text: /* fe_low_mem.c
   *
   * Copyright (C) 2006-2017 wolfSSL Inc.
   *
   * This file is part of wolfSSL.
   *
   * wolfSSL is free software; you can redistribute it and/or modify
   * it under the terms of the GNU General Public License as published by
   * the Free Software Foundation; either version 2 of the License, or
   * (at your option) any later version.
   *
   * wolfSSL is distributed in the hope that it will be useful,
   * but WITHOUT ANY WARRANTY; without even the implied warranty of
   * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
   * GNU General Public License for more details.
   *
   * You should have received a copy of the GNU General Public License
   * along with this program; if not, write to the Free Software
   * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1335, USA
   */
  
  
  /* Based from Daniel Beer's public domain work. */
  
  #ifdef HAVE_CONFIG_H
      #include <config.h>
  #endif
  
  #include <wolfssl/wolfcrypt/settings.h>
  
  #if defined(HAVE_CURVE25519) || defined(HAVE_ED25519)
  #if defined(CURVE25519_SMALL) || defined(ED25519_SMALL)  || defined(ESP8266_LOWMEM_FORCE_IMPL)/* use slower code that takes less memory */
  
  #include <wolfssl/wolfcrypt/fe_operations.h>
  
  #ifdef NO_INLINE
      #include <wolfssl/wolfcrypt/misc.h>
  #else
      #define WOLFSSL_MISC_INCLUDED
      #include <wolfcrypt/src/misc.c>
  #endif
  
  void fprime_copy(byte *x, const byte *a)
  {
      int i;
      for (i = 0; i < F25519_SIZE; i++)
          x[i] = a[i];
  }
  
  
  void lm_copy(byte* x, const byte* a)
  {
      int i;
      for (i = 0; i < F25519_SIZE; i++)
          x[i] = a[i];
  }
  
  
  #ifdef CURVE25519_SMALL
  #ifndef FREESCALE_LTC_ECC
  void fe_init()
  {
  }
  #endif
  
  /* Double an X-coordinate */
  static void xc_double(byte *x3, byte *z3,
  		      const byte *x1, const byte *z1)
  {
  	/* Explicit formulas database: dbl-1987-m
  	 *
  	 * source 1987 Montgomery "Speeding the Pollard and elliptic
  	 *   curve methods of factorization", page 261, fourth display
  	 * compute X3 = (X1^2-Z1^2)^2
  	 * compute Z3 = 4 X1 Z1 (X1^2 + a X1 Z1 + Z1^2)
  	 */
  	byte x1sq[F25519_SIZE];
  	byte z1sq[F25519_SIZE];
  	byte x1z1[F25519_SIZE];
  	byte a[F25519_SIZE];
  
  	fe_mul__distinct(x1sq, x1, x1);
  	fe_mul__distinct(z1sq, z1, z1);
  	fe_mul__distinct(x1z1, x1, z1);
  
  	lm_sub(a, x1sq, z1sq);
  	fe_mul__distinct(x3, a, a);
  
  	fe_mul_c(a, x1z1, 486662);
  	lm_add(a, x1sq, a);
  	lm_add(a, z1sq, a);
  	fe_mul__distinct(x1sq, x1z1, a);
  	fe_mul_c(z3, x1sq, 4);
  }
  
  
  /* Differential addition */
  static void xc_diffadd(byte *x5, byte *z5,
  		       const byte *x1, const byte *z1,
  		       const byte *x2, const byte *z2,
  		       const byte *x3, const byte *z3)
  {
  	/* Explicit formulas database: dbl-1987-m3
  	 *
  	 * source 1987 Montgomery "Speeding the Pollard and elliptic curve
  	 *   methods of factorization", page 261, fifth display, plus
  	 *   common-subexpression elimination
  	 * compute A = X2+Z2
  	 * compute B = X2-Z2
  	 * compute C = X3+Z3
  	 * compute D = X3-Z3
  	 * compute DA = D A
  	 * compute CB = C B
  	 * compute X5 = Z1(DA+CB)^2
  	 * compute Z5 = X1(DA-CB)^2
  	 */
  	byte da[F25519_SIZE];
  	byte cb[F25519_SIZE];
  	byte a[F25519_SIZE];
  	byte b[F25519_SIZE];
  
  	lm_add(a, x2, z2);
  	lm_sub(b, x3, z3); /* D */
  	fe_mul__distinct(da, a, b);
  
  	lm_sub(b, x2, z2);
  	lm_add(a, x3, z3); /* C */
  	fe_mul__distinct(cb, a, b);
  
  	lm_add(a, da, cb);
  	fe_mul__distinct(b, a, a);
  	fe_mul__distinct(x5, z1, b);
  
  	lm_sub(a, da, cb);
  	fe_mul__distinct(b, a, a);
  	fe_mul__distinct(z5, x1, b);
  }
  
