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2101 y Fu(X)472 2108 y Fr(i)500 2101 y Fw(:)f(16-bit)i(plain)o(text)f(subblo)q(c)o(k)432 2162 y Fu(Y)460 2169 y Fr(i)488 2162 y Fw(:)e(16-bit)k(ciphertext)e(subblo)q(c)o(k) 432 2230 y Fu(Z)469 2205 y Fl(\()p Fr(r)q Fl(\))465 2242 y Fr(i)529 2230 y Fw(:)e(16-bit)k(k)o(ey)e(subblo)q(c)o(k)432 2257 y Fj(L)492 2290 y Fw(:)e(bit-b)o(y-bit)j(exclusiv)o(e-OR)e(of)i(16-bit)h (subblo)q(c)o(ks)p 434 2321 37 2 v 434 2354 2 33 v 434 2350 a(+)p 469 2354 V 434 2355 37 2 v 15 w(:)d(addition)i(mo)q(dulo)g(2)906 2332 y Fl(16)960 2350 y Fw(of)g(16-bit)h(in)o(tegers)432 2377 y Fj(J)492 2411 y Fw(:)c(m)o(ultiplic)o(ation)h(mo)q(dulo)i(2)1031 2393 y Fl(16)1080 2411 y Fw(+)11 b(1)16 b(of)h(16-bit)g(in)o(tegers)497 2471 y(with)f(the)g(zero)g(subblo)q(c)o(k)g(corresp)q(onding)h(to)g(2)1394 2453 y Fl(16)62 2588 y Fw(Figure)e(3.1:)22 b(Computational)16 b(graph)h(for)g(the)f(encryption)f(pro)q(cess)i(of)g(the)f(IDEA)f(cipher.)p eop %%Page: 23 3 bop 36 103 a Fq(3.1.)38 b(DESCRIPTION)16 b(OF)g(IDEA)1035 b Fw(23)60 275 y({)12 b(m)o(ultipli)o(cation)c(of)j(in)o(tegers)f(mo)q(dulo)h (2)822 257 y Fl(16)860 275 y Fw(+1)g(where)f(the)h(16-bit)g(subblo)q(c)o(k)g (is)g(treated)f(as)i(the)158 345 y(usual)17 b(radix-t)o(w)o(o)f(represen)o (tation)g(of)g(an)h(in)o(teger)f(except)f(that)i(the)f(all-zero)g(subblo)q(c) o(k)158 416 y(is)g(treated)g(as)h(represen)o(ting)e(2)734 398 y Fl(16)772 416 y Fw(;)h(the)g(resulting)g(op)q(eration)h(is)f(denoted)g(as) 1598 383 y Fj(J)1644 416 y Fw(.)36 524 y(As)g(an)h(example)d(of)i(these)g (group)i(op)q(erations,)e(note)h(that)540 640 y(\(0)p Fu(;)8 b(:::;)g Fw(0\))712 606 y Fj(J)757 640 y Fw(\(1)p Fu(;)g Fw(0)p Fu(;)g(:::;)g Fw(0\))13 b(=)h(\(1)p Fu(;)8 b Fw(0)p Fu(;)g(:::;)g Fw(0)p Fu(;)g Fw(1\))36 755 y(b)q(ecause)592 826 y(2)616 805 y Fl(16)654 826 y Fw(2)678 805 y Fl(15)732 826 y Fw(mo)q(d)16 b(\(2)885 805 y Fl(16)934 826 y Fw(+)11 b(1\))j(=)g(2)1116 805 y Fl(15)1164 826 y Fw(+)d(1)p Fu(:)109 921 y Fw(The)h(64-bit)h(plain)o (text)d(blo)q(c)o(k)i Fu(X)k Fw(is)c(partitioned)g(in)o(to)f(four)i(16-bit)f (subblo)q(c)o(ks)g Fu(X)1602 928 y Fl(1)1623 921 y Fu(;)f(X)1688 928 y Fl(2)1708 921 y Fu(;)h(X)1774 928 y Fl(3)1794 921 y Fu(;)36 992 y(X)76 999 y Fl(4)96 992 y Fu(;)18 b Fw(i.e.,)e Fu(X)k Fw(=)d(\()p Fu(X)396 999 y Fl(1)416 992 y Fu(;)8 b(X)478 999 y Fl(2)498 992 y Fu(;)g(X)560 999 y Fl(3)580 992 y Fu(;)g(X)642 999 y Fl(4)662 992 y Fw(\).)26 b(The)17 b(four)i(plain)o(text)d(subblo)q(c)o (ks)i(are)g(then)f(transformed)36 1062 y(in)o(to)22 b(four)g(16-bit)g (ciphertext)f(subblo)q(c)o(ks)h Fu(Y)887 1069 y Fl(1)907 1062 y Fu(;)f(Y)970 1069 y Fl(2)990 1062 y Fu(;)h(Y)1054 1069 y Fl(3)1074 1062 y Fu(;)f(Y)1137 1069 y Fl(4)1179 1062 y Fw([)g(i.e.,)h(the)f (ciphertext)g(blo)q(c)o(k)g(is)36 1133 y Fu(Y)k Fw(=)14 b(\()p Fu(Y)188 1140 y Fl(1)208 1133 y Fu(;)8 b(Y)258 1140 y Fl(2)278 1133 y Fu(;)g(Y)328 1140 y Fl(3)347 1133 y Fu(;)g(Y)397 1140 y Fl(4)417 1133 y Fw(\))14 b(])f(under)g(the)g(con)o(trol)g(of)g(52)h(k)o(ey) 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b(computational)f (graph)i(of)f(the)g(decryption)f(pro)q(cess)i(is)f(essen)o(tially)e(the)i (same)f(as)h(that)36 1732 y(of)e(the)g(encryption)f(pro)q(cess)i(\(cf.)j (Sec.3.4.1\),)15 b(the)h(only)f(c)o(hange)h(b)q(eing)g(that)h(the)e (decryption)36 1803 y(k)o(ey)e(subblo)q(c)o(ks)i Fu(K)387 1777 y Fl(\()p Fr(r)q Fl(\))383 1814 y Fr(i)448 1803 y Fw(are)g(computed)e(from)h (the)g(encryption)g(k)o(ey)f(subblo)q(c)o(ks)i Fu(Z)1531 1777 y Fl(\()p Fr(r)q Fl(\))1527 1814 y Fr(i)1592 1803 y Fw(as)g(follo)o(ws:)66 1922 y(\()p Fu(K)130 1896 y Fl(\()p Fr(r)q Fl(\))126 1933 y(1)176 1922 y Fu(;)8 b(K)243 1896 y Fl(\()p Fr(r)q Fl(\))239 1933 y(2)289 1922 y Fu(;)g(K)356 1896 y Fl(\()p Fr(r)q Fl(\))352 1933 y(3)403 1922 y Fu(;)g(K)470 1896 y Fl(\()p Fr(r)q Fl(\))466 1933 y(4)516 1922 y Fw(\))e(=)14 b(\()p Fu(Z)649 1896 y Fl(\(10)p Fh(\000)p Fr(r)q Fl(\))645 1933 y(1)758 1885 y Fh(\000)p Fl(1)805 1922 y Fu(;)8 b Fs(\000)p Fu(Z)903 1896 y Fl(\(10)p Fh(\000)p Fr(r)q Fl(\))899 1933 y(3)1012 1922 y Fu(;)g Fs(\000)p Fu(Z)1110 1896 y Fl(\(10)p Fh(\000)p Fr(r)q Fl(\))1106 1933 y(2)1219 1922 y Fu(;)g(Z)1278 1896 y Fl(\(10)p Fh(\000)p Fr(r)q Fl(\))1274 1933 y(4)1387 1885 y Fh(\000)p Fl(1)1434 1922 y Fw(\))16 b(for)g Fu(r)q Fw(=2,3,..,8;)66 1999 y(\()p Fu(K)130 1973 y Fl(\()p Fr(r)q Fl(\))126 2010 y(1)176 1999 y Fu(;)8 b(K)243 1973 y Fl(\()p Fr(r)q Fl(\))239 2010 y(2)289 1999 y Fu(;)g(K)356 1973 y Fl(\()p Fr(r)q Fl(\))352 2010 y(3)403 1999 y Fu(;)g(K)470 1973 y Fl(\()p Fr(r)q Fl(\))466 2010 y(4)516 1999 y Fw(\))e(=)14 b(\()p Fu(Z)649 1973 y Fl(\(10)p Fh(\000)p Fr(r)q Fl(\))645 2010 y(1)758 1961 y Fh(\000)p Fl(1)805 1999 y Fu(;)8 b Fs(\000)p Fu(Z)903 1973 y Fl(\(10)p Fh(\000)p Fr(r)q Fl(\))899 2010 y(2)1012 1999 y Fu(;)g Fs(\000)p Fu(Z)1110 1973 y Fl(\(10)p Fh(\000)p Fr(r)q Fl(\))1106 2010 y(3)1219 1999 y Fu(;)g(Z)1278 1973 y Fl(\(10)p Fh(\000)p Fr(r)q Fl(\))1274 2010 y(4)1387 1961 y Fh(\000)p Fl(1)1434 1999 y Fw(\))16 b(for)g Fu(r)q Fw(=1)h(and)g(9;)292 2067 y(\()p Fu(K)356 2042 y Fl(\()p Fr(r)q Fl(\))352 2078 y(5)403 2067 y Fu(;)8 b(K)470 2042 y Fl(\()p Fr(r)q Fl(\))466 2078 y(6)516 2067 y Fw(\))e(=)14 b(\()p Fu(Z)649 2042 y Fl(\()p Fr(r)q Fl(\))645 2078 y(5)695 2067 y Fu(;)8 b(Z)754 2042 y Fl(\()p Fr(r)q Fl(\))750 2078 y(6)800 2067 y Fw(\))669 b(for)16 b Fu(r)q Fw(=1,2,..,8;)36 2174 y(where)d Fu(Z)211 2156 y Fh(\000)p Fl(1)271 2174 y Fw(denotes)g(the)g(m)o(ultiplicati)o(v)o(e)d(in)o(v)o(erse)h (\(mo)q(dulo)i(2)1203 2156 y Fl(16)1245 2174 y Fw(+)5 b(1\))13 b(of)h Fu(Z)t Fw(,)f(i.e.,)e Fu(Z)1587 2141 y Fj(J)1634 2174 y Fu(Z)1671 2156 y Fh(\000)p Fl(1)1732 2174 y Fw(=)i(1)36 2244 y(and)k Fs(\000)p Fu(Z)j Fw(denotes)c(the)g(additiv)o(e)f(in)o(v)o(erse)g (\(mo)q(dulo)g(2)1049 2226 y Fl(16)1087 2244 y Fw(\))h(of)h Fu(Z)t Fw(,)f(i.e.,)e Fs(\000)p Fu(Z)p 1415 2215 37 2 v 1415 2248 2 33 v 6 w Fw(+)p 1450 2248 V 1415 2249 37 2 v 1 w Fu(Z)k Fw(=)c(0.)109 2315 y(The)g(computation)e(of)i(decryption)f(k)o(ey)f(subblo)q (c)o(ks)h(from)g(the)g(encryption)f(k)o(ey)h(subblo)q(c)o(ks)36 2385 y(is)j(also)h(sho)o(wn)g(in)f(table)g(3.1.)36 2529 y Fk(3.1.3)70 b(The)22 b(k)n(ey)i(sc)n(hedule)36 2632 y Fw(The)17 b(52)h(k)o(ey)e(subblo)q (c)o(ks)i(of)f(16)h(bits)g(used)f(in)g(the)g(encryption)f(pro)q(cess)i(are)g (generated)f(from)36 2702 y(the)e(128-bit)h(user-selected)e(k)o(ey)f(as)j (follo)o(ws:)21 b(The)15 b(128-bit)h(user-selected)d(k)o(ey)h(is)h (partitioned)36 2773 y(in)o(to)k(8)g(subblo)q(c)o(ks)g(that)g(are)g(directly) f(used)h(as)g(the)g(\014rst)g(eigh)o(t)f(k)o(ey)g(subblo)q(c)o(ks,)h(where)g (the)36 2844 y(ordering)i(of)f(the)h(k)o(ey)e(subblo)q(c)o(ks)i(is)f (de\014ned)g(as)h(follo)o(ws:)30 b Fu(Z)1213 2818 y Fl(\(1\))1209 2854 y(1)1260 2844 y Fw(,)p Fu(Z)1311 2818 y Fl(\(1\))1307 2854 y(2)1358 2844 y Fw(,..,)p Fu(Z)1451 2818 y Fl(\(1\))1447 2854 y(6)1496 2844 y Fw(,)21 b Fu(Z)1568 2818 y Fl(\(2\))1564 2854 y(1)1615 2844 y Fw(,..,)p Fu(Z)1708 2818 y Fl(\(2\))1704 2854 y(6)1753 2844 y Fw(,..,)p eop %%Page: 24 4 bop 36 103 a Fw(24)900 b Fq(CH.)16 b(3.)38 b(THE)16 b(BLOCK)g(CIPHER)g(IDEA) 211 826 y Fw(Encryption)g(k)o(ey)g(subblo)q(c)o(ks)327 b(Decryption)16 b(k)o(ey)f(subblo)q(c)o(ks)p 132 900 658 2 v 839 900 874 2 v 131 975 2 77 v 228 957 a(1-st)p 402 975 V 121 w Fu(Z)464 932 y Fl(\(1\))460 968 y(1)512 957 y Fu(Z)549 932 y Fl(\(1\))545 968 y(2)596 957 y Fu(Z)633 932 y Fl(\(1\))629 968 y(3)680 957 y Fu(Z)717 932 y Fl(\(1\))713 968 y(4)p 788 975 V 838 975 V 935 957 a Fw(1-st)p 1109 975 V 122 w Fu(Z)1172 932 y Fl(\(9\))1168 968 y(1)1219 920 y Fh(\000)p Fl(1)1277 957 y Fs(\000)c Fu(Z)1364 932 y Fl(\(9\))1360 968 y(2)1422 957 y Fs(\000)g Fu(Z)1509 932 y Fl(\(9\))1505 968 y(3)1556 957 y Fu(Z)1593 932 y Fl(\(9\))1589 968 y(4)1640 920 y Fh(\000)p Fl(1)p 1711 975 V 131 1044 2 69 v 205 1025 a Fw(round)p 402 1044 V 183 w Fu(Z)549 1000 y Fl(\(1\))545 1036 y(5)596 1025 y Fu(Z)633 1000 y Fl(\(1\))629 1036 y(6)p 788 1044 V 838 1044 V 912 1025 a Fw(round)p 1109 1044 V 291 w Fu(Z)1364 1000 y Fl(\(8\))1360 1036 y(5)1411 1025 y Fu(Z)1448 1000 y Fl(\(8\))1444 1036 y(6)p 1711 1044 V 132 1045 658 2 v 839 1045 874 2 v 131 1120 2 77 v 220 1102 a Fw(2-nd)p 402 1120 V 113 w Fu(Z)464 1077 y Fl(\(2\))460 1113 y(1)512 1102 y Fu(Z)549 1077 y Fl(\(2\))545 1113 y(2)596 1102 y Fu(Z)633 1077 y Fl(\(2\))629 1113 y(3)680 1102 y Fu(Z)717 1077 y Fl(\(2\))713 1113 y(4)p 788 1120 V 838 1120 V 927 1102 a Fw(2-nd)p 1109 1120 V 114 w Fu(Z)1172 1077 y Fl(\(8\))1168 1113 y(1)1219 1065 y Fh(\000)p Fl(1)1277 1102 y Fs(\000)g Fu(Z)1364 1077 y Fl(\(8\))1360 1113 y(3)1422 1102 y Fs(\000)g Fu(Z)1509 1077 y Fl(\(8\))1505 1113 y(2)1556 1102 y Fu(Z)1593 1077 y Fl(\(8\))1589 1113 y(4)1640 1065 y Fh(\000)p Fl(1)p 1711 1120 V 131 1189 2 69 v 205 1171 a Fw(round)p 402 1189 V 183 w Fu(Z)549 1145 y Fl(\(2\))545 1181 y(5)596 1171 y Fu(Z)633 1145 y Fl(\(2\))629 1181 y(6)p 788 1189 V 838 1189 V 912 1171 a Fw(round)p 1109 1189 V 291 w Fu(Z)1364 1145 y Fl(\(7\))1360 1181 y(5)1411 1171 y Fu(Z)1448 1145 y Fl(\(7\))1444 1181 y(6)p 1711 1189 V 132 1190 658 2 v 839 1190 874 2 v 131 1265 2 77 v 224 1247 a Fw(3-rd)p 402 1265 V 117 w Fu(Z)464 1222 y Fl(\(3\))460 1258 y(1)512 1247 y Fu(Z)549 1222 y Fl(\(3\))545 1258 y(2)596 1247 y Fu(Z)633 1222 y Fl(\(3\))629 1258 y(3)680 1247 y Fu(Z)717 1222 y Fl(\(3\))713 1258 y(4)p 788 1265 V 838 1265 V 931 1247 a Fw(3-rd)p 1109 1265 V 118 w Fu(Z)1172 1222 y Fl(\(7\))1168 1258 y(1)1219 1210 y Fh(\000)p Fl(1)1277 1247 y Fs(\000)g Fu(Z)1364 1222 y Fl(\(7\))1360 1258 y(3)1422 1247 y Fs(\000)g Fu(Z)1509 1222 y Fl(\(7\))1505 1258 y(2)1556 1247 y Fu(Z)1593 1222 y Fl(\(7\))1589 1258 y(4)1640 1210 y Fh(\000)p Fl(1)p 1711 1265 V 131 1334 2 69 v 205 1316 a Fw(round)p 402 1334 V 183 w Fu(Z)549 1290 y Fl(\(3\))545 1327 y(5)596 1316 y Fu(Z)633 1290 y Fl(\(3\))629 1327 y(6)p 788 1334 V 838 1334 V 912 1316 a Fw(round)p 1109 1334 V 291 w Fu(Z)1364 1290 y Fl(\(6\))1360 1327 y(5)1411 1316 y Fu(Z)1448 1290 y Fl(\(6\))1444 1327 y(6)p 1711 1334 V 132 1335 658 2 v 839 1335 874 2 v 131 1410 2 77 v 224 1392 a Fw(4-th)p 402 1410 V 117 w Fu(Z)464 1367 y Fl(\(4\))460 1403 y(1)512 1392 y Fu(Z)549 1367 y Fl(\(4\))545 1403 y(2)596 1392 y Fu(Z)633 1367 y Fl(\(4\))629 1403 y(3)680 1392 y Fu(Z)717 1367 y Fl(\(4\))713 1403 y(4)p 788 1410 V 838 1410 V 931 1392 a Fw(4-th)p 1109 1410 V 118 w Fu(Z)1172 1367 y Fl(\(6\))1168 1403 y(1)1219 1355 y Fh(\000)p Fl(1)1277 1392 y Fs(\000)g Fu(Z)1364 1367 y Fl(\(6\))1360 1403 y(3)1422 1392 y Fs(\000)g Fu(Z)1509 1367 y Fl(\(6\))1505 1403 y(2)1556 1392 y Fu(Z)1593 1367 y Fl(\(6\))1589 1403 y(4)1640 1355 y Fh(\000)p Fl(1)p 1711 1410 V 131 1479 2 69 v 205 1461 a Fw(round)p 402 1479 V 183 w Fu(Z)549 1435 y Fl(\(4\))545 1472 y(5)596 1461 y Fu(Z)633 1435 y Fl(\(4\))629 1472 y(6)p 788 1479 V 838 1479 V 912 1461 a Fw(round)p 1109 1479 V 291 w Fu(Z)1364 1435 y Fl(\(5\))1360 1472 y(5)1411 1461 y Fu(Z)1448 1435 y Fl(\(5\))1444 1472 y(6)p 1711 1479 V 132 1480 658 2 v 839 1480 874 2 v 131 1556 2 77 v 224 1537 a Fw(5-th)p 