(it)i(encoun)-6 b(ters)81 b(a)463 9559 y(collision)56 b(b)6 b(et)-6 b(w)g(een)55 b Fk(D)6 b Fp(\()p Fk(\013)q(;)33 b(\014)11 b(;)33 b(H)16 b Fp(\()p Fk(k)3016 9589 y Fo(i)3083 9559 y Fp(\)\))57 b(and)e Fk(D)6 b Fp(\()p Fk(\013)q(;)33 b(\014)11 b(;)33 b(r)4437 9589 y Fo(j)4522 9559 y Fp(\),)57 b(the)f(mac)-6 b(hine)55 b(pauses)f(to)i(redo)f(the)463 9798 y Fk(D)6 b Fp(\()p Fk(\013)q(;)33 b(\014)11 b(;)33 b(r)1229 9828 y Fo(j)1314 9798 y Fp(\))64 b(computation.)g(If)g(an)-6 b(y)64 b(in)-6 b(termediate)64 b(v)-11 b(alue)63 b(\()p Fk(\013)q(;)33 b(\014)11 b(;)33 b(s)p Fp(\))66 b(satis\014es)d Fk(H)16 b Fp(\()p Fk(\014)50 b Fh(\010)40 b Fk(s)p Fp(\))56 b(=)463 10037 y Fk(H)16 b Fp(\()p Fk(k)826 10067 y Fo(i)893 10037 y Fp(\))67 b(then)e(the)i(mac)-6 b(hine)65 b(prin)-6 b(ts)67 b Fk(\014)54 b Fh(\010)44 b Fk(s)67 b Fp(as)f(a)g(guess)e(for)i Fk(k)5054 10067 y Fo(i)5121 10037 y Fp(.)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 463 -364 a Fe(Heuristic)77 b(analysis)463 34 y Fp(What's)66 b(the)g(c)-6 b(hance)65 b(that)i(a)f(particular)g(k) -6 b(ey)-17 b(,)67 b(sa)-6 b(y)66 b Fk(k)4511 64 y Fi(1)4601 34 y Fp(,)g(is)h(found)e(b)-6 b(y)66 b(this)h(mac)-6 b(hine?)762 274 y(The)82 b(crude)g(estimate)g(is)h(that)g(there)f(are)g (2)4202 201 y Fi(36)4453 274 y Fp(v)-11 b(alues)82 b Fk(D)6 b Fp(\()p Fk(\013)q(;)33 b(\014)11 b(;)33 b(r)5829 304 y Fo(j)5913 274 y Fp(\),)84 b(eac)-6 b(h)82 b(in)-6 b(v)g(olving)463 513 y(2)563 440 y Fi(23)814 513 y Fp(in)g(termediate) 82 b(v)-11 b(alues)81 b(\()p Fk(\013)q(;)33 b(\014)11 b(;)33 b(s)p Fp(\))84 b(and)d(2)3858 440 y Fi(23)4108 513 y Fp(inputs)h Fk(\014)65 b Fh(\010)54 b Fk(s)83 b Fp(to)f Fk(H)16 b Fp(,)82 b(for)g(a)f(total)i(of)e(2)7211 440 y Fi(59)463 752 y Fp(inputs)53 b(to)h Fk(H)16 b Fp(.)53 b(If)h(an)-6 b(y)53 b(of)g(those)f(inputs)h(bump)6 b(ed)52 b(in)-6 b(to)54 b Fk(k)4702 782 y Fi(1)4844 752 y Fp(then)f(the)g(mac) -6 b(hine)52 b(will)j(\014nd)d Fk(k)7235 782 y Fi(1)7324 752 y Fp(.)463 991 y(Sp)6 b(eci\014cally)-17 b(,)73 b(if)g(an)g(in)-6 b(termediate)73 b(v)-11 b(alue)73 b(\()p Fk(\013)q(;)33 b(\014)11 b(;)33 b(s)p Fp(\))75 b(in)f(the)f(computation)f(of)h Fk(D)6 b Fp(\()p Fk(\013)q(;)33 b(\014)11 b(;)33 b(r)7218 1021 y Fo(j)7302 991 y Fp(\))463 1230 y(satis\014es)87 b Fk(\014)69 b Fh(\010)58 b Fk(s)92 b Fp(=)g Fk(k)2150 1260 y Fi(1)2239 1230 y Fp(,)c(then)f Fk(D)6 b Fp(\()p Fk(\013)q(;)33 b(\014)11 b(;)33 b(r)3623 1260 y Fo(j)3708 1230 y Fp(\))92 b(=)f Fk(D)6 b Fp(\()p Fk(\013)q(;)33 b(\014)11 b(;)33 b(H)16 b Fp(\()p Fk(\014)70 b Fh(\010)58 b Fk(s)p Fp(\)\))93 b(=)f Fk(D)6 b Fp(\()p Fk(\013)q(;)33 b(\014)11 b(;)33 b(H)16 b Fp(\()p Fk(k)7080 