(***********************************************************************
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(*CacheID: 232*)
(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[ 18376, 710]*)
(*NotebookOutlinePosition[ 19227, 740]*)
(* CellTagsIndexPosition[ 19183, 736]*)
(*WindowFrame->Normal*)
Notebook[{
Cell["\<\
(* curverecords
Recorded data for Apple ECC curves.
R. Crandall
3 Apr 2001
*)
pointQ[x_] := (JacobiSymbol[x^3 + c x^2 + a x + b, p] > -1);
(* Next, binary expansion for very old M'ca versions,
otherwise use IntegerDigits[.,2]. *)
bitList[k_] := Block[{li = {}, j = k},
\tWhile[j > 0,
\t li = Append[li, Mod[j,2]];
\t j = Floor[j/2];
\t];
\tReturn[Reverse[li]];
\t];
\t
ellinv[n_] := PowerMod[n,-1,p];
(* Next, obtain actual x,y coords via normalization:
{x,y,z} := {X/Z^2, Y/Z^3, 1}. *)
normalize[pt_] := Block[{z,z2,z3},
\t\tIf[pt[[3]] == 0, Return[pt]];
\t\tz = ellinv[pt[[3]]];
\t\tz2 = Mod[z^2,p];
\t\tz3 = Mod[z z2,p];
\t\tReturn[{Mod[pt[[1]] z2, p], Mod[pt[[2]] z3, p], 1}];
\t\t];
ellneg[pt_] := Mod[pt * {1,-1,1}, p];
ellsub[pt1_, pt2_] := elladd[pt1, ellneg[pt2]];
elldouble[pt_] := Block[{x,y,z,m,y2,s},
\tx = pt[[1]]; y = pt[[2]]; z = pt[[3]];
\tIf[(y==0) || (z==0), Return[{1,1,0}]];
\tm = Mod[3 x^2 + a Mod[Mod[z^2,p]^2,p],p];
\tz = Mod[2 y z, p];
\ty2 = Mod[y^2,p];
\ts = Mod[4 x y2,p];
\tx = Mod[m^2 - 2s,p];
\ty = Mod[m(s - x) - 8 y2^2,p];
\tReturn[{x,y,z}];
];
elladd[pt0_, pt1_] := Block[
\t{x0,y0,z0,x1,y1,z1,
\tt1,t2,t3,t4,t5,t6,t7},
\tx0 = pt0[[1]]; y0 = pt0[[2]]; z0 = pt0[[3]];
\tx1 = pt1[[1]]; y1 = pt1[[2]]; z1 = pt1[[3]];
\tIf[z0 == 0, Return[pt1]];
\tIf[z1 == 0, Return[pt0]];
\tt1 = x0;
\tt2 = y0;
\tt3 = z0;
\tt4 = x1;
\tt5 = y1;
\tIf[(z1 != 1),
\t\tt6 = z1;
\t\tt7 = Mod[t6^2, p];
\t\tt1 = Mod[t1 t7, p];
\t\tt7 = Mod[t6 t7, p];
\t\tt2 = Mod[t2 t7, p];
\t];
\tt7 = Mod[t3^2, p];
\tt4 = Mod[t4 t7, p];
\tt7 = Mod[t3 t7, p];
\tt5 = Mod[t5 t7, p];
\tt4 = Mod[t1-t4, p];
\tt5 = Mod[t2 - t5, p];
\tIf[t4 == 0, If[t5 == 0,
\t\t\t\t Return[elldouble[pt0]],
\t \t\t\t\tReturn[{1,1,0}]
\t \t\t\t]
\t];
\tt1 = Mod[2t1 - t4,p];
\tt2 = Mod[2t2 - t5, p];
\tIf[z1 != 1, t3 = Mod[t3 t6, p]];
\tt3 = Mod[t3 t4, p];
\tt7 = Mod[t4^2, p];
\tt4 = Mod[t4 t7, p];
\tt7 = Mod[t1 t7, p];
\tt1 = Mod[t5^2, p];
\tt1 = Mod[t1-t7, p];
\tt7 = Mod[t7 - 2t1, p];
\tt5 = Mod[t5 t7, p];
\tt4 = Mod[t2 t4, p];
\tt2 = Mod[t5-t4, p];
\tIf[EvenQ[t2], t2 = t2/2, t2 = (p+t2)/2];
\tReturn[{t1, t2, t3}];
];
\t\t
(* Next, elliptic-multiply a normalized pt by k. *)
elliptic[pt_, k_] := Block[{pt2, hh, kk, hb, kb, lenh, lenk},
\tIf[k==0, Return[{1,1,0}]];
\thh = Reverse[bitList[3k]];
\tkk = Reverse[bitList[k]];
\tpt2 = pt;
\tlenh = Length[hh];
\tlenk = Length[kk];
\tDo[
\t\tpt2 = elldouble[pt2];
\t\thb = hh[[b]];
\t\tIf[b <= lenk, kb = kk[[b]], kb = 0];
\t\tIf[{hb,kb} == {1,0},
\t\t\tpt2 = elladd[pt2, pt],
\t\t\tIf[{hb, kb} == {0,1},
\t\t\tpt2 = ellsub[pt2, pt]]
\t\t]
\t ,{b, lenh-1, 2,-1}
\t ];
\tReturn[pt2];
];
(* Next, provide point-finding functions. *)
(* Next, perform (a + b w)^n (mod p), where pair = {a,b}, w2 = w^2. *)
pow[pair_, w2_, n_, p_] := Block[{bitlist, z},
bitlist = bitList[n];
z = pair;
\tDo[\t
\t zi = Mod[z[[2]]^2,p];
\t z = {Mod[z[[1]]^2 + w2 zi, p], Mod[2 z[[1]] z[[2]], p]};
\t If[bitlist[[q]] == 1,
\t zi = Mod[pair[[2]] z[[2]], p];
\t \t z = {Mod[pair[[1]] z[[1]] + w2 zi, p],
\t \t Mod[pair[[1]] z[[2]] + pair[[2]] z[[1]], p]};
\t ],
\t {q,2,Length[bitlist]}
];
Return[z]
];
sqrt[x_, p_] := Module[{t, b, w2},
If[Mod[x,p] == 0, Return[0]];
\tIf[Mod[p,4] == 3, Return[PowerMod[x, (p+1)/4, p]]];
\tIf[Mod[p,8] == 5,
\t\tb = PowerMod[x, (p-1)/4, p];
\t\tIf[b==1, Return[PowerMod[x, (p+3)/8, p]],
\t\t\tReturn[Mod[2x PowerMod[4x, (p-5)/8,p],p]]
\t\t]
\t];
\tt = 2;
While[True,
w2 = Mod[t^2 - x, p];
If[JacobiSymbol[w2,p] == -1, Break[]];
++t
];
(* Next, raise (t + Sqrt[w2])^((p+1)/2). *)
t = pow[{t,1},w2, (p+1)/2, p];
Return[t[[1]]];
];
findpoint[start_] := Block[{x = start, y, s},
\tWhile[True,
\t s = Mod[x(Mod[x^2+a,p]) + b, p];
\t y = sqrt[s, p];
\t If[Mod[y^2, p] == s, Break[]];
\t ++x;
\t];
\tReturn[{x, y, 1}]
];
report[a_] := Module[{ia = IntegerDigits[a,65536]},
Prepend[Reverse[ia], Length[ia]]
];
\
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AspectRatioFixed->True],
Cell[CellGroupData[{
Cell["\<\
report[a_] := Module[{ia = IntegerDigits[a,65536]},
Prepend[Reverse[ia], Length[ia]]
];
(* Case of Weierstrass/feemod curve. *)
p = 2^127 + 57675
report[p]
r = 512000; s = 512001;
a = Mod[-3 r s^3, p]
report[a]
b = Mod[-2 r s^5, p]
report[b]
pt = findpoint[3];
pt
plusOrd = 170141183460469231756943134065055014407
report[plusOrd]
PrineQ[plusOrd]
minusOrd = 170141183460469231706431473366713312401
report[minusOrd]
PrimeQ[minusOrd]
elliptic[pt, plusOrd]
elliptic[pt, minusOrd]\
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\( (*\ Case\ of\ Weierstrass/gen . \ mod\ \(curve . \)\ *) \n
p\ = \ 1654338658923174831024422729553880293604080853451; \n
Mod[p, 4]\),
\(Length[IntegerDigits[p, 2]]\),
\(report[p]\),
\(PrimeQ[p]\n\n\),
\(a\ = \ \(-152\); \nreport[a]\),
\(b\ = \ Mod[722, \ p]\),
\(report[b]\),
\(ptplus\ = \
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\(ptminus\ = \
findpoint[1173563507729187954550227059395955904200719019884]\),
\(plusOrd\ = \ \ 1654338658923174831024425147405519522862430265804; \n
report[plusOrd]\),
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\(minusOrd\ = \ 2 p + 2\ - \ plusOrd\),
\(report[minusOrd]\),
\(PrimeQ[minusOrd]\n\),
\(pt2\ = \
elliptic[ptplus, \ plusOrd/\((2^2\ *\ 23\ *\ 359\ *\ 479\ *\ 102107)\)]
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Cell[CellGroupData[{
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(* Case of Weierstrass/feemod curve. *)
p = 2^160 + 5875
report[p]
PrimeQ[p]
r = 512; s = 513;
a = Mod[-3 r s^3, p]
report[a]
b = Mod[2 r s^5, p]
report[b]
pt = findpoint[3];
pt
plusOrd = 1461501637330902918203687223801810245920805144027
report[plusOrd]
PrimeQ[plusOrd]
minusOrd = 1461501637330902918203682441630755793391059953677
report[minusOrd]
PrimeQ[minusOrd]
elliptic[pt, plusOrd]
elliptic[pt, minusOrd]\
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\( (*\ Case\ of\ NIST\ P - 192. \ *) \n
p\ = \ 6277101735386680763835789423207666416083908700390324961279; \n
Mod[p, 4]\),
\(Length[IntegerDigits[p, 2]]\),
\(report[p]\),
\(PrimeQ[p]\n\n\),
\(a\ = \ \(-3\); \nreport[a]\),
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p]\),
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6277101735386680763835789423176059013767194773182842284081\),
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\(PrimeQ[minusOrd]\),
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