402 1556 V 117 w Fu(Z)464 1512 y Fl(\(5\))460 1548 y(1)512 1537 y Fu(Z)549 1512 y Fl(\(5\))545 1548 y(2)596 1537 y Fu(Z)633 1512 y Fl(\(5\))629 1548 y(3)680 1537 y Fu(Z)717 1512 y Fl(\(5\))713 1548 y(4)p 788 1556 V 838 1556 V 931 1537 a Fw(5-th)p 1109 1556 V 118 w Fu(Z)1172 1512 y Fl(\(5\))1168 1548 y(1)1219 1500 y Fh(\000)p Fl(1)1277 1537 y Fs(\000)g Fu(Z)1364 1512 y Fl(\(5\))1360 1548 y(3)1422 1537 y Fs(\000)g Fu(Z)1509 1512 y Fl(\(5\))1505 1548 y(2)1556 1537 y Fu(Z)1593 1512 y Fl(\(5\))1589 1548 y(4)1640 1500 y Fh(\000)p Fl(1)p 1711 1556 V 131 1624 2 69 v 205 1606 a Fw(round)p 402 1624 V 183 w Fu(Z)549 1580 y Fl(\(5\))545 1617 y(5)596 1606 y Fu(Z)633 1580 y Fl(\(5\))629 1617 y(6)p 788 1624 V 838 1624 V 912 1606 a Fw(round)p 1109 1624 V 291 w Fu(Z)1364 1580 y Fl(\(4\))1360 1617 y(5)1411 1606 y Fu(Z)1448 1580 y Fl(\(4\))1444 1617 y(6)p 1711 1624 V 132 1626 658 2 v 839 1626 874 2 v 131 1701 2 77 v 224 1683 a Fw(6-th)p 402 1701 V 117 w Fu(Z)464 1657 y Fl(\(6\))460 1693 y(1)512 1683 y Fu(Z)549 1657 y Fl(\(6\))545 1693 y(2)596 1683 y Fu(Z)633 1657 y Fl(\(6\))629 1693 y(3)680 1683 y Fu(Z)717 1657 y Fl(\(6\))713 1693 y(4)p 788 1701 V 838 1701 V 931 1683 a Fw(6-th)p 1109 1701 V 118 w Fu(Z)1172 1657 y Fl(\(4\))1168 1693 y(1)1219 1645 y Fh(\000)p Fl(1)1277 1683 y Fs(\000)g Fu(Z)1364 1657 y Fl(\(4\))1360 1693 y(3)1422 1683 y Fs(\000)g Fu(Z)1509 1657 y Fl(\(4\))1505 1693 y(2)1556 1683 y Fu(Z)1593 1657 y Fl(\(4\))1589 1693 y(4)1640 1645 y Fh(\000)p Fl(1)p 1711 1701 V 131 1769 2 69 v 205 1751 a Fw(round)p 402 1769 V 183 w Fu(Z)549 1726 y Fl(\(6\))545 1762 y(5)596 1751 y Fu(Z)633 1726 y Fl(\(6\))629 1762 y(6)p 788 1769 V 838 1769 V 912 1751 a Fw(round)p 1109 1769 V 291 w Fu(Z)1364 1726 y Fl(\(3\))1360 1762 y(5)1411 1751 y Fu(Z)1448 1726 y Fl(\(3\))1444 1762 y(6)p 1711 1769 V 132 1771 658 2 v 839 1771 874 2 v 131 1846 2 77 v 224 1828 a Fw(7-th)p 402 1846 V 117 w Fu(Z)464 1802 y Fl(\(7\))460 1839 y(1)512 1828 y Fu(Z)549 1802 y Fl(\(7\))545 1839 y(2)596 1828 y Fu(Z)633 1802 y Fl(\(7\))629 1839 y(3)680 1828 y Fu(Z)717 1802 y Fl(\(7\))713 1839 y(4)p 788 1846 V 838 1846 V 931 1828 a Fw(7-th)p 1109 1846 V 118 w Fu(Z)1172 1802 y Fl(\(3\))1168 1839 y(1)1219 1790 y Fh(\000)p Fl(1)1277 1828 y Fs(\000)g Fu(Z)1364 1802 y Fl(\(3\))1360 1839 y(3)1422 1828 y Fs(\000)g Fu(Z)1509 1802 y Fl(\(3\))1505 1839 y(2)1556 1828 y Fu(Z)1593 1802 y Fl(\(3\))1589 1839 y(4)1640 1790 y Fh(\000)p Fl(1)p 1711 1846 V 131 1914 2 69 v 205 1896 a Fw(round)p 402 1914 V 183 w Fu(Z)549 1871 y Fl(\(7\))545 1907 y(5)596 1896 y Fu(Z)633 1871 y Fl(\(7\))629 1907 y(6)p 788 1914 V 838 1914 V 912 1896 a Fw(round)p 1109 1914 V 291 w Fu(Z)1364 1871 y Fl(\(2\))1360 1907 y(5)1411 1896 y Fu(Z)1448 1871 y Fl(\(2\))1444 1907 y(6)p 1711 1914 V 132 1916 658 2 v 839 1916 874 2 v 131 1991 2 77 v 224 1973 a Fw(8-th)p 402 1991 V 117 w Fu(Z)464 1947 y Fl(\(8\))460 1984 y(1)512 1973 y Fu(Z)549 1947 y Fl(\(8\))545 1984 y(2)596 1973 y Fu(Z)633 1947 y Fl(\(8\))629 1984 y(3)680 1973 y Fu(Z)717 1947 y Fl(\(8\))713 1984 y(4)p 788 1991 V 838 1991 V 931 1973 a Fw(8-th)p 1109 1991 V 118 w Fu(Z)1172 1947 y Fl(\(2\))1168 1984 y(1)1219 1936 y Fh(\000)p Fl(1)1277 1973 y Fs(\000)g Fu(Z)1364 1947 y Fl(\(2\))1360 1984 y(3)1422 1973 y Fs(\000)g Fu(Z)1509 1947 y Fl(\(2\))1505 1984 y(2)1556 1973 y Fu(Z)1593 1947 y Fl(\(2\))1589 1984 y(4)1640 1936 y Fh(\000)p Fl(1)p 1711 1991 V 131 2059 2 69 v 205 2041 a Fw(round)p 402 2059 V 183 w Fu(Z)549 2016 y Fl(\(8\))545 2052 y(5)596 2041 y Fu(Z)633 2016 y Fl(\(8\))629 2052 y(6)p 788 2059 V 838 2059 V 912 2041 a Fw(round)p 1109 2059 V 291 w Fu(Z)1364 2016 y Fl(\(1\))1360 2052 y(5)1411 2041 y Fu(Z)1448 2016 y Fl(\(1\))1444 2052 y(6)p 1711 2059 V 132 2061 658 2 v 839 2061 874 2 v 131 2136 2 77 v 195 2118 a Fw(output)p 402 2136 V 89 w Fu(Z)464 2092 y Fl(\(9\))460 2129 y(1)512 2118 y Fu(Z)549 2092 y Fl(\(9\))545 2129 y(2)596 2118 y Fu(Z)633 2092 y Fl(\(9\))629 2129 y(3)680 2118 y Fu(Z)717 2092 y Fl(\(9\))713 2129 y(4)p 788 2136 V 838 2136 V 902 2118 a Fw(output)p 1109 2136 V 90 w Fu(Z)1172 2092 y Fl(\(1\))1168 2129 y(1)1219 2081 y Fh(\000)p Fl(1)1277 2118 y Fs(\000)g Fu(Z)1364 2092 y Fl(\(1\))1360 2129 y(2)1422 2118 y Fs(\000)g Fu(Z)1509 2092 y Fl(\(1\))1505 2129 y(3)1556 2118 y Fu(Z)1593 2092 y Fl(\(1\))1589 2129 y(4)1640 2081 y Fh(\000)p Fl(1)p 1711 2136 V 131 2196 2 61 v 157 2178 a Fw(transform.)p 402 2196 V 788 2196 V 838 2196 V 485 w(transform.)p 1109 2196 V 1711 2196 V 132 2198 658 2 v 839 2198 874 2 v 313 2276 a(T)l(able)16 b(3.1:)21 b(The)c(encryption)e(and)i(decryption)e(k)o(ey)g (subblo)q(c)o(ks.)p eop %%Page: 25 5 bop 36 103 a Fq(3.2.)38 b(GR)o(OUP)16 b(OPERA)l(TIONS)f(AND)h(THEIR)f(INTERA) o(CTION)410 b Fw(25)36 284 y Fu(Z)73 258 y Fl(\(8\))69 295 y(1)120 284 y Fw(,..,)p Fu(Z)213 258 y Fl(\(8\))209 295 y(6)258 284 y Fw(,)20 b Fu(Z)329 258 y Fl(\(9\))325 295 y(1)376 284 y Fw(,)p Fu(Z)427 258 y Fl(\(9\))423 295 y(2)474 284 y Fw(,)p Fu(Z)525 258 y Fl(\(9\))521 295 y(3)571 284 y Fw(,)p Fu(Z)622 258 y Fl(\(9\))618 295 y(4)669 284 y Fw(.)30 b(The)19 b(128-bit)h (user-selected)e(k)o(ey)f(is)i(then)g(cyclic)e(shifted)36 354 y(to)12 b(the)g(left)g(b)o(y)f(25)i(p)q(ositions,)g(after)f(whic)o(h)g(the)g (resulting)f(128-bit)j(blo)q(c)o(k)d(is)h(again)h(partitioned)36 425 y(in)o(to)18 b(eigh)o(t)f(subblo)q(c)o(ks)h(that)g(are)g(tak)o(en)f(as)i (the)e(next)h(eigh)o(t)f(k)o(ey)f(subblo)q(c)o(ks.)27 b(The)17 b(obtained)36 495 y(128-bit)h(blo)q(c)o(k)f(is)g(again)h(cyclic)e(shifted)g (to)i(the)f(left)g(b)o(y)f(25)j(p)q(ositions)f(to)f(pro)q(duce)h(the)f(next) 36 566 y(eigh)o(t)f(k)o(ey)g(subblo)q(c)o(ks,)h(and)h(this)f(pro)q(cedure)g (is)f(rep)q(eated)h(un)o(til)f(all)h(52)h(k)o(ey)d(subblo)q(c)o(ks)i(ha)o(v)o (e)36 637 y(b)q(een)f(generated.)36 800 y Fp(3.2)83 b(Group)27 b(Op)r(erations)g(and)g(their)h(In)n(teraction)36 920 y Fw(The)15 b(IDEA)f(cipher)g(is)g(based)h(on)h(the)e(new)h(design)g(concept)f(of)h (mixing)d(op)q(erations)k(from)e(dif-)36 990 y(feren)o(t)e(algebraic)h (groups)i(ha)o(ving)e(the)g(same)f(n)o(um)o(b)q(er)g(of)h(elemen)o(ts.)18 b(Group)c(op)q(erations)g(w)o(ere)36 1061 y(c)o(hosen)g(b)q(ecause)g(the)f (statistical)h(relation)f(of)h(an)o(y)g(three)f(random)g(v)m(ariables)h Fu(U;)8 b(V)s(;)g(W)20 b Fw(related)36 1131 y(b)o(y)c(a)i(group)g(op)q (eration)g(as)f Fu(W)22 b Fw(=)15 b Fu(U)i Fs(\003)12 b Fu(V)28 b Fw(has)18 b(the)e(\\p)q(erfect)h(secrecy")f(prop)q(ert)o(y)h(that)g(if)g (an)o(y)36 1202 y(one)e(of)g(the)g(three)g(random)f(v)m(ariables)h(is)g(c)o (hosen)g(indep)q(enden)o(tly)e(of)i(the)g(others)g(and)h(equally)36 1273 y(lik)o(ely)11 b(to)k(b)q(e)f(an)o(y)g(group)g(elemen)o(t,)d(then)j(the) g(other)g(t)o(w)o(o)g(random)f(v)m(ariables)h(are)g(statistically)36 1343 y(indep)q(enden)o(t.)31 b(The)20 b(in)o(teraction)f(of)h(the)f (di\013eren)o(t)g(group)i(op)q(erations)g(con)o(tributes)e(to)h(the)36 1414 y(\\confusion")g(required)e(for)h(a)g(secure)g(cipher,)f(as)h(will)f(b)q (e)h(explained)f(in)h(the)f(follo)o(wing)h(t)o(w)o(o)36 1484 y(sections.)109 1555 y(The)14 b(in)o(teraction)f(of)h(the)f(di\013eren)o(t)g (group)i(op)q(erations)g(will)d(no)o(w)j(b)q(e)f(considered)f(in)g(terms)36 1625 y(of)19 b(isotopism)e(of)i(quasigroups)g(and)g(in)f(terms)f(of)i(p)q (olynomial)e(expressions.)27 b(T)l(o)19 b(generalize)36 1696 y(the)k(discussion)g(b)q(ey)o(ond)g(the)g(case)g(of)g(16-bit)h(subblo)q(c)o (ks,)g(let)e Fu(n)h Fw(b)q(e)g(one)g(of)h(the)e(in)o(tegers)36 1767 y(1,2,4,8)d(or)g(16)h(so)g(that)f(the)g(in)o(teger)f(2)769 1748 y Fr(n)805 1767 y Fw(+)13 b(1)19 b(is)g(a)h(prime,)d(and)i(let)f Fb(Z)-21 b(Z)1359 1774 y Fl(2)1377 1765 y Fi(n)1419 1767 y Fw(denote)19 b(the)f(ring)h(of)36 1837 y(in)o(tegers)e(mo)q(dulo)h(2)419 1819 y Fr(n)443 1837 y Fw(.)26 b(Let)18 b(\()p Fb(Z)-21 b(Z)630 1816 y Fh(\003)630 1849 y Fl(2)648 1840 y Fi(n)669 1849 y Fl(+1)716 1837 y Fu(;)8 b Fs(\001)p Fw(\))18 b(denote)g(the)g(m)o(ultipli)o(cativ)n(e)d (group)k(of)f(the)g(non-zero)36 1908 y(elemen)o(ts)c(of)j(the)g(\014eld)f Fb(Z)-21 b(Z)522 1915 y Fl(2)540 1906 y Fi(n)561 1915 y Fl(+1)608 1908 y Fw(,)17 b(let)f(\()p Fb(Z)-21 b(Z)767 1915 y Fl(2)785 1906 y Fi(n)808 1908 y Fu(;)8 b Fw(+\))17 b(denote)g(the)f(additiv)o(e)g (group)i(of)f(the)f(ring)h Fb(Z)-21 b(Z)1753 1915 y Fl(2)1771 1906 y Fi(n)1794 1908 y Fw(,)36 1978 y(and)14 b(let)f(\()215 1980 y Fm(I)-12 b(F)263 1950 y Fr(n)263 1991 y Fl(2)287 1978 y Fu(;)309 1945 y Fj(L)355 1978 y Fw(\))13 b(denote)h(the)f(group)h(of)g Fu(n)p Fw(-tuples)g(o)o(v)o(er)1100 1980 y Fm(I)-12 b(F)1148 1985 y Fl(2)1181 1978 y Fw(under)14 b(the)f(bit)o(wise)f(exclusiv)o(e-or)36 2049 y(op)q(eration.)22 b(De\014ne)16 b(the)g Fo(dir)n(e)n(ct)g Fw(mapping)f Fu(d)i Fw(from)e Fb(Z)-21 b(Z)1037 2027 y Fh(\003)1037 2061 y Fl(2)1055 2052 y Fi(n)1076 2061 y Fl(+1)1139 2049 y Fw(on)o(to)17 b Fb(Z)-21 b(Z)1288 2056 y Fl(2)1306 2047 y Fi(n)1345 2049 y Fw(as)567 2148 y Fu(d)p Fw(\()p Fu(i)p Fw(\))14 b(=)g Fu(i)i Fw(for)g Fu(i)e Fs(6)p Fw(=)g(2)927 2127 y Fr(n)967 2148 y Fw(and)j Fu(d)p Fw(\(2)1130 2127 y Fr(n)1154 2148 y Fw(\))d(=)f(0)p Fu(:)431 b Fw(\(3)p Fu(:)p Fw(1\))36 2289 y Fk(3.2.1)70 b(The)22 b(three)h(op)r(erations)g(as)g(quasigroup)i(op)r (erations)36 2392 y Fw(Let)14 b Fu(S)j Fw(b)q(e)d(a)g(non-empt)o(y)f(set)h (and)g(let)f Fs(\003)h Fw(denote)g(an)g(op)q(eration)h(from)e(pairs)h(\()p Fu(a;)8 b(b)p Fw(\))13 b(of)h(elemen)o(ts)36 2462 y(of)20 b Fu(S)j Fw(to)e(an)f(elemen)o(t)d Fu(a)c Fs(\003)h Fu(b)19 b Fw(of)i Fu(S)s Fw(.)32 b(Then)20 b(\()p Fu(S;)8 b Fs(\003)p Fw(\))20 b(is)g(said)g(to)g(b)q(e)g(a)h Fo(quasigr)n(oup)e Fw(if,)h(for)h(all)e Fu(a)36 2533 y Fw(and)h Fu(b)f Fw(in)f Fu(S)s Fw(,)i(the)f(equations)g Fu(a)13 b Fs(\003)g Fu(x)18 b Fw(=)h Fu(b)g Fw(and)g Fu(y)c Fs(\003)e Fu(a)18 b Fw(=)h Fu(b)g Fw(b)q(oth)h(ha)o(v)o(e)e(exactly)g(one)h(solution)36 2604 y(in)f Fu(S)s Fw(.)26 b(A)17 b Fo(gr)n(oup)h Fw(is)f(a)i(quasigroup)f (in)g(whic)o(h)f(the)h(op)q(eration)h(is)e(asso)q(ciativ)o(e,)h(i.e.,)e(for)i (whic)o(h)36 2674 y Fu(a)12 b Fs(\003)g Fw(\()p Fu(b)g Fs(\003)g Fu(c)p Fw(\))k(=)g(\()p Fu(a)c Fs(\003)g Fu(b)p Fw(\))g Fs(\003)g Fu(c)17 b Fw(for)h(all)g Fu(a;)8 b(b)17 b Fw(and)h Fu(c)g Fw(in)f(S.)h(The)f (quasigroups)i(\()p Fu(S)1434 2681 y Fl(1)1454 2674 y Fu(;)8 b Fs(\003)1501 2681 y Fl(1)1520 2674 y Fw(\))18 b(and)h(\()p Fu(S)1703 2681 y Fl(2)1722 2674 y Fu(;)8 b Fs(\003)1769 2681 y Fl(2)1789 2674 y Fw(\))36 2745 y(are)16 b(said)h(to)f(b)q(e)h Fo(isotopic)f Fw(if)g(there)g(are)g(bijectiv)o(e)e(mappings)h Fu(\022)q(;)8 b(\036;)g( )15 b Fw(:)f Fu(S)1389 2752 y Fl(1)1422 2745 y Fs(!)g Fu(S)1516 2752 y Fl(2)1536 2745 y Fw(,)h(suc)o(h)i(that,)390 2844 y Fu(\022)q Fw(\()p Fu(x)p Fw(\))11 b Fs(\003)516 2851 y Fl(2)547 2844 y Fu(\036)p Fw(\()p Fu(y)r Fw(\))i(=)h Fu( )r Fw(\()p Fu(x)c Fs(\003)821 2851 y Fl(1)852 2844 y Fu(y)r Fw(\))113 b(for)17 b(all)e Fu(x)h Fw(and)h Fu(y)h Fw(in)e Fu(S)1420 2851 y Fl(1)1440 2844 y Fu(:)p eop %%Page: 26 6 bop 36 103 a Fw(26)900 b Fq(CH.)16 b(3.)38 b(THE)16 b(BLOCK)g(CIPHER)g(IDEA) 36 275 y Fw(Suc)o(h)k(a)g(triple)f(\()p Fu(\022)q(;)8 b(\036;)g( )r