crypto/fipsmodule/bn/sqrt.c.inc - boringssl - Git at Google

 /* Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de>
  * and Bodo Moeller for the OpenSSL project. */
 /* ====================================================================
  * Copyright (c) 1998-2000 The OpenSSL Project.  All rights reserved.
  *
  * Redistribution and use in source and binary forms, with or without
  * modification, are permitted provided that the following conditions
  * are met:
  *
  * 1. Redistributions of source code must retain the above copyright
  *    notice, this list of conditions and the following disclaimer.
  *
  * 2. Redistributions in binary form must reproduce the above copyright
  *    notice, this list of conditions and the following disclaimer in
  *    the documentation and/or other materials provided with the
  *    distribution.
  *
  * 3. All advertising materials mentioning features or use of this
  *    software must display the following acknowledgment:
  *    "This product includes software developed by the OpenSSL Project
  *    for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
  *
  * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
  *    endorse or promote products derived from this software without
  *    prior written permission. For written permission, please contact
  *    openssl-core@openssl.org.
  *
  * 5. Products derived from this software may not be called "OpenSSL"
  *    nor may "OpenSSL" appear in their names without prior written
  *    permission of the OpenSSL Project.
  *
  * 6. Redistributions of any form whatsoever must retain the following
  *    acknowledgment:
  *    "This product includes software developed by the OpenSSL Project
  *    for use in the OpenSSL Toolkit (http://www.openssl.org/)"
  *
  * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
  * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
  * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE OpenSSL PROJECT OR
  * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
  * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
  * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
  * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
  * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
  * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
  * OF THE POSSIBILITY OF SUCH DAMAGE.
  * ====================================================================
  *
  * This product includes cryptographic software written by Eric Young
  * (eay@cryptsoft.com).  This product includes software written by Tim
  * Hudson (tjh@cryptsoft.com). */

 #include <openssl/bn.h>

 #include <openssl/err.h>

 #include "internal.h"


 BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) {
   // Compute a square root of |a| mod |p| using the Tonelli/Shanks algorithm
   // (cf. Henri Cohen, "A Course in Algebraic Computational Number Theory",
   // algorithm 1.5.1). |p| is assumed to be a prime.

   BIGNUM *ret = in;
   int err = 1;
   int r;
   BIGNUM *A, *b, *q, *t, *x, *y;
   int e, i, j;

   if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) {
     if (BN_abs_is_word(p, 2)) {
       if (ret == NULL) {
         ret = BN_new();
       }
       if (ret == NULL ||
           !BN_set_word(ret, BN_is_bit_set(a, 0))) {
         if (ret != in) {
           BN_free(ret);
         }
         return NULL;
       }
       return ret;
     }

     OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
     return NULL;
   }

   if (BN_is_zero(a) || BN_is_one(a)) {
     if (ret == NULL) {
       ret = BN_new();
     }
     if (ret == NULL ||
         !BN_set_word(ret, BN_is_one(a))) {
       if (ret != in) {
         BN_free(ret);
       }
       return NULL;
     }
     return ret;
   }

   BN_CTX_start(ctx);
   A = BN_CTX_get(ctx);
   b = BN_CTX_get(ctx);
   q = BN_CTX_get(ctx);
   t = BN_CTX_get(ctx);
   x = BN_CTX_get(ctx);
   y = BN_CTX_get(ctx);
   if (y == NULL) {
     goto end;
   }

   if (ret == NULL) {
     ret = BN_new();
   }
   if (ret == NULL) {
     goto end;
   }

   // A = a mod p
   if (!BN_nnmod(A, a, p, ctx)) {
     goto end;
   }

   // now write  |p| - 1  as  2^e*q  where  q  is odd
   e = 1;
   while (!BN_is_bit_set(p, e)) {
     e++;
   }
   // we'll set  q  later (if needed)

   if (e == 1) {
     // The easy case:  (|p|-1)/2  is odd, so 2 has an inverse
     // modulo  (|p|-1)/2,  and square roots can be computed
     // directly by modular exponentiation.
     // We have
     //     2 * (|p|+1)/4 == 1   (mod (|p|-1)/2),
     // so we can use exponent  (|p|+1)/4,  i.e.  (|p|-3)/4 + 1.
     if (!BN_rshift(q, p, 2)) {
       goto end;
     }
     q->neg = 0;
     if (!BN_add_word(q, 1) ||
         !BN_mod_exp_mont(ret, A, q, p, ctx, NULL)) {
       goto end;
     }
     err = 0;
     goto vrfy;
   }

   if (e == 2) {
     // |p| == 5  (mod 8)
     //
     // In this case  2  is always a non-square since
     // Legendre(2,p) = (-1)^((p^2-1)/8)  for any odd prime.
     // So if  a  really is a square, then  2*a  is a non-square.
     // Thus for
     //      b := (2*a)^((|p|-5)/8),
     //      i := (2*a)*b^2
     // we have
     //     i^2 = (2*a)^((1 + (|p|-5)/4)*2)
     //         = (2*a)^((p-1)/2)
     //         = -1;
     // so if we set
     //      x := a*b*(i-1),
     // then
     //     x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
     //         = a^2 * b^2 * (-2*i)
     //         = a*(-i)*(2*a*b^2)
     //         = a*(-i)*i
     //         = a.
     //
     // (This is due to A.O.L. Atkin,
     // <URL:
     //http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
     // November 1992.)

     // t := 2*a
     if (!bn_mod_lshift1_consttime(t, A, p, ctx)) {
       goto end;
     }

     // b := (2*a)^((|p|-5)/8)
     if (!BN_rshift(q, p, 3)) {
       goto end;
     }
     q->neg = 0;
     if (!BN_mod_exp_mont(b, t, q, p, ctx, NULL)) {
       goto end;
     }

     // y := b^2
     if (!BN_mod_sqr(y, b, p, ctx)) {
       goto end;
     }

     // t := (2*a)*b^2 - 1
     if (!BN_mod_mul(t, t, y, p, ctx) ||
         !BN_sub_word(t, 1)) {
       goto end;
     }

     // x = a*b*t
     if (!BN_mod_mul(x, A, b, p, ctx) ||
         !BN_mod_mul(x, x, t, p, ctx)) {
       goto end;
     }

     if (!BN_copy(ret, x)) {
       goto end;
     }
     err = 0;
     goto vrfy;
   }

   // e > 2, so we really have to use the Tonelli/Shanks algorithm.
   // First, find some  y  that is not a square.
   if (!BN_copy(q, p)) {
     goto end;  // use 'q' as temp
   }
   q->neg = 0;
   i = 2;
   do {
     // For efficiency, try small numbers first;
     // if this fails, try random numbers.
     if (i < 22) {
       if (!BN_set_word(y, i)) {
         goto end;
       }
     } else {
       if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) {
         goto end;
       }
       if (BN_ucmp(y, p) >= 0) {
         if (BN_usub(y, y, p)) {
           goto end;
         }
       }
       // now 0 <= y < |p|
       if (BN_is_zero(y)) {
         if (!BN_set_word(y, i)) {
           goto end;
         }
       }
     }

     r = bn_jacobi(y, q, ctx);  // here 'q' is |p|
     if (r < -1) {
       goto end;
     }
     if (r == 0) {
       // m divides p
       OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
       goto end;
     }
   } while (r == 1 && ++i < 82);

   if (r != -1) {
     // Many rounds and still no non-square -- this is more likely
     // a bug than just bad luck.
     // Even if  p  is not prime, we should have found some  y
     // such that r == -1.
     OPENSSL_PUT_ERROR(BN, BN_R_TOO_MANY_ITERATIONS);
     goto end;
   }

   // Here's our actual 'q':
   if (!BN_rshift(q, q, e)) {
     goto end;
   }

   // Now that we have some non-square, we can find an element
   // of order  2^e  by computing its q'th power.
   if (!BN_mod_exp_mont(y, y, q, p, ctx, NULL)) {
     goto end;
   }
   if (BN_is_one(y)) {
     OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
     goto end;
   }

   // Now we know that (if  p  is indeed prime) there is an integer
   // k,  0 <= k < 2^e,  such that
   //
   //      a^q * y^k == 1   (mod p).
   //
   // As  a^q  is a square and  y  is not,  k  must be even.
   // q+1  is even, too, so there is an element
   //
   //     X := a^((q+1)/2) * y^(k/2),
   //
   // and it satisfies
   //
   //     X^2 = a^q * a     * y^k
   //         = a,
   //
   // so it is the square root that we are looking for.

   // t := (q-1)/2  (note that  q  is odd)
   if (!BN_rshift1(t, q)) {
     goto end;
   }

   // x := a^((q-1)/2)
   if (BN_is_zero(t)) {  // special case: p = 2^e + 1
     if (!BN_nnmod(t, A, p, ctx)) {
       goto end;
     }
     if (BN_is_zero(t)) {
       // special case: a == 0  (mod p)
       BN_zero(ret);
       err = 0;
       goto end;
     } else if (!BN_one(x)) {
       goto end;
     }
   } else {
     if (!BN_mod_exp_mont(x, A, t, p, ctx, NULL)) {
       goto end;
     }
     if (BN_is_zero(x)) {
       // special case: a == 0  (mod p)
       BN_zero(ret);
       err = 0;
       goto end;
     }
   }

   // b := a*x^2  (= a^q)
   if (!BN_mod_sqr(b, x, p, ctx) ||
       !BN_mod_mul(b, b, A, p, ctx)) {
     goto end;
   }

   // x := a*x    (= a^((q+1)/2))
   if (!BN_mod_mul(x, x, A, p, ctx)) {
     goto end;
   }

   while (1) {
     // Now  b  is  a^q * y^k  for some even  k  (0 <= k < 2^E
     // where  E  refers to the original value of  e,  which we
     // don't keep in a variable),  and  x  is  a^((q+1)/2) * y^(k/2).
     //
     // We have  a*b = x^2,
     //    y^2^(e-1) = -1,
     //    b^2^(e-1) = 1.
     if (BN_is_one(b)) {
       if (!BN_copy(ret, x)) {
         goto end;
       }
       err = 0;
       goto vrfy;
     }

     // Find the smallest i, 0 < i < e, such that b^(2^i) = 1
     for (i = 1; i < e; i++) {
       if (i == 1) {
         if (!BN_mod_sqr(t, b, p, ctx)) {
           goto end;
         }
       } else {
         if (!BN_mod_mul(t, t, t, p, ctx)) {
           goto end;
         }
       }
       if (BN_is_one(t)) {
         break;
       }
     }
     // If not found, a is not a square or p is not a prime.
     if (i >= e) {
       OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
       goto end;
     }

     // t := y^2^(e - i - 1)
     if (!BN_copy(t, y)) {
       goto end;
     }
     for (j = e - i - 1; j > 0; j--) {
       if (!BN_mod_sqr(t, t, p, ctx)) {
         goto end;
       }
     }
     if (!BN_mod_mul(y, t, t, p, ctx) ||
         !BN_mod_mul(x, x, t, p, ctx) ||
         !BN_mod_mul(b, b, y, p, ctx)) {
       goto end;
     }

     // e decreases each iteration, so this loop will terminate.
     assert(i < e);
     e = i;
   }

 vrfy:
   if (!err) {
     // Verify the result. The input might have been not a square.
     if (!BN_mod_sqr(x, ret, p, ctx)) {
       err = 1;
     }

     if (!err && 0 != BN_cmp(x, A)) {
       OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
       err = 1;
     }
   }

 end:
   if (err) {
     if (ret != in) {
       BN_clear_free(ret);
     }
     ret = NULL;
   }
   BN_CTX_end(ctx);
   return ret;
 }

 int BN_sqrt(BIGNUM *out_sqrt, const BIGNUM *in, BN_CTX *ctx) {
   BIGNUM *estimate, *tmp, *delta, *last_delta, *tmp2;
   int ok = 0, last_delta_valid = 0;

   if (in->neg) {
     OPENSSL_PUT_ERROR(BN, BN_R_NEGATIVE_NUMBER);
     return 0;
   }
   if (BN_is_zero(in)) {
     BN_zero(out_sqrt);
     return 1;
   }

   BN_CTX_start(ctx);
   if (out_sqrt == in) {
     estimate = BN_CTX_get(ctx);
   } else {
     estimate = out_sqrt;
   }
   tmp = BN_CTX_get(ctx);
   last_delta = BN_CTX_get(ctx);
   delta = BN_CTX_get(ctx);
   if (estimate == NULL || tmp == NULL || last_delta == NULL || delta == NULL) {
     goto err;
   }

   // We estimate that the square root of an n-bit number is 2^{n/2}.
   if (!BN_lshift(estimate, BN_value_one(), BN_num_bits(in)/2)) {
     goto err;
   }

   // This is Newton's method for finding a root of the equation |estimate|^2 -
   // |in| = 0.
   for (;;) {
     // |estimate| = 1/2 * (|estimate| + |in|/|estimate|)
     if (!BN_div(tmp, NULL, in, estimate, ctx) ||
         !BN_add(tmp, tmp, estimate) ||
         !BN_rshift1(estimate, tmp) ||
         // |tmp| = |estimate|^2
         !BN_sqr(tmp, estimate, ctx) ||
         // |delta| = |in| - |tmp|
         !BN_sub(delta, in, tmp)) {
       OPENSSL_PUT_ERROR(BN, ERR_R_BN_LIB);
       goto err;
     }

     delta->neg = 0;
     // The difference between |in| and |estimate| squared is required to always
     // decrease. This ensures that the loop always terminates, but I don't have
     // a proof that it always finds the square root for a given square.
     if (last_delta_valid && BN_cmp(delta, last_delta) >= 0) {
       break;
     }

     last_delta_valid = 1;

     tmp2 = last_delta;
     last_delta = delta;
     delta = tmp2;
   }

   if (BN_cmp(tmp, in) != 0) {
     OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
     goto err;
   }

   ok = 1;

 err:
   if (ok && out_sqrt == in && !BN_copy(out_sqrt, estimate)) {
     ok = 0;
   }
   BN_CTX_end(ctx);
   return ok;
 }
	/* Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de>
	* and Bodo Moeller for the OpenSSL project. */
	/* ====================================================================
	* Copyright (c) 1998-2000 The OpenSSL Project. All rights reserved.
	*
	* Redistribution and use in source and binary forms, with or without
	* modification, are permitted provided that the following conