OSX/libsecurity_cryptkit/lib/CurveParamDocs/curverecords.nb

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  36
  37 (*CacheID: 232*)
  38
  39
  40 (*NotebookFileLineBreakTest
  41 NotebookFileLineBreakTest*)
  42 (*NotebookOptionsPosition[     18376,        710]*)
  43 (*NotebookOutlinePosition[     19227,        740]*)
  44 (*  CellTagsIndexPosition[     19183,        736]*)
  45 (*WindowFrame->Normal*)
  46
  47
  48
  49 Notebook[{
  50 Cell["\<\
  51 (* curverecords
  52
  53    Recorded data for Apple ECC curves.
  54
  55    R. Crandall
  56    3 Apr 2001
  57
  58
  59  *)
  60
  61 pointQ[x_] := (JacobiSymbol[x^3 + c x^2 + a x + b, p] > -1);
  62
  63 (* Next, binary expansion for very old M'ca versions,
  64    otherwise use IntegerDigits[.,2]. *)
  65 bitList[k_] := Block[{li = {}, j = k},
  66 \tWhile[j > 0,
  67 \t    li = Append[li, Mod[j,2]];
  68 \t    j = Floor[j/2];
  69 \t];
  70 \tReturn[Reverse[li]];
  71 \t];
  72 \t
  73 ellinv[n_] := PowerMod[n,-1,p];
  74 (* Next, obtain actual x,y coords via normalization:
  75    {x,y,z} := {X/Z^2, Y/Z^3, 1}. *)
  76 normalize[pt_] := Block[{z,z2,z3},
  77 \t\tIf[pt[[3]] == 0, Return[pt]];
  78 \t\tz = ellinv[pt[[3]]];
  79 \t\tz2 = Mod[z^2,p];
  80 \t\tz3 = Mod[z z2,p];
  81 \t\tReturn[{Mod[pt[[1]] z2, p], Mod[pt[[2]] z3, p], 1}];
  82 \t\t];
  83
  84 ellneg[pt_] := Mod[pt * {1,-1,1}, p];
  85 ellsub[pt1_, pt2_] := elladd[pt1, ellneg[pt2]];
  86 elldouble[pt_] := Block[{x,y,z,m,y2,s},
  87 \tx = pt[[1]]; y = pt[[2]]; z = pt[[3]];
  88 \tIf[(y==0) || (z==0), Return[{1,1,0}]];
  89 \tm = Mod[3 x^2 + a Mod[Mod[z^2,p]^2,p],p];
  90 \tz = Mod[2 y z, p];
  91 \ty2 = Mod[y^2,p];
  92 \ts = Mod[4 x y2,p];
  93 \tx = Mod[m^2 - 2s,p];
  94 \ty = Mod[m(s - x) - 8 y2^2,p];
  95 \tReturn[{x,y,z}];
  96 ];
  97
  98 elladd[pt0_, pt1_] := Block[
  99 \t{x0,y0,z0,x1,y1,z1,
 100 \tt1,t2,t3,t4,t5,t6,t7},
 101 \tx0 = pt0[[1]]; y0 = pt0[[2]]; z0 = pt0[[3]];
 102 \tx1 = pt1[[1]]; y1 = pt1[[2]]; z1 = pt1[[3]];
 103 \tIf[z0 == 0, Return[pt1]];
 104 \tIf[z1 == 0, Return[pt0]];
 105
 106 \tt1 = x0;
 107 \tt2 = y0;
 108 \tt3 = z0;
 109 \tt4 = x1;
 110 \tt5 = y1;
 111 \tIf[(z1 != 1),
 112 \t\tt6 = z1;
 113 \t\tt7 = Mod[t6^2, p];
 114 \t\tt1 = Mod[t1 t7, p];
 115 \t\tt7 = Mod[t6 t7, p];
 116 \t\tt2 = Mod[t2 t7, p];
 117 \t];
 118 \tt7 = Mod[t3^2, p];
 119 \tt4 = Mod[t4 t7, p];
 120 \tt7 = Mod[t3 t7, p];
 121 \tt5 = Mod[t5 t7, p];
 122 \tt4 = Mod[t1-t4, p];
 123 \tt5 = Mod[t2 - t5, p];
 124 \tIf[t4 == 0, If[t5 == 0,
 125 \t\t\t\t    Return[elldouble[pt0]],
 126 \t   \t\t\t\tReturn[{1,1,0}]
 127 \t   \t\t\t]
 128 \t];
 129 \tt1 = Mod[2t1 - t4,p];
 130 \tt2 = Mod[2t2 - t5, p];
 131 \tIf[z1 != 1, t3 = Mod[t3 t6, p]];
 132 \tt3 = Mod[t3 t4, p];
 133 \tt7 = Mod[t4^2, p];
 134 \tt4 = Mod[t4 t7, p];
 135 \tt7 = Mod[t1 t7, p];
 136 \tt1 = Mod[t5^2, p];
 137 \tt1 = Mod[t1-t7, p];
 138 \tt7 = Mod[t7 - 2t1, p];
 139 \tt5 = Mod[t5 t7, p];
 140 \tt4 = Mod[t2 t4, p];
 141 \tt2 = Mod[t5-t4, p];
 142 \tIf[EvenQ[t2], t2 = t2/2, t2 = (p+t2)/2];
 143 \tReturn[{t1, t2, t3}];
 144 ];
 145 \t\t
 146 (* Next, elliptic-multiply a normalized pt by k. *)
 147 elliptic[pt_, k_] := Block[{pt2, hh, kk, hb, kb, lenh, lenk},
 148 \tIf[k==0, Return[{1,1,0}]];
 149 \thh = Reverse[bitList[3k]];
 150 \tkk = Reverse[bitList[k]];
 151 \tpt2 = pt;
 152 \tlenh = Length[hh];
 153 \tlenk = Length[kk];
 154 \tDo[
 155 \t\tpt2 = elldouble[pt2];
 156 \t\thb = hh[[b]];
 157 \t\tIf[b <= lenk, kb = kk[[b]], kb = 0];
 158 \t\tIf[{hb,kb} == {1,0},
 159 \t\t\tpt2 = elladd[pt2, pt],
 160 \t\t\tIf[{hb, kb} == {0,1},
 161 \t\t\tpt2 = ellsub[pt2, pt]]
 162 \t\t]
 163 \t   ,{b, lenh-1, 2,-1}
 164 \t ];
 165 \tReturn[pt2];
 166 ];
 167
 168 (* Next, provide point-finding functions. *)
 169
 170 (* Next, perform (a + b w)^n (mod p), where pair = {a,b}, w2 = w^2. *)
 171 pow[pair_, w2_, n_, p_] := Block[{bitlist, z},
 172     bitlist = bitList[n];
 173     z = pair;
 174 \tDo[\t
 175 \t   zi = Mod[z[[2]]^2,p];
 176 \t   z = {Mod[z[[1]]^2 + w2 zi, p], Mod[2 z[[1]] z[[2]], p]};
 177 \t   If[bitlist[[q]] == 1,
 178 \t       zi = Mod[pair[[2]] z[[2]], p];
 179 \t   \t   z = {Mod[pair[[1]] z[[1]] + w2 zi, p],
 180 \t   \t         Mod[pair[[1]] z[[2]] + pair[[2]] z[[1]], p]};
 181 \t   ],
 182 \t   {q,2,Length[bitlist]}
 183     ];
 184     Return[z]
 185 ];
 186
 187
 188 sqrt[x_, p_] := Module[{t, b, w2},
 189     If[Mod[x,p] == 0, Return[0]];
 190 \tIf[Mod[p,4] == 3, Return[PowerMod[x, (p+1)/4, p]]];
 191 \tIf[Mod[p,8] == 5,
 192 \t\tb = PowerMod[x, (p-1)/4, p];
 193 \t\tIf[b==1, Return[PowerMod[x, (p+3)/8, p]],
 194 \t\t\tReturn[Mod[2x PowerMod[4x, (p-5)/8,p],p]]
 195 \t\t]
 196 \t];
 197 \tt = 2;
 198     While[True,
 199       w2 = Mod[t^2 - x, p];
 200       If[JacobiSymbol[w2,p] == -1, Break[]];
 201       ++t
 202     ];
 203     (* Next, raise (t + Sqrt[w2])^((p+1)/2). *)
 204     t = pow[{t,1},w2, (p+1)/2, p];
 205     Return[t[[1]]];
 206     ];
 207
 208 findpoint[start_] := Block[{x = start, y, s},
 209 \tWhile[True,
 210 \t   s = Mod[x(Mod[x^2+a,p]) + b, p];
 211 \t   y = sqrt[s, p];
 212 \t   If[Mod[y^2, p] == s, Break[]];
 213 \t   ++x;
 214 \t];
 215 \tReturn[{x, y, 1}]
 216 ];
 217
 218 report[a_] := Module[{ia = IntegerDigits[a,65536]},
 219       Prepend[Reverse[ia], Length[ia]]
 220       ];
 221     \
 222 \>", "Input",
 223   AspectRatioFixed->True],
 224
 225 Cell[CellGroupData[{
 226
 227 Cell["\<\
 228
 229
 230
 231 report[a_] := Module[{ia = IntegerDigits[a,65536]},
 232       Prepend[Reverse[ia], Length[ia]]
 233       ];
 234
 235
 236 (* Case of Weierstrass/feemod curve. *)
 237 p = 2^127 + 57675
 238 report[p]
 239 r = 512000; s = 512001;
 240
 241 a = Mod[-3 r s^3, p]
 242 report[a]
 243 b = Mod[-2 r s^5, p]
 244 report[b]
 245 pt = findpoint[3];
 246 pt
 247 plusOrd = 170141183460469231756943134065055014407
 248 report[plusOrd]
 249 PrineQ[plusOrd]
 250 minusOrd = 170141183460469231706431473366713312401
 251 report[minusOrd]
 252 PrimeQ[minusOrd]
 253 elliptic[pt, plusOrd]
 254 elliptic[pt, minusOrd]\
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 296 Cell[BoxData[{
 297     \( (*\ Case\ of\ Weierstrass/gen . \ mod\ \(curve . \)\ *) \n
 298     p\  = \ 1654338658923174831024422729553880293604080853451; \n
 299     Mod[p, 4]\),
 300     \(Length[IntegerDigits[p, 2]]\),
 301     \(report[p]\),
 302     \(PrimeQ[p]\n\n\),
 303     \(a\  = \ \(-152\); \nreport[a]\),
 304     \(b\  = \ Mod[722, \ p]\),
 305     \(report[b]\),
 306     \(ptplus\  = \
 307       findpoint[1245904487553815885170631576005220733978383542270]\),
 308     \(ptminus\  = \
 309       findpoint[1173563507729187954550227059395955904200719019884]\),
 310     \(plusOrd\  = \ \ 1654338658923174831024425147405519522862430265804; \n
 311     report[plusOrd]\),
 312     \(PrimeQ[plusOrd]\),
 313     \(minusOrd\  = \ 2  p + 2\  - \ plusOrd\),
 314     \(report[minusOrd]\),
 315     \(PrimeQ[minusOrd]\n\),
 316     \(pt2\  = \
 317       elliptic[ptplus, \ plusOrd/\((2^2\ *\ 23\ *\ 359\ *\ 479\ *\ 102107)\)]
 318         \),
 319     \(pt3\  = \ elliptic[ptminus, \ minusOrd/\((2^2\ *\ 5^2\ *\ 17^2)\)]\)}],
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 376 }, Open  ]]
 377 }, Open  ]],
 378
 379 Cell[CellGroupData[{
 380
 381 Cell["\<\
 382
 383
 384 (* Case of Weierstrass/feemod curve. *)
 385 p = 2^160 + 5875
 386 report[p]
 387 PrimeQ[p]
 388 r = 512; s = 513;
 389
 390 a = Mod[-3 r s^3, p]
 391 report[a]
 392 b = Mod[2 r s^5, p]
 393 report[b]
 394 pt = findpoint[3];
 395 pt
 396 plusOrd =  1461501637330902918203687223801810245920805144027
 397 report[plusOrd]
 398 PrimeQ[plusOrd]
 399 minusOrd = 1461501637330902918203682441630755793391059953677
 400 report[minusOrd]
 401 PrimeQ[minusOrd]
 402
 403 elliptic[pt, plusOrd]
 404 elliptic[pt, minusOrd]\
 405 \>", "Input",
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 459 }, Open  ]],
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 461 Cell[CellGroupData[{
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 463 Cell[BoxData[{
 464     \( (*\ Case\ of\ NIST\ P - 192. \ *) \n
 465     p\  = \ 6277101735386680763835789423207666416083908700390324961279; \n
 466     Mod[p, 4]\),
 467     \(Length[IntegerDigits[p, 2]]\),
 468     \(report[p]\),
 469     \(PrimeQ[p]\n\n\),
 470     \(a\  = \ \(-3\); \nreport[a]\),
 471     \(b\  = \
 472       Mod[\(-2455155546008943817740293915197451784769108058161191238065\), \
 473         p]\),
 474     \(report[b]\),
 475     \(pt\  = \ findpoint[3]; \npt\),
 476     \(plusOrd\  = \ \
 477       6277101735386680763835789423176059013767194773182842284081\),
 478     \(report[plusOrd]\),
 479     \(PrimeQ[plusOrd]\),
 480     \(minusOrd\  = \ 2  p + 2\  - \ plusOrd\),
 481     \(report[minusOrd]\),
 482     \(PrimeQ[minusOrd]\),
 483     \(elliptic[pt, \ plusOrd]\),
 484     \(pt2\  = \ elliptic[pt, \ 23]\),
 485     \(pt\  = \ elliptic[pt2, \ minusOrd/23]\),
 486     \(report[minusOrd/23]\)}], "Input"],
 487
 488 Cell[BoxData[
 489     \(3\)], "Output"],
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