Fw(\))19 b(of)i(bijections)e(is)h(called)f(an)h Fo(isotopism)g Fw(of)g(\()p Fu(S)1401 282 y Fl(1)1421 275 y Fu(;)8 b Fs(\003)1468 282 y Fl(1)1487 275 y Fw(\))20 b(on)o(to)h(\()p Fu(S)1689 282 y Fl(2)1709 275 y Fu(;)8 b Fs(\003)1756 282 y Fl(2)1775 275 y Fw(\).)36 345 y(Tw)o(o)20 b(groups)g(are)g(said)f(to)h(b)q(e)g Fo(isomorphic)e Fw(if)h(they)g(are)g(isotopic)g(as)h(quasigroups)h(and)f(the) 36 416 y(isotopism)f(has)i(the)e(form)g(\()p Fu(\022)q(;)8 b(\022)q(;)g(\022)q Fw(\).)32 b(It)20 b(can)g(b)q(e)g(sho)o(wn)h(that)f(t)o (w)o(o)g(groups)h(are)f(isomorphic)36 487 y(if)i(and)i(only)e(if)g(they)h (are)f(isotopic)h([18].)41 b(Note)22 b(that)h(ev)o(ery)e(isomorphism)g(b)q (et)o(w)o(een)h(t)o(w)o(o)36 557 y(groups)d(is)f(also)h(an)f(isotopism,)g (but)g(the)g(con)o(v)o(erse)e(is)i(not)h(true)f(in)f(general.)27 b(In)18 b(general)f(for)36 628 y(t)o(w)o(o)i(isomorphic)f(groups,)i(there)f (will)f(b)q(e)h(man)o(y)f(more)g(isotopisms)g(b)q(et)o(w)o(een)h(these)g (groups)36 698 y(than)j(there)g(will)f(b)q(e)h(isomorphisms.)36 b(F)l(or)22 b(this)g(reason,)h(w)o(e)f(consider)f(isotopisms)h(rather)36 769 y(than)i(isomorphisms)e(although)j(our)g(ob)s(jects)e(are)h(all)f (groups.)46 b(The)24 b(follo)o(wing)f(theorem)36 839 y(states)16 b(some)e(\\incompatibilit)o(y")f(prop)q(erties)j(of)f(the)h(three)f(groups)h (\()1348 841 y Fm(I)-12 b(F)1396 811 y Fr(n)1396 852 y Fl(2)1420 839 y Fu(;)1442 806 y Fj(L)1488 839 y Fw(\),)15 b(\()p Fb(Z)-21 b(Z)1594 846 y Fl(2)1612 837 y Fi(n)1635 839 y Fu(;)8 b Fw(+\))15 b(and)36 910 y(\()p Fb(Z)-21 b(Z)94 889 y Fh(\003)94 922 y Fl(2)112 913 y Fi(n)133 922 y Fl(+1)180 910 y Fu(;)8 b Fs(\001)p Fw(\))16 b(when)g Fu(n)e Fs(\025)g Fw(2.)36 1035 y Ft(Theorem)j(2)73 b Fq(F)l(or)16 b Fu(n)e Fs(2)g(f)p Fw(1)p Fu(;)8 b Fw(2)p Fu(;)g Fw(4)p Fu(;)g Fw(8)p Fu(;)g Fw(16)p Fs(g)p Fq(:)36 1106 y(1\))17 b(The)f(quasigroups)h Fw(\()482 1108 y Fm(I)-12 b(F)530 1078 y Fr(n)530 1118 y Fl(2)554 1106 y Fu(;)576 1073 y Fj(L)622 1106 y Fw(\))16 b Fq(and)h Fw(\()p Fb(Z)-21 b(Z)810 1113 y Fl(2)828 1104 y Fi(n)851 1106 y Fu(;)8 b Fw(+\))16 b Fq(are)g(not)h(isotopic) f(for)g Fu(n)e Fs(\025)g Fw(2)p Fq(.)36 1177 y(2\))j(The)f(quasigroups)h Fw(\()482 1179 y Fm(I)-12 b(F)530 1148 y Fr(n)530 1189 y Fl(2)554 1177 y Fu(;)576 1143 y Fj(L)622 1177 y Fw(\))16 b Fq(and)h Fw(\()p Fb(Z)-21 b(Z)810 1155 y Fh(\003)810 1189 y Fl(2)828 1179 y Fi(n)849 1189 y Fl(+1)896 1177 y Fu(;)8 b Fs(\001)p Fw(\))16 b Fq(are)g(not)h(isotopic)f(for)g Fu(n)e Fs(\025)g Fw(2)p Fq(.)36 1247 y(3\))k(The)g(triple)f Fw(\()p Fu(\022)q(;)8 b(\036;)g( )r Fw(\))17 b Fq(of)h(bijections)f(from)g Fb(Z)-21 b(Z)950 1226 y Fh(\003)950 1259 y Fl(2)968 1250 y Fi(n)989 1259 y Fl(+1)1054 1247 y Fq(to)18 b Fb(Z)-21 b(Z)1154 1254 y Fl(2)1172 1245 y Fi(n)1213 1247 y Fq(is)18 b(an)g(isotopism)f(of)h Fw(\()p Fb(Z)-21 b(Z)1667 1226 y Fh(\003)1667 1259 y Fl(2)1685 1250 y Fi(n)1706 1259 y Fl(+1)1753 1247 y Fu(;)8 b Fs(\001)p Fw(\))36 1318 y Fq(on)o(to)20 b Fw(\()p Fb(Z)-21 b(Z)206 1325 y Fl(2)224 1316 y Fi(n)247 1318 y Fu(;)8 b Fw(+\))19 b Fq(if)g(and)h(only)f (if)f(there)h(exist)f Fu(c)915 1325 y Fl(1)954 1318 y Fq(and)i Fu(c)1073 1325 y Fl(2)1111 1318 y Fq(in)f Fb(Z)-21 b(Z)1210 1325 y Fl(2)1228 1316 y Fi(n)1270 1318 y Fq(and)20 b(a)f(generator)h Fu(\013)g Fq(of)f(the)36 1388 y(cyclic)14 b(group)j Fb(Z)-21 b(Z)347 1367 y Fh(\003)347 1401 y Fl(2)365 1391 y Fi(n)386 1401 y Fl(+1)450 1388 y Fq(suc)o(h)16 b(that,)g(for)g(all)g Fu(x)g Fq(in)g Fb(Z)-21 b(Z)961 1367 y Fh(\003)961 1401 y Fl(2)979 1391 y Fi(n)1000 1401 y Fl(+1)1047 1388 y Fq(,)372 1509 y Fu(\022)q Fw(\()p Fu(x)p Fw(\))11 b Fs(\000)g Fu(c)544 1516 y Fl(1)578 1509 y Fw(=)i Fu(\036)p Fw(\()p Fu(x)p Fw(\))e Fs(\000)g Fu(c)806 1516 y Fl(2)839 1509 y Fw(=)j Fu( )r Fw(\()p Fu(x)p Fw(\))c Fs(\000)h Fw(\()p Fu(c)1091 1516 y Fl(1)1122 1509 y Fw(+)g Fu(c)1192 1516 y Fl(2)1212 1509 y Fw(\))i(=)h(log)1359 1521 y Fr(\013)1384 1509 y Fw(\()p Fu(x)p Fw(\))p Fu(;)243 b Fw(\(3)p Fu(:)p Fw(2\))36 1630 y Fq(i.e.,)13 b(an)o(y)i(isotopism)f(b)q(et)o(w)o(een)g (these)g(groups)j(is)d(essen)o(tially)g(the)g(discrete)g(logarithm.)20 b(More-)36 1700 y(o)o(v)o(er,)c(when)h Fu(n)e Fs(\025)g Fw(2)p Fq(,)i(none)g(of)g(the)g(three)f(bijections)g(in)h(an)g(isotopism)f Fw(\()p Fu(\022)q(;)8 b(\036;)g( )r Fw(\))16 b Fq(from)g Fb(Z)-21 b(Z)1722 1679 y Fh(\003)1722 1713 y Fl(2)1740 1703 y Fi(n)1761 1713 y Fl(+1)36 1771 y Fq(on)o(to)17 b Fb(Z)-21 b(Z)185 1778 y Fl(2)203 1769 y Fi(n)242 1771 y Fq(can)16 b(b)q(e)h(the)f(direct)f(mapping) g Fu(d)i Fq(de\014ned)f(in)g(\(3.1\).)36 1896 y Ft(Pro)r(of.)36 1967 y Fw(1\))f(F)l(or)f Fu(n)g Fs(\025)g Fw(2,)h(the)f(groups)h(\()584 1969 y Fm(I)-12 b(F)632 1938 y Fr(n)632 1979 y Fl(2)656 1967 y Fu(;)678 1934 y Fj(L)724 1967 y Fw(\))14 b(and)h(\()p Fb(Z)-21 b(Z)908 1974 y Fl(2)926 1965 y Fi(n)949 1967 y Fu(;)8 b Fw(+\))14 b(are)h(not)g(isomorphic)e(b)q(ecause)h(\()p Fb(Z)-21 b(Z)1688 1974 y Fl(2)1706 1965 y Fi(n)1729 1967 y Fu(;)8 b Fw(+\))36 2037 y(is)16 b(a)h(cyclic)d(group)j(while)e(\()544 2039 y Fm(I)-12 b(F)593 2009 y Fr(n)593 2050 y Fl(2)616 2037 y Fu(;)638 2004 y Fj(L)684 2037 y Fw(\))16 b(is)g(not.)22 b(Th)o(us,)16 b(they)g(are)g(not)h (isotopic)f(as)h(quasigroups.)36 2108 y(2\))h(\()p Fb(Z)-21 b(Z)155 2087 y Fh(\003)155 2120 y Fl(2)173 2111 y Fi(n)194 2120 y Fl(+1)241 2108 y Fu(;)8 b Fs(\001)p Fw(\))18 b(and)h(\()p Fb(Z)-21 b(Z)469 2115 y Fl(2)487 2106 y Fi(n)510 2108 y Fu(;)8 b Fw(+\))17 b(are)h(isomorphic)f(groups)i(for)g Fu(n)e Fw(=)g(1)p Fu(;)8 b Fw(2)p Fu(;)g Fw(4)p Fu(;)g Fw(8)p Fu(;)g Fw(16)19 b(b)q(ecause)g(b)q(oth)36 2179 y(groups)g(are)f(cyclic.)25 b(Th)o(us,)18 b(\()p Fb(Z)-21 b(Z)634 2157 y Fh(\003)634 2191 y Fl(2)652 2181 y Fi(n)673 2191 y Fl(+1)720 2179 y Fu(;)8 b Fs(\001)p Fw(\))17 b(is)h(isotopic)g(to)g(\()1104 2181 y Fm(I)-12 b(F)1153 2150 y Fr(n)1153 2191 y Fl(2)1176 2179 y Fu(;)1198 2145 y Fj(L)1244 2179 y Fw(\))18 b(if)f(and)i(only)f(if)f(\()p Fb(Z)-21 b(Z)1637 2186 y Fl(2)1655 2177 y Fi(n)1678 2179 y Fu(;)8 b Fw(+\))18 b(is)36 2249 y(isotopic)e(to)h(\()294 2251 y Fm(I)-12 b(F)342 2221 y Fr(n)342 2261 y Fl(2)365 2249 y Fu(;)387 2216 y Fj(L)433 2249 y Fw(\),)16 b(whic)o(h)g(is)g(not)g(the)g(case)h(for)f Fu(n)e Fs(\025)g Fw(2.)36 2320 y(3\))j(Supp)q(ose)g(that)g(\()p Fu(\022)q(;)8 b(\036;)g( )r Fw(\))15 b(satis\014es)i(\(3.2\))g(for)f(all)g Fu(x)g Fw(in)g Fb(Z)-21 b(Z)1156 2298 y Fh(\003)1156 2332 y Fl(2)1174 2323 y Fi(n)1195 2332 y Fl(+1)1242 2320 y Fw(,)16 b(then)g(for)g(ev)o(ery)f Fu(x)h Fw(and)h Fu(y)h Fw(in)36 2390 y Fb(Z)-21 b(Z)75 2369 y Fh(\003)75 2403 y Fl(2)93 2393 y Fi(n)114 2403 y Fl(+1)161 2390 y Fw(,)124 2511 y Fu( )r Fw(\()p Fu(x)10 b Fs(\001)h Fu(y)r Fw(\))i(=)h(log)413 2523 y Fr(\013)438 2511 y Fw(\()p Fu(x)d Fs(\001)g Fu(y)r Fw(\))f(+)h Fu(c)646 2518 y Fl(1)677 2511 y Fw(+)g Fu(c)747 2518 y Fl(2)780 2511 y Fw(=)j(log)895 2523 y Fr(\013)920 2511 y Fw(\()p Fu(x)p Fw(\))d(+)g(log)1109 2523 y Fr(\013)1133 2511 y Fw(\()p Fu(y)r Fw(\))g(+)g Fu(c)1278 2518 y Fl(1)1309 2511 y Fw(+)g Fu(c)1379 2518 y Fl(2)1412 2511 y Fw(=)j Fu(\022)q Fw(\()p Fu(x)p Fw(\))d(+)g Fu(\036)p Fw(\()p Fu(y)r Fw(\))p Fu(:)36 2632 y Fw(Th)o(us,)16 b(\()p Fu(\022)q(;)8 b(\036;)g( )r Fw(\))15 b(is)i(indeed)e(an)i(isotopism.)109 2702 y(Con)o(v)o(ersely)l(,)d(if)h(\()p Fu(\022)q(;)8 b(\036;)g( )r Fw(\))15 b(is)h(an)g(isotopism)f(from)g(\()p Fb(Z)-21 b(Z)1099 2681 y Fh(\003)1099 2715 y Fl(2)1117 2705 y Fi(n)1138 2715 y Fl(+1)1185 2702 y Fu(;)8 b Fs(\001)p Fw(\))15 b(on)o(to)h(\()p Fb(Z)-21 b(Z)1422 2709 y Fl(2)1440 2700 y Fi(n)1463 2702 y Fu(;)8 b Fw(+\),)15 b(then)h(for)g(all)36 2773 y Fu(x)d Fw(and)g Fu(y)i Fw(in)d Fb(Z)-21 b(Z)299 2752 y Fh(\003)299 2785 y Fl(2)317 2776 y Fi(n)338 2785 y Fl(+1)385 2773 y Fw(,)13 b Fu(\022)q Fw(\()p Fu(x)p Fw(\))t(+)t Fu(\036)p Fw(\()p Fu(y)r Fw(\))h(=)f Fu( )r Fw(\()p Fu(x)t Fs(\001)t Fu(y)r Fw(\).)19 b(Let)13 b Fu(\022)994 2780 y Fl(1)1014 2773 y Fw(\()p Fu(x)p Fw(\))g(=)h Fu(\022)q Fw(\()p Fu(x)p Fw(\))t Fs(\000)t Fu(\022)q Fw(\(1\))p Fu(;)f(\036)1424 2780 y Fl(1)1444 2773 y Fw(\()p Fu(x)p Fw(\))g(=)h Fu(\036)p Fw(\()p Fu(x)p Fw(\))t Fs(\000)t Fu(\036)p Fw(\(1\))36 2844 y(and)j Fu( )163 2851 y Fl(1)182 2844 y Fw(\()p Fu(x)p Fw(\))d(=)g Fu( )r Fw(\()p Fu(x)p Fw(\))c Fs(\000)h Fu( )r Fw(\(1\),)16 b(then)g(\()p Fu(\022)753 2851 y Fl(1)773 2844 y Fu(;)8 b(\036)824 2851 y Fl(1)843 2844 y Fu(;)g( )897 2851 y Fl(1)916 2844 y Fw(\))17 b(is)f(also)h(an)g(isotopism)e(from)g(\()p Fb(Z)-21 b(Z)1557 2822 y Fh(\003)1557 2856 y Fl(2)1575 2846 y Fi(n)1596 2856 y Fl(+1)1643 2844 y Fu(;)8 b Fs(\001)p Fw(\))16 b(on)o(to)p eop %%Page: 27 7 bop 36 103 a Fq(3.2.)38 b(GR)o(OUP)16 b(OPERA)l(TIONS)f(AND)h(THEIR)f(INTERA) o(CTION)410 b Fw(27)36 275 y(\()p Fb(Z)-21 b(Z)94 282 y Fl(2)112 273 y Fi(n)135 275 y Fu(;)8 b Fw(+\))13 b(as)h(is)f(easily)f(c)o(hec)o(k)o (ed.)18 b(Moreo)o(v)o(er,)12 b Fu(\022)906 282 y Fl(1)925 275 y Fw(\(1\))j(=)e Fu(\036)1082 282 y Fl(1)1102 275 y Fw(\(1\))h(=)g Fu( )1262 282 y Fl(1)1281 275 y Fw(\(1\))g(=)g(0.)21 b(In)13 b(the)g(isotopism)36 345 y(equation)649 419 y Fu(\022)672 426 y Fl(1)691 419 y Fw(\()p Fu(x)p Fw(\))e(+)g Fu(\036)846 426 y Fl(1)865 419 y Fw(\()p Fu(y)r Fw(\))j(=)g Fu( )1027 426 y Fl(1)1046 419 y Fw(\()p Fu(x)d Fs(\001)g Fu(y)r Fw(\))p Fu(;)519 b Fw(\(3)p Fu(:)p Fw(3\))36 520 y(setting)19 b Fu(x)g Fw(to)g(1)h(results)f (in)g Fu(\036)599 527 y Fl(1)618 520 y Fw(\()p Fu(y)r Fw(\))f(=)h Fu( )789 527 y Fl(1)809 520 y Fw(\()p Fu(y)r Fw(\))f(for)i(all)e Fu(y)j Fw(in)e Fb(Z)-21 b(Z)1183 499 y Fh(\003)1183 533 y Fl(2)1201 523 y Fi(n)1221 533 y Fl(+1)1269 520 y Fu(;)18 b Fw(and)i(then)f(setting)g Fu(y)i Fw(to)e(1)36 591 y(in)d(\(3.3\))g(results)f(in)h Fu(\022)443 598 y Fl(1)462 591 y Fw(\()p Fu(x)p Fw(\))e(=)g Fu( )626 598 y Fl(1)645 591 y Fw(\()p Fu(x)p Fw(\))i(for)g(all)f Fu(x)h Fw(in)f Fb(Z)-21 b(Z)1007 569 y Fh(\003)1007 603 y Fl(2)1025 594 y Fi(n)1046 603 y Fl(+1)1093 591 y Fu(:)16 b Fw(Th)o(us,)g(the)f(three)h (mappings)f Fu(\022)1710 598 y Fl(1)1730 591 y Fw(,)g Fu(\036)1788 598 y Fl(1)36 661 y Fw(and)i Fu( )163 668 y Fl(1)199 661 y Fw(are)f(iden)o(tical.)j(Equation)e(\(3.3\))f(can)h(th)o(us)f(b)q(e)h (written)e(as)647 785 y Fu( )679 792 y Fl(1)698 785 y Fw(\()p Fu(x)c Fs(\001)g Fu(y)r Fw(\))i(=)h Fu( )923 792 y Fl(1)943 785 y Fw(\()p Fu(x)p Fw(\))c(+)h Fu( )1100 792 y Fl(1)1120 785 y Fw(\()p Fu(y)r Fw(\))p Fu(:)509 b Fw(\(3)p Fu(:)p Fw(4\))36 908 y(Let)12 b Fu(\013)g Fw(b)q(e)g(the)f(elemen)o(t)e(of)j Fb(Z)-21 b(Z)569 887 y Fh(\003)569 921 y Fl(2)587 911 y Fi(n)608 921 y Fl(+1)666 908 y Fw(suc)o(h)12 b(that)g Fu( )905 915 y Fl(1)925 908 y Fw(\()p Fu(\013)p Fw(\))i(=)f(1,)g(then)f(\(3.4\))f(implies)f (that)i Fu( )1623 915 y Fl(1)1642 908 y Fw(\()p Fu(\013)1692 890 y Fr(i)1706 908 y Fw(\))i(=)g Fu(i)36 979 y Fw(for)h Fu(i)f Fw(=)f(1)p Fu(;)8 b Fw(2)p Fu(;)g(::)p Fw(2)335 961 y Fr(n)367 979 y Fs(\000)g Fw(1)15 b(and)g Fu( )578 986 y Fl(1)598 979 y Fw(\()p Fu(\013)648 961 y Fl(2)666 949 y Fi(n)689 979 y Fw(\))f(=)g(0)p Fu(:)g Fw(This)h(implies)e(that)i Fu(\013)g Fw(is)g(a)g(generator)g(of)g(the) g(cyclic)36 