416 lines
8.2 KiB
C++
416 lines
8.2 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=16 ake2sst.cpp zzn2.cpp big.cpp zzn.cpp ecn.cpp miracl.lib
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using COMBA build
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Super-Singular curve - Tate pairing
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Requires file k2ss.ecs which contains details of supersingular
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elliptic curve, with order divisible by q=2^159+2^17+1, and security
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multiplier k=2. The prime p is 512 bits
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CHANGES: Use twisted curve to avoid ECn2 arithmetic completely
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Use Lucas functions to evaluate powers.
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Output of tate pairing is now a ZZn (half-size)
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Modified to prevent sub-group confinement attack
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*/
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#include <iostream>
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#include <fstream>
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#include "ecn.h"
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#include <ctime>
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#include "zzn2.h"
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using namespace std;
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Miracl precision(16,0);
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// Using SHA-512 as basic hash algorithm
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#define HASH_LEN 64
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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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ZZn2 line(ECn& A,ECn& C,ECn& B,int type,ZZn& slope,ZZn& ex1,ZZn& ex2,ZZn& a,ZZn& d)
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{
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ZZn2 w;
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ZZn x,y,z,t,n=a;
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#ifdef AFFINE
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extract(A,x,y);
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n+=x; n*=slope;
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w.set(y,-d); w-=n;
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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.set(slope*(a+x2)-z3*y2,z3*d);
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return w;
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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.set(-(slope*ex2)*a-slope*x+ex1,-(z3*ex2)*d);
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return w;
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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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// Bump up num
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//
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ZZn2 g(ECn& A,ECn& B,ZZn& a,ZZn& d)
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{
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int type;
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ZZn lam,extra1,extra2;
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ECn P=A;
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big ptr,ex1,ex2;
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// Evaluate line from A - lam is line slope
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type=A.add(B,&ptr,&ex1,&ex2);
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if (!type) return (ZZn2)1;
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lam=ptr; // in projective case slope = lam/A.z
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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,a,d);
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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 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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// curve E(Fp^2) -> Q([Qx,0],[0,Qy])
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// Here we have morphed Q onto the twisted curve E'(Fp)
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//
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BOOL tate(ECn& P,ECn& Q,Big& q,ZZn& r)
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{
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int i,nb,qnr;
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ZZn2 res;
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ZZn a,d;
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Big p,x,y,n;
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ECn A;
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p=get_modulus();
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// Note that q is fixed - q.P=2^17*(2^142.P + P) + P
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normalise(Q); // make sure z=1
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extract(Q,a,d);
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qnr=get_mip()->qnr;
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if (qnr==-2)
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{
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a=a/2; /* Convert off twist */
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d=d/4;
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}
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normalise(P);
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A=P; // remember A
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n=q-1;
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nb=bits(n);
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res=1;
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for (i=nb-2;i>=0;i--)
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{
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res*=res; // 2 modmul
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res*=g(A,A,a,d);
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if (bit(n,i))
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res*=g(A,P,a,d); // executed just once
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}
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if (A != -P || res.iszero()) return FALSE;
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res=conj(res)/res; // raise to power of (p-1)
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r=powl(real(res),(p+1)/q); // raise to power of (p+1)/q
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if (r==1) 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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sha512 sh;
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shs512_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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shs512_process(&sh,string[i]);
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}
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shs512_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(ZZn x)
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{ // Hash an Fp to a big number
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sha sh;
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Big a,h,p;
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char s[20];
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int m;
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shs_init(&sh);
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a=(Big)x;
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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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h=from_binary(20,s);
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return h;
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}
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// Hash an Identity to a curve point
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ECn Hash(char *ID)
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{
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ECn T;
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Big a=H1(ID);
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while (!is_on_curve(a)) a+=1;
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T.set(a); // Make sure its on E(F_p)
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return T;
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}
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// Hash an Identity to a curve point and map to point of order q
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ECn hash_and_map(char *ID,Big cof)
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{
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ECn T;
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Big a=H1(ID);
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while (!is_on_curve(a)) a+=1;
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T.set(a);
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T*=cof;
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return T;
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}
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/* Note that if #E(Fp) = p+1-t
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then #E(Fp2) = (p+1-t)(p+1+t) (a multiple of #E(Fp))
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(Weil's Theorem)
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*/
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int main()
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{
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ifstream common("k2ss.ecs"); // elliptic curve parameters
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miracl* mip=&precision;
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ECn Alice,Bob,sA,sB;
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ECn Server,sS;
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ZZn res,sp,ap,bp;
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Big r,a,b,s,ss,p,q,x,y,B,cof;
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int i,nbits,A,qnr;
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time_t seed;
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common >> nbits;
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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;
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common >> q;
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//cout << "p= " << p << endl;
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//cout << "q%cof= " << q*cof << endl;
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time(&seed);
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irand((long)seed);
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//
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// initialise twisted curve...
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// Server ID is hashed to points on this
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//
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modulo(p);
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qnr=mip->qnr;
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#ifdef AFFINE
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ecurve(qnr*qnr*A,qnr*qnr*qnr*B,p,MR_AFFINE);
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#endif
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#ifdef PROJECTIVE
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ecurve(qnr*qnr*A,qnr*qnr*qnr*B,p,MR_PROJECTIVE);
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#endif
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mip->IOBASE=16;
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// hash Identity to curve point
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// Server does not need to be of order q (its order is a multiple of q)
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ss=rand(q); // TA's super-secret
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cout << "Mapping Server ID to point on twisted curve" << endl;
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Server=Hash((char *)"Server");
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cout << "Server visits trusted authority" << endl;
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sS=ss*Server;
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// initialise curve...
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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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cout << "Mapping Alice & Bob ID's to points on curve" << 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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// Alice, Bob are points of order q
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cout << "Alice and Bob visit Trusted Authority" << endl;
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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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//for (i=0;i<10000;i++)
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if (!tate(sA,Server,q,res)) cout << "Trouble" << endl;
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if (powl(res,q)!=(ZZn)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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if (!tate(Alice,sS,q,res)) cout << "Trouble" << endl;
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if (powl(res,q)!=(ZZn)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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cout << "Alice Key= " << H2(powl(sp,a)) << endl;
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cout << "Server Key= " << H2(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 (!tate(sB,Server,q,res)) cout << "Trouble" << endl;
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if (powl(res,q)!=(ZZn)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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if (!tate(Bob,sS,q,res)) cout << "Trouble" << endl;
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sp=powl(res,s);
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cout << "Bob Key= " << H2(powl(sp,b)) << endl;
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cout << "Server Key= " << H2(powl(bp,s)) << endl;
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cout << "Alice and Bob's attempted Key exchange" << endl;
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if (!tate(Alice,sB,q,res)) cout << "Trouble" << endl;
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if (powl(res,q)!=(ZZn)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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if (!tate(Bob,sA,q,res)) cout << "Trouble" << endl;
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ap=powl(res,a);
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cout << "Alice Key= " << H2(powl(bp,a)) << endl;
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cout << "Bob Key= " << H2(powl(ap,b)) << endl;
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return 0;
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}
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