626 lines
12 KiB
C++
626 lines
12 KiB
C++
/*
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Scott's AKE Client/Server testbed
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See http://eprint.iacr.org/2002/164
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Compile as
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cl /O2 /GX /DZZNS=5 ake6mntx.cpp zzn6a.cpp ecn3.cpp zzn3.cpp zzn2.cpp big.cpp zzn.cpp ecn.cpp miracl.lib
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using COMBA build
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MNT Curve - Tate pairing
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Thanks to Drew Sutherland for providing the MNT curve
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Irreducible binomial MUST be of the form x^6+2. This excludes many of the curves
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found using the mnt utility!
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Modified to prevent sub-group confinement attack
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NOTE: Key exchange bandwidth could be reduced further using ideas from
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"Doing more with Fewer Bits", Brouwer, Pellikaan & Verheul, Asiacrypt
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'99
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NOTE: This version uses a "compositum". That is the ZZn6 class is a cubic tower over ZZn2, but can
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also be considered as a quadratic tower over ZZn3. The routine shuffle converts from one form to the other.
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The former is fastest for ZZn6 arithmetic, the latter form is required for handling the second parameter
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to the pairing, which is on the quadratic twist E(Fp3)
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*/
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#include <iostream>
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#include <fstream>
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#include <string.h>
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#include "ecn.h"
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#include <ctime>
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#include "ecn3.h"
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#include "zzn6a.h"
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Miracl precision(5,0);
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#ifdef MR_COUNT_OPS
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extern "C"
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{
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int fpc,fpa,fpx,fpm2,fpi2;
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}
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#endif
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// Using SHA-1 as basic hash algorithm
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#define HASH_LEN 20
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//
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// Define one or the other of these
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//
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// Which is faster depends on the I/M ratio - See imratio.c
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// Roughly if I/M ratio > 16 use PROJECTIVE, otherwise use AFFINE
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//
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#ifdef MR_AFFINE_ONLY
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#define AFFINE
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#else
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#define PROJECTIVE
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#endif
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//
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// Tate Pairing Code
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//
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// Extract ECn point in internal ZZn format
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//
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void extract(ECn& A,ZZn& x,ZZn& y)
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{
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x=(A.get_point())->X;
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y=(A.get_point())->Y;
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}
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#ifdef PROJECTIVE
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void extract(ECn& A,ZZn& x,ZZn& y,ZZn& z)
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{
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big t;
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x=(A.get_point())->X;
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y=(A.get_point())->Y;
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t=(A.get_point())->Z;
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if (A.get_status()!=MR_EPOINT_GENERAL) z=1;
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else z=t;
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}
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#endif
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void set_frobenius_constant(ZZn2 &X)
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{
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Big p=get_modulus();
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switch (get_mip()->pmod8)
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{
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case 5:
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X.set((Big)0,(Big)1); // = (sqrt(-2)^(p-1)/2
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break;
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case 3: // = (1+sqrt(-1))^(p-1)/2
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X.set((Big)1,(Big)1);
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break;
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case 7:
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X.set((Big)2,(Big)1); // = (2+sqrt(-1))^(p-1)/2
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default: break;
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}
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X=pow(X,(p-1)/3);
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}
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ZZn6 shuffle(const ZZn3 &first, const ZZn3 &second)
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{ // shuffle from a pair ZZn3's to three ZZn2's, as required by ZZn6
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ZZn6 w;
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ZZn x0,x1,x2,x3,x4,x5;
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ZZn2 t0,t1,t2;
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first.get(x0,x2,x4);
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second.get(x1,x3,x5);
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t0.set(x0,x3);
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t1.set(x1,x4);
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t2.set(x2,x5);
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w.set(t0,t1,t2);
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return w;
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}
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void unshuffle(ZZn6 &S,ZZn3 &first,ZZn3 &second)
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{ // unshuffle a ZZn6 into two ZZn3's
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ZZn x0,x1,x2,x3,x4,x5;
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ZZn2 t0,t1,t2;
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S.get(t0,t1,t2);
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t0.get(x0,x3);
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t1.get(x1,x4);
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t2.get(x2,x5);
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first.set(x0,x2,x4);
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second.set(x1,x3,x5);
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}
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//
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// Line from A to destination C. Let A=(x,y)
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// Line Y-slope.X-c=0, through A, so intercept c=y-slope.x
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// Line Y-slope.X-y+slope.x = (Y-y)-slope.(X-x) = 0
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// Now evaluate at Q -> return (Qy-y)-slope.(Qx-x)
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//
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ZZn6 line(ECn& A,ECn& C,ECn& B,int type,ZZn& slope,ZZn& ex1,ZZn& ex2,ZZn3& Qx,ZZn3& Qy)
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{
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ZZn6 w;
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#ifdef AFFINE
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ZZn3 nn=Qx;
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ZZn x,y;
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extract(A,x,y);
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nn-=x;
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nn*=slope;
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nn+=y;
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nn=-nn;
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w=shuffle(nn,Qy);
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#endif
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#ifdef PROJECTIVE
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if (type==MR_ADD)
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{
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ZZn x2,y2,x3,z3;
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extract(B,x2,y2);
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extract(C,x3,x3,z3);
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w=shuffle(slope*(Qx-x2)+z3*y2,-z3*Qy);
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}
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if (type==MR_DOUBLE)
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{
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ZZn x,y,x3,z3;
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extract(A,x,y);
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extract(C,x3,x3,z3);
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w=shuffle((slope*ex2)*Qx-slope*x+ex1,-(z3*ex2)*Qy);
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}
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#endif
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return w;
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}
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//
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// Add A=A+B (or A=A+A)
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// Return line function value
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//
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ZZn6 g(ECn& A,ECn& B,ZZn3& Qx,ZZn3& Qy)
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{
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ZZn lam,extra1,extra2;
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int type;
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big ptr,ex1,ex2;
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ECn P=A;
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// Evaluate line from A
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type=A.add(B,&ptr,&ex1,&ex2);
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if (!type) return (ZZn6)1;
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lam=ptr;
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extra1=ex1;
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extra2=ex2;
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return line(P,A,B,type,lam,extra1,extra2,Qx,Qy);
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}
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//
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// Tate Pairing - note denominator elimination has been applied
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//
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// P is a point of order q. Q(x,y) is a point of order m.q.
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// Note that P is a point on the curve over Fp, Q(x,y) a point on the
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// twisted curve over the extension field Fp^3
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//
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#define WINDOW_SIZE 4
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#define PRECOMP (1<<(WINDOW_SIZE-1))
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BOOL fast_tate_pairing(ECn& P,ZZn3& Qx,ZZn3& Qy,Big &x,ZZn2& X,ZZn6& res)
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{
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int i,j,n,nb,nbw,nzs;
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ECn A,P2,t[PRECOMP];
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ZZn6 w,hc,z2n,zn[PRECOMP];
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Big q=x*x-x+1;
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res=zn[0]=1;
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t[0]=P2=A=P;
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z2n=g(P2,P2,Qx,Qy); // P2=P+P
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normalise(P2);
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//
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// Build windowing table
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//
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for (i=1;i<PRECOMP;i++)
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{
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hc=g(A,P2,Qx,Qy);
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t[i]=A;
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zn[i]=z2n*zn[i-1]*hc;
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}
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multi_norm(PRECOMP,t); // make t points Affine
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/*
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A=P; // reset A
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nb=bits(q);
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for (i=nb-2;i>=0;i--)
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{
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res*=res;
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res*=g(A,A,Qx,Qy);
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if (bit(q,i)==1)
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res*=g(A,P,Qx,Qy);
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if (res.iszero()) return FALSE;
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}
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*/
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A=P; // reset A
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nb=bits(q);
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for (i=nb-2;i>=0;i-=(nbw+nzs))
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{ // windowing helps a little..
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n=window(q,i,&nbw,&nzs,WINDOW_SIZE); // standard MIRACL windowing
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for (j=0;j<nbw;j++)
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{
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res*=res;
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res*=g(A,A,Qx,Qy);
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}
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if (n>0)
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{
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res*=zn[n/2];
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res*=g(A,t[n/2],Qx,Qy);
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}
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for (j=0;j<nzs;j++)
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{
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res*=res;
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res*=g(A,A,Qx,Qy);
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}
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if (res.iszero()) return FALSE;
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}
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#ifdef MR_COUNT_OPS
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printf("After Miller fpc= %d fpa= %d fpx= %d\n",fpc,fpa,fpx);
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#endif
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// if (!A.iszero() || res.iszero()) return FALSE;
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w=res;
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w.powq(X);
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res*=w; // ^(p+1)
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w=res;
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w.powq(X); w.powq(X); w.powq(X);
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res=w/res; // ^(p^3-1)
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// exploit the clever "trick" for a half-length exponentiation!
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res.mark_as_unitary();
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w=res;
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res.powq(X); // res*=res; // res=pow(res,CF);
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if (x<0) res/=powu(w,-x);
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else res*=powu(w,x);
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if (res==(ZZn6)1) return FALSE;
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return TRUE;
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}
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//
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// ecap(.) function
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//
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BOOL ecap(ECn& P,ECn3& Q,Big& x,ZZn2 &X,ZZn6& res)
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{
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BOOL Ok;
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ECn PP=P;
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ZZn3 Qx,Qy;
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int qnr=get_mip()->cnr;
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normalise(PP);
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Q.get(Qx,Qy);
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// untwist
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Qx=Qx/qnr;
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Qy=tx(Qy);
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Qy=Qy/(qnr*qnr);
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#ifdef MR_COUNT_OPS
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fpc=fpa=fpx=0;
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#endif
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Ok=fast_tate_pairing(PP,Qx,Qy,x,X,res);
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#ifdef MR_COUNT_OPS
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printf("After pairing fpc= %d fpa= %d fpx= %d\n",fpc,fpa,fpx);
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fpa=fpc=fpx=0;
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#endif
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if (Ok) return TRUE;
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return FALSE;
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}
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//
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// Hash functions
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//
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Big H1(char *string)
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{ // Hash a zero-terminated string to a number < modulus
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Big h,p;
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char s[HASH_LEN];
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int i,j;
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sha sh;
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shs_init(&sh);
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for (i=0;;i++)
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{
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if (string[i]==0) break;
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shs_process(&sh,string[i]);
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}
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shs_hash(&sh,s);
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p=get_modulus();
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h=1; j=0; i=1;
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forever
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{
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h*=256;
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if (j==HASH_LEN) {h+=i++; j=0;}
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else h+=s[j++];
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if (h>=p) break;
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}
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h%=p;
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return h;
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}
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Big H2(ZZn6 y)
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{ // Hash and compress an Fp6 to a big number
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sha sh;
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ZZn u,v,w;
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ZZn2 x;
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Big a,h,p,xx[2];
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char s[HASH_LEN];
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int i,j,m;
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shs_init(&sh);
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y.get(x);
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x.get(u,v);
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xx[0]=u; xx[1]=v;
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for (i=0;i<2;i++)
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{
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a=xx[i];
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while (a>0)
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{
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m=a%256;
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shs_process(&sh,m);
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a/=256;
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}
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}
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shs_hash(&sh,s);
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h=from_binary(HASH_LEN,s);
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return h;
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}
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// Hash and map a Server Identity to a curve point E_(Fp3)
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ECn3 hash_and_map3(char *ID)
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{
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int i;
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ECn3 S;
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ZZn3 X;
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Big x0=H1(ID);
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forever
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{
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x0+=1;
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X.set2((ZZn)x0);
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if (!S.set(X)) continue;
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break;
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}
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// cout << "S= " << S << endl;
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return S;
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}
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// Hash and map a Client Identity to a curve point E_(Fp) of order q
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ECn hash_and_map(char *ID)
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{
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ECn Q;
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Big x0=H1(ID);
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while (!Q.set(x0,x0)) x0+=1;
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return Q;
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}
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BOOL member(ZZn6 r,Big &x,ZZn2 &X)
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{ // check its an element of order q
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ZZn6 w=r;
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w.powq(X);
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if (x<0) r=powu(inverse(r),-x);
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else r=powu(r,x);
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if (r==w) return TRUE;
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return FALSE;
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}
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void q_power_frobenius(ECn3 &S,ZZn2& X)
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{
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ZZn6 X1,X2,Y1,Y2;
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ZZn3 Sx,Sy,T;
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int qnr=get_mip()->cnr;
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S.get(Sx,Sy);
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// untwist
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Sx=Sx/qnr;
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Sy=tx(Sy);
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Sy=Sy/(qnr*qnr);
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X1=shuffle(Sx,(ZZn3)0); Y1=shuffle((ZZn3)0,Sy);
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X1.powq(X); Y1.powq(X);
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unshuffle(X1,Sx,T); unshuffle(Y1,T,Sy);
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// twist
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Sx=qnr*Sx;
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Sy=txd(Sy*qnr*qnr);
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S.set(Sx,Sy);
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}
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// Use Scott et al. idea - http://eprint.iacr.org/2008/530.pdf
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void cofactor(ECn3 &S,Big &x, ZZn2& X)
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{ // S=Phi(2xP)+phi^2(2xP)
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ZZn6 X1,X2,Y1,Y2;
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ZZn3 Sx,Sy,T;
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ECn3 S2;
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int qnr=get_mip()->cnr;
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S*=x; S+=S; // hard work done here
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S.get(Sx,Sy);
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// untwist
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Sx=Sx/qnr;
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Sy=tx(Sy);
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Sy=Sy/(qnr*qnr);
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X1=shuffle(Sx,(ZZn3)0); Y1=shuffle((ZZn3)0,Sy);
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X1.powq(X); Y1.powq(X);
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X2=X1; Y2=Y1;
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X2.powq(X); Y2.powq(X);
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unshuffle(X1,Sx,T); unshuffle(Y1,T,Sy);
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// twist
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Sx=qnr*Sx;
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Sy=txd(Sy*qnr*qnr);
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S.set(Sx,Sy);
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unshuffle(X2,Sx,T); unshuffle(Y2,T,Sy);
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//twist (again, like we did last summer...)
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Sx=qnr*Sx;
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Sy=txd(Sy*qnr*qnr);
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S2.set(Sx,Sy);
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S+=S2;
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}
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// GLV + Galbraith-Scott
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ZZn6 GT_pow(ZZn6& u,Big& k,Big& x,ZZn2& X)
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{
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ZZn6 v=u;
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v.powq(X);
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v=powu(v,k/x,u,k%x);
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return v;
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}
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ECn3 G2_mul(ECn3& P,Big& k,Big& x,ZZn2& X)
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{
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ECn3 V=P;
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q_power_frobenius(V,X);
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V=mul(V,k/x,P,k%x);
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return V;
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}
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int main()
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{
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miracl* mip=&precision;
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ECn Alice,Bob,sA,sB;
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ECn3 B6,Server,sS;
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ZZn6 sp,ap,bp;
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ZZn6 res;
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ZZn2 X;
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Big a,b,s,ss,p,q,x,y,B,cf,t,sru,T;
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int i,A;
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time_t seed;
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mip->IOBASE=16;
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x="-D285DA0CFEF02F06F812"; // MNT elliptic curve parameters (Thanks to Drew Sutherland)
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p=x*x+1;
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q=x*x-x+1;
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t=x+1;
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cf=x*x+x+1;
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T=t-1;
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// cout << "t-1= " << T << endl;
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// cout << "p%24= " << p%24 << endl;
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time(&seed);
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irand((long)seed);
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A=-3;
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B="77479D33943B5B1F590B54258B72F316B3261D45";
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#ifdef AFFINE
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ecurve(A,B,p,MR_AFFINE);
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#endif
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#ifdef PROJECTIVE
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ecurve(A,B,p,MR_PROJECTIVE);
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#endif
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set_frobenius_constant(X);
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sru=pow((ZZn)-2,(p-1)/6); // x^6+2 is irreducible
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set_zzn3(-2,sru);
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mip->IOBASE=16;
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mip->TWIST=MR_QUADRATIC; // map Server to point on twisted curve E(Fp3)
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ss=rand(q); // TA's super-secret
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cout << "Mapping Server ID to point" << endl;
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Server=hash_and_map3((char *)"Server");
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cofactor(Server,x,X);
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cout << "Mapping Alice & Bob ID's to points" << endl;
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Alice=hash_and_map((char *)"Alice");
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Bob= hash_and_map((char *)"Robert");
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cout << "Alice, Bob and the Server visit Trusted Authority" << endl;
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sS=G2_mul(Server,ss,x,X);
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sA=ss*Alice;
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sB=ss*Bob;
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cout << "Alice and Server Key Exchange" << endl;
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a=rand(q); // Alice's random number
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s=rand(q); // Server's random number
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if (!ecap(sA,Server,x,X,res)) cout << "Trouble" << endl;
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if (!member(res,x,X))
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{
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cout << "Wrong group order - aborting" << endl;
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exit(0);
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}
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ap=GT_pow(res,a,x,X);//powu(res,a);
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if (!ecap(Alice,sS,x,X,res)) cout << "Trouble" << endl;
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if (!member(res,x,X))
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{
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cout << "Wrong group order - aborting" << endl;
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exit(0);
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}
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sp=GT_pow(res,s,x,X);
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cout << "Alice Key= " << H2(powu(sp,a)) << endl;
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cout << "Server Key= " << H2(powu(ap,s)) << endl;
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cout << "Bob and Server Key Exchange" << endl;
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b=rand(q); // Bob's random number
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s=rand(q); // Server's random number
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if (!ecap(sB,Server,x,X,res)) cout << "Trouble" << endl;
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if (!member(res,x,X))
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{
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cout << "Wrong group order - aborting" << endl;
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exit(0);
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}
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bp=GT_pow(res,b,x,X);
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if (!ecap(Bob,sS,x,X,res)) cout << "Trouble" << endl;
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if (!member(res,x,X))
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{
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cout << "Wrong group order - aborting" << endl;
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exit(0);
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}
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sp=GT_pow(res,s,x,X);
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cout << "Bob's Key= " << H2(powu(sp,b)) << endl;
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cout << "Server Key= " << H2(powu(bp,s)) << endl;
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return 0;
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}
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