int ecCheck (const ecPoint *p)
/* confirm that y^2 + x*y = x^3 + EC_B for point p */
{
gfPoint t1, t2, t3, b;
b[0] = 1; b[1] = EC_B;
gfSquare (t1, p->y);
gfMultiply (t2, p->x, p->y);
gfAdd (t1, t1, t2); /* t1 := y^2 + x*y */
gfSquare (t2, p->x);
gfMultiply (t3, t2, p->x);
gfAdd (t2, t3, b); /*/ t2 := x^3 + EC_B */
return gfEqual (t1, t2);
} /* ecCheck */
void RSCoder16::MakeEncoderMatrix()
{
// Create Cauchy encoder generator matrix. Skip trivial "1" diagonal rows,
// which would just copy source data to destination.
for (uint I = 0; I < NR; I++)
for (uint J = 0; J < ND; J++)
MX[I * ND + J] = gfInv( gfAdd( (I+ND), J) );
}
void ecSub (ecPoint *p, const ecPoint *r)
/* sets p := p - r */
{
ecPoint t;
gfCopy (t.x, r->x);
gfAdd (t.y, r->x, r->y);
ecAdd (p, &t);
} /* ecSub */
int ecCalcY (ecPoint *p, int ybit)
/* given the x coordinate of p, evaluate y such that y^2 + x*y = x^3 + EC_B */
{
gfPoint a, b, t;
b[0] = 1; b[1] = EC_B;
if (p->x[0] == 0) {
/* elliptic equation reduces to y^2 = EC_B: */
gfSquareRoot (p->y, EC_B);
return 1;
}
/* evaluate alpha = x^3 + b = (x^2)*x + EC_B: */
gfSquare (t, p->x); /* keep t = x^2 for beta evaluation */
gfMultiply (a, t, p->x);
gfAdd (a, a, b); /* now a == alpha */
if (a[0] == 0) {
p->y[0] = 0;
/* destroy potentially sensitive data: */
gfClear (a); gfClear (t);
return 1;
}
/* evaluate beta = alpha/x^2 = x + EC_B/x^2 */
gfSmallDiv (t, EC_B);
gfInvert (a, t);
gfAdd (a, p->x, a); /* now a == beta */
/* check if a solution exists: */
if (gfTrace (a) != 0) {
/* destroy potentially sensitive data: */
gfClear (a); gfClear (t);
return 0; /* no solution */
}
/* solve equation t^2 + t + beta = 0 so that gfYbit(t) == ybit: */
gfQuadSolve (t, a);
if (gfYbit (t) != ybit) {
t[1] ^= 1;
}
/* compute y = x*t: */
gfMultiply (p->y, p->x, t);
/* destroy potentially sensitive data: */
gfClear (a); gfClear (t);
return 1;
} /* ecCalcY */
void ecDouble (ecPoint *p)
/* sets p := 2*p */
{
gfPoint lambda, t1, t2;
/* evaluate lambda = x + y/x: */
gfInvert (t1, p->x);
gfMultiply (lambda, p->y, t1);
gfAdd (lambda, lambda, p->x);
/* evaluate x3 = lambda^2 + lambda: */
gfSquare (t1, lambda);
gfAdd (t1, t1, lambda); /* now t1 = x3 */
/* evaluate y3 = x^2 + lambda*x3 + x3: */
gfSquare (p->y, p->x);
gfMultiply (t2, lambda, t1);
gfAdd (p->y, p->y, t2);
gfAdd (p->y, p->y, t1);
/* deposit the value of x3: */
gfCopy (p->x, t1);
} /* ecDouble */
static int gfSlowTrace (const gfPoint p)
/* slowly evaluates to the trace of p (or an error code) */
{
int i;
gfPoint t;
assert (logt != NULL && expt != NULL);
assert (p != NULL);
gfCopy (t, p);
for (i = 1; i < GF_M; i++) {
gfSquare (t, t);
gfAdd (t, t, p);
}
return t[0] != 0;
} /* gfSlowTrace */
void RSCoder16::MakeDecoderMatrix()
{
// Create Cauchy decoder matrix. Skip trivial rows matching valid data
// units and containing "1" on main diagonal. Such rows would just copy
// source data to destination and they have no real value for us.
// Include rows only for broken data units and replace them by first
// available valid recovery code rows.
for (uint Flag=0, R=ND, Dest=0; Flag < ND; Flag++)
if (!ValidFlags[Flag]) // For every broken data unit.
{
while (!ValidFlags[R]) // Find a valid recovery unit.
R++;
for (uint J = 0; J < ND; J++) // And place its row to matrix.
MX[Dest*ND + J] = gfInv( gfAdd(R,J) );
Dest++;
R++;
}
}
void ecAdd (ecPoint *p, const ecPoint *q)
/* sets p := p + q */
{
gfPoint lambda, t, tx, ty, x3;
/* first check if there is indeed work to do (q != 0): */
if (q->x[0] != 0 || q->y[0] != 0) {
if (p->x[0] != 0 || p->y[0] != 0) {
/* p != 0 and q != 0 */
if (gfEqual (p->x, q->x)) {
/* either p == q or p == -q: */
if (gfEqual (p->y, q->y)) {
/* points are equal; double p: */
ecDouble (p);
} else {
/* must be inverse: result is zero */
/* (should assert that q->y = p->x + p->y) */
p->x[0] = p->y[0] = 0;
}
} else {
/* p != 0, q != 0, p != q, p != -q */
/* evaluate lambda = (y1 + y2)/(x1 + x2): */
gfAdd (ty, p->y, q->y);
gfAdd (tx, p->x, q->x);
gfInvert (t, tx);
gfMultiply (lambda, ty, t);
/* evaluate x3 = lambda^2 + lambda + x1 + x2: */
gfSquare (x3, lambda);
gfAdd (x3, x3, lambda);
gfAdd (x3, x3, tx);
/* evaluate y3 = lambda*(x1 + x3) + x3 + y1: */
gfAdd (tx, p->x, x3);
gfMultiply (t, lambda, tx);
gfAdd (t, t, x3);
gfAdd (p->y, t, p->y);
/* deposit the value of x3: */
gfCopy (p->x, x3);
}
} else {
/* just copy q into p: */
gfCopy (p->x, q->x);
gfCopy (p->y, q->y);
}
}
} /* ecAdd */
// -----------------------------------------------------------------------------
int main ()
// -----------------------------------------------------------------------------
{
// verify basic operations (mul, add, div):
// a+b = b+a
// a*b = b*a
// a+(b+c) = (a+b)+c
// a*(b*c) = (a*b)*c
// a*c + b*c = (a+b)*c
// a*(b/a) = b (if a != 0)
gfInit();
for (int a=0; a<GF_N; a++) {
for (int b=0; b<GF_N; b++) {
if (gfAdd(a, b) != gfAdd(b, a))
return 1;
if (gfMul(a, b) != gfMul(b, a))
return 2;
if (gfMul(a, b) != gfMul(b, a))
return 2;
for (int c=0; c<GF_N; c++) {
if (gfAdd(a, gfAdd(b, c)) != gfAdd(gfAdd(a, b), c))
return 3;
if (gfMul(a, gfMul(b, c)) != gfMul(gfMul(a, b), c))
return 4;
if (gfAdd(gfMul(a, c), gfMul(b, c)) != gfMul(gfAdd(a, b), c))
return 5;
}
if (a != GF_0)
if (gfMul(a, gfDiv(b, a)) != b)
return 6;
}
}
// verify polynomial operations:
// A = BQ + R
const int M = GF_N; // max deg
gfExp A[M+1]; int nA;
gfExp B[M+1]; int nB;
gfExp Q[M+1]; int nQ;
gfExp R1[M+2];
gfExp* R = R1 + 1; int nR;
gfExp Z[M+1]; int nZ;
gfExp Z1[2*M]; int nZ1;
gfExp Z2[2*M]; int nZ2;
gfExp* P = R; int nP; // alias
gfExp* N = B; int nN; // alias
gfExp* Y = Q; int nY; // alias
gfExp Mem[8*(M+1) + 3];
// // -------------------- test polDiv, polMul, gfPolAdd(): --------------------
// for (int test=0; test<100000; test++) {
// // // clear all -- should not be required:
// // for (int i=0; i<=M; i++)
// // A[i] = B[i] = Q[i] = R[i] = Z[i] = 0;
// // R[-1] = 0;
// nB = randInt(0, M);
// nA = randInt(nB, M);
// randPol(A, nA);
// randPol(B, nB);
// if (gfPolDeg(B, nB) == -1) // avoid dividing by B=0
// continue;
// nB = polDiv(A, nA, B, nB, Q, &nQ, R, &nR);
// nZ = gfPolMul(Q, nQ, B, nB, Z); // B * Q
// if (nZ > nA)
// return 7;
// nZ = gfPolAdd(Z, nZ, R, nR, Z); // + R
// if (! polCmp(Z, A, nZ, nA))
// return 8;
// }
// // -------------------- test gfPolEEA(): --------------------
// for (int test=0; test<100000; test++) {
// nN = randInt(1, M);
// nA = randInt(0, nN-1);
// randPol(A, nA);
// randPol(N, nN);
// if (gfPolDeg(A, nA) == -1) // avoid dividing by A=0
// continue;
// gfPolEEA(N, nN, A, nA, P, &nP, Q, &nQ, Mem);
// nZ1 = gfPolMul(P, nP, N, nN, Z1);
// nZ2 = gfPolMul(Q, nQ, A, nA, Z2);
// if (! polCmp(Z1, Z2, nZ1, nZ2))
// return 9;
// }
// -------------------- test gfPolEvalSeq() against gfPolEval(): --------------------
for (int test=0; test<10000; test++) {
nA = randInt(0, GF_N - 2); // limit of gfPolEvalSeq()
nY = randInt(0, M);
gfVec* Yv = Y;
randPol(A, nA);
gfExp x = GF_Z(1);//randE1();
gfPolEvalSeq(A, nA, Yv, nY, x);
for (int i=0; i<=nY; i++) {
gfExp y2 = gfPolEval(A, nA, x);
gfExp y1 = gfV2E[Yv[nY-i]];
if (y2 != y1)
return 10;
x = gfMul(x, GF_Z(1));
}
}
return 0;
}
void ecNegate (ecPoint *p)
/* sets p := -p */
{
gfAdd (p->y, p->x, p->y);
} /* ecNegate */