+/* Copyright (c) 1998,2011,2014 Apple Inc. All Rights Reserved.
+ *
+ * NOTICE: USE OF THE MATERIALS ACCOMPANYING THIS NOTICE IS SUBJECT
+ * TO THE TERMS OF THE SIGNED "FAST ELLIPTIC ENCRYPTION (FEE) REFERENCE
+ * SOURCE CODE EVALUATION AGREEMENT" BETWEEN APPLE, INC. AND THE
+ * ORIGINAL LICENSEE THAT OBTAINED THESE MATERIALS FROM APPLE,
+ * INC. ANY USE OF THESE MATERIALS NOT PERMITTED BY SUCH AGREEMENT WILL
+ * EXPOSE YOU TO LIABILITY.
+ ***************************************************************************
+
+ giantFFT.c
+ Library for large-integer arithmetic via FFT. Currently unused
+ in CryptKit.
+
+
+ Revision History
+ ----------------
+ 19 Jan 1998 at Apple
+ Split off from NSGiantIntegers.c.
+
+*/
+
+/*
+ * FIXME - make sure platform-specific math lib has floor(), fmod(),
+ * sin(), pow()
+ */
+#include <math.h>
+#include "NSGiantIntegers.h"
+
+#define AUTO_MUL 0
+#define GRAMMAR_MUL 1
+#define FFT_MUL 2
+
+#define TWOPI (double)(2*3.1415926535897932384626433)
+#define SQRT2 (double)(1.414213562373095048801688724209)
+#define SQRTHALF (double)(0.707106781186547524400844362104)
+#define TWO16 (double)(65536.0)
+#define TWOM16 (double)(0.0000152587890625)
+#define BREAK_SHORTS 400 // Number of shorts at which FFT breaks over.
+
+static int lpt(int n, int *lambda);
+static void mul_hermitian(double *a, double *b, int n) ;
+static void square_hermitian(double *b, int n);
+static void addsignal(giant x, double *zs, int n);
+static void scramble_real(double *x, int n);
+static void fft_real_to_hermitian(double *zs, int n);
+static void fftinv_hermitian_to_real(double *zs, int n);
+static void GiantFFTSquare(giant gx);
+static void GiantFFTMul(giant,giant);
+static void giant_to_double(giant x, int sizex, double *zs, int L);
+
+static int mulmode = AUTO_MUL;
+
+void mulg(giant a, giant b) { /* b becomes a*b. */
+ PROF_START;
+ INCR_MULGS;
+ GiantAuxMul(a,b);
+ #if FEE_DEBUG
+ (void)bitlen(b); // XXX
+ #endif FEE_DEBUG
+ PROF_END(mulgTime);
+ PROF_INCR(numMulg);
+}
+
+static void GiantAuxMul(giant a, giant b) {
+/* Optimized general multiply, b becomes a*b. Modes are:
+ AUTO_MUL: switch according to empirical speed criteria.
+ GRAMMAR_MUL: force grammar-school algorithm.
+ FFT_MUL: force floating point FFT method.
+*/
+ int square = (a==b);
+
+ if (isZero(b)) return;
+ if (isZero(a)) {
+ gtog(a, b);
+ return;
+ }
+ switch(mulmode) {
+ case GRAMMAR_MUL:
+ GiantGrammarMul(a,b);
+ break;
+ case FFT_MUL:
+ if (square) {
+ GiantFFTSquare(b);
+ }
+ else {
+ GiantFFTMul(a,b);
+ }
+ break;
+ case AUTO_MUL: {
+ int sizea, sizeb;
+ float grammartime;
+ sizea = abs(a->sign);
+ sizeb = abs(b->sign);
+ grammartime = sizea; grammartime *= sizeb;
+ if(grammartime < BREAK_SHORTS*BREAK_SHORTS) {
+ GiantGrammarMul(a,b);
+ }
+ else {
+ if (square) GiantFFTSquare(b);
+ else GiantFFTMul(a,b);
+ }
+ break;
+ }
+ }
+}
+
+/***************** Commence FFT multiply routines ****************/
+
+static int CurrentRun = 0;
+double *sincos = NULL;
+static void init_sincos(int n) {
+ int j;
+ double e = TWOPI/n;
+
+ if (n <= CurrentRun) return;
+ CurrentRun = n;
+ if (sincos) free(sincos);
+ sincos = (double *)malloc(sizeof(double)*(1+(n>>2)));
