452 lines
8.9 KiB
C++
452 lines
8.9 KiB
C++
/*
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Scott's AKE Client/Server testbed
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See http://www.compapp.dcu.ie/research/CA_Working_Papers/wp02.html#0202
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Compile as
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cl /O2 /GX /DZZNS=16 ake8cpt.cpp zzn8.cpp zzn4.cpp zzn2.cpp ecn4.cpp big.cpp zzn.cpp ecn.cpp miracl.lib
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Fastest using COMBA build for 512-bit mod-mul
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Cocks-Pinch Curve - Tate pairing
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The file k8.ecs is required
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Security is G224/F4096 (224-bit group, 4096-bit field)
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Modified to prevent sub-group confinement attack
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**** NEW **** Based on the observation by R. Granger and D. Page and N.P. Smart in "High Security
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Pairing-Based Cryptography Revisited" that multi-exponentiation can be used for the final exponentiation
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of the Tate pairing, we suggest the Power Pairing, which calculates E(P,Q,e) = e(P,Q)^e, where the
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exponentiation by e is basically for free, as it can be folded into the multi-exponentiation.
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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 "ecn4.h"
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#include "zzn8.h"
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using namespace std;
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Miracl precision(16,0);
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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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//
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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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ZZn8 line(ECn& A,ECn& C,ZZn& slope,ZZn4& Qx,ZZn4& Qy)
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{
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ZZn8 w;
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ZZn4 m=Qx;
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ZZn x,y,z,t;
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#ifdef AFFINE
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extract(A,x,y);
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m-=x; m*=slope;
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w.set((ZZn4)-y,Qy); w-=m;
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#endif
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#ifdef PROJECTIVE
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extract(A,x,y,z);
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x*=z; t=z; z*=z; z*=t;
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x*=slope; t=slope*z;
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m*=t; m-=x; t=z;
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extract(C,x,x,z);
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m+=(z*y); t*=z;
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w.set(m,-Qy*t);
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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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// Bump up num
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//
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ZZn8 g(ECn& A,ECn& B,ZZn4& Qx,ZZn4& Qy)
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{
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int type;
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ZZn lam;
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big ptr;
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ECn P=A;
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// Evaluate line from A
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type=A.add(B,&ptr);
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if (!type) return (ZZn8)1;
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lam=ptr;
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return line(P,A,lam,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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// extension field Fp^2
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//
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// New! Power Pairing calculates E(P,Q,e) = e(P,Q)^e at no extra cost!
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//
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BOOL power_tate(ECn& P,ECn4 Q,Big& q,Big *cf,ZZn2 &Fr,Big &e,ZZn4& r)
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{
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int i,nb;
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ECn A;
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ZZn8 w,res,a[4];
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ZZn4 Qx,Qy;
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ZZn2 x,y;
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Big carry,ex[4];
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Big p=get_modulus();
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Q.get(Qx,Qy);
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Qx=txd(Qx);
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Qy=txd(txd(Qy));
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res=1;
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/* Left to right method */
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A=P;
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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))
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res*=g(A,P,Qx,Qy);
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}
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if (!A.iszero() || res.iszero()) return FALSE;
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w=res;
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w.powq(Fr); w.powq(Fr); // ^(p^4-1)
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w.powq(Fr); w.powq(Fr);
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res=w/res;
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res.mark_as_unitary();
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a[3]=res;
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a[2]=a[3]; a[2].powq(Fr);
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a[1]=a[2]; a[1].powq(Fr);
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a[0]=a[1]; a[0].powq(Fr);
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if (e.isone()) for (i=0;i<4;i++) ex[i]=cf[i];
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else
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{ // cf *= e
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carry=0;
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for (i=3;i>=0;i--)
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carry=mad(cf[i],e,carry,p,ex[i]);
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}
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res=pow(4,a,ex);
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r=real(res); // compression
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// r=powl(real(res),cf); // ^(p*p*p*p+1)/q
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if (r.isunity()) return FALSE;
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return TRUE;
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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 H4(ZZn4 x)
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{ // Hash an Fp2 to a big number
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sha sh;
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Big a,u,v;
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ZZn2 X,Y;
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char s[HASH_LEN];
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int m;
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shs_init(&sh);
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x.get(X,Y);
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X.get(u,v);
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a=u;
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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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a=v;
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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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Y.get(u,v);
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a=u;
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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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a=v;
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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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shs_hash(&sh,s);
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a=from_binary(HASH_LEN,s);
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return a;
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}
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// Hash and map a Server Identity to a curve point E(Fp4)
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ECn4 hash4(char *ID)
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{
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ECn4 T;
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ZZn4 x;
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ZZn2 X,Y=0;
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Big x0,y0=0;
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x0=H1(ID);
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do
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{
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X.set(x0,y0);
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x.set(X,Y);
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x0+=1;
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}
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while (!is_on_curve(x));
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T.set(x);
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return T;
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}
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// Hash and map a Client Identity to a curve point E(Fp)
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ECn hash_and_map(char *ID,Big cof)
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{
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ECn Q;
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Big x0=H1(ID);
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while (!is_on_curve(x0)) x0+=1;
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Q.set(x0); // Make sure its on E(F_p)
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Q*=cof;
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return Q;
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}
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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(sqrt(-2))^(p-1)/4
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X=pow(X,(p-1)/4);
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break;
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case 3:
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X.set((Big)1,(Big)1);
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X=pow(X,(p-3)/4);
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break;
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case 7:
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X.set((Big)2,(Big)1);
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X=pow(X,(p-3)/4); // note that 4 does not divide p-1, so this is the best we can do...
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default: break;
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}
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}
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int main()
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{
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ifstream common("k8.ecs"); // elliptic curve parameters
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miracl* mip=&precision;
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ECn Alice,Bob,sA,sB;
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ECn4 Server,sS;
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ZZn4 res,sp,ap,bp;
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ZZn2 Fr;
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Big a,b,s,ss,p,q,B,cof;
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Big cf[4];
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int i,bitz,A;
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time_t seed;
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cout << "Started" << endl;
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common >> bitz;
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mip->IOBASE=16;
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common >> p;
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common >> A;
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common >> B;
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common >> cof; // #E/q
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common >> q; // low hamming weight q
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common >> cf[0]; // [(p^4+1)/q]/(p*p*p)
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common >> cf[1]; // [(p^4+1)/q]/(p*p)
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common >> cf[2]; // [(p^4+1)/q]/p
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common >> cf[3]; // [(p^4+1)/q]%p
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cout << "Initialised... " << p%8 << endl;
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time(&seed);
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irand((long)seed);
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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(Fr);
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//cout << "Fr= " << Fr << endl;
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mip->IOBASE=16;
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mip->TWIST=MR_QUADRATIC; // map Server to point on twisted curve E(Fp4)
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// hash Identities to curve point
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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=hash4((char *)"Server");
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cout << "Mapping Alice & Bob ID's to points" << endl;
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Alice=hash_and_map((char *)"Alice",cof);
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Bob= hash_and_map((char *)"Robert",cof);
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cout << "Alice, Bob and the Server visit Trusted Authority" << endl;
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sS=ss*Server;
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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 (!power_tate(sA,Server,q,cf,Fr,a,res)) cout << "Trouble" << endl;
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if (powl(res,q)!=(ZZn4)1)
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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=powl(res,a);
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ap=res;
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if (!power_tate(Alice,sS,q,cf,Fr,s,res)) cout << "Trouble" << endl;
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if (powl(res,q)!=(ZZn4)1)
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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=powl(res,s);
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sp=res;
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cout << "Alice Key= " << H4(powl(sp,a)) << endl;
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cout << "Server Key= " << H4(powl(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 (!power_tate(sB,Server,q,cf,Fr,b,res)) cout << "Trouble" << endl;
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if (powl(res,q)!=(ZZn4)1)
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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=powl(res,b);
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bp=res;
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if (!power_tate(Bob,sS,q,cf,Fr,s,res)) cout << "Trouble" << endl;
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if (powl(res,q)!=(ZZn4)1)
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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=powl(res,s);
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sp=res;
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cout << "Bob's Key= " << H4(powl(sp,b)) << endl;
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cout << "Server Key= " << H4(powl(bp,s)) << endl;
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return 0;
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}
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