1050 y(group)i(\()p Fb(Z)-21 b(Z)232 1028 y Fh(\003)232 1062 y Fl(2)250 1052 y Fi(n)271 1062 y Fl(+1)318 1050 y Fu(;)8 b Fs(\001)p Fw(\))16 b(and)h(that)g Fu( )622 1057 y Fl(1)642 1050 y Fw(\()p Fu(x)p Fw(\))c(=)h(log)836 1061 y Fr(\013)861 1050 y Fw(\()p Fu(x)p Fw(\))i(for)h(all)f Fu(x)g Fw(in)g Fb(Z)-21 b(Z)1226 1028 y Fh(\003)1226 1062 y Fl(2)1244 1052 y Fi(n)1265 1062 y Fl(+1)1312 1050 y Fw(.)22 b(Letting)16 b Fu(c)1540 1057 y Fl(1)1574 1050 y Fw(=)e Fu(\022)q Fw(\(1\))j(and)36 1120 y Fu(c)57 1127 y Fl(2)91 1120 y Fw(=)c Fu(\036)p Fw(\(1\),)j(w)o(e)g(arriv)o (e)f(at)i(\(3.2\).)109 1192 y(Finally)l(,)f(if)i(\()p Fu(\022)q(;)8 b(\036;)g( )r Fw(\))16 b(is)i(an)g(isotopism)f(from)f(\()p Fb(Z)-21 b(Z)1031 1170 y Fh(\003)1031 1204 y Fl(2)1049 1195 y Fi(n)1070 1204 y Fl(+1)1117 1192 y Fu(;)8 b Fs(\001)p Fw(\))17 b(on)o(to)i(\()p Fb(Z)-21 b(Z)1358 1199 y Fl(2)1376 1190 y Fi(n)1399 1192 y Fu(;)8 b Fw(+\))18 b(and)g(one)g(of)g(the)36 1262 y(mappings)h Fu(\022)q Fw(,)h Fu(\036)f Fw(and)h Fu( )h Fw(is)e(the)g(direct)f(mapping)h Fu(d)p Fw(,)h(then)f(there)f(exist)h Fu(c)g Fw(in)g Fb(Z)-21 b(Z)1558 1269 y Fl(2)1576 1260 y Fi(n)1618 1262 y Fw(and)20 b Fu(\013)g Fw(in)36 1333 y Fb(Z)-21 b(Z)75 1311 y Fh(\003)75 1345 y Fl(2)93 1336 y Fi(n)114 1345 y Fl(+1)177 1333 y Fw(suc)o(h)16 b(that)527 1407 y Fu(d)p Fw(\()p Fu(x)p Fw(\))e(=)g(log)747 1418 y Fr(\013)772 1407 y Fw(\()p Fu(x)p Fw(\))c(+)h Fu(c)17 b Fw(for)f(all)g Fu(x)g Fw(in)g Fb(Z)-21 b(Z)1217 1385 y Fh(\003)1217 1419 y Fl(2)1235 1409 y Fi(n)1256 1419 y Fl(+1)1303 1407 y Fu(:)390 b Fw(\(3)p Fu(:)p Fw(5\))36 1508 y(But)20 b(then)g Fu(d)p Fw(\(1\))i(=)f(1)f(implies)e(that)j(1)g(=)g (log)911 1519 y Fr(\013)936 1508 y Fw(\(1\))14 b(+)g Fu(c)21 b Fw(=)g Fu(c)f Fw(so)h(that)g Fu(d)p Fw(\()p Fu(x)p Fw(\))g(=)g(log)1614 1519 y Fr(\013)1638 1508 y Fw(\()p Fu(x)p Fw(\))14 b(+)g(1)p Fu(:)36 1578 y Fw(Moreo)o(v)o(er,)k(for)h Fu(n)f Fs(\025)g Fw(2,)h Fu(d)p Fw(\(2\))g(=)f(2)p Fu(;)h Fw(whic)o(h)f(implies)e(that)j(2)g (=)f(log)1304 1590 y Fr(a)1325 1578 y Fw(\(2\))13 b(+)f(1)20 b(so)f(that)g Fu(\013)g Fw(=)f(2.)36 1649 y(But)f(then)g Fu(d)p Fw(\(2)314 1631 y Fr(n)339 1649 y Fw(\))f(=)f(0)j(implies)d(that)j(log)806 1661 y Fl(2)826 1649 y Fw(\(2)869 1631 y Fr(n)893 1649 y Fw(\))11 b(+)h(1)17 b(=)e Fu(n)d Fw(+)g(1)17 b(=)e(0)j(whic)o(h)f(is)g(a)h(con)o (tradiction)36 1719 y(b)q(ecause)g Fu(n)e(<)h Fw(2)342 1701 y Fr(n)378 1719 y Fs(\000)11 b Fw(1)18 b(for)g Fu(n)f Fs(\025)f Fw(2.)26 b(Th)o(us,)18 b(none)g(of)g(the)f(mappings)g Fu(\022)q Fw(,)h Fu(\036)g Fw(and)g Fu( )h Fw(can)f(b)q(e)g(the)36 1790 y(direct)d(mapping)h Fu(d)g Fw(if)g Fu(n)e Fs(\025)g Fw(2.)1175 b Fc(2)36 1941 y Fk(3.2.2)70 b(P)n(olynomial)23 b(expressions)i(for)g(m)n (ultipli)o(cati)o(on)d(and)k(addi-)246 2015 y(tion)36 2120 y Fw(In)21 b(the)g(encryption)f(pro)q(cess)i(of)f(the)g(cipher)f(IDEA,)g(m)o (ultiplic)o(ation)f(mo)q(dulo)h(2)1593 2102 y Fr(n)1631 2120 y Fw(+)15 b(1)21 b(and)36 2191 y(addition)c(mo)q(dulo)f(2)428 2173 y Fr(n)469 2191 y Fw(are)h(related)f(via)h(the)g(direct)f(mapping)g Fu(d)h Fw(and)g(its)g(in)o(v)o(erse)e Fu(d)1613 2173 y Fh(\000)p Fl(1)1661 2191 y Fw(.)23 b(More)36 2261 y(precisely)l(,)16 b(m)o(ultiplication)f(mo)q(dulo)j(2)763 2243 y Fr(n)799 2261 y Fw(+)13 b(1)18 b(induces)g(the)g(function)g Fu(g)i Fw(:)c Fb(Z)-21 b(Z)1459 2268 y Fl(2)1477 2259 y Fi(n)1513 2261 y Fs(\002)12 b Fb(Z)-21 b(Z)1603 2268 y Fl(2)1621 2259 y Fi(n)1661 2261 y Fs(!)17 b Fb(Z)-21 b(Z)1767 2268 y Fl(2)1785 2259 y Fi(n)36 2332 y Fw(de\014ned)16 b(b)o(y)152 2456 y Fu(g)r Fw(\()p Fu(x;)8 b(y)r Fw(\))13 b(=)h Fu(d)p Fw([\()p Fu(d)439 2435 y Fh(\000)p Fl(1)486 2456 y Fw(\()p Fu(x)p Fw(\))d Fs(\001)g Fu(d)613 2435 y Fh(\000)p Fl(1)661 2456 y Fw(\()p Fu(y)r Fw(\)\))k(mo)q(d)h (\(2)912 2435 y Fr(n)947 2456 y Fw(+)11 b(1\)])65 b(for)16 b(all)g Fu(x)g Fw(and)h Fu(y)h Fw(in)e Fb(Z)-21 b(Z)1537 2463 y Fl(2)1555 2454 y Fi(n)1578 2456 y Fu(:)115 b Fw(\(3)p Fu(:)p Fw(6\))36 2579 y(Note)14 b(that)g Fu(g)r Fw(\()p Fu(x;)8 b(y)r Fw(\))13 b(is)h(the)f(function)h(that)g(w)o(e)g(denoted)g(as)g Fu(x)1164 2546 y Fj(J)1210 2579 y Fu(y)i Fw(in)d(Section)g(3.1.1.)21 b(Similarly)l(,)36 2650 y(addition)11 b(mo)q(dulo)g(2)417 2631 y Fr(n)452 2650 y Fw(\(the)g(op)q(eration)p 766 2620 37 2 v 766 2653 2 33 v 766 2649 a(+)p 800 2653 V 766 2654 37 2 v 805 2650 a(\))g(induces)f(a)i(function)f Fu(f)1254 2631 y Fh(\003)1288 2650 y Fw(:)i Fb(Z)-21 b(Z)1354 2628 y Fh(\003)1354 2662 y Fl(2)1372 2652 y Fi(n)1393 2662 y Fl(+1)1441 2650 y Fs(\002)q Fb(Z)g(Z)1519 2628 y Fh(\003)1519 2662 y Fl(2)1537 2652 y Fi(n)1558 2662 y Fl(+1)1619 2650 y Fs(!)14 b Fb(Z)-21 b(Z)1722 2628 y Fh(\003)1722 2662 y Fl(2)1740 2652 y Fi(n)1761 2662 y Fl(+1)36 2720 y Fw(de\014ned)16 b(as)241 2844 y Fu(f)270 2823 y Fh(\003)290 2844 y Fw(\()p Fu(x;)8 b(y)r Fw(\))13 b(=)g Fu(d)493 2823 y Fh(\000)p Fl(1)541 2844 y Fw([\()p Fu(d)p Fw(\()p Fu(x)p Fw(\))e(+)g Fu(d)p Fw(\()p Fu(y)r Fw(\)\))16 b(mo)q(d)f(2)982 2823 y Fr(n)1006 2844 y Fw(])65 b(for)16 b(all)g Fu(x)g Fw(and)h Fu(y)h Fw(in)e Fb(Z)-21 b(Z)1503 2822 y Fh(\003)1503 2856 y Fl(2)1521 2846 y Fi(n)1542 2856 y Fl(+1)1589 2844 y Fu(:)104 b Fw(\(3)p Fu(:)p Fw(7\))p eop %%Page: 28 8 bop 36 103 a Fw(28)900 b Fq(CH.)16 b(3.)38 b(THE)16 b(BLOCK)g(CIPHER)g(IDEA) 36 275 y Fw(W)l(e)g(can)h(and)g(do)f(extend)g(the)g(function)g Fu(f)835 257 y Fh(\003)872 275 y Fw(to)g(a)h(function)f Fu(f)k Fw(:)13 b Fb(Z)-21 b(Z)1273 282 y Fl(2)1291 273 y Fi(n)1312 282 y Fl(+1)1370 275 y Fs(\002)11 b Fb(Z)-21 b(Z)1459 282 y Fl(2)1477 273 y Fi(n)1498 282 y Fl(+1)1559 275 y Fs(!)14 b Fb(Z)-21 b(Z)1662 282 y Fl(2)1680 273 y Fi(n)1701 282 y Fl(+1)1764 275 y Fw(as)36 345 y(follo)o(ws:)167 478 y Fu(f)5 b Fw(\()p Fu(x;)j(y)r Fw(\))13 b(=)375 405 y Fj(\()429 447 y Fu(d)454 429 y Fh(\000)p Fl(1)501 447 y Fw([\()p Fu(d)p Fw(\()p Fu(x)p Fw(\))e(+)g Fu(d)p Fw(\()p Fu(y)r Fw(\)\))16 b(mo)q(d)g(2)943 429 y Fr(n)967 447 y Fw(])57 b(for)17 b(all)e Fu(x)h Fw(and)h Fu(y)h Fw(in)e Fb(Z)-21 b(Z)1457 426 y Fh(\003)1457 459 y Fl(2)1475 450 y Fi(n)1496 459 y Fl(+1)429 507 y Fw(0)585 b(otherwise.)1707 478 y(\(3)p Fu(:)p Fw(8\))36 620 y(F)l(or)16 b(example,)e(when)i Fu(n)e Fw(=)g(1,)i(the)g(function)g Fu(f)22 b Fw(induced)15 b(b)o(y)h(addition)h(mo)q(dulo)e(2)i(is)445 743 y Fu(f)5 b Fw(\()p Fu(x;)j(y)r Fw(\))14 b(=)f(2)p Fu(xy)18 b Fw(mo)q(d)e(3)65 b(for)17 b(all)f Fu(x)g Fw(and)h Fu(y)g Fw(in)f Fb(Z)-21 b(Z)1365 750 y Fl(3)1385 743 y Fu(:)36 867 y Fw(Similarly)l(,)13 b(the)j(function)g Fu(g)i Fw(induced)e(b)o(y)g(m)o(ultipli)o(cation)e(mo)q(dulo)h(3)i(is)387 991 y Fu(g)r Fw(\()p Fu(x;)8 b(y)r Fw(\))13 b(=)h Fu(x)d Fw(+)g Fu(y)i Fw(+)e(1)16 b(mo)q(d)g(2)65 b(for)17 b(all)f Fu(x)g Fw(and)g Fu(y)i Fw(in)e Fb(Z)-21 b(Z)1423 998 y Fl(2)1443 991 y Fu(:)36 1115 y Fw(In)19 b(what)g(follo)o(ws)g(in)g(this)g(section,)g(w)o(e) f(sho)o(w)i(the)f(\\nonlinearit)o(y")f(of)i(the)e(function)h Fu(f)25 b Fw(o)o(v)o(er)36 1186 y(the)17 b(\014eld)f Fb(Z)-21 b(Z)266 1193 y Fl(2)284 1184 y Fi(n)305 1193 y Fl(+1)369 1186 y Fw(and)17 b(the)g(\\nonlinearit)o(y")f(of)h(the)g(function)g Fu(g)i Fw(o)o(v)o(er)c(the)i(ring)g Fb(Z)-21 b(Z)1573 1193 y Fl(2)1591 1184 y Fi(n)1631 1186 y Fw(in)16 b(terms)36 1256 y(of)h(their)e(p)q(olynomial)g(expressions)h(when)h Fu(n)d Fs(\025)f Fw(2.)36 1390 y Ft(Theorem)k(3)73 b Fq(F)l(or)16 b Fu(n)e Fs(2)g(f)p Fw(2)p Fu(;)8 b Fw(4)p Fu(;)g Fw(8)p Fu(;)g Fw(16)p Fs(g)18 b Fq(:)36 1461 y(F)l(or)i(ev)o(ery)e Fu(a)h Fq(in)h Fb(Z)-21 b(Z)404 1468 y Fl(2)422 1459 y Fi(n)442 1468 y Fl(+1)503 1461 y Fs(\000)13 b(f)p Fw(0)p Fu(;)8 b Fw(2)650 1443 y Fr(n)674 1461 y Fs(g)p Fu(;)19 b Fq(the)h(function)f Fu(f)5 b Fw(\()p Fu(a;)j(y)r Fw(\))19 b Fq(is)h(a)g(p)q(olynomial)e(in)i Fu(y)h Fq(o)o(v)o(er)e(the)36 1531 y(\014eld)14 b Fb(Z)-21 b(Z)179 1538 y Fl(2)197 1529 y Fi(n)218 1538 y Fl(+1)279 1531 y Fq(with)14 b(degree)g Fw(2)562 1513 y Fr(n)593 1531 y Fs(\000)7 b Fw(1)p Fu(:)14 b Fq(Similarly)l(,)e(for)i(ev)o(ery)f Fu(a)h Fq(in)g Fb(Z)-21 b(Z)1240 1538 y Fl(2)1258 1529 y Fi(n)1279 1538 y Fl(+1)1333 1531 y Fs(\000)7 b(f)p Fw(0)p Fu(;)h Fw(2)1474 1513 y Fr(n)1498 1531 y Fs(g)p Fu(;)14 b Fq(the)g(function)36 1602 y Fu(f)5 b Fw(\()p Fu(x;)j(a)p Fw(\))16 b Fq(is)g(a)g(p)q(olynomial)f (in)h Fu(x)g Fq(o)o(v)o(er)g Fb(Z)-21 b(Z)781 1609 y Fl(2)799 1600 y Fi(n)819 1609 y Fl(+1)883 1602 y Fq(with)16 b(degree)g Fw(2)1170 1584 y Fr(n)1205 1602 y Fs(\000)10 b Fw(1)p Fq(.)36 1735 y Ft(Example)35 b(3)24 b Fq(F)l(or)18 b Fu(n)f Fw(=)f(2)p Fq(,)i(the)f(function)h Fu(f)5 b Fw(\()p Fu(x;)j(y)r Fw(\))17 b Fq(o)o(v)o(er)g Fb(Z)-21 b(Z)1166 1742 y Fl(5)1203 1735 y Fq(induced)17 b(b)o(y)g(addition)h(mo)q(dulo)36 1805 y(4)f(is)240 1879 y Fu(f)5 b Fw(\()p Fu(x;)j(y)r Fw(\))13 b(=)h(3\()p Fu(x)519 1859 y Fl(3)539 1879 y Fu(y)565 1859 y Fl(2)595 1879 y Fw(+)d Fu(x)672 1859 y Fl(2)692 1879 y Fu(y)718 1859 y Fl(3)737 1879 y Fw(\))g(+)g(3\()p Fu(x)887 1859 y Fl(3)907 1879 y Fu(y)h Fw(+)f Fu(xy)1046 1859 y Fl(3)1065 1879 y Fw(\))g(+)h(2)p Fu(x)1197 1859 y Fl(2)1216 1879 y Fu(y)1242 1859 y Fl(2)1273 1879 y Fw(+)f(4\()p Fu(x)1393 1859 y Fl(2)1412 1879 y Fu(y)i Fw(+)e Fu(xy)1552 1859 y Fl(2)1571 1879 y Fw(\))p Fu(:)36 2012 y Ft(Pro)r(of)21 b(of)h(Theorem)d(3.)28 b Fw(F)l(or)19 b(an)o(y)f(\014nite)g(\014eld)991 2014 y Fm(I)-12 b(F)1057 2012 y Fw(=)18 b Fu(GF)7 b Fw(\()p Fu(q)r Fw(\))18 b(and)h(for)g(ev)o(ery)e Fu(\013)i Fw(in)1684 2014 y Fm(I)-12 b(F)1732 1984 y Fh(\003)1770 2012 y Fw(=)36 2083 y Fu(GF)7 b Fw(\()p Fu(q)r Fw(\))j Fs(\000)h(f)p Fw(0)p Fs(g)p Fw(,)429 2178 y(\()p Fs(\000)p Fu(\013)p Fw(\))610 2136 y Fj(Y)546 2231 y Fr(\014)r Fh(2)592 2235 y Ff(I)-8 b(F)624 2217 y Fd(\003)642 2231 y Fh(\000f)p Fr(\013)p Fh(g)728 2178 y Fw(\()p Fu(x)11 b Fs(\000)f Fu(\014)s Fw(\))k(=)950 2104 y Fj(\()1004 2147 y Fw(1)42 b Fu(x)14 b Fw(=)g Fu(\013)i Fw(or)h Fu(x)c Fw(=)h(0)1004 2207 y(0)59 b(otherwise,)1707 2178 y(\(3)p Fu(:)p Fw(9\))36 2313 y(as)15 b(follo)o(ws)f(from)g(the)g(fact)g(that,)h(in)f (an)o(y)g(\014nite)g(\014eld,)g(the)g(pro)q(duct)h(of)g(all)f(non-zero)h (elemen)n(ts)36 2384 y(equals)h Fs(\000)p Fw(1)g(so)h(that)517 2458 y(\()p Fs(\000)p Fu(\013)p Fw(\))697 2416 y Fj(Y)633 2511 y Fr(\014)r Fh(2)679 2515 y Ff(I)-8 b(F)712 2497 y Fd(\003)730 2511 y Fh(\000f)p Fr(\013)p Fh(g)815 2458 y Fw(\()p Fu(\013)11 b Fs(\000)g Fu(\014)s Fw(\))i(=)h Fs(\000)1110 2416 y Fj(Y)1088 2511 y Fr(\014)r Fh(2)1134 2515 y Ff(I)-8 b(F)1167 2497 y Fd(\003)1193 2458 y Fu(\014)16 b Fw(=)e(1)p Fu(:)36 2600 y Fw(Th)o(us,)k(ev)o(ery)f (function)g Fu(h)p Fw(\()p Fs(\001)p Fw(\))h(from)713 2602 y Fm(I)-12 b(F)779 2600 y Fw(to)841 2602 y Fm(I)g(F)907 2600 y Fw(can)18 b(b)q(e)g(written)f(as)i(a)f(p)q(olynomial)f(o)o(v)o(er)1702 2602 y Fm(I)-12 b(F)1768 2600 y Fw(of)36 2671 y(degree)16 b(at)g(most)g Fu(q)c Fs(\000)f Fw(1)17 b(as)g(follo)o(ws:)115 2795 y Fu(h)p Fw(\()p Fu(x)p Fw(\))d(=)293 2753 y Fj(X)274 2848 y Fr(\013)p Fh(2)321 2852 y Ff(I)-8 b(F)354 2834 y Fd(\003)381 2795 y Fu(h)p Fw(\()p Fu(\013)p Fw(\)\()p Fs(\000)p Fu(\013)p Fw(\))659 2753 y Fj(Y)594 2848 y Fr(\014)r