+ for(j=0;j<=(n>>2);j++) {
+ sincos[j] = sin(e*j);
+ }
+}
+
+static double s_sin(int n) {
+ int seg = n/(CurrentRun>>2);
+
+ switch(seg) {
+ case 0: return(sincos[n]);
+ case 1: return(sincos[(CurrentRun>>1)-n]);
+ case 2: return(-sincos[n-(CurrentRun>>1)]);
+ case 3:
+ default: return(-sincos[CurrentRun-n]);
+ }
+}
+
+static double s_cos(int n) {
+ int quart = (CurrentRun>>2);
+
+ if (n < quart) return(s_sin(n+quart));
+ return(-s_sin(n-quart));
+}
+
+
+static int lpt(int n, int *lambda) {
+/* returns least power of two greater than n */
+ register int i = 1;
+
+ *lambda = 0;
+ while(i<n) {
+ i<<=1;
+ ++(*lambda);
+ }
+ return(i);
+}
+
+static void addsignal(giant x, double *zs, int n) {
+ register int j, k, m, car;
+ register double f, g;
+ /*double err, maxerr = 0.0;*/
+
+ for(j=0;j<n;j++) {
+ f = floor(zs[j]+0.5);
+
+ /* err = fabs(zs[j]-f);
+ if(err>maxerr) maxerr = err;
+ */
+
+ zs[j] =0;
+ k = 0;
+ do{
+ g = floor(f*TWOM16);
+ zs[j+k] += f-g*TWO16;
+ ++k;
+ f=g;
+ } while(f != 0.0);
+ }
+ car = 0;
+ for(j=0;j<n;j++) {
+ m = zs[j]+car;
+ x->n[j] = m & 0xffff;
+ car = (m>>16);
+ }
+ if(car) x->n[j] = car;
+ else --j;
+ while(!(x->n[j])) --j;
+ x->sign = j+1;
+ if (abs(x->sign) > x->capacity) NSGiantRaise("addsignal overflow");
+}
+
+static void GiantFFTSquare(giant gx) {
+ int j,size = abs(gx->sign);
+ register int L;
+
+ if(size<4) { GiantGrammarMul(gx,gx); return; }
+ L = lpt(size+size, &j);
+ {
+ //was...double doubles[L];
+ //is...
+ double *doubles = malloc(sizeof(double) * L);
+ // end
+ giant_to_double(gx, size, doubles, L);
+ fft_real_to_hermitian(doubles, L);
+ square_hermitian(doubles, L);
+ fftinv_hermitian_to_real(doubles, L);
+ addsignal(gx, doubles, L);
+ // new
+ free(doubles);
+ }
+ gx->sign = abs(gx->sign);
+ bitlen(gx); // XXX
+ if (abs(gx->sign) > gx->capacity) NSGiantRaise("GiantFFTSquare overflow");
+}
+
+static void GiantFFTMul(giant y, giant x) { /* x becomes y*x. */
+ int lambda, size, sizex = abs(x->sign), sizey = abs(y->sign);
+ int finalsign = gsign(x)*gsign(y);
+ register int L;
+
+ if((sizex<=4)||(sizey<=4)) { GiantGrammarMul(y,x); return; }
+ size = sizex; if(size<sizey) size=sizey;
+ L = lpt(size+size, &lambda);
+ {
+ //double doubles1[L];
+ //double doubles2[L];
+ double *doubles1 = malloc(sizeof(double) * L);
+ double *doubles2 = malloc(sizeof(double) * L);
+
+ giant_to_double(x, sizex, doubles1, L);
+ giant_to_double(y, sizey, doubles2, L);
+ fft_real_to_hermitian(doubles1, L);
+ fft_real_to_hermitian(doubles2, L);
+ mul_hermitian(doubles2, doubles1, L);
+ fftinv_hermitian_to_real(doubles1, L);
+ addsignal(x, doubles1, L);
+
+ free(doubles1);
+ free(doubles2);
+ }
+ x->sign = finalsign*abs(x->sign);
+ bitlen(x); // XXX
+ if (abs(x->sign) > x->capacity) NSGiantRaise("GiantFFTMul overflow");
+}
+
+static void scramble_real(double *x, int n) {
+ register int i,j,k;
+ register double tmp;
+
+ for(i=0,j=0;i<n-1;i++) {
+ if(i<j) {
+ tmp = x[j];
+ x[j]=x[i];
+ x[i]=tmp;
+ }
+ k = n/2;
+ while(k<=j) {
+ j -= k;
+ k>>=1;
+ }
+ j += k;
+ }
+}
+
+static void fft_real_to_hermitian(double *zs, int n) {
+/* Output is {Re(z^[0]),...,Re(z^[n/2),Im(z^[n/2-1]),...,Im(z^[1]).
+ This is a decimation-in-time, split-radix algorithm.
+ */
+ register double cc1, ss1, cc3, ss3;
+ register int is, iD, i0, i1, i2, i3, i4, i5, i6, i7, i8,
+ a, a3, b, b3, nminus = n-1, dil, expand;
+ register double *x, e;
+ int nn = n>>1;
+ double t1, t2, t3, t4, t5, t6;
+ register int n2, n4, n8, i, j;
+
+ init_sincos(n);
+ expand = CurrentRun/n;
+ scramble_real(zs, n);
+ x = zs-1; /* FORTRAN compatibility. */
+ is = 1;
+ iD = 4;
+ do{
+ for(i0=is;i0<=n;i0+=iD) {
+ i1 = i0+1;
+ e = x[i0];
+ x[i0] = e + x[i1];
+ x[i1] = e - x[i1];
+ }
+ is = (iD<<1)-1;
+ iD <<= 2;
+ } while(is<n);
+ n2 = 2;
+ while(nn>>=1) {
+ n2 <<= 1;
+ n4 = n2>>2;
+ n8 = n2>>3;
+ is = 0;
+ iD = n2<<1;
+ do {
+ for(i=is;i<n;i+=iD) {
+ i1 = i+1;
+ i2 = i1 + n4;
+ i3 = i2 + n4;
+ i4 = i3 + n4;
+ t1 = x[i4]+x[i3];
+ x[i4] -= x[i3];
+ x[i3] = x[i1] - t1;
+ x[i1] += t1;
+ if(n4==1) continue;
+ i1 += n8;
+ i2 += n8;
+ i3 += n8;
+ i4 += n8;
+ t1 = (x[i3]+x[i4])*SQRTHALF;
+ t2 = (x[i3]-x[i4])*SQRTHALF;
+ x[i4] = x[i2] - t1;
+ x[i3] = -x[i2] - t1;
+ x[i2] = x[i1] - t2;
+ x[i1] += t2;
+ }
+ is = (iD<<1) - n2;
+ iD <<= 2;
+ } while(is<n);
+ dil = n/n2;
+ a = dil;
+ for(j=2;j<=n8;j++) {
+ a3 = (a+(a<<1))&nminus;
+ b = a*expand;
+ b3 = a3*expand;
+ cc1 = s_cos(b);
+ ss1 = s_sin(b);
+ cc3 = s_cos(b3);
+ ss3 = s_sin(b3);
+ a = (a+dil)&nminus;
+ is = 0;
+ iD = n2<<1;
+ do {
+ for(i=is;i<n;i+=iD) {
+ i1 = i+j;
+ i2 = i1 + n4;
+ i3 = i2 + n4;
+ i4 = i3 + n4;
+ i5 = i + n4 - j + 2;
+ i6 = i5 + n4;
+ i7 = i6 + n4;
+ i8 = i7 + n4;
+ t1 = x[i3]*cc1 + x[i7]*ss1;
+ t2 = x[i7]*cc1 - x[i3]*ss1;
+ t3 = x[i4]*cc3 + x[i8]*ss3;
+ t4 = x[i8]*cc3 - x[i4]*ss3;
+ t5 = t1 + t3;
+ t6 = t2 + t4;
+ t3 = t1 - t3;
+ t4 = t2 - t4;
+ t2 = x[i6] + t6;
+ x[i3] = t6 - x[i6];
+ x[i8] = t2;
+ t2 = x[i2] - t3;
+ x[i7] = -x[i2] - t3;
+ x[i4] = t2;
+ t1 = x[i1] + t5;
+ x[i6] = x[i1] - t5;
+ x[i1] = t1;
+ t1 = x[i5] + t4;
+ x[i5] -= t4;
+ x[i2] = t1;
+ }
+ is = (iD<<1) - n2;
+ iD <<= 2;
+ } while(is<n);
+ }
+ }
+}
+
+static void fftinv_hermitian_to_real(double *zs, int n) {
+/* Input is {Re(z^[0]),...,Re(z^[n/2),Im(z^[n/2-1]),...,Im(z^[1]).
+ This is a decimation-in-frequency, split-radix algorithm.
+ */
+ register double cc1, ss1, cc3, ss3;
+ register int is, iD, i0, i1, i2, i3, i4, i5, i6, i7, i8,
+ a, a3, b, b3, nminus = n-1, dil, expand;
+ register double *x, e;
+ int nn = n>>1;
+ double t1, t2, t3, t4, t5;
+ int n2, n4, n8, i, j;
+
+ init_sincos(n);
+ expand = CurrentRun/n;
+ x = zs-1;
+ n2 = n<<1;
+ while(nn >>= 1) {
+ is = 0;
+ iD = n2;
+ n2 >>= 1;
+ n4 = n2>>2;
+ n8 = n4>>1;
+ do {
+ for(i=is;i<n;i+=iD) {
+ i1 = i+1;
+ i2 = i1 + n4;
+ i3 = i2 + n4;
+ i4 = i3 + n4;
+ t1 = x[i1] - x[i3];
+ x[i1] += x[i3];
+ x[i2] += x[i2];
+ x[i3] = t1 - 2.0*x[i4];
+ x[i4] = t1 + 2.0*x[i4];
+ if(n4==1) continue;
+ i1 += n8;
+ i2 += n8;
+ i3 += n8;
+ i4 += n8;
+ t1 = (x[i2]-x[i1])*SQRTHALF;
+ t2 = (x[i4]+x[i3])*SQRTHALF;
+ x[i1] += x[i2];
+ x[i2] = x[i4]-x[i3];
+ x[i3] = -2.0*(t2+t1);
+ x[i4] = 2.0*(t1-t2);
+ }
+ is = (iD<<1) - n2;
+ iD <<= 2;
+ } while(is<n-1);
+ dil = n/n2;
+ a = dil;
+ for(j=2;j<=n8;j++) {
+ a3 = (a+(a<<1))&nminus;
+ b = a*expand;
+ b3 = a3*expand;
+ cc1 = s_cos(b);
+ ss1 = s_sin(b);
+ cc3 = s_cos(b3);
+ ss3 = s_sin(b3);
+ a = (a+dil)&nminus;
+ is = 0;
+ iD = n2<<1;
+ do {
+ for(i=is;i<n;i+=iD) {
+ i1 = i+j;
+ i2 = i1+n4;
+ i3 = i2+n4;
+ i4 = i3+n4;
+ i5 = i+n4-j+2;
+ i6 = i5+n4;
+ i7 = i6+n4;
+ i8 = i7+n4;
+ t1 = x[i1] - x[i6];
+ x[i1] += x[i6];
+ t2 = x[i5] - x[i2];
+ x[i5] += x[i2];
+ t3 = x[i8] + x[i3];
+ x[i6] = x[i8] - x[i3];
+ t4 = x[i4] + x[i7];
+ x[i2] = x[i4] - x[i7];
+ t5 = t1 - t4;
+ t1 += t4;
+ t4 = t2 - t3;
+ t2 += t3;
+ x[i3] = t5*cc1 + t4*ss1;
+ x[i7] = -t4*cc1 + t5*ss1;
+ x[i4] = t1*cc3 - t2*ss3;
+ x[i8] = t2*cc3 + t1*ss3;
+ }
+ is = (iD<<1) - n2;
+ iD <<= 2;
+ } while(is<n-1);
+ }
+ }
+ is = 1;
+ iD = 4;
+ do {
+ for(i0=is;i0<=n;i0+=iD){
+ i1 = i0+1;
+ e = x[i0];
+ x[i0] = e + x[i1];
+ x[i1] = e - x[i1];
+ }
+ is = (iD<<1) - 1;
+ iD <<= 2;
+ } while(is<n);
+ scramble_real(zs, n);
+ e = 1/(double)n;
+ for(i=0;i<n;i++) zs[i] *= e;
+}
+
+
+static void mul_hermitian(double *a, double *b, int n) {
+ register int k, half = n>>1;
+ register double aa, bb, am, bm;
+
+ b[0] *= a[0];
+ b[half] *= a[half];
+ for(k=1;k<half;k++) {
+ aa = a[k]; bb = b[k];
+ am = a[n-k]; bm = b[n-k];
+ b[k] = aa*bb - am*bm;
+ b[n-k] = aa*bm + am*bb;
+ }
+}
+
+static void square_hermitian(double *b, int n) {
+ register int k, half = n>>1;
+ register double c, d;
+
+ b[0] *= b[0];
+ b[half] *= b[half];
+ for(k=1;k<half;k++) {
+ c = b[k]; d = b[n-k];
+ b[n-k] = 2.0*c*d;
+ b[k] = (c+d)*(c-d);
+ }
+}
+
+static void giant_to_double(giant x, int sizex, double *zs, int L) {
+ register int j;
+ for(j=sizex;j<L;j++) zs[j]=0.0;
+ for(j=0;j<sizex;j++) {
+ zs[j] = x->n[j];
+ }
+}
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