1260 y Fi(1)7169 1230 y Fp(\)\);)463 1469 y(the)81 b(sorting)f(will)h(disco)-6 b(v)g(er)81 b(this)f(collision,)h(will)h(lo)6 b(cate)79 b(the)h(same)g(in)-6 b(termediate)81 b(v)-11 b(alue)463 1708 y(\()p Fk(\013)q(;)33 b(\014)11 b(;)33 b(s)p Fp(\),)53 b(and)d(will)i(prin)-6 b(t)51 b Fk(\014)23 b Fh(\010)13 b Fk(s)56 b Fp(=)f Fk(k)3223 1738 y Fi(1)3363 1708 y Fp(as)50 b(its)h(guess)e(for)h Fk(k)4754 1738 y Fi(1)4843 1708 y Fp(.)h(This)g(o)6 b(ccurs)48 b(with)j(probabilit)-6 b(y)463 1947 y(2)563 1875 y Fi(59)732 1947 y Fk(=)p Fp(2)932 1875 y Fi(128)1235 1947 y Fp(=)55 b(2)1545 1875 y Fd(\000)p Fi(69)1838 1947 y Fp(.)762 2186 y(This)73 b(estimate)f(is)h(sligh)-6 b(tly)74 b(o)-6 b(v)g(eroptimistic,)74 b(for)e(all)h(the)g(reasons)e (in)i(Section)e(2)i(and)463 2426 y(more:)81 b(for)g(example,)f(some)g Fk(r)2806 2456 y Fo(j)2890 2426 y Fp('s)h(fail)g(to)g(pro)6 b(duce)79 b(v)-11 b(alues)80 b Fk(D)6 b Fp(\()p Fk(\013)q(;)33 b(\014)11 b(;)33 b(r)5863 2456 y Fo(j)5947 2426 y Fp(\).)82 b(These)e(failures)463 2665 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b(e)g(disc)g(overy)798 6530 y(of)53 b(time)e(memory)g(data)h(tr)-9 b(ade)g(o\013s)45 b Fn(\(2005\).)i(URL:)f Fm (http://www.ecrypt.eu.org/stream/)p Fn(.)555 6749 y(3.)100 b(Niels)84 b(F)-15 b(erguson,)84 b(Doug)g(Whiting,)h(Bruce)f(Sc)-5 b(hneier,)85 b(John)f(Kelsey)-15 b(,)83 b(Stefan)i(Luc)-5 b(ks,)83 b(T)-15 b(a-)798 6968 y(da)-5 b(y)g(oshi)57 b(Kohno,)f Fb(Helix:)61 b(fast)g(encryption)e(and)h(authentic)-9 b(ation)60 b(in)g(a)g(single)g(crypto)-9 b(gr)g(aphic)798 7188 y(primitive)p Fn(,)62 b(in)g([4])g(\(2003\),)g(330{346.)h(URL:)e Fm(http://www.macfergus.com/helix/)p Fn(.)555 7407 y(4.)100 b(Thomas)43 b(Johansson)h(\(editor\),)f Fb(F)-14 b(ast)50 b(softwar)-9 b(e)49 b(encryption:)f(10th)i(international)e(workshop,) 798 7626 y(FSE)66 b(2003,)g(Lund,)h(Swe)-9 b(den,)64 b(F)-14 b(ebruary)65 b(24{26,)i(2003,)f(r)-9 b(evise)g(d)65 b(p)-9 b(ap)g(ers)p Fn(,)61 b(Lecture)g(Notes)g(in)798 7845 y(Computer)h(Science,)h(2887,)f(Springer-V)-15 b(erlag,)64 b(Berlin,)e(2003.)g(ISBN)f(3{540{20449{0.)555 8064 y(5.)100 b(Philipp)5 b(e)98 b(Oec)-5 b(hslin,)96 b Fb(Making)h(a)g(faster)g (cryptanalytic)g(time-memory)f(tr)-9 b(ade-o\013)p Fn(,)94 b(in)i([1])798 8284 y(\(2003\),)62 b(617{630.)555 8503 y(6.)100 b(Mic)-5 b(hael)98 b(J.)f(Wiener,)h Fb(The)g(ful)9 b(l)99 b(c)-9 b(ost)99 b(of)g(cryptanalytic)g(attacks)p Fn(,)e(Journal)h(of)f(Cryptol-)798 8722 y(ogy)107 b Fa(17)g Fn(\(2004\),)f(105{124.)i(ISSN)f(0933{2790.)h(URL:)e Fm(http://www3.sympatico.ca/)798 8941 y(wienerfamily/Michael/)p Fn(.)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF