OSX/libsecurity_cryptkit/lib/CurveParamDocs/FEEDsansY.nb

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  51     \(\( (*\n\tNo - Y - coordinate\ version\ of\ Algorithm\ 8.1  .10; \n\t
  52       see\ program\ 8.1  .10 . directembed . nb\n\t\t\t\n\n\ Support\ code\
  53         for\n\ R . \ Crandall\ and\ C . \ Pomerance, \n\
  54       "\<Prime Numbers: a Computational Perspective,\>"\n\ Springer -
  55         Verlag\ 2001. \n\ c . \ 2000\ Perfectly\ Scientific, \
  56       Inc . \n\ All\ Rights\ Reserved . \n\t\n\t20\ Apr\ 2001\ RC\
  57         \((revamped\ for\ simplicity)\)\n\ 10\ Dec\ 2000\ AH\
  58         \((Formatting)\)\n\t14\ Sep\ 2000\ RT\ \((Creation)\)\n*) \n\)\)],
  59   "Input"],
  60
  61 Cell[CellGroupData[{
  62
  63 Cell[BoxData[
  64     \(\( (*\ CODE\ *) \n
  65     \n (*\ First, \ a\ function\ for\ inverting\ n\ mod\ \(p . \)\ *) \n
  66     ellinv[n_]\  := \ If[n == 0, 0, PowerMod[n, \(-1\), p]]; \n
  67     \n (*\ Next, \
  68       a\ function\ for\ normalizing\ the\ x\ \(coordinate . \)\ *) \n
  69     ex[pt_]\  := \ Mod[pt[\([1]\)]\ *\ ellinv[pt[\([2]\)]], \ p]; \n
  70     \n (*\ Next, \
  71       the\ doubleh \(()\)\ function\ for\ doubling\ a\ \(point . \)\ *) \n
  72     elleven[pt_]\  := \ \n\t
  73       Block[{x1\  = \ pt[\([1]\)], \ z1\  = \ pt[\([2]\)], \ e, \ f\ }, \n
  74         \ \ \t\te\  = \
  75           Mod[\((x1^2\  - \ a\ z1^2)\)^2\  - \
  76               4\ b\ \((2\ x1\  + \ c\ z1)\)\ z1^3, \ p]; \n\ \ \t\t
  77         f\  = \ Mod[
  78             4\ z1\ \((x1^3\  + \ c\ x1^2\ z1\  + \ a\ x1\ z1^2\  + \ b\ z1^3)
  79                 \), \ p]; \n\ \ \t\t{e, f}\n\t]; \n
  80     \n (*\ Next, \
  81       the\ addh \(()\)\ function\ for\ adding\ pt\ and\ pu\ with\ pv\  = \
  82         pt - pu\ known\ \n
  83             \(\((x\ and\ z\ coordinates\ only\ of\ course)\) . \)\ *) \n
  84     ellodd[pt_, \ pu_, \ pv_]\  := \ \n\t
  85       Block[\n\t\t{x1\  = \ pt[\([1]\)], \ z1\  = \ pt[\([2]\)], \n\t\t\
  86           x2\  = \ pu[\([1]\)], \ z2\  = \ pu[\([2]\)], \n\t\t\
  87           xx\  = \ pv[\([1]\)], \ zz\  = \ pv[\([2]\)], \ i, \ j\n\t\t\ }, \n
  88         \ \ \t\ \ \ \ \
  89         i\  = \ Mod[
  90             zz\ \((\((x1\ x2\  - \ a\ z1\ z2)\)^2\  - \n
  91                   \ \ \t\ \ \ \ \ \ \ \ \ \ \t
  92                   4\ b \((x1\ z2\  + \ x2\ z1\  + \ c\ z1\ z2)\)\ z1\ z2)\),
  93             \ \n\ \ \t\ \ \ \ \ \ \ \ \ \ \tp\n\ \ \t\ \ \ \ \ \ \ \ \ ]; \n
  94         \ \ \t\ \ \ \ \ j\  = \ Mod[xx\ \((x1\ z2\  - \ x2\ z1)\)^2, \ p]; \n
  95         \ \ \t\t\ {i, j}\n\t]; \n