  #ifndef FREESCALE_LTC_ECC
  int curve25519(byte *result, byte *e, byte *q)
  {
  	/* Current point: P_m */
  	byte xm[F25519_SIZE];
  	byte zm[F25519_SIZE] = {1};
  
  	/* Predecessor: P_(m-1) */
  	byte xm1[F25519_SIZE] = {1};
  	byte zm1[F25519_SIZE] = {0};
  
  	int i;
  
  	/* Note: bit 254 is assumed to be 1 */
  	lm_copy(xm, q);
  
  	for (i = 253; i >= 0; i--) {
  		const int bit = (e[i >> 3] >> (i & 7)) & 1;
  		byte xms[F25519_SIZE];
  		byte zms[F25519_SIZE];
  
  		/* From P_m and P_(m-1), compute P_(2m) and P_(2m-1) */
  		xc_diffadd(xm1, zm1, q, f25519_one, xm, zm, xm1, zm1);
  		xc_double(xm, zm, xm, zm);
  
  		/* Compute P_(2m+1) */
  		xc_diffadd(xms, zms, xm1, zm1, xm, zm, q, f25519_one);
  
  		/* Select:
  		 *   bit = 1 --> (P_(2m+1), P_(2m))
  		 *   bit = 0 --> (P_(2m), P_(2m-1))
  		 */
  		fe_select(xm1, xm1, xm, bit);
  		fe_select(zm1, zm1, zm, bit);
  		fe_select(xm, xm, xms, bit);
  		fe_select(zm, zm, zms, bit);
  	}
  
  	/* Freeze out of projective coordinates */
  	fe_inv__distinct(zm1, zm);
  	fe_mul__distinct(result, zm1, xm);
  	fe_normalize(result);
      return 0;
  }
  #endif /* !FREESCALE_LTC_ECC */
  #endif /* CURVE25519_SMALL */
  
  
  static void raw_add(byte *x, const byte *p)
  {
  	word16 c = 0;
  	int i;
  
  	for (i = 0; i < F25519_SIZE; i++) {
  		c += ((word16)x[i]) + ((word16)p[i]);
  		x[i] = (byte)c;
  		c >>= 8;
  	}
  }
  
  
  static void raw_try_sub(byte *x, const byte *p)
  {
  	byte minusp[F25519_SIZE];
  	word16 c = 0;
  	int i;
  
  	for (i = 0; i < F25519_SIZE; i++) {
  		c = ((word16)x[i]) - ((word16)p[i]) - c;
  		minusp[i] = (byte)c;
  		c = (c >> 8) & 1;
  	}
  
  	fprime_select(x, minusp, x, (byte)c);
  }
  
  
  static int prime_msb(const byte *p)
  {
      int i;
      byte x;
      int shift = 1;
      int z     = F25519_SIZE - 1;
  
     /*
         Test for any hot bits.
         As soon as one instance is encountered set shift to 0.
      */
  	for (i = F25519_SIZE - 1; i >= 0; i--) {
          shift &= ((shift ^ ((-p[i] | p[i]) >> 7)) & 1);
          z -= shift;
      }
  	x = p[z];
  	z <<= 3;
      shift = 1;
      for (i = 0; i < 8; i++) {
          shift &= ((-(x >> i) | (x >> i)) >> (7 - i) & 1);
          z += shift;
      }
  
  	return z - 1;
  }
  
  
  void fprime_select(byte *dst, const byte *zero, const byte *one, byte condition)
  {
  	const byte mask = -condition;
  	int i;
  
  	for (i = 0; i < F25519_SIZE; i++)
  		dst[i] = zero[i] ^ (mask & (one[i] ^ zero[i]));
  }
  
  
  void fprime_add(byte *r, const byte *a, const byte *modulus)
  {
  	raw_add(r, a);
  	raw_try_sub(r, modulus);
  }
  
  
  void fprime_sub(byte *r, const byte *a, const byte *modulus)
  {
  	raw_add(r, modulus);
  	raw_try_sub(r, a);
  	raw_try_sub(r, modulus);
  }
  
  
  void fprime_mul(byte *r, const byte *a, const byte *b,
  		const byte *modulus)
  {
  	word16 c = 0;
  	int i,j;
  
  	XMEMSET(r, 0, F25519_SIZE);
  
  	for (i = prime_msb(modulus); i >= 0; i--) {
  		const byte bit = (b[i >> 3] >> (i & 7)) & 1;
  		byte plusa[F25519_SIZE];
  
  	    for (j = 0; j < F25519_SIZE; j++) {
  		    c |= ((word16)r[j]) << 1;
  		    r[j] = (byte)c;
  		    c >>= 8;
  	    }
  		raw_try_sub(r, modulus);
  
  		fprime_copy(plusa, r);
  		fprime_add(plusa, a, modulus);
  
  		fprime_select(r, r, plusa, bit);
  	}
  }
  
  
  void fe_load(byte *x, word32 c)
  {
  	word32 i;
  
  	for (i = 0; i < sizeof(c); i++) {
  		x[i] = c;
  		c >>= 8;
  	}
  
  	for (; i < F25519_SIZE; i++)
  		x[i] = 0;
  }
  
  
  void fe_normalize(byte *x)
  {
  	byte minusp[F25519_SIZE];
  	word16 c;
  	int i;
  
  	/* Reduce using 2^255 = 19 mod p */
  	c = (x[31] >> 7) * 19;
  	x[31] &= 127;
  
  	for (i = 0; i < F25519_SIZE; i++) {
  		c += x[i];
  		x[i] = (byte)c;
  		c >>= 8;
  	}
  
  	/* The number is now less than 2^255 + 18, and therefore less than
  	 * 2p. Try subtracting p, and conditionally load the subtracted
  	 * value if underflow did not occur.
  	 */
  	c = 19;
  
  	for (i = 0; i + 1 < F25519_SIZE; i++) {
  		c += x[i];
  		minusp[i] = (byte)c;
  		c >>= 8;
  	}
  
  	c += ((word16)x[i]) - 128;
  	minusp[31] = (byte)c;
  
  	/* Load x-p if no underflow */
  	fe_select(x, minusp, x, (c >> 15) & 1);
  }
  
  
  void fe_select(byte *dst,
  		   const byte *zero, const byte *one,
  		   byte condition)
  {
  	const byte mask = -condition;
  	int i;
  
  	for (i = 0; i < F25519_SIZE; i++)
  		dst[i] = zero[i] ^ (mask & (one[i] ^ zero[i]));
  }
  
  
  void lm_add(byte* r, const byte* a, const byte* b)
  {
  	word16 c = 0;
  	int i;
  
  	/* Add */
  	for (i = 0; i < F25519_SIZE; i++) {
  		c >>= 8;
  		c += ((word16)a[i]) + ((word16)b[i]);
  		r[i] = (byte)c;
  	}
  
  	/* Reduce with 2^255 = 19 mod p */
  	r[31] &= 127;
  	c = (c >> 7) * 19;
  
  	for (i = 0; i < F25519_SIZE; i++) {
  		c += r[i];
  		r[i] = (byte)c;
  		c >>= 8;
  	}
  }
  
  
  void lm_sub(byte* r, const byte* a, const byte* b)
  {
  	word32 c = 0;
  	int i;
  
  	/* Calculate a + 2p - b, to avoid underflow */
  	c = 218;
  	for (i = 0; i + 1 < F25519_SIZE; i++) {
  		c += 65280 + ((word32)a[i]) - ((word32)b[i]);
  		r[i] = c;
  		c >>= 8;
  	}
  
  	c += ((word32)a[31]) - ((word32)b[31]);
  	r[31] = c & 127;
  	c = (c >> 7) * 19;
  
  	for (i = 0; i < F25519_SIZE; i++) {
  		c += r[i];
  		r[i] = c;
  		c >>= 8;
  	}
  }
  
  
  void lm_neg(byte* r, const byte* a)
  {
  	word32 c = 0;
  	int i;
  
  	/* Calculate 2p - a, to avoid underflow */
  	c = 218;
  	for (i = 0; i + 1 < F25519_SIZE; i++) {
  		c += 65280 - ((word32)a[i]);
  		r[i] = c;
  		c >>= 8;
  	}
  
  	c -= ((word32)a[31]);
  	r[31] = c & 127;
  	c = (c >> 7) * 19;
  
  	for (i = 0; i < F25519_SIZE; i++) {
  		c += r[i];
  		r[i] = c;
  		c >>= 8;
  	}
  }
  
  
  void fe_mul__distinct(byte *r, const byte *a, const byte *b)
  {
  	word32 c = 0;
  	int i;
  
  	for (i = 0; i < F25519_SIZE; i++) {
  		int j;
  
  		c >>= 8;
  		for (j = 0; j <= i; j++)
  			c += ((word32)a[j]) * ((word32)b[i - j]);
  
  		for (; j < F25519_SIZE; j++)
  			c += ((word32)a[j]) *
  			     ((word32)b[i + F25519_SIZE - j]) * 38;
  
  		r[i] = c;
  	}
  
  	r[31] &= 127;
  	c = (c >> 7) * 19;
  
  	for (i = 0; i < F25519_SIZE; i++) {
  		c += r[i];
  		r[i] = c;
  		c >>= 8;
  	}
  }
  
  
  void lm_mul(byte *r, const byte* a, const byte *b)
  {
  	byte tmp[F25519_SIZE];
  
  	fe_mul__distinct(tmp, a, b);
  	lm_copy(r, tmp);
  }
  
  
  void fe_mul_c(byte *r, const byte *a, word32 b)
  {
  	word32 c = 0;
  	int i;
  
  	for (i = 0; i < F25519_SIZE; i++) {
  		c >>= 8;
  		c += b * ((word32)a[i]);
  		r[i] = c;
  	}
  
  	r[31] &= 127;
  	c >>= 7;
  	c *= 19;
  
  	for (i = 0; i < F25519_SIZE; i++) {
  		c += r[i];
  		r[i] = c;
  		c >>= 8;
  	}
  }
  
  
  void fe_inv__distinct(byte *r, const byte *x)
  {
  	byte s[F25519_SIZE];
  	int i;
  
  	/* This is a prime field, so by Fermat's little theorem:
  	 *
  	 *     x^(p-1) = 1 mod p
  	 *
  	 * Therefore, raise to (p-2) = 2^255-21 to get a multiplicative
  	 * inverse.
  	 *
  	 * This is a 255-bit binary number with the digits:
  	 *
  	 *     11111111... 01011
  	 *
  	 * We compute the result by the usual binary chain, but
  	 * alternate between keeping the accumulator in r and s, so as
  	 * to avoid copying temporaries.
  	 */
  
  	/* 1 1 */
  	fe_mul__distinct(s, x, x);
  	fe_mul__distinct(r, s, x);
  
  	/* 1 x 248 */
  	for (i = 0; i < 248; i++) {
  		fe_mul__distinct(s, r, r);
  		fe_mul__distinct(r, s, x);
  	}
  
  	/* 0 */
  	fe_mul__distinct(s, r, r);
  
  	/* 1 */
  	fe_mul__distinct(r, s, s);
  	fe_mul__distinct(s, r, x);
  
  	/* 0 */
  	fe_mul__distinct(r, s, s);
  
  	/* 1 */
  	fe_mul__distinct(s, r, r);
  	fe_mul__distinct(r, s, x);
  
  	/* 1 */
  	fe_mul__distinct(s, r, r);
  	fe_mul__distinct(r, s, x);
  }
  
  
  void lm_invert(byte *r, const byte *x)
  {
  	byte tmp[F25519_SIZE];
  
  	fe_inv__distinct(tmp, x);
  	lm_copy(r, tmp);
  }
  
  
  /* Raise x to the power of (p-5)/8 = 2^252-3, using s for temporary
   * storage.
   */
  static void exp2523(byte *r, const byte *x, byte *s)
  {
  	int i;
  
  	/* This number is a 252-bit number with the binary expansion:
  	 *
  	 *     111111... 01
  	 */
  
  	/* 1 1 */
  	fe_mul__distinct(r, x, x);
  	fe_mul__distinct(s, r, x);
  
  	/* 1 x 248 */
  	for (i = 0; i < 248; i++) {
  		fe_mul__distinct(r, s, s);
  		fe_mul__distinct(s, r, x);
  	}
  
  	/* 0 */
  	fe_mul__distinct(r, s, s);
  
  	/* 1 */
  	fe_mul__distinct(s, r, r);
  	fe_mul__distinct(r, s, x);
  }
  
  
  void fe_sqrt(byte *r, const byte *a)
  {
  	byte v[F25519_SIZE];
  	byte i[F25519_SIZE];
  	byte x[F25519_SIZE];
  	byte y[F25519_SIZE];
  
  	/* v = (2a)^((p-5)/8) [x = 2a] */
  	fe_mul_c(x, a, 2);
  	exp2523(v, x, y);
  
  	/* i = 2av^2 - 1 */
  	fe_mul__distinct(y, v, v);
  	fe_mul__distinct(i, x, y);
  	fe_load(y, 1);
  	lm_sub(i, i, y);
  
  	/* r = avi */
  	fe_mul__distinct(x, v, a);
  	fe_mul__distinct(r, x, i);
  }
  
  #endif /* CURVE25519_SMALL || ED25519_SMALL */
  #endif /* HAVE_CURVE25519 || HAVE_ED25519 */
  