Fh(2)640 2852 y Ff(I)g(F)673 2834 y Fd(\003)691 2848 y Fh(\000f)p Fr(\013)p Fh(g)777 2795 y Fw(\()p Fu(x)10 b Fs(\000)h Fu(\014)s Fw(\))f(+)h(\()p Fu(x)1040 2774 y Fr(q)q Fh(\000)p Fl(1)1115 2795 y Fs(\000)g Fw(1\)[)p Fu(h)p Fw(\(0\))g Fs(\000)1392 2753 y Fj(X)1373 2848 y Fr(\013)p Fh(2)1420 2852 y Ff(I)-8 b(F)1453 2834 y Fd(\003)1480 2795 y Fu(h)p Fw(\()p Fu(\013)p Fw(\)])p Fu(:)78 b Fw(\(3)p Fu(:)p Fw(10\))p eop %%Page: 29 9 bop 36 103 a Fq(3.2.)38 b(GR)o(OUP)16 b(OPERA)l(TIONS)f(AND)h(THEIR)f(INTERA) o(CTION)410 b Fw(29)36 275 y(Note)17 b(that)h Fu(f)5 b Fw(\(0)p Fu(;)j(y)r Fw(\))16 b(=)g(0)i(for)g(all)f Fu(y)j Fw(in)760 277 y Fm(I)-12 b(F)808 275 y Fw(,)17 b(and)i(that)f Fu(f)5 b Fw(\()p Fu(a;)j Fs(\001)p Fw(\))17 b(for)h(ev)o(ery)e Fu(a)f Fs(6)p Fw(=)h(0)i(is)g(a)g(bijection)36 345 y(from)151 347 y Fm(I)-12 b(F)199 317 y Fh(\003)235 345 y Fw(to)295 347 y Fm(I)g(F)343 317 y Fh(\003)363 345 y Fw(,)16 b(so)g(that)242 424 y Fj(X)223 518 y Fr(\013)p Fh(2)270 522 y Ff(I)-8 b(F)303 504 y Fd(\003)329 465 y Fu(f)5 b Fw(\()p Fu(a;)j(\013)p Fw(\))14 b(=)560 424 y Fj(X)541 518 y Fr(\013)p Fh(2)588 522 y Ff(I)-8 b(F)621 504 y Fd(\003)647 465 y Fu(\013)14 b Fw(=)g(0)66 b(for)16 b(ev)o(ery)f Fu(a)e Fs(6)p Fw(=)h(0)j(and)33 b(for)1354 467 y Fm(I)-12 b(F)1416 465 y Fs(6)p Fw(=)14 b Fu(GF)7 b Fw(\(2\))p Fu(:)36 617 y Fw(F)l(rom)15 b(the)g(de\014nition)g(of)h Fu(f)5 b Fw(\()p Fu(x;)j(y)r Fw(\))16 b(and)g(from)f(equation)g(\(3.10\),)h(the)g (function)g Fu(f)5 b Fw(\()p Fu(a;)j(y)r Fw(\))15 b(can)h(b)q(e)36 688 y(written)g(for)g(ev)o(ery)f Fu(a)e Fs(6)p Fw(=)h(0)j(as)221 826 y Fu(f)5 b Fw(\()p Fu(a;)j(y)r Fw(\))13 b(=)427 753 y Fj(\()481 795 y Fu(a)e Fw(+)g Fu(y)127 b Fw(1)15 b Fs(\024)e Fu(y)j Fs(\024)d Fw(2)925 777 y Fr(n)960 795 y Fs(\000)e Fu(a)481 855 y(a)g Fw(+)g Fu(y)h Fw(+)f(1)42 b(2)742 837 y Fr(n)777 855 y Fs(\000)11 b Fu(a)j(<)f(y)j Fs(\024)d Fw(2)1034 837 y Fr(n)221 973 y Fw(=)273 919 y Fl(2)291 907 y Fi(n)312 919 y Fh(\000)p Fr(a)285 931 y Fj(X)287 1022 y Fr(i)p Fl(=1)358 973 y Fw(\()p Fu(a)e Fw(+)g Fu(i)p Fw(\)\()p Fs(\000)p Fu(i)p Fw(\))637 931 y Fj(Y)639 1022 y Fi(j)q Fd(6)p Fe(=)p Fi(i)600 1053 y Fl(1)p Fh(\024)p Fr(j)r Fh(\024)p Fl(2)706 1044 y Fi(n)727 973 y Fw(\()p Fu(y)i Fs(\000)e Fu(j)s Fw(\))g(+)1000 919 y Fl(2)1018 907 y Fi(n)990 931 y Fj(X)935 1022 y Fr(i)p Fl(=2)992 1013 y Fi(n)1013 1022 y Fh(\000)p Fr(a)p Fl(+1)1104 973 y Fw(\()p Fu(a)g Fw(+)g Fu(i)g Fw(+)g(1\)\()p Fs(\000)p Fu(i)p Fw(\))1468 931 y Fj(Y)1470 1022 y Fi(j)q Fd(6)p Fe(=)p Fi(i)1431 1053 y Fl(1)p Fh(\024)p Fr(j)r Fh(\024)p Fl(2)1537 1044 y Fi(n)1558 973 y Fw(\()p Fu(y)i Fs(\000)e Fu(j)s Fw(\))221 1161 y(=)283 1107 y Fl(2)301 1095 y Fi(n)273 1120 y Fj(X)274 1211 y Fr(i)p Fl(=1)333 1161 y Fw(\()p Fu(a)f Fw(+)h Fu(i)p Fw(\)\()p Fs(\000)p Fu(i)p Fw(\))612 1120 y Fj(Y)614 1210 y Fi(j)q Fd(6)p Fe(=)p Fi(i)575 1242 y Fl(1)p Fh(\024)p Fr(j)r Fh(\024)p Fl(2)681 1232 y Fi(n)702 1161 y Fw(\()p Fu(y)i Fs(\000)e Fu(j)s Fw(\))g(+)975 1107 y Fl(2)993 1095 y Fi(n)964 1120 y Fj(X)910 1211 y Fr(i)p Fl(=2)967 1201 y Fi(n)988 1211 y Fh(\000)p Fr(a)p Fl(+1)1079 1161 y Fw(\()p Fs(\000)p Fu(i)p Fw(\))1218 1120 y Fj(Y)1220 1210 y Fi(j)q Fd(6)p Fe(=)p Fi(i)1181 1242 y Fl(1)p Fh(\024)p Fr(j)r Fh(\024)p Fl(2)1287 1232 y Fi(n)1309 1161 y Fw(\()p Fu(y)h Fs(\000)f Fu(j)s Fw(\))p Fu(:)36 1343 y Fw(That)25 b(is,)h(the)e(function)g Fu(f)5 b Fw(\()p Fu(a;)j(y)r Fw(\))24 b(is)g(a)h(p)q(olynomial)e(in)h Fu(y)i Fw(with)e(degree)g(at)g(most)g(2)1674 1325 y Fr(n)1714 1343 y Fs(\000)17 b Fw(1.)36 1414 y(Moreo)o(v)o(er,)e(the)h(co)q(e\016cien)o (t)e(of)j Fu(y)657 1396 y Fl(2)675 1384 y Fi(n)695 1396 y Fh(\000)p Fl(1)759 1414 y Fw(in)f Fu(f)5 b Fw(\()p Fu(a;)j(y)r Fw(\))15 b(is)371 1506 y Fl(2)389 1494 y Fi(n)360 1518 y Fj(X)362 1610 y Fr(i)p Fl(=1)420 1560 y Fw(\()p Fu(a)c Fw(+)g Fu(i)p Fw(\)\()p Fs(\000)p Fu(i)p Fw(\))f(+)779 1506 y Fl(2)797 1494 y Fi(n)769 1518 y Fj(X)714 1610 y Fr(i)p Fl(=2)771 1600 y Fi(n)792 1610 y Fh(\000)p Fr(a)p Fl(+1)883 1560 y Fw(\()p Fs(\000)p Fu(i)p Fw(\))k(=)f Fs(\000)p Fu(a)1126 1506 y Fl(2)1144 1494 y Fi(n)1115 1518 y Fj(X)1116 1610 y Fr(i)p Fl(=1)1183 1560 y Fu(i)e Fs(\000)1271 1506 y Fl(2)1289 1494 y Fi(n)1261 1518 y Fj(X)1262 1610 y Fr(i)p Fl(=1)1329 1560 y Fu(i)1346 1539 y Fl(2)1376 1560 y Fw(+)1446 1506 y Fh(\000)p Fl(1)1438 1518 y Fj(X)1425 1610 y Fr(i)p Fl(=)p Fh(\000)p Fr(a)1511 1560 y Fw(\()p Fs(\000)p Fu(i)p Fw(\))281 1713 y(=)380 1659 y Fh(\000)p Fl(1)373 1672 y Fj(X)360 1763 y Fr(i)p Fl(=)p Fh(\000)p Fr(a)446 1713 y Fw(\()p Fs(\000)p Fu(i)p Fw(\))i(=)625 1659 y Fr(a)605 1672 y Fj(X)606 1763 y Fr(i)p Fl(=1)673 1713 y Fu(i)h Fw(=)760 1680 y Fu(a)p Fw(\()p Fu(a)d Fw(+)g(1\))p 760 1702 174 2 v 835 1748 a(2)939 1713 y Fu(;)36 1858 y Fw(whic)o(h)20 b(is)g(zero)g(if)g(and)h(only)f(if)g Fu(a)h Fw(=)g(0)g(or)g Fu(a)f Fw(=)h Fs(\000)p Fw(1)g(=)g(2)1135 1840 y Fr(n)1159 1858 y Fw(,)g(whic)o(h)f(cases)h(are)f(excluded)f(b)o(y)36 1929 y(h)o(yp)q(othesis.)27 b(\(W)l(e)17 b(ha)o(v)o(e)h(used)g(the)f(facts)i (that)943 1896 y Fj(P)987 1909 y Fl(2)1005 1897 y Fi(n)987 1941 y Fr(i)p Fl(=1)1054 1929 y Fu(i)e Fw(=)g(0)f(mo)q(d)g(\(2)1336 1911 y Fr(n)1372 1929 y Fw(+)c(1\),)19 b(that)1605 1896 y Fj(P)1649 1909 y Fl(2)1667 1897 y Fi(n)1649 1941 y Fr(i)p Fl(=1)1717 1929 y Fu(i)1734 1911 y Fl(2)1770 1929 y Fw(=)41 1980 y Fl(1)p 41 1988 18 2 v 41 2017 a(6)64 2000 y Fw(2)88 1982 y Fr(n)112 2000 y Fw(\(2)155 1982 y Fr(n)188 2000 y Fw(+)9 b(1\)\(2)i Fs(\002)e Fw(2)404 1982 y Fr(n)438 2000 y Fw(+)g(1\),)16 b(that)g(2)p Fs(j)p Fw(2)725 1982 y Fr(n)749 2000 y Fw(,)f(and)h(that)g(3)p Fs(j)p Fw(2)10 b Fs(\002)g Fw(2)1122 1982 y Fr(n)1155 2000 y Fw(+)g(1)15 b(for)h Fu(n)e Fs(2)g(f)p Fw(2)p Fu(;)8 b Fw(4)p Fu(;)g Fw(8)p Fu(;)g Fw(16)p Fs(g)17 b Fw(so)f(that)36 2037 y Fj(P)80 2050 y Fl(2)98 2038 y Fi(n)80 2082 y Fr(i)p Fl(=1)147 2070 y Fu(i)164 2052 y Fl(2)198 2070 y Fw(=)d(0)k(mo)q(d)f(\(2)443 2052 y Fr(n)476 2070 y Fw(+)9 b(1\))p Fu(:)15 b Fw(\))h(Th)o(us,)f(w)o(e)g (ha)o(v)o(e)f(sho)o(wn)i(that)g(the)f(degree)g(of)h(the)f(p)q(olynomial)36 2141 y Fu(f)5 b Fw(\()p Fu(a;)j(y)r Fw(\))16 b(is)g(indeed)f(2)420 2123 y Fr(n)455 2141 y Fs(\000)c Fw(1.)109 2211 y(Note)i(that)h Fu(f)5 b Fw(\()p Fu(x;)j(y)r Fw(\))13 b(=)h Fu(f)5 b Fw(\()p Fu(y)r(;)j(x)p Fw(\))13 b(for)h(all)f Fu(x)g Fw(and)h Fu(y)h Fw(in)e Fb(Z)-21 b(Z)1093 2218 y Fl(2)1111 2209 y Fi(n)1132 2218 y Fl(+1)1192 2211 y Fw(so)14 b(that,)g(for)g(ev)o(ery)e Fu(a)19 b(=)-30 b Fs(2)14 b(f)p Fw(0)p Fu(;)8 b Fw(2)1745 2193 y Fr(n)1769 2211 y Fs(g)p Fw(,)36 2282 y Fu(f)d Fw(\()p Fu(x;)j(a)p Fw(\))16 b(is)g(a)g(p)q(olynomial)f(in)h Fu(x)g Fw(of)h(degree)f(2)869 2264 y Fr(n)904 2282 y Fs(\000)10 b Fw(1.)779 b Fc(2)36 2406 y Ft(Theorem)17 b(4)24 b Fq(If)f Fu(n)i Fs(2)h(f)p Fw(2)p Fu(;)8 b Fw(4)p Fu(;)g Fw(8)p Fu(;)g Fw(16)p Fs(g)p Fq(,)26 b(then,)e(for)f(ev)o (ery)f Fu(a)g Fq(in)h Fb(Z)-21 b(Z)1272 2413 y Fl(2)1290 2404 y Fi(n)1329 2406 y Fs(\000)15 b(f)p Fw(0)p Fu(;)8 b Fw(1)p Fs(g)p Fq(,)25 b(the)e(function)36 2476 y Fu(g)r Fw(\()p Fu(a;)8 b(x)p Fw(\))13 b(=)h Fu(a)266 2443 y Fj(J)312 2476 y Fu(x)f Fw(=)h Fu(x)433 2443 y Fj(J)479 2476 y Fu(a)i Ft(cannot)g Fq(b)q(e)h(written) e(as)i(a)g(p)q(olynomial)e(in)h Fu(x)g Fq(o)o(v)o(er)f(the)h(ring)g Fb(Z)-21 b(Z)1719 2483 y Fl(2)1737 2474 y Fi(n)1760 2476 y Fu(:)109 2600 y Fw(W)l(e)16 b(sho)o(w)h(\014rst)g(the)f(follo)o(wing)f (lemma:)36 2711 y Ft(Lemma)h(1)24 b Fq(If)16 b Fu(p)p Fw(\()p Fu(x)p Fw(\))g Fq(is)g(a)h(p)q(olynomial)e(o)o(v)o(er)g Fb(Z)-21 b(Z)927 2718 y Fl(2)945 2709 y Fi(n)968 2711 y Fu(;)16 b Fq(then,)f(for)i (all)f Fu(\014)i Fq(in)e Fb(Z)-21 b(Z)1407 2718 y Fl(2)1425 2709 y Fi(n)1448 2711 y Fu(;)635 2831 y(p)p Fw(\(2)p Fu(\014)s Fw(\))14 b(mo)q(d)f(2)h(=)g Fu(p)p Fw(\(0\))h(mo)q(d)e(2)p Fu(:)p eop %%Page: 30 10 bop 36 103 a Fw(30)900 b Fq(CH.)16 b(3.)38 b(THE)16 b(BLOCK)g(CIPHER)g(IDEA) 36 275 y Ft(Pro)r(of.)119 b Fw(Let)16 b Fu(p)p Fw(\()p Fu(x)p Fw(\))e(=)g Fu(a)576 282 y Fr(k)597 275 y Fu(x)625 257 y Fr(k)657 275 y Fw(+)d Fu(a)732 282 y Fr(k)q Fh(\000)p Fl(1)798 275 y Fu(x)826 257 y Fr(k)q Fh(\000)p Fl(1)903 275 y Fw(+)g Fs(\001)d(\001)g(\001)j Fw(+)g Fu(a)1096 282 y Fl(1)1116 275 y Fu(x)f Fw(+)h Fu(a)1229 282 y Fl(0)1249 275 y Fu(:)36 345 y Fw(Then)16 b(for)h(all)f Fu(\014)i Fw(in)e Fb(Z)-21 b(Z)448 352 y Fl(2)466 343 y Fi(n)489 345 y Fw(,)396 479 y Fu(p)p Fw(\(2)p Fu(\014)s Fw(\))14 b(=)g Fu(a)605 486 y Fr(k)626 479 y Fw(\(2)p Fu(\014)s Fw(\))719 458 y Fr(k)751 479 y Fw(+)d Fu(a)826 486 y Fr(k)q Fh(\000)p Fl(1)892 479 y Fw(\(2)p Fu(\014)s Fw(\))985 458 y Fr(k)q Fh(\000)p Fl(1)1062 479 y Fw(+)g Fs(\001)d(\001)g(\001)j Fw(+)g Fu(a)1255 486 y Fl(1)1274 479 y Fw(2)p Fu(\014)j Fw(+)d Fu(a)1415 486 y Fl(0)1434 479 y Fu(:)36 612 y Fw(Th)o(us,)16 b Fu(p)p Fw(\(2)p Fu(x)p Fw(\))e(mo)q(d)g(2)g(=)g Fu(a)525 619 y Fl(0)558 612 y Fw(mo)q(d)f(2)i(=)e Fu(f)5 b Fw(\(0\))17 b(mo)q(d)f(2)p Fu(:)759 b Fc(2)36 711 y Ft(Pro)r(of)17 b(of)h(Theorem)d(4)32 b Fw(Let)16 b Fu(n)e(>)f Fw(1)p Fu(;)i Fw(then)g(for)h(ev)o(ery)d(in)o(teger)h Fu(a)p Fw(,)h(1)f Fu(<)g(a)f(<)h Fw(2)1513 692 y Fr(n)1537 711 y Fw(,)h(there)f(exists)36 781 y(an)j(in)o(teger)e Fu(x)293 788 y Fl(0)326 781 y Fs(2)f(f)p Fw(1)p Fu(;)8 b Fw(2)p Fu(;)g(:)g(:)g(:)g(;)g Fw(2)602 763 y Fr(n)626 781 y Fs(g)16 b Fw(suc)o(h)g(that)h(the)f(follo)o (wing)g(three)f(inequalities)g(all)g(satis\014ed:)633 914 y(2)657 894 y Fr(n)692 914 y Fw(+)c(1)j Fu(<)g Fw(2)p Fu(ax)909 921 y Fl(0)942 914 y Fu(<)g Fw(2\(2)1061 894 y Fr(n)1096 914 y Fw(+)d(1\))p Fu(;)481 b Fw(\(3)p Fu(:)p Fw(11\))667 1061 y(0)14 b Fs(\024)g Fw(2)p Fu(a)p Fw(\()p Fu(x)855 1068 y Fl(0)885 1061 y Fs(\000)d Fw(1\))j Fu(<)g Fw(2)1068 1040 y Fr(n)1103 1061 y Fw(+)d(1)507 b(\(3)p Fu(:)p Fw(12\))36 1173 y(and)777 1257 y(0)14 b Fs(\024)f Fw(2)p Fu(x)919 1264 y Fl(0)953 1257 y Fs(\024)h Fw(2)1030 1236 y Fr(n)1054 1257 y Fu(:)615 b Fw(\(3)p Fu(:)p Fw(13\))36 1369 y(Inequalit)o(y)16 b(\(3.11\))i(is)f(equiv)m(alen)o(t) f(to)h(the)h(inequalit)o(y)d(0)h Fu(<)g Fw(2)p Fu(ax)1238 1376 y Fl(0)1269 1369 y Fs(\000)c Fw(\(2)1363 1351 y Fr(n)1399 1369 y Fw(+)f(1\))17 b Fu(<)e Fw(2)1585 1351 y Fr(n)1621 1369 y Fw(+)d(1)18 b(with)36 1440 y(the)d(condition)f(that)h(2)p Fu(ax)513 1447 y Fl(0)541 1440 y Fs(\000)8 b Fw(\(2)631 1422 y Fr(n)662 1440 y Fw(+)g(1\))15 b(is)g(an)g(o)q(dd)h(in)o(teger.)j(Because)14 b(of)i(\(3.13\))f(and)g(from)f(the)36 1510 y(de\014nition)i(of)g(the)g (function)g Fu(g)r Fw(,)503 1644 y Fu(g)r Fw(\()p Fu(a;)8 b Fw(2)p Fu(x)647 1651 y Fl(0)667 1644 y Fw(\))13 b(=)h Fu(a)777 1611 y Fj(J)823 1644 y Fw(\(2)p Fu(x)894 1651 y Fl(0)914 1644 y Fw(\))g(=)f(2)p Fu(ax)1076 1651 y Fl(0)1107 1644 y Fs(\000)e Fw(\(2)1200 1623 y Fr(n)1235 1644 y Fw(+)g(1\))p Fu(:)36 1777 y Fw(Th)o(us,)415 1861 y Fu(g)r Fw(\()p Fu(a;)d Fw(2)p Fu(x)559 1868 y Fl(0)578 1861 y Fw(\))14 b(mo)q(d)f(2)i(=)e(\(2)p Fu(ax)905 1868 y Fl(0)936 1861 y Fs(\000)e Fw(\(2)1029 1840 y Fr(n)1064 1861 y Fw(+)g(1\)\))16 b(mo)q(d)g(2)e(=)g(1)p Fu(:)36 1973 y Fw(On)19 b(the)g(other)g(hand,)g(inequalit)o(y)e(\(3.12\))j(implies)c(that) j(2)p Fu(a)p Fw(\()p Fu(x)1221 1980 y Fl(0)1254 1973 y Fs(\000)12 b Fw(1\))19 b(is)g(an)g(ev)o(en)f(in)o(teger)g(in)36 2044 y Fs(f)p Fw(0)p Fu(;)8 b Fw(1)p Fu(;)g(::;)g Fw(2)227 2026 y Fr(n)250 2044 y Fs(g)16 b Fw(so)h(that)417 2177 y Fu(g)r Fw(\()p Fu(a;)8 b Fw(\(2\()p Fu(x)599 2184 y Fl(0)629 2177 y Fs(\000)j Fw(1\)\))j(mo)q(d)f(2)i(=)e(2)p Fu(a)p Fw(\()p Fu(x)1049 2184 y Fl(0)1080 2177 y Fs(\000)e Fw(1\))16 b(mo)q(d)g(2)e(=)g(0)p Fu(:)36 2310 y Fw(Hence,)i(it)h(follo)o(ws)h(from)e(Lemma)g(1)i(that)g Fu(g)r Fw(\()p Fu(a;)8 b(x)p Fw(\))15 b(=)i Fu(a)1084 2277 y Fj(J)1129 2310 y Fu(x)h Fw(is)f(not)h(a)g(p)q(olynomial)f(o)o(v)o(er)f Fb(Z)-21 b(Z)1753 2317 y Fl(2)1771 2308 y Fi(n)1794 2310 y Fw(.)36 2381 y Fc(2)36 2574 y Fp(3.3)83 b(Securit)n(y)26 b(F)-7 b(eatures)26 b(of)i(IDEA)36 2702 y Fw(In)18 b(this)h(section,)f(w)o(e)g (state)h(some)f(pro)o(v)m(able)g(securit)o(y)f(features)i(of)g(the)f(IDEA)g (cipher.)28 b(The)36 2773 y(securit)o(y)19 b(of)h(the)g(IDEA)g(cipher)f (against)j(di\013eren)o(tial)d(cryptanalysis)g(will)h(b)q(e)g(discussed)g(in) 36 2844 y(detail)c(in)f(Chapter)i(5.)p eop %%Page: 31 11 bop 36 103 a Fq(3.3.)38 b(SECURITY)15 b(FEA)l(TURES)h(OF)g(IDEA)843 b Fw(31)36 275 y Fk(3.3.1)70 b(Confusion)36 378 y Fw(The)20 b(confusion)g(\(see)g(page)g(12\))g(required)f(for)h(a)g(secure)g(cipher)e (is)i(ac)o(hiev)o(ed)e(in)i(the)f(IDEA)36 449 y(cipher)c(b)o(y)h(mixing)e (three)h(incompatible)f(group)j(op)q(erations.)22 b(In)16 b(the)g (computational)f(graph)36 519 y(of)24 b(the)f(encryption)g(pro)q(cess)i(for)f (IDEA,)f(the)g(three)g(di\013eren)o(t)g(group)i(op)q(erations)f(are)g(so)36 590 y(arranged)19 b(that)g Fo(the)h(output)g(of)g(an)f(op)n(er)n(ation)g(of)g (one)i(typ)n(e)e(is)g(never)i(use)n(d)e(as)h(the)g(input)g(to)36 661 y(an)e(op)n(er)n(ation)e(of)i(the)g(same)f(typ)n(e)p Fw(.)109 731 y(The)f(three)g(op)q(erations)h(are)g Fo(inc)n(omp)n(atible)34 b Fw(in)16 b(the)g(sense)g(that:)36 874 y Ft(1.)48 b Fw(No)18 b(pair)f(of)g(the)g(3)h(op)q(erations)g(satis\014es)g(a)f(\\distributiv)o(e") g(la)o(w.)23 b(F)l(or)18 b(instance,)e(for)i(the)36 944 y(op)q(erations)273 911 y Fj(J)336 944 y Fw(and)p 433 915 37 2 v 433 948 2 33 v 19 w(+)p 468 948 V 433 949 37 2 v 1 w(,)e(there)g(exist)f Fu(a)p Fw(,)g Fu(b)p Fw(,)h(and)h Fu(c)f Fw(in)1037 946 y Fm(I)-12 b(F)1085 916 y Fl(16)1085 956 y(2)1122 944 y Fw(,)16 b(suc)o(h)g(that,)637 1065 y Fu(a)p 665 1036 V 665 1069 2 33 v 2 w Fw(+)p 700 1069 V 665 1070 37 2 v 1 w(\()p Fu(b)744 1032 y Fj(J)790 1065 y Fu(c)p Fw(\))e Fs(6)p Fw(=)f(\()p Fu(a)p 942 1036 V 942 1069 2 33 v 2 w Fw(+)p 977 1069 V 942 1070 37 2 v 1 w Fu(b)p Fw(\))1021 1032 y Fj(J)1067 1065 y Fw(\()p Fu(a)p 1114 1036 V 1114 1069 2 33 v 2 w Fw(+)p 1149 1069 V 1114 1070 37 2 v 1 w Fu(c)p Fw(\))p Fu(:)36 1187 y Fw(F)l(or)g(example,)d(when)j Fu(a)g Fw(=)h Fu(b)f Fw(=)h Fu(c)g Fw(=)g(1)g(=)g(\(0)p Fu(;)8 b Fw(0)p Fu(;)g(::;)g Fw(0)p Fu(;)g Fw(1\),)13 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1596 37 2 v 1 w Fu(b)p Fw(\))1074 1558 y Fj(L)1120 1591 y Fu(c:)36 1713 y Fw(F)l(or)20 b(example,)e(for)i Fu(a)g Fw(=)g Fu(b)g Fw(=)g Fu(c)g Fw(=)g(1)h(=)f(\(0)p Fu(;)8 b Fw(0)p Fu(;)g(:::)p Fw(0)p Fu(;)g Fw(1\))19 b(in)1141 1715 y Fm(I)-12 b(F)1189 1684 y Fl(16)1189 1725 y(2)1226 1713 y Fw(,)21 b(the)e(left)g(side)h(of)g (the)g(ab)q(o)o(v)o(e)36 1783 y(inequalit)o(y)15 b(is)i(1)f(=)g(\(0)p Fu(;)8 b Fw(0)p Fu(;)g(:::)p Fw(0)p Fu(;)g Fw(1\),)16 b(while)h(the)g(righ)o (t)f(side)h(equals)g(3)f(=)f(\(0)p Fu(;)8 b Fw(0)p Fu(;)g(::;)g Fw(0)p Fu(;)g Fw(1)p Fu(;)g Fw(1\).)25 b(Th)o(us,)36 1854 y(one)16 b(cannot)h(arbitrarily)f(c)o(hange)g(the)g(order)g(of)h(op)q(erations)g(to)g (simplify)c(analysis.)36 1996 y Ft(3.)48 b Fw(The)17 b(3)f(op)q(erations)h (are)g(connected)e(b)o(y)h(the)g(direct)f(mapping)g Fu(d)i Fw(and)f(its)g(in)o(v)o(erse,)e(whic)o(h)36 2066 y(inhibits)k(isotopisms)f (as)i(w)o(as)g(sho)o(wn)g(in)f(Theorem)f(2.)28 b(The)18 b(cryptographic)h (signi\014cance)e(of)36 2137 y(this)f(fact)g(is)g(that,)g(if)g(there)g(w)o (ere)f(an)i(isotopism)e(b)q(et)o(w)o(een)g(t)o(w)o(o)i(op)q(erations,)f(then) g(one)h(could)36 2207 y(replace)f(one)i(op)q(eration)g(with)f(the)g(other)g (b)o(y)g(applying)g(bijectiv)o(e)d(mappings)j(on)h(the)f(inputs)36 2278 y(and)23 b(on)g(the)f(output.)40 b(It)22 b(follo)o(ws)g(from)f(Theorem)g (2)i(that)f(\()1233 2280 y Fm(I)-12 b(F)1281 2250 y Fl(16)1281 2290 y(2)1319 2278 y Fu(;)1341 2245 y Fj(J)1387 2278 y Fw(\))22 b(and)h(\()1548 2280 y Fm(I)-12 b(F)1596 2250 y Fl(16)1596 2290 y(2)1633 2278 y Fu(;)1655 2245 y Fj(L)1701 2278 y Fw(\))23 b(are)36 2349 y(not)18 b(isotopic)f(and)h(that)g(\()526 2351 y Fm(I)-12 b(F)574 2320 y Fl(16)574 2361 y(2)612 2349 y Fu(;)p 636 2319 V 636 2352 2 33 v 636 2348 a Fw(+)p 671 2352 V 636 2353 37 2 v 675 2349 a(\))18 b(and)g(\()827 2351 y Fm(I)-12 b(F)875 2320 y Fl(16)875 2361 y(2)912 2349 y Fu(;)934 2315 y Fj(L)980 2349 y Fw(\))18 b(are)f(not)h(isotopic.)25 b(The)17 b(isotopism)g(from)36 2419 y(\()55 2421 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(ey)f(subblo)q(c)o(ks)h Fu(Z)917 2510 y Fl(5)957 2503 y Fw(and)g Fu(Z)1087 2510 y Fl(6)1107 2503 y Fw(,)h Fu(M)5 b(A)p Fw(\()p Fs(\001)p Fu(;)j Fs(\001)p Fu(;)g(Z)1354 2510 y Fl(5)1373 2503 y Fu(;)g(Z)1428 2510 y Fl(6)1448 2503 y Fw(\))19 b(is)g(an)h(in)o(v)o (ertible)158 2574 y(transformation;)15 b(for)h(an)o(y)g(c)o(hoice)e(of)i Fu(U)901 2581 y Fl(1)937 2574 y Fw(and)g Fu(U)1064 2581 y Fl(2)1084 2574 y Fw(,)g Fu(M)5 b(A)p Fw(\()p Fu(U)1255 2581 y Fl(1)1274 2574 y Fu(;)j(U)1329 2581 y Fl(2)1349 2574 y Fu(;)g Fs(\001)p Fu(;)g Fs(\001)p Fw(\))15 b(is)h(also)g(an)g(in)o(v)o(ert-)158 2644 y(ible)f(transformation;)60 2773 y({)22 b(this)g(structure)f(has)i(a)f (\\complete)d(di\013usion")k(e\013ect)e(in)g(the)g(sense)h(that)g(eac)o(h)f (output)158 2844 y(subblo)q(c)o(k)16 b(dep)q(ends)h(on)f(ev)o(ery)f(input)h (subblo)q(c)o(k,)g(and)p eop %%Page: 33 13 bop 36 103 a Fq(3.3.)38 b(SECURITY)15 b(FEA)l(TURES)h(OF)g(IDEA)843 b Fw(33)60 275 y({)24 b(this)g(structure)f(uses)h(the)f(least)g(n)o(um)o(b)q (er)f(of)i(op)q(erations)h(\(four\))e(required)g(to)h(ac)o(hiev)o(e)158 345 y(suc)o(h)16 b(complete)e(di\013usion.)21 b([T)l(o)c(giv)o(e)e(a)h (formal)f(pro)q(of)j(of)e(this)g(prop)q(ert)o(y)l(,)g(w)o(e)f(need)h(the)158 416 y(follo)o(wing)g(de\014nitions.)158 507 y(An)11 b Fo(op)n(er)n(ation)23 b Fw(is)11 b(a)h(mapping)e(from)h(t)o(w)o(o)g(v)m(ariables)g(to)h(one)f(v)m (ariable.)19 b(A)11 b Fo(c)n(omputational)158 577 y(gr)n(aph)36 b Fw(of)19 b(a)f(function)g(is)g(a)h(directed)d(graph)k(in)d(whic)o(h)h(the)g (v)o(ertices)e(are)i(op)q(erations,)158 648 y(the)13 b(edges)h(en)o(tering)f (a)h(v)o(ertex)e(are)i(the)f(inputs)h(to)g(the)g(op)q(eration,)g(the)g(edges) g(lea)o(ving)e(a)158 719 y(v)o(ertex)e(are)h(the)h(output)g(v)m(ariable)f(of) h(the)f(op)q(eration,)i(the)e(edges)h(en)o(tering)f(no)h(v)o(ertex)e(are)158 789 y(the)15 b(output)h(v)m(ariables,)e(and)i(the)f(edges)g(lea)o(ving)f(no)i (v)o(ertex)d(are)i(the)g(inputs)g(v)m(ariables.)158 860 y(An)f(algorithm)g (to)h(compute)f(a)h(function)g(determines)d(a)j(computational)f(graph)i (where)158 930 y(the)d(input)h(v)m(ariables)f(are)h(the)g(input)f(to)h(the)f (algorithm)g(and)h(the)g(output)g(v)m(ariables)g(are)158 1001 y(the)i(outputs)h(of)g(the)f(algorithm.)158 1092 y(Consider)g(a)h(function)f (ha)o(ving)g(the)g(form)316 1209 y(\()p Fu(Y)363 1216 y Fl(1)383 1209 y Fu(;)8 b(Y)433 1216 y Fl(2)452 1209 y Fw(\))14 b(=)g Fu(E)s Fw(\()p Fu(X)635 1216 y Fl(1)655 1209 y Fu(;)8 b(X)717 1216 y Fl(2)737 1209 y Fu(;)g(Z)792 1216 y Fl(1)812 1209 y Fu(;)g(Z)867 1216 y Fl(2)887 1209 y Fw(\))p Fu(;)106 b(X)1066 1216 y Fr(i)1080 1209 y Fu(;)8 b(Y)1130 1216 y Fr(i)1158 1209 y Fs(2)1205 1211 y Fm(I)-12 b(F)1253 1181 y Fr(n)1253 1221 y Fl(2)1277 1209 y Fu(;)57 b(Z)1381 1216 y Fr(i)1409 1209 y Fs(2)1456 1211 y Fm(I)-12 b(F)1504 1181 y Fr(k)1504 1221 y Fl(2)1683 1209 y Fw(\(3)p Fu(:)p Fw(14\))158 1326 y(and)17 b(suc)o(h)g(that,)g(for)g(ev)o(ery)e(c)o(hoice)h(of)h(\()p Fu(Z)942 1333 y Fl(1)962 1326 y Fu(;)8 b(Z)1017 1333 y Fl(2)1037 1326 y Fw(\),)16 b Fu(E)s Fw(\()p Fs(\001)p Fu(;)8 b Fs(\001)p Fu(;)g(Z)1249 1333 y Fl(1)1269 1326 y Fu(;)g(Z)1324 1333 y Fl(2)1344 1326 y Fw(\))16 b(is)h(in)o(v)o(ertible.)j(Suc)o(h)c(a)158 1397 y(function)h(will)e(b)q(e)i(called)f(a)h Fo(2-blo)n(ck)j(cipher.)j Fw(A)17 b(2-blo)q(c)o(k)g(cipher)e(will)h(b)q(e)h(said)g(to)g(ha)o(v)o(e)158 1467 y Fo(c)n(omplete)24 b(di\013usion)f Fw(if)f(eac)o(h)g(of)h(its)f(output) i(v)m(ariable)e(dep)q(ends)h(non-idly)f(on)h(ev)o(ery)158 1538 y(input)16 b(v)m(ariable.)158 1679 y Ft(Lemma)g(2)24 b Fq(If)16 b(a)g(2-blo)q(c)o(k)h(cipher)e(of)i(the)f(form)f(\(3.14\))i(has)g(complete)d (di\013usion,)i(then)158 1750 y(the)h(computational)f(graph)j(determined)14 b(b)o(y)j(an)o(y)g(algorithm)f(that)i(computes)e(the)h(ci-)158 1820 y(pher)f(function)g(con)o(tains)g(at)h(least)f(4)h(op)q(erations.)158 1962 y Ft(Pro)r(of.)k Fw(Let)15 b Fu(Y)445 1969 y Fl(1)479 1962 y Fw(=)e Fu(E)566 1969 y Fl(1)586 1962 y Fw(\()p Fu(X)645 1969 y Fl(1)665 1962 y Fu(;)8 b(X)727 1969 y Fl(2)747 1962 y Fu(;)g(Z)802 1969 y Fl(1)822 1962 y Fu(;)g(Z)877 1969 y Fl(2)897 1962 y Fw(\))p Fu(;)15 b Fw(and)g Fu(Y)1066 1969 y Fl(2)1100 1962 y Fw(=)f Fu(E)1188 1969 y Fl(2)1208 1962 y Fw(\()p Fu(X)1267 1969 y Fl(1)1287 1962 y Fu(;)8 b(X)1349 1969 y Fl(2)1369 1962 y Fu(;)g(Z)1424 1969 y Fl(1)1444 1962 y Fu(;)g(Z)1499 1969 y Fl(2)1519 1962 y Fw(\))p Fu(:)15 b Fw(Because)f Fu(E)1788 1969 y Fl(1)158 2032 y Fw(has)i(complete)d(di\013usion,)j(its)f (computational)f(graph)j(con)o(tains)e(at)h(least)f(3)h(op)q(erations)158 2103 y(b)q(ecause)k(this)f(function)g(has)h(four)g(input)f(v)m(ariables.)30 b(Supp)q(ose)20 b Fu(E)1422 2110 y Fl(1)1462 2103 y Fw(con)o(tains)f(exactly) 158 2173 y(3)j(op)q(erations.)39 b(The)22 b(in)o(v)o(ertibili)o(t)o(y)c(of)k (the)g(2-blo)q(c)o(k)g(cipher)f(implies)e(that)j Fu(E)1648 2180 y Fl(2)1691 2173 y Fs(6)p Fw(=)h Fu(E)1788 2180 y Fl(1)158 2244 y Fw(and)15 b(complete)c(di\013usion)k(requires)e(that)h Fu(E)969 2251 y Fl(2)1003 2244 y Fw(not)g(equal)g(an)o(y)g(in)o(termediate)d (result)i(that)158 2315 y(app)q(ears)j(in)f Fu(E)429 2322 y Fl(1)449 2315 y Fu(:)g Fw(Th)o(us,)g(at)h(least)f(one)g(op)q(eration)h(not)g (app)q(earing)g(in)f Fu(E)1494 2322 y Fl(1)1529 2315 y Fw(is)g(required)f(in) 158 2385 y(the)i(computational)f(graph)j(of)e Fu(E)794 2392 y Fl(2)814 2385 y Fw(.)21 b(This)c(pro)o(v)o(es)e(the)h(lemma.)i(])36 2529 y Fk(3.3.3)70 b(P)n(erfect)22 b(secrecy)f(for)j(a)f(\\one-time")e(k)n (ey)36 2632 y Fw(P)o(erfect)d(secrecy)g(\(see)i(page)g(7\))f(in)h(the)f (sense)g(of)h(Shannon)h(is)e(obtained)h(in)f(eac)o(h)g(round)h(of)36 2702 y(encryption)c(if)h(a)g(\\one-time")f(k)o(ey)g(\(see)h(page)g(7\))g(is)g (used.)24 b(In)16 b(fact,)h(suc)o(h)g(p)q(erfect)f(secrecy)g(is)36 2773 y(ac)o(hiev)o(ed)f(at)i(the)f(input)h(transformation)f(in)h(the)f (\014rst)h(round)g(b)q(ecause)g(eac)o(h)f(op)q(eration)i(is)e(a)36 2844 y(group)i(op)q(eration.)24 b(In)17 b(addition,)g(for)g(ev)o(ery)f(c)o (hoice)f(of)j(\()p Fu(p)1133 2851 y Fl(1)1153 2844 y Fu(;)8 b(p)1199 2851 y Fl(2)1219 2844 y Fu(;)g(p)1265 2851 y Fl(3)1285 2844 y Fu(;)g(p)1331 2851 y Fl(4)1351 2844 y Fw(\))17 b(and)h(of)f(\()p Fu(q)1580 2851 y Fl(1)1599 2844 y Fu(;)8 b(q)1643 2851 y Fl(2)1662 2844 y Fu(;)g(q)1706 2851 y Fl(3)1726 2844 y Fu(;)g(q)1770 2851 y Fl(4)1789 2844 y Fw(\))p eop %%Page: 34 14 bop 36 103 a Fw(34)900 b Fq(CH.)16 b(3.)38 b(THE)16 b(BLOCK)g(CIPHER)g(IDEA) 36 283 y Fw(in)94 285 y Fm(I)-12 b(F)142 255 y Fl(64)142 295 y(2)180 283 y Fw(,)17 b(there)g(are)h(exactly)e(2)613 265 y Fl(32)668 283 y Fw(di\013eren)o(t)h(c)o(hoices)f(of)i(the)f(k)o(ey)g(subblo)q (c)o(ks)g(\()p Fu(Z)1533 290 y Fl(1)1554 283 y Fu(;)8 b(::;)g(Z)1659 290 y Fl(6)1677 283 y Fw(\))18 b(suc)o(h)36 354 y(that)f(the)f(\014rst)g (round)h(of)g(the)f(cipher)f(transforms)h(\()p Fu(p)1039 361 y Fl(1)1059 354 y Fu(;)8 b(p)1105 361 y Fl(2)1125 354 y Fu(;)g(p)1171 361 y Fl(3)1191 354 y Fu(;)g(p)1237 361 y Fl(4)1257 354 y Fw(\))17 b(in)o(to)f(\()p Fu(q)1433 361 y Fl(1)1452 354 y Fu(;)8 b(q)1496 361 y Fl(2)1515 354 y Fu(;)g(q)1559 361 y Fl(3)1578 354 y Fu(;)g(q)1622 361 y Fl(4)1641 354 y Fw(\).)36 525 y Fp(3.4)83 b(Implemen)n(tations)26 b(of)i(the)f(Cipher)36 647 y Fw(The)17 b(cipher)g(IDEA)g(can)g(b)q(e)h (easily)e(impleme)o(n)o(ted)e(in)j(soft)o(w)o(are)g(b)q(ecause)h(only)f (basic)h(op)q(era-)36 717 y(tions)e(on)g(pairs)g(of)f(16-bit)i(subblo)q(c)o (ks)e(are)h(used)f(in)g(the)h(encryption)f(pro)q(cess.)21 b(A)15 b(C-language)36 788 y(program)e(implem)o(e)o(n)o(ting)d(the)j(cipher)e(and)j (some)e(sample)f(data)j(for)f(c)o(hec)o(king)e(the)h(correctness)36 859 y(of)h(implem)o(en)o(tation)d(are)j(giv)o(en)f(in)g(Section)h(3.4.3.)20 b(This)13 b(C-program)g(can)g(ac)o(hiev)o(e)e(data-rates)36 929 y(from)16 b(ab)q(out)i(200)f(Kbits)g(p)q(er)g(second)g(on)g(an)g(IBM-PC)f (to)h(ab)q(out)h(3.2)f(Mbits)g(p)q(er)f(second)h(on)36 1000 y(a)g(V)-5 b(AX-9000.)109 1071 y(The)22 b(regular)f(mo)q(dular)g(structure)g (of)h(the)f(cipher)f(facilitates)h(hardw)o(are)g(impleme)o(n)o(ta-)36 1142 y(tions.)28 b(The)19 b(similarit)o(y)c(of)k(encryption)f(and)h (decryption)f(for)h(the)f(IDEA)g(cipher,)g(sho)o(wn)h(in)36 1212 y(next)f(section,)h(mak)o(es)e(it)h(p)q(ossible)h(to)g(use)f(the)h(same) e(device)h(in)g(b)q(oth)h(encryption)f(and)i(de-)36 1283 y(cryption.)g(An)13 b(algorithm)g(for)h(computing)e(the)i(op)q(eration)1152 1250 y Fj(J)1212 1283 y Fw(is)f(describ)q(ed)g(in)h(Section)f(3.4.2.)36 1432 y Fk(3.4.1)70 b(Similarit)m(y)21 b(of)i(encryption)f(and)i(decryption)36 1537 y Fw(The)14 b Fo(similarity)e Fw(of)i(encryption)f(and)h(decryption)f (means)f(that)i(decryption)f(is)g(essen)o(tially)f(the)36 1607 y(same)18 b(pro)q(cess)i(as)f(encryption,)g(the)g(only)g(di\013erence)f(b)q (eing)h(that)g(di\013eren)o(t)g(k)o(ey)f(subblo)q(c)o(ks)36 1678 y(are)h(used.)28 b(Th)o(us,)19 b(the)f(same)g(device)f(can)h(b)q(e)h (used)g(for)f(b)q(oth)i(encryption)e(and)h(decryption,)36 1749 y(the)k(only)h(\\extra")g(cost)g(b)q(eing)g(the)g(pre-computation)e(of)j(the) e(k)o(ey)f(subblo)q(c)o(ks)i(from)f(the)36 1819 y(128-bit)d(secret)d(k)o(ey)l (.)27 b(In)18 b(the)g(follo)o(wing)g(w)o(e)g(sho)o(w)i(that)e(the)h(round)g (function)f(of)h(the)f(IDEA)36 1890 y(cipher)c(has)i(the)f(form)f(\(2.2\))h (on)g(page)h(17,)f(that)h(is,)e(the)h(round)h(function)f(consists)g(of)g(a)h (group)36 1960 y(cipher)g(follo)o(w)o(ed)f(b)o(y)h(an)h(in)o(v)o(olution)e (cipher)h(plus)g(an)h(in)o(v)o(olutary)f(p)q(erm)o(utation)f(whic)o(h)h(is)g (an)36 2031 y(automorphism)d(of)h(the)g(group)h(\()647 2033 y Fm(I)-12 b(F)695 2002 y Fl(64)695 2043 y(2)733 2031 y Fu(;)8 b Fs(\012)p Fw(\).)20 b(Then)14 b(it)g(follo)o(ws)g(from)f(Theorem)g(1)h (\(see)g(page)h(17\))36 2101 y(that)i(IDEA)f(cipher)f(has)i(similarit)o(y)c (of)j(encryption)g(and)h(decryption.)109 2173 y(F)l(or)g(the)f(encryption)f (pro)q(cess)i(of)g(the)f(IDEA)f(cipher)h(sho)o(wn)h(in)f(Fig.3.1,)f(de\014ne) 433 2296 y Fu(X)g Fs(\012)c Fu(Z)571 2303 y Fr(A)614 2296 y Fw(=)i(\()p Fu(X)724 2303 y Fl(1)745 2262 y Fj(J)791 2296 y Fu(Z)824 2303 y Fl(1)844 2296 y Fu(;)8 b(X)906 2303 y Fl(2)p 928 2266 37 2 v 928 2299 2 33 v 928 2295 a Fw(+)p 963 2299 V 928 2300 37 2 v 967 2296 a Fu(Z)1000 2303 y Fl(2)1021 2296 y Fu(;)g(X)1083 2303 y Fl(3)p 1105 2266 V 1105 2299 2 33 v 1105 2295 a Fw(+)p 1140 2299 V 1105 2300 37 2 v 1144 2296 a Fu(Z)1177 2303 y Fl(3)1197 2296 y Fu(;)g(X)1259 2303 y Fl(4)1279 2262 y Fj(J)1325 2296 y Fu(Z)1358 2303 y Fl(4)1378 2296 y Fw(\))p Fu(;)36 2419 y Fw(then)16 b(it)g(is)g(easy)g(to)h(see)f(that)g(\()615 2421 y Fm(I)-12 b(F)664 2390 y Fl(64)664 2431 y(2)701 2419 y Fu(;)8 b Fs(\012)p Fw(\))16 b(is)g(a)g(group.)109 2490 y(Let)21 b Fu(P)232 2497 y Fr(I)253 2490 y Fw(\()p Fu(X)t Fw(\))g(b)q(e)g(the)g(p)q (erm)o(utation)f(on)i Fu(X)j Fw(that)d(in)o(terc)o(hanges)e(the)h(subblo)q(c) o(ks)g Fu(X)1688 2497 y 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Fr(B)1067 2773 y Fw(\),)15 b(sho)o(wn)g(in)g(Fig.3.3,)f(with)h(the)f (64-bit)36 2844 y(input)19 b(\()p Fu(S)218 2851 y Fl(1)238 2844 y Fu(;)8 b(S)290 2851 y Fl(2)309 2844 y Fu(;)g(S)361 2851 y Fl(3)381 2844 y Fu(;)g(S)433 2851 y Fl(4)452 2844 y Fw(\))19 b(and)h(the)f(64-bit)g(output)h(\()p Fu(T)1029 2851 y Fl(1)1048 2844 y Fu(;)8 b(T)1099 2851 y Fl(2)1118 2844 y Fu(;)g(T)1169 2851 y Fl(3)1189 2844 y Fu(;)g(T)1240 2851 y Fl(4)1259 2844 y Fw(\))19 b(con)o(trolled)f(b)o(y)g(the)h(32-bit)p eop %%Page: 35 15 bop 36 103 a Fq(3.4.)38 b(IMPLEMENT)l(A)l(TIONS)15 b(OF)h(THE)g(CIPHER)702 b Fw(35)36 275 y(k)o(ey)18 b Fu(Z)160 282 y Fr(B)210 275 y Fw(=)h(\()p Fu(Z)319 282 y Fl(5)339 275 y Fu(;)8 b(Z)394 282 y Fl(6)414 275 y Fw(\),)20 b(is)f(an)h(in)o(v)o(olution.)29 b(That)20 b(is,)f(for)h(an)o(y)f(\014xed)g Fu(Z)1365 282 y Fr(B)1396 275 y Fw(,)h(the)f(in)o(v)o(erse)e(of)j(the)36 345 y(function)f Fu(I)t(n)p Fw(\()p Fs(\001)p Fu(;)8 b(Z)373 352 y Fr(B)403 345 y Fw(\))20 b(is)f(itself.)29 b(This)20 b(self-in)o(v)o(erse)d (prop)q(ert)o(y)i(is)h(a)f(consequence)g(of)h(the)f(fact)36 416 y(that)13 b(the)g(exclusiv)o(e-OR)e(of)i(\()p Fu(S)612 423 y Fl(1)632 416 y Fu(;)8 b(S)684 423 y Fl(2)703 416 y Fw(\))13 b(and)h(\()p Fu(S)876 423 y Fl(3)896 416 y Fu(;)8 b(S)948 423 y Fl(4)967 416 y Fw(\))13 b(is)g(equal)f(to)i(the)e(exclusiv)o(e-OR)f(of)i (\()p Fu(T)1699 423 y Fl(1)1719 416 y Fu(;)8 b(T)1770 423 y Fl(2)1789 416 y Fw(\))36 487 y(and)15 b(\()p Fu(T)177 494 y Fl(3)196 487 y Fu(;)8 b(T)247 494 y Fl(4)266 487 y Fw(\);)15 b(Th)o(us,)f(the)h(input)f(to)g(the)g(MA)g(structure)g(in)g(Fig.)g(3.2)g(is)g (unc)o(hanged)h(when)f Fu(S)1774 494 y Fl(1)1794 487 y Fu(;)36 557 y(S)66 564 y Fl(2)86 557 y Fu(;)i(S)146 564 y Fl(3)182 557 y Fw(and)g Fu(S)306 564 y Fl(4)342 557 y Fw(are)h(replaced)e(b)o(y)h Fu(T)713 564 y Fl(1)732 557 y Fu(;)g(T)791 564 y Fl(2)810 557 y Fu(;)g(T)869 564 y Fl(3)905 557 y Fw(and)g Fu(T)1028 564 y Fl(4)1048 557 y Fw(.)606 687 y(\()p Fu(S)655 694 y Fl(1)675 687 y Fu(;)8 b(S)727 694 y Fl(2)746 687 y Fw(\))266 b(\()p Fu(S)1080 694 y Fl(3)1100 687 y Fu(;)8 b(S)1152 694 y Fl(4)1171 687 y Fw(\))p 685 899 2 189 v 686 899 a Fn(?)p 1110 899 V 383 w(?)875 772 y Fj(L)686 792 y Fg(r)413 b(r)p 686 793 195 2 v -284 w Fn(-)p 916 793 V 35 w(\033)p 897 863 2 54 v 898 863 a(?)p 838 862 122 2 v 838 957 2 95 v 858 927 a Fw(MA)p 957 957 V 838 959 122 2 v 663 890 a Fj(L)379 b(L)p 703 911 136 2 v 703 910 a Fn(\033)p 957 911 V 307 w(-)815 1076 y Fw(\()p Fu(Z)867 1083 y Fl(5)887 1076 y Fu(;)8 b(Z)942 1083 y Fl(6)962 1076 y Fw(\))p 897 1029 2 71 v 898 999 a Fn(6)p 685 1105 2 178 v 686 1105 a(?)p 1110 1105 V 383 w(?)607 1136 y Fw(\()p Fu(T)655 1143 y Fl(1)675 1136 y Fu(;)g(T)726 1143 y Fl(2)745 1136 y Fw(\))269 b(\()p Fu(T)1081 1143 y Fl(3)1100 1136 y Fu(;)8 b(T)1151 1143 y Fl(4)1170 1136 y Fw(\))272 1367 y(Figure)16 b(3.3:)21 b(Computational)16 b(graph)i(of)e(the)g(in)o(v)o(olution)f Fu(I)t(n)p Fw(\()p Fs(\001)p Fu(;)8 b(Z)1509 1374 y Fr(B)1539 1367 y Fw(\).)36 1560 y Fk(3.4.2)70 b(Lo)n(w-High)23 b(algorithm)f(for)h(m)n (ultipli)o(cati)o(on)36 1662 y Fw(The)16 b(most)g(di\016cult)e(step)i(in)g (the)g(implem)o(en)o(tation)d(of)k(the)f(IDEA)f(cipher)h(is)g(the)f(impleme)o (n-)36 1733 y(tation)22 b(of)f(m)o(ultiplicati)o(on)e(mo)q(dulo)i(\(2)786 1715 y Fl(16)838 1733 y Fw(+)15 b(1\),)22 b(whic)o(h)f(can)g(b)q(e)g(impleme) o(n)o(ted)d(in)j(the)g(w)o(a)o(y)36 1803 y(suggested)c(b)o(y)f(the)g(follo)o (wing)g(lemm)o(a.)36 1901 y Ft(Lemma)g(3)24 b([Lo)n(w-High)c(algorithm)e(for) 906 1868 y Fj(J)952 1901 y Ft(])g Fq(Let)f Fu(a;)8 b(b)17 b Fq(b)q(e)h(t)o(w)o(o)g Fu(n)p Fq(-bit)f(non-zero)h(in)o(tegers)36 1971 y(in)e Fb(Z)-21 b(Z)132 1978 y Fl(2)150 1970 y Fi(n)171 1978 y Fl(+1)218 1971 y Fq(,)16 b(then)140 2075 y Fu(ab)g Fq(mo)q(d)f Fw(\(2)355 2054 y Fr(n)390 2075 y Fw(+)c(1\))k(=)223 2106 y Fj(\()277 2148 y Fw(\()p Fu(ab)h Fq(mo)q(d)g Fw(2)493 2130 y Fr(n)516 2148 y Fw(\))c Fs(\000)e Fw(\()p Fu(ab)16 b Fq(div)g Fw(2)785 2130 y Fr(n)808 2148 y Fw(\))251 b Fq(if)15 b Fw(\()p Fu(ab)h Fq(mo)q(d)g Fw(2)1338 2130 y Fr(n)1361 2148 y Fw(\))e Fs(\025)g Fw(\()p Fu(ab)h Fq(div)h Fw(2)1635 2130 y Fr(n)1659 2148 y Fw(\))277 2208 y(\()p Fu(ab)g Fq(mo)q(d)g Fw(2)493 2190 y Fr(n)516 2208 y Fw(\))c Fs(\000)e Fw(\()p Fu(ab)16 b Fq(div)g Fw(2)785 2190 y Fr(n)808 2208 y Fw(\))11 b(+)g(2)911 2190 y Fr(n)946 2208 y Fw(+)g(1)59 b Fq(if)15 b Fw(\()p Fu(ab)h Fq(mo)q(d)g Fw(2)1338 2190 y Fr(n)1361 2208 y Fw(\))e Fu(<)g Fw(\()p Fu(ab)i Fq(div)f Fw(2)1634 2190 y Fr(n)1658 2208 y Fw(\))36 2303 y Fq(where)h Fw(\()p Fu(ab)f Fq(div)h Fw(2)365 2285 y Fr(n)389 2303 y Fw(\))g Fq(denotes)h(the)f(quotien)o(t)f(when)h(ab)h(is)f(divided)f(b) o(y)h Fw(2)1385 2285 y Fr(n)1409 2303 y Fq(.)109 2400 y Fw(Note)24 b(that)g(\()p Fu(ab)15 b Fw(mo)q(d)h(2)563 2382 y Fr(n)587 2400 y Fw(\))24 b(corresp)q(onds)h(to)f(the)g Fu(n)g Fw(least)f(signi\014can) o(t)h(bits)f(of)h Fu(ab)p Fw(,)h(and)36 2471 y(\()p Fu(ab)16 b Fw(div)f(2)224 2453 y Fr(n)248 2471 y Fw(\))22 b(is)f(just)h(the)f(righ)o (t-shift)h(of)g Fu(ab)f Fw(b)o(y)g Fu(n)h Fw(bits.)37 b(Note)21 b(also)h(that)h(\()p Fu(ab)15 b Fw(mo)q(d)h(2)1704 2453 y Fr(n)1728 2471 y Fw(\))23 b(=)36 2541 y(\()p Fu(ab)16 b Fw(div)f(2)224 2523 y Fr(n)248 2541 y Fw(\))j(implies)d(that)j Fu(ab)e Fw(mo)q(d)f(\(2)774 2523 y Fr(n)810 2541 y Fw(+)d(1\))17 b(=)g(0)h(and)g(hence)f(cannot)h(o)q (ccur)g(when)g(2)1697 2523 y Fr(n)1733 2541 y Fw(+)12 b(1)36 2612 y(is)k(a)h(prime.)36 2702 y Ft(Pro)r(of.)k Fw(F)l(or)16 b(an)o(y)f(non-zero)h Fu(a)f Fw(and)h Fu(b)g Fw(in)f Fb(Z)-21 b(Z)851 2709 y Fl(2)869 2700 y Fi(n)889 2709 y Fl(+1)937 2702 y Fw(,)15 b(there)g(exist)f(unique)h(in)o(tegers)g Fu(q)i Fw(and)f Fu(r)h Fw(suc)o(h)36 2773 y(that)373 2844 y Fu(ab)c Fw(=)h Fu(q)r Fw(\(2)552 2823 y Fr(n)587 2844 y Fw(+)d(1\))g(+)g Fu(r)o(;)105 b Fw(0)14 b Fs(\024)g Fu(r)h(<)f Fw(2)1083 2823 y Fr(n)1118 2844 y Fw(+)d(1)p Fu(;)24 b Fw(0)14 b Fs(\024)g Fu(q)h(<)f Fw(2)1433 2823 y Fr(n)1457 2844 y Fu(:)p eop %%Page: 36 16 bop 36 103 a Fw(36)906 b Fq(CH.)15 b(3.)33 b(THE)16 b(BLOCK)g(CIPHER)g(IDEA) 36 275 y Fw(Moreo)o(v)o(er,)f Fu(q)d Fw(+)f Fu(r)k(<)f