  96     \n (*\ Now, \ the\ main\ routine, \ elliptic\ multiply\ [k] \(pt . \)\ *)
  97       \nelliptic[pt_, \ k_]\  := \ \n\t
  98       Block[{porg, \ ps, \ pp, \ q}, \n\t\tIf[k\  == 1, \ Return[pt]]; \n\t\t
  99         If[k\  == 2, \ Return[elleven[pt]]]; \n\t\tporg\  = \ pt; \n\t\t
 100         ps\  = \ elleven[pt]; \n\t\tpp\  = \ pt; \n\t\t
 101         bitlist\  = \ Reverse[IntegerDigits[k, 2]]; \n\t\t
 102         Do[\t\ \ \ \n\t\ \ \ \t\t
 103           If[bitlist[\([q]\)]\  == \ 1, \n\t\ \ \ \t\ \ \ \t\t
 104             pp\  = \ ellodd[ps, \ pp, \ porg]; \n\t\ \ \ \t\ \ \ \t\t
 105             ps\  = \ elleven[ps]\n\t\ \ \ \t\ \ \ \t\t, \n
 106             \t\ \ \ \t\ \ \ \ \ \ \tps\  = \ ellodd[pp, \ ps, \ porg]; \n
 107             \t\t\ \ \ \ \ \tpp\  = \ elleven[pp]\n\t\ \ \ \t\t]\n
 108           \t\ \ \ \t\t, \n
 109           \t\ \ \ \t\t{q, \ Length[bitlist] - 1, \ 1, \ \(-1\)}\n\ \ \ \ \t];
 110         \n\ \ \ \ \tReturn[Mod[pp, p]]\n\t]; \n
 111     \n (*\ Next, \
 112       we\ include\ algorithm\ 2.3  .8\ for\ finding\ square\ roots\ \nmodulo\
 113         a\ prime\ \(p . \)\ *) \n\n
 114     sqrtmod[b_, p_] := \ \n\t
 115       Module[{a, x, c, d, cd, m, t, tst}, \n\ \ \ \t\ta\  = \ Mod[b, p]; \n
 116         \ \ \ \t\tIf[p\  == \ 2, \ Return[a]]; \n\ \ \ \ \t
 117         If[MemberQ[{3, 7}, Mod[p, 8]], \n\ \ \ \ \ \ \t\t
 118           Return[PowerMod[a, \((p + 1)\)/4, p]]\n\ \ \ \ \ \ \t]; \n\ \ \ \ \t
 119         If[Mod[p, 8]\  == \ 5, \n\ \ \ \ \ \ \t\t
 120           x\  = \ PowerMod[a, \((p + 3)\)/8, p]; \n\ \ \ \ \ \ \t\t
 121           c\  = \ Mod[x^2, p]; \n\ \ \ \ \ \ \t\t
 122           If[Not[c\  == \ a], \n\ \ \ \ \ \ \ \ \t\t
 123             Return[Mod[x\ PowerMod[2, \((p - 1)\)/4, p], \ p]]\n
 124             \ \ \ \ \ \ \ \ \t]; \n\ \ \ \ \ \ \t]; \n\ \ \ \ \t\n
 125         \ \ \ \ \t (*\ Here, \ p\  = \ 1\ \(\((mod\ 8)\) . \)\ *) \n
 126         \ \ \ \ \ \ \ttst\  = \ 1; \n\ \ \ \ \ \ \t
 127         While[Not[tst\  == \ \(-1\)], \n\ \ \ \ \ \ \ \ \t
 128           d\  = \ Random[Integer, {1, p}]; \n\ \ \ \ \ \ \ \ \t
 129           tst\  = \ JacobiSymbol[d, p]\n\ \ \ \ \ \ \ \ ]; \n\ \ \ \ \ \ \t
 130         t\  = \ \((p - 1)\)/2; \ s\  = \ 1; \n\ \ \ \ \ \ \t
 131         While[EvenQ[t], \ t\  = \ t/2; \ \(++s\)]; \n\ \ \ \ \ \ \t
 132         ca\  = \ PowerMod[a, t, p]; \n\ \ \ \ \ \ \t
 133         cd\  = \ PowerMod[d, t, p]; \n\ \ \ \ \ \ \tm\  = \ 0; \n
 134         \ \ \ \ \ \ \t
 135         Do[\n\ \ \ \ \ \ \t\ \ \
 136           If[PowerMod[Mod[ca\ PowerMod[cd, \ m, \ p], p], \
 137                 2^\((s - 1 - i)\), \ p]\n\ \ \ \ \ \ \t\ \ \ \t\t == \ p - 1,
 138             \ m\  += \ 2^i]\n\ \ \ \ \ \ \t\ \ \ , {i, 0, s - 1}\n
 139           \ \ \ \ \ \ \t]; \ \ \ \ \ \ \t\ \ \ \ \n\ \ \ \ \ \ \t
 140         Return[Mod[PowerMod[a, \ \((t + 1)\)/2, p]\ PowerMod[cd, \ m/2, p],
 141             p]]; \ \n\t]; \n
 142     \n (*\ Next, \ a\ function\ relevant\ to\ Algorithm\ 7.2 \( .8 . \)\ *) \n
 143     \nellXadd[x1_, x2_] := \n\t
 144       Module[{u2, v, g}, \[IndentingNewLine]\t\tg = x1 - x2;
 145         \[IndentingNewLine]\t\tden = PowerMod[g, \(-2\), p];
 146         \[IndentingNewLine]\t\t
 147         alpha = Mod[
 148             \((\((x1\ x2 + a)\) \((x1 + x2)\) + 2  c\ x1\ x2 + 2  b)\), p];
 149         \[IndentingNewLine]\t\t
 150         beta = Mod[\((\((x1\ x2 - a)\)^2 - 4  b \((x1 + x2 + c)\))\), p];
 151         \[IndentingNewLine]\t\tdisc = Mod[alpha^2 - beta\ g^2, p];
 152         \[IndentingNewLine]\t\t{\ \
 153           Mod[\ den*\((alpha + sqrtmod[disc, p])\), p], \ \n\t\t\ \ \ \
 154           Mod[den*\((alpha - sqrtmod[disc, p])\), p]\n\t\t}
 155           \[IndentingNewLine]\t]; \n
 156     \n (*\ Now, \
 157       the\ main\ routine . \ Parameters\ are\ given\ for\ 161 -
 158         bit\ prime\ field\n\t\t\tand\ specific\ curve; \n\t\ \
 159       direct\ embedding\ proceeds\ on\ "\<plaintext\>"\ integers\ x\
 160         \((mod\ p)\) . \ \n\ \ \ We\ start\ with\ relevant\ global\
 161         \((and\ public, \ except\ for\ kb)\)\n\ \ \ \(parameters . \)\n\ *) \n
 162     \[IndentingNewLine]p\  = \
 163       1654338658923174831024422729553880293604080853451; \na\  = \ \(-152\);
 164     \nb = \ 722; \nc\  = \ 0; \ \  (*\ Montgomery\ \(parameter . \)\ *) \n
 165     \n (*\ Next, \
 166       create\ public\ point\ P\ of\ prime\ order\ on\ main\ \(curve . \)\ *)
 167       \npubpoint\  =
 168       \ {124590448755381588517063157600522073397838354227, \ 1}; \ \ \n
 169     pubpointtwist\  =
 170       \ {1173563507729187954550227059395955904200719019884, 1}; \n\n
 171     kb\  = \ 968525826201321079923232842886222248;
 172     \ \  (*\ Private\ key\ \(K_B . \)\ *) \n\n
 173     pubkey\  = \ \ \ elliptic[pubpoint, \ kb];
 174     \ \ \ \ \ \ \ \  (*\ Public\ key\ \(P_B . \)\ *) \n
 175     pubkeytwist\  = \ \telliptic[pubpointtwist, \ kb];
 176     \ \ \ \ \  (*\ Public\ key\ \(P_B' . \)\ *) \n\ \n\t\t\n
 177     encryptEmbed[x_] := \
 178       Module[{cubic, \ curve, \ X\  = \ x, \ pbk, \ pbp, \ clueX, \ X2, \ uX,
 179           \n\t\t\ \ piece, \ try, \ sign},
 180         \[IndentingNewLine] (*\ First, \
 181           let\ us\ determine\ which\ curve . \ \n\t\t\ \ \ EITHER\ X\ lies\ in
 182               \ the\ curve\ y^2\  = \ X^3\  + \ c\ X^2\  + \ a\ X\  + \ b, \n
 183           \t\t\ \ \
 184           or\ on\ g\ y^2\  = \ X^3\  + \ c\ X^2\  + \ a\ X\  + \ b\ *) \n
 185         \t\t\ cubic\  = \ Mod[X\ Mod[X^2\  + c\ X\  + \ \ a, p]\  + \ b, p];
 186         \n\t\t\ If[JacobiSymbol[cubic, \ p]\  > \ \(-1\), \ \n
 187           \t\t\t\ \ \ \ \ \ curve\  = \ 1; \ pbk\  = \ pubkey; \
 188           pbp\  = \ pubpoint, \t\t\t\ \ \ \ \ \ \n\t\t\t\t\ \ \ \ \
 189           curve\  = \ \(-1\)\ ; \ pbk\  = \ pubkeytwist; \
 190           pbp\  = \ pubpointtwist; \ \n\t\t\ \ ]; \n\t\t\n\t\t\t
 191         r\  = \ Random[Integer, \ {2, p - 2}]; \t\t\ \ \n\t\t\t
 192         clueX\  = \ ex[elliptic[pbp, \ r]]; \n\t\t\ \
 193         X2\  = \ ex[elliptic[pbk, \ r]];
 194         \  (*\ We\ shall\ be\ adding\ the\ points\ having\ X, \ X2, \
 195           and\n\t\t\t\t\t\ \ \ there\ is\ a\ sign\ ambiguity\ a\ la\ Algorithm
 196               \ 7.2  .8\ because\ Y -
 197             coordinates\n\t\t\t\t\t\t\ \ are\ being\ \(avoided . \)\ *) \ \n
 198         \t\t\ \ \ uX\  = \ \(ellXadd[X, \ X2]\)[\([1]\)]; \n\t\t\n
 199         \t\t (*\ Next, \
 200           feedback\ loop\ to\ determine\ which\ value\ of\ sign\ recovers\
 201             \(plaintext . \)\ *) \n\t\t\n\t\t\ \ \
 202         piece\  = \ ex[elliptic[{clueX, 1}, \ kb]]; \t\t\ \n\t\t\ \ \
 203         try\  = \ ellXadd[uX, \ piece]; \n\n\t\t\t\
 204         If[\ttry[\([1]\)]\  == \ X, \ sign\  = \ 1, \n
 205           \t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
 206           If[try[\([2]\)]\  == \ X, \ sign\  = \ \(-1\), \ Print["\<TILT!\>"]]
 207             \n\t\t\t]; \t\t\t\t\ \ \ \ \ \ \ \ \n
 208         \t\t\ \ {uX, \ clueX, \ curve, \ sign}\[IndentingNewLine]];
 209     \[IndentingNewLine]\n
 210     decryptEmbed[cipherList_] := \
 211       Module[{uX\  = \ cipherList[\([1]\)], \
 212           clueX\  = \ cipherList[\([2]\)], \ curve\  = \ cipherList[\([3]\)],
 213           \ sign\  = \ cipherList[\([4]\)]}, \n\t\t\ \ \
 214         piece\  = \ ex[elliptic[{clueX, 1}, \ kb]]; \t\t\ \n\t\t\ \ \
 215         try\  = \ ellXadd[uX, \ piece]; \n\t\t\ \ \
 216         X\  = \ try[\([\((3 - sign)\)/2]\)]; \n\t\t\tX\[IndentingNewLine]];
 217     \[IndentingNewLine]\n\)\)], "Input"],
 218
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 230 Cell[CellGroupData[{
 231
 232 Cell[BoxData[
 233     \(\( (*\ EXAMPLE\ *) \ \n\n
 234       ciph\  = \ encryptEmbed[plain\  = \ Random[Integer, p - 1]]; \n
 235       If[decryptEmbed[ciph]\  != \ plain, \ Print["\<TILT!\>"]], {ct, 1, 10}]
 236       \)\)], "Input"],
 237
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