Security-55471.14.18.tar.gz

[apple/security.git] / libsecurity_cryptkit / lib / CurveParamDocs / FEEDsansY.nb

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    \(\( (*\n\tNo - Y - coordinate\ version\ of\ Algorithm\ 8.1  .10; \n\t
      see\ program\ 8.1  .10 . directembed . nb\n\t\t\t\n\n\ Support\ code\ 
        for\n\ R . \ Crandall\ and\ C . \ Pomerance, \n\ 
      "\<Prime Numbers: a Computational Perspective,\>"\n\ Springer - 
        Verlag\ 2001. \n\ c . \ 2000\ Perfectly\ Scientific, \ 
      Inc . \n\ All\ Rights\ Reserved . \n\t\n\t20\ Apr\ 2001\ RC\ 
        \((revamped\ for\ simplicity)\)\n\ 10\ Dec\ 2000\ AH\ 
        \((Formatting)\)\n\t14\ Sep\ 2000\ RT\ \((Creation)\)\n*) \n\)\)], 
  "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\( (*\ CODE\ *) \n
    \n (*\ First, \ a\ function\ for\ inverting\ n\ mod\ \(p . \)\ *) \n
    ellinv[n_]\  := \ If[n == 0, 0, PowerMod[n, \(-1\), p]]; \n
    \n (*\ Next, \ 
      a\ function\ for\ normalizing\ the\ x\ \(coordinate . \)\ *) \n
    ex[pt_]\  := \ Mod[pt[\([1]\)]\ *\ ellinv[pt[\([2]\)]], \ p]; \n
    \n (*\ Next, \ 
      the\ doubleh \(()\)\ function\ for\ doubling\ a\ \(point . \)\ *) \n
    elleven[pt_]\  := \ \n\t
      Block[{x1\  = \ pt[\([1]\)], \ z1\  = \ pt[\([2]\)], \ e, \ f\ }, \n
        \ \ \t\te\  = \ 
          Mod[\((x1^2\  - \ a\ z1^2)\)^2\  - \ 
              4\ b\ \((2\ x1\  + \ c\ z1)\)\ z1^3, \ p]; \n\ \ \t\t
        f\  = \ Mod[
            4\ z1\ \((x1^3\  + \ c\ x1^2\ z1\  + \ a\ x1\ z1^2\  + \ b\ z1^3)
                \), \ p]; \n\ \ \t\t{e, f}\n\t]; \n
    \n (*\ Next, \ 
      the\ addh \(()\)\ function\ for\ adding\ pt\ and\ pu\ with\ pv\  = \ 
        pt - pu\ known\ \n
            \(\((x\ and\ z\ coordinates\ only\ of\ course)\) . \)\ *) \n
    ellodd[pt_, \ pu_, \ pv_]\  := \ \n\t
      Block[\n\t\t{x1\  = \ pt[\([1]\)], \ z1\  = \ pt[\([2]\)], \n\t\t\ 
          x2\  = \ pu[\([1]\)], \ z2\  = \ pu[\([2]\)], \n\t\t\ 
          xx\  = \ pv[\([1]\)], \ zz\  = \ pv[\([2]\)], \ i, \ j\n\t\t\ }, \n
        \ \ \t\ \ \ \ \ 
        i\  = \ Mod[
            zz\ \((\((x1\ x2\  - \ a\ z1\ z2)\)^2\  - \n
                  \ \ \t\ \ \ \ \ \ \ \ \ \ \t
                  4\ b \((x1\ z2\  + \ x2\ z1\  + \ c\ z1\ z2)\)\ z1\ z2)\), 
            \ \n\ \ \t\ \ \ \ \ \ \ \ \ \ \tp\n\ \ \t\ \ \ \ \ \ \ \ \ ]; \n
        \ \ \t\ \ \ \ \ j\  = \ Mod[xx\ \((x1\ z2\  - \ x2\ z1)\)^2, \ p]; \n
        \ \ \t\t\ {i, j}\n\t]; \n
    \n (*\ Now, \ the\ main\ routine, \ elliptic\ multiply\ [k] \(pt . \)\ *) 
      \nelliptic[pt_, \ k_]\  := \ \n\t
      Block[{porg, \ ps, \ pp, \ q}, \n\t\tIf[k\  == 1, \ Return[pt]]; \n\t\t
        If[k\  == 2, \ Return[elleven[pt]]]; \n\t\tporg\  = \ pt; \n\t\t
        ps\  = \ elleven[pt]; \n\t\tpp\  = \ pt; \n\t\t
        bitlist\  = \ Reverse[IntegerDigits[k, 2]]; \n\t\t
        Do[\t\ \ \ \n\t\ \ \ \t\t
          If[bitlist[\([q]\)]\  == \ 1, \n\t\ \ \ \t\ \ \ \t\t
            pp\  = \ ellodd[ps, \ pp, \ porg]; \n\t\ \ \ \t\ \ \ \t\t
            ps\  = \ elleven[ps]\n\t\ \ \ \t\ \ \ \t\t, \n
            \t\ \ \ \t\ \ \ \ \ \ \tps\  = \ ellodd[pp, \ ps, \ porg]; \n
            \t\t\ \ \ \ \ \tpp\  = \ elleven[pp]\n\t\ \ \ \t\t]\n
          \t\ \ \ \t\t, \n
          \t\ \ \ \t\t{q, \ Length[bitlist] - 1, \ 1, \ \(-1\)}\n\ \ \ \ \t]; 
        \n\ \ \ \ \tReturn[Mod[pp, p]]\n\t]; \n
    \n (*\ Next, \ 
      we\ include\ algorithm\ 2.3  .8\ for\ finding\ square\ roots\ \nmodulo\ 
        a\ prime\ \(p . \)\ *) \n\n
    sqrtmod[b_, p_] := \ \n\t
      Module[{a, x, c, d, cd, m, t, tst}, \n\ \ \ \t\ta\  = \ Mod[b, p]; \n
        \ \ \ \t\tIf[p\  == \ 2, \ Return[a]]; \n\ \ \ \ \t
        If[MemberQ[{3, 7}, Mod[p, 8]], \n\ \ \ \ \ \ \t\t
          Return[PowerMod[a, \((p + 1)\)/4, p]]\n\ \ \ \ \ \ \t]; \n\ \ \ \ \t
        If[Mod[p, 8]\  == \ 5, \n\ \ \ \ \ \ \t\t
          x\  = \ PowerMod[a, \((p + 3)\)/8, p]; \n\ \ \ \ \ \ \t\t
          c\  = \ Mod[x^2, p]; \n\ \ \ \ \ \ \t\t
          If[Not[c\  == \ a], \n\ \ \ \ \ \ \ \ \t\t
            Return[Mod[x\ PowerMod[2, \((p - 1)\)/4, p], \ p]]\n
            \ \ \ \ \ \ \ \ \t]; \n\ \ \ \ \ \ \t]; \n\ \ \ \ \t\n
        \ \ \ \ \t (*\ Here, \ p\  = \ 1\ \(\((mod\ 8)\) . \)\ *) \n
        \ \ \ \ \ \ \ttst\  = \ 1; \n\ \ \ \ \ \ \t
        While[Not[tst\  == \ \(-1\)], \n\ \ \ \ \ \ \ \ \t
          d\  = \ Random[Integer, {1, p}]; \n\ \ \ \ \ \ \ \ \t
          tst\  = \ JacobiSymbol[d, p]\n\ \ \ \ \ \ \ \ ]; \n\ \ \ \ \ \ \t
        t\  = \ \((p - 1)\)/2; \ s\  = \ 1; \n\ \ \ \ \ \ \t
        While[EvenQ[t], \ t\  = \ t/2; \ \(++s\)]; \n\ \ \ \ \ \ \t
        ca\  = \ PowerMod[a, t, p]; \n\ \ \ \ \ \ \t
        cd\  = \ PowerMod[d, t, p]; \n\ \ \ \ \ \ \tm\  = \ 0; \n
        \ \ \ \ \ \ \t
        Do[\n\ \ \ \ \ \ \t\ \ \ 
          If[PowerMod[Mod[ca\ PowerMod[cd, \ m, \ p], p], \ 
                2^\((s - 1 - i)\), \ p]\n\ \ \ \ \ \ \t\ \ \ \t\t == \ p - 1, 
            \ m\  += \ 2^i]\n\ \ \ \ \ \ \t\ \ \ , {i, 0, s - 1}\n
          \ \ \ \ \ \ \t]; \ \ \ \ \ \ \t\ \ \ \ \n\ \ \ \ \ \ \t
        Return[Mod[PowerMod[a, \ \((t + 1)\)/2, p]\ PowerMod[cd, \ m/2, p], 
            p]]; \ \n\t]; \n
    \n (*\ Next, \ a\ function\ relevant\ to\ Algorithm\ 7.2 \( .8 . \)\ *) \n
    \nellXadd[x1_, x2_] := \n\t
      Module[{u2, v, g}, \[IndentingNewLine]\t\tg = x1 - x2; 
        \[IndentingNewLine]\t\tden = PowerMod[g, \(-2\), p]; 
        \[IndentingNewLine]\t\t
        alpha = Mod[
            \((\((x1\ x2 + a)\) \((x1 + x2)\) + 2  c\ x1\ x2 + 2  b)\), p]; 
        \[IndentingNewLine]\t\t
        beta = Mod[\((\((x1\ x2 - a)\)^2 - 4  b \((x1 + x2 + c)\))\), p]; 
        \[IndentingNewLine]\t\tdisc = Mod[alpha^2 - beta\ g^2, p]; 
        \[IndentingNewLine]\t\t{\ \ 
          Mod[\ den*\((alpha + sqrtmod[disc, p])\), p], \ \n\t\t\ \ \ \ 
          Mod[den*\((alpha - sqrtmod[disc, p])\), p]\n\t\t}
          \[IndentingNewLine]\t]; \n
    \n (*\ Now, \ 
      the\ main\ routine . \ Parameters\ are\ given\ for\ 161 - 
        bit\ prime\ field\n\t\t\tand\ specific\ curve; \n\t\ \ 
      direct\ embedding\ proceeds\ on\ "\<plaintext\>"\ integers\ x\ 
        \((mod\ p)\) . \ \n\ \ \ We\ start\ with\ relevant\ global\ 
        \((and\ public, \ except\ for\ kb)\)\n\ \ \ \(parameters . \)\n\ *) \n
    \[IndentingNewLine]p\  = \ 
      1654338658923174831024422729553880293604080853451; \na\  = \ \(-152\); 
    \nb = \ 722; \nc\  = \ 0; \ \  (*\ Montgomery\ \(parameter . \)\ *) \n
    \n (*\ Next, \ 
      create\ public\ point\ P\ of\ prime\ order\ on\ main\ \(curve . \)\ *) 
      \npubpoint\  = 
      \ {124590448755381588517063157600522073397838354227, \ 1}; \ \ \n
    pubpointtwist\  = 
      \ {1173563507729187954550227059395955904200719019884, 1}; \n\n
    kb\  = \ 968525826201321079923232842886222248; 
    \ \  (*\ Private\ key\ \(K_B . \)\ *) \n\n
    pubkey\  = \ \ \ elliptic[pubpoint, \ kb]; 
    \ \ \ \ \ \ \ \  (*\ Public\ key\ \(P_B . \)\ *) \n
    pubkeytwist\  = \ \telliptic[pubpointtwist, \ kb]; 
    \ \ \ \ \  (*\ Public\ key\ \(P_B' . \)\ *) \n\ \n\t\t\n
    encryptEmbed[x_] := \ 
      Module[{cubic, \ curve, \ X\  = \ x, \ pbk, \ pbp, \ clueX, \ X2, \ uX, 
          \n\t\t\ \ piece, \ try, \ sign}, 
        \[IndentingNewLine] (*\ First, \ 
          let\ us\ determine\ which\ curve . \ \n\t\t\ \ \ EITHER\ X\ lies\ in
              \ the\ curve\ y^2\  = \ X^3\  + \ c\ X^2\  + \ a\ X\  + \ b, \n
          \t\t\ \ \ 
          or\ on\ g\ y^2\  = \ X^3\  + \ c\ X^2\  + \ a\ X\  + \ b\ *) \n
        \t\t\ cubic\  = \ Mod[X\ Mod[X^2\  + c\ X\  + \ \ a, p]\  + \ b, p]; 
        \n\t\t\ If[JacobiSymbol[cubic, \ p]\  > \ \(-1\), \ \n
          \t\t\t\ \ \ \ \ \ curve\  = \ 1; \ pbk\  = \ pubkey; \ 
          pbp\  = \ pubpoint, \t\t\t\ \ \ \ \ \ \n\t\t\t\t\ \ \ \ \ 
          curve\  = \ \(-1\)\ ; \ pbk\  = \ pubkeytwist; \ 
          pbp\  = \ pubpointtwist; \ \n\t\t\ \ ]; \n\t\t\n\t\t\t
        r\  = \ Random[Integer, \ {2, p - 2}]; \t\t\ \ \n\t\t\t
        clueX\  = \ ex[elliptic[pbp, \ r]]; \n\t\t\ \ 
        X2\  = \ ex[elliptic[pbk, \ r]]; 
        \  (*\ We\ shall\ be\ adding\ the\ points\ having\ X, \ X2, \ 
          and\n\t\t\t\t\t\ \ \ there\ is\ a\ sign\ ambiguity\ a\ la\ Algorithm
              \ 7.2  .8\ because\ Y - 
            coordinates\n\t\t\t\t\t\t\ \ are\ being\ \(avoided . \)\ *) \ \n
        \t\t\ \ \ uX\  = \ \(ellXadd[X, \ X2]\)[\([1]\)]; \n\t\t\n
        \t\t (*\ Next, \ 
          feedback\ loop\ to\ determine\ which\ value\ of\ sign\ recovers\ 
            \(plaintext . \)\ *) \n\t\t\n\t\t\ \ \ 
        piece\  = \ ex[elliptic[{clueX, 1}, \ kb]]; \t\t\ \n\t\t\ \ \ 
        try\  = \ ellXadd[uX, \ piece]; \n\n\t\t\t\ 
        If[\ttry[\([1]\)]\  == \ X, \ sign\  = \ 1, \n
          \t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
          If[try[\([2]\)]\  == \ X, \ sign\  = \ \(-1\), \ Print["\<TILT!\>"]]
            \n\t\t\t]; \t\t\t\t\ \ \ \ \ \ \ \ \n
        \t\t\ \ {uX, \ clueX, \ curve, \ sign}\[IndentingNewLine]]; 
    \[IndentingNewLine]\n
    decryptEmbed[cipherList_] := \ 
      Module[{uX\  = \ cipherList[\([1]\)], \ 
          clueX\  = \ cipherList[\([2]\)], \ curve\  = \ cipherList[\([3]\)], 
          \ sign\  = \ cipherList[\([4]\)]}, \n\t\t\ \ \ 
        piece\  = \ ex[elliptic[{clueX, 1}, \ kb]]; \t\t\ \n\t\t\ \ \ 
        try\  = \ ellXadd[uX, \ piece]; \n\t\t\ \ \ 
        X\  = \ try[\([\((3 - sign)\)/2]\)]; \n\t\t\tX\[IndentingNewLine]]; 
    \[IndentingNewLine]\n\)\)], "Input"],

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Cell[CellGroupData[{

Cell[BoxData[
    \(\( (*\ EXAMPLE\ *) \ \n\n
      ciph\  = \ encryptEmbed[plain\  = \ Random[Integer, p - 1]]; \n
      If[decryptEmbed[ciph]\  != \ plain, \ Print["\<TILT!\>"]], {ct, 1, 10}]
      \)\)], "Input"],

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