Fw(2)460 257 y Fr(n)p Fl(+1)529 275 y Fu(:)i Fw(Note)g(that)g Fu(r)g Fw(=)d Fu(ab)j Fw(mo)q(d)g(\(2)1087 257 y Fr(n)1122 275 y Fw(+)11 b(1\))p Fu(:)16 b Fw(W)l(e)g(ha)o(v)o(e)518 414 y(\()p Fu(ab)f Fw(div)h(2)706 394 y Fr(n)730 414 y Fw(\))e(=)814 341 y Fj(\()869 384 y Fu(q)143 b Fw(if)16 b Fu(q)d Fw(+)e Fu(r)k(<)e Fw(2)1275 366 y Fr(n)869 444 y Fu(q)f Fw(+)f(1)58 b(if)16 b Fu(q)d Fw(+)e Fu(r)k Fs(\025)e Fw(2)1276 426 y Fr(n)36 551 y Fw(and)444 630 y(\()p Fu(ab)i Fw(mo)q(d)h(2)659 609 y Fr(n)683 630 y Fw(\))e(=)767 557 y Fj(\()822 599 y Fu(q)e Fw(+)f Fu(r)168 b Fw(if)16 b Fu(q)c Fw(+)f Fu(r)k(<)f Fw(2)1336 581 y Fr(n)822 659 y Fu(q)e Fw(+)f Fu(r)h Fs(\000)f Fw(2)1013 641 y Fr(n)1095 659 y Fw(if)16 b Fu(q)c Fw(+)f Fu(r)k Fs(\025)f Fw(2)1337 641 y Fr(n)1361 659 y Fu(:)36 745 y Fw(Th)o(us)298 823 y Fu(r)h Fw(=)387 750 y Fj(\()441 793 y Fw(\()p Fu(ab)g Fw(mo)q(d)h(2)656 775 y Fr(n)680 793 y Fw(\))11 b Fs(\000)g Fw(\()p Fu(ab)k Fw(div)h(2)948 775 y Fr(n)972 793 y Fw(\))250 b(if)16 b Fu(q)c Fw(+)f Fu(r)k(<)f Fw(2)1482 775 y Fr(n)441 853 y Fw(\()p Fu(ab)h Fw(mo)q(d)h(2)656 835 y Fr(n)680 853 y Fw(\))11 b Fs(\000)g Fw(\()p Fu(ab)k Fw(div)h(2)948 835 y Fr(n)972 853 y Fw(\))11 b(+)g(2)1075 835 y Fr(n)1110 853 y Fw(+)g(1)58 b(if)16 b Fu(q)c Fw(+)f Fu(r)k Fs(\025)f Fw(2)1483 835 y Fr(n)1507 853 y Fu(:)36 941 y Fw(But)h Fu(q)10 b Fw(+)f Fu(r)15 b(<)f Fw(2)324 923 y Fr(n)363 941 y Fw(if)h(and)h(only)f(if) f(\()p Fu(ab)i Fw(mo)q(d)g(2)866 923 y Fr(n)890 941 y Fw(\))d Fs(\025)h Fw(\()p Fu(ab)i Fw(div)f(2)1163 923 y Fr(n)1187 941 y Fw(\))p Fu(:)g Fw(This)g(pro)o(v)o(es)g(the)g(Lemma.)j Fc(2)36 1081 y Ft(Remark.)46 b Fw(There)15 b(are)g(of)g(course)g(other)f(w)o(a)o(ys)h (to)g(compute)f(the)g(op)q(eration)1512 1048 y Fj(J)1558 1081 y Fw(.)21 b(F)l(or)15 b(exam-)36 1152 y(ple,)g(based)i(on)g(the)f(fact)g (that)455 1270 y Fu(x)11 b Fs(\001)g Fu(y)k Fw(=)f Fu(\013)641 1250 y Fl(\(log)701 1258 y Fi(\013)723 1250 y Fl(\()p Fr(x)p Fl(\)+log)843 1258 y Fi(\013)865 1250 y Fl(\()p Fr(y)q Fl(\))j Fw(mo)q(d)e Fl(2)1055 1238 y Fi(n)1076 1250 y Fl(\))1108 1270 y Fw(mo)q(d)h(\(2)1261 1250 y Fr(n)1296 1270 y Fw(+)11 b(1\))36 1389 y(for)18 b(all)f Fu(x)h Fw(and)g Fu(y)h Fw(in)f Fb(Z)-21 b(Z)463 1367 y Fh(\003)463 1401 y Fl(2)481 1392 y Fi(n)502 1401 y Fl(+1)567 1389 y Fw(where)18 b Fu(\013)g Fw(is)f(a)h(generator)g(of)g (the)g(cyclic)d(group)k Fb(Z)-21 b(Z)1526 1367 y Fh(\003)1526 1401 y Fl(2)1544 1392 y Fi(n)1565 1401 y Fl(+1)1612 1389 y Fw(,)18 b(one)g(can)36 1459 y(compute)233 1426 y Fj(J)294 1459 y Fw(b)o(y)d(using)p 490 1430 37 2 v 490 1463 2 33 v 18 w(+)p 525 1463 V 490 1464 37 2 v 16 w(together)h(with)f(lo)q(ok-up)h(tables)f(for)h (computing)e(log)1538 1471 y Fr(\013)1563 1459 y Fw(\()p Fs(\001)p Fw(\))h(and)g Fu(\013)1754 1441 y Fl(\()p Fh(\001)p Fl(\))1794 1459 y Fw(.)36 1530 y(F)l(or)e(small)e Fu(n)p Fw(,)i(i.e.,)e(for)i Fu(n)h Fw(=)g(2)p Fu(;)8 b Fw(4)13 b(or)h(8,)f(this)g(is)f(more)f(e\016cien)o (t)g(than)i(the)g(Lo)o(w-High)g(algorithm.)36 1600 y(Ho)o(w)o(ev)o(er,)f(for) j Fu(n)e Fw(=)h(16,)h(this)f(metho)q(d)f(requires)g(more)g(memory)l(.)k(More) d(details)g(can)g(b)q(e)g(found)36 1671 y(in)i([10].)36 1815 y Fk(3.4.3)70 b(C-program)24 b(of)f(IDEA)g(cipher)f(and)h(sample)f(data)36 1897 y Fa(/*)f(C)h(-)g(program)42 b(of)21 b(block)g(cipher)f(IDEA)h(*/)36 1997 y(#include)f()36 2047 y(#)i(define)e(maxim)h(65537)36 2096 y(#)h(define)e(fuyi)43 b(65536)36 2146 y(#)22 b(define)e(one)65 b(65535)36 2196 y(#)22 b(define)e(round)43 b(8)36 2246 y(void)65 b(cip\(unsigned)19 b(IN[5],unsigned)g(OUT[5],unsigned)f(Z[7][10]\);)36 2296 y(void)65 b(key\()21 b(short)f(unsigned)g(uskey[9],unsigned)f(Z[7][10])h (\);)36 2345 y(void)65 b(de_key\(unsigned)18 b(Z[7][10],unsigned)h (DK[7][10]\);)36 2395 y(unsigned)h(inv\(unsigned)f(xin\);)36 2445 y(unsigned)h(mul\(unsigned)f(a,)j(unsigned)e(b\);)36 2545 y(main\(\))36 2595 y({)80 2644 y(int)152 b(i,)21 b(j,)g(k,)h(x;)80 2694 y(unsigned)42 b(Z[7][10],)19 b(DK[7][10],)h(XX[5],TT[5],)g(YY[5];)80 2744 y(short)g(unsigned)h(uskey[9];)101 2794 y(for\()g(i=1;)g(i<=8;)g(i++)g (\))h(uskey[i]=)e(i;)101 2844 y(key\(uskey,Z\);)434 b(/*)21 b(generate)f(encryption)g(subkeys)g(Z[i][r])h(*/)p eop %%Page: 37 17 bop 36 103 a Fq(3.4.)32 b(IMPLEMENT)l(A)l(TIONS)15 b(OF)h(THE)h(CIPHER)707 b Fw(37)101 275 y Fa(printf\("\\n)20 b(encryption)g(keys)86 b(Z1)109 b(Z2)f(Z3)h(Z4)f(Z5)h(Z6"\);)101 325 y(for\()21 b(j=1;)g(j<=9;)g (j++)g(\))h({)87 b(printf\("\\n)20 b(\0453d-th)g(round)43 b(",)21 b(j\);)232 375 y(if)h(\(j==9\))42 b(for\()21 b(i=1;)g(i<=4;)g(i++)g(\))109 b(printf\(")20 b(\0456d",Z[i][j]\);)319 424 y(else)65 b(for\()21 b(i=1;)g(i<=6;)g(i++)g(\))109 b(printf\(")20 b(\0456d",Z[i][j]\);)145 474 y(})101 524 y(de_key\(Z,DK\);)434 b(/*)21 b(compute)f(decryption)g (subkeys)g(DK[i][r])h(*/)101 624 y(printf\("\\n)f(\\n)i(decryption)d(keys)65 b(DK1)86 b(DK2)h(DK3)f(DK4)h(DK5)g(DK6)21 b("\);)101 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b(x4)21 b(=)h(x4^t2;)210 2417 y(a)g(=)g(x2^t2;)151 b(x2)21 b(=)h(x3^t1;)129 b(x3)22 b(=)f(a;)210 2467 y(printf\("\\n)107 b(\0451u-th)21 b(rnd)43 b(\0456u)g(\0456u)g(\0456u)g (\0456u)g(",)21 b(r,)g(x1,)h(x2,)f(x3,)g(x4\);)167 2516 y(})319 2566 y(/*)65 b(the)21 b(output)g(transformation)128 b(*/)123 2616 y(OUT[1])21 b(=)g(mul\()g(x1,Z[1][round+1])e(\);)123 2666 y(OUT[4])i(=)g(mul\()g(x4,Z[4][round+1])e(\);)123 2716 y(OUT[2])i(=)43 b(\()22 b(x3)f(+)h(Z[2][round)d(+1])j(\)&)f(one;)123 2766 y(OUT[3])g(=)43 b(\()22 b(x2)f(+)h(Z[3][round+1])d(\))i(&)h(one;)58 2815 y(})p eop %%Page: 38 18 bop 36 103 a Fw(38)906 b Fq(CH.)15 b(3.)33 b(THE)16 b(BLOCK)g(CIPHER)g(IDEA) 101 375 y Fa(/*)44 b(multiplication)18 b(using)j(the)g(Low-High)g(algorithm) 41 b(*/)36 424 y(unsigned)20 b(mul\(unsigned)f(a,)j(unsigned)e(b\))36 474 y({)80 524 y(long)h(int)g(p;)80 574 y(long)g(unsigned)f(q;)145 624 y(if)h(\(a==0\))86 b(p)22 b(=)f(maxim-b;)145 673 y(else)g(if)g(\()h(b==0) f(\))43 b(p)22 b(=)f(maxim-a;)42 b(else)189 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y(void)g(key\()21 b(short)f(unsigned)g(uskey[9],)g (unsigned)42 b(Z[7][10])20 b(\))36 1919 y({)80 1969 y(short)g(unsigned)h (S[54];)80 2018 y(int)43 b(i,j,r;)58 2068 y(for)21 b(\(i)g(=)h(1;)f(i<9;)g (i++\))43 b(S[i-1])21 b(=)g(uskey[i];)101 2118 y(/*)65 b(shifts)g(*/)58 2168 y(for)21 b(\(i)g(=)h(8;)f(i<)h(54;)f(i++)g(\))80 2218 y({)145 2267 y(if)g(\()h(\(i+2\)\0458)e(==)i(0)f(\))480 b(/*)21 b(for)g(S[14],S[22],..)41 b(*/)210 2317 y(S[i])21 b(=)h(\()g(\()f(S[i-7])g (<<9)g(\)^\()g(S[i-14])f(>>7)i(\))f(\))h(&)f(one;)145 2367 y(else)43 b(if)21 b(\()h(\(i+1\)\0458)e(==0)h(\))371 b(/*)21 b(for)g(S[15],S[23],..)63 b(*/)210 2417 y(S[i])21 b(=\()h(\()f(S[i-15])g(<<9) g(\)^\()g(S[i-14])f(>>7)i(\))f(\))h(&)f(one)g(;)145 2467 y(else)210 2516 y(S[i])g(=)h(\()g(\()f(S[i-7])g(<<9)g(\)^\()g(S[i-6])g(>>7)g(\)\))g(&)h (one;)101 2566 y(})123 2616 y(/*)g(get)f(subkeys)107 b(*/)167 2666 y(for)21 b(\(r=)g(1;)h(r<=round+1;)d(r++\))65 b(for\(j=)20 b(1;j<7;)h(j++\))385 2716 y(Z[j][r])f(=)i(S[6*\(r-1\))e(+)h(j-1];)58 2766 y(})p eop %%Page: 39 19 bop 36 103 a Fq(3.4.)32 b(IMPLEMENT)l(A)l(TIONS)15 b(OF)h(THE)h(CIPHER)707 b Fw(39)145 375 y Fa(/*)21 b(compute)g(decryption)e(subkeys)i(DK's)43 b(*/)36 424 y(void)g(de_key\(unsigned)18 b(Z[7][10],unsigned)h(DK[7][10]\))36 474 y({)101 524 y(int)43 b(j;)101 574 y(for)22 b(\(j)f(=)h(1;)f(j<=round+1;)e (j++\))123 624 y({)65 b(DK[1][round-j+2])19 b(=)i(inv\(Z[1][j]\);)210 673 y(DK[4][round-j+2])e(=)i(inv\(Z[4][j]\);)145 723 y(if)g(\()h(j==1)f(||)g (j==round+1)f(\){)210 773 y(DK[2][round-j+2])f(=)i(\()h(fuyi-Z[2][j])d(\))j (&)g(one;)210 823 y(DK[3][round-j+2])d(=)i(\()h(fuyi-Z[3][j])d(\))j(&)g(one;) 210 873 y(})145 922 y(else)f({)210 972 y(DK[2][round-j+2])e(=)i(\()h (fuyi-Z[3][j])d(\))j(&)g(one;)210 1022 y(DK[3][round-j+2])d(=)i(\()h (fuyi-Z[2][j])d(\))j(&)g(one;)210 1072 y(})145 1122 y(})101 1172 y(for)g(\(j=)f(1;j<=round+1;j++)o(\))167 1221 y({)g(DK[5][round+1-j])e (=)i(Z[5][j];)42 b(DK[6][round+1-j])19 b(=)i(Z[6][j];})58 1271 y(})36 1535 y Ft(Sample)26 b(Data.)41 b Ff(All)20 b(the)h(n)o(um)o(b)q(ers)f (are)h(16-bit)e(in)o(tegers)i(with)f(the)h(leftmost)e(bit)h(b)q(eing)h(the)g (most)36 1585 y(signi\014can)o(t)13 b(bit.)58 1798 y Fa(encryption)20 b(keys)86 b(Z1)108 b(Z2)h(Z3)g(Z4)f(Z5)h(Z6)101 1847 y(1-th)21 b(round)174 b(1)130 b(2)h(3)f(4)h(5)f(6)101 1897 y(2-th)21 b(round)174 b(7)130 b(8)65 b(1024)g(1536)g(2048)f(2560)101 1947 y(3-th)21 b(round)108 b(3072)65 b(3584)f(4096)87 b(512)108 b(16)h(20)101 1997 y(4-th)21 b(round)152 b(24)108 b(28)h(32)130 b(4)h(8)109 b(12)101 2047 y(5-th)21 b(round)87 b(10240)42 b(12288)h(14336)f (16384)65 b(2048)f(4096)101 2096 y(6-th)21 b(round)108 b(6144)65 b(8192)86 b(112)h(128)108 b(16)h(32)101 2146 y(7-th)21 b(round)152 b(48)108 b(64)h(80)g(96)130 b(0)65 b(8192)101 2196 y(8-th)21 b(round)87 b(16384)42 b(24576)h(32768)f(40960)h(49152)f(57345)101 2246 y(9-th)21 b(round)130 b(128)87 b(192)f(256)h(320)58 2345 y(decryption)20 b(keys)64 b(DK1)87 b(DK2)f(DK3)h(DK4)f(DK5)h(DK6)101 2395 y(1-th)21 b(round)87 b(65025)42 b(65344)h(65280)f(26010)h(49152)f(57345) 101 2445 y(2-th)21 b(round)87 b(65533)42 b(32768)h(40960)f(52428)130 b(0)65 b(8192)101 2495 y(3-th)21 b(round)87 b(42326)42 b(65456)h(65472)f (21163)108 b(16)h(32)101 2545 y(4-th)21 b(round)87 b(21835)42 b(65424)h(57344)f(65025)65 b(2048)f(4096)101 2595 y(5-th)21 b(round)87 b(13101)42 b(51200)h(53248)f(65533)130 b(8)109 b(12)101 2644 y(6-th)21 b(round)87 b(19115)42 b(65504)h(65508)f(49153)108 b(16)h(20)101 2694 y(7-th)21 b(round)87 b(43670)42 b(61440)h(61952)f(65409)65 b(2048)f(2560)101 2744 y(8-th)21 b(round)87 b(18725)42 b(64512)h(65528)f (21803)130 b(5)g(6)101 2794 y(9-th)21 b(round)174 b(1)43 b(65534)g(65533)f (49153)p eop %%Page: 40 20 bop 36 103 a Fw(40)900 b Fq(CH.)16 b(3.)38 b(THE)16 b(BLOCK)g(CIPHER)g(IDEA) 58 275 y Fa(plaintext)k(X)152 b(0)g(1)h(2)f(3)145 325 y(after)145 375 y(1-th)21 b(rnd)108 b(240)h(245)f(266)g(261)145 424 y(2-th)21 b(rnd)87 b(8751)f(8629)64 b(62558)h(59737)145 474 y(3-th)21 b(rnd)87 b(3974)64 b(14782)g(36584)87 b(4467)145 524 y(4-th)21 b(rnd)65 b(22495)f(44120)g(50779)h(47693)145 574 y(5-th)21 b(rnd)65 b(36481)f(47772)g(63359)h(14922)145 624 y(6-th)21 b(rnd)65 b(26946)f(37897)g(57883)87 b(7268)145 673 y(7-th)21 b(rnd)65 b(39376)f(51190)g(21297)h(25102)145 723 y(8-th)21 b(rnd)87 b(2596)108 b(152)64 b(60523)h(18725)58 823 y(ciphertext)20 b(Y)87 b(4603)64 b(60715)108 b(408)65 b(28133)145 873 y(after)145 922 y(1-th)21 b(rnd)65 b(55693)f(54065)g(10230)h(33464)145 972 y(2-th)21 b(rnd)65 b(48205)f(57963)g(37961)h(42358)145 1022 y(3-th)21 b(rnd)87 b(2724)64 b(63471)g(55964)87 b(9443)145 1072 y(4-th)21 b(rnd)65 b(51782)f(65115)g(56408)87 b(4461)145 1122 y(5-th)21 b(rnd)65 b(29839)f(36616)g(14810)h(17868)145 1172 y(6-th)21 b(rnd)65 b(12902)86 b(1118)64 b(12213)h(45102)145 1221 y(7-th)21 b(rnd)87 b(1680)f(1290)108 b(253)87 b(7674)145 1271 y(8-th)21 b(rnd)152 b(0)g(5)h(3)130 b(12)58 1371 y(result)108 b(T)152 b(0)g(1)h(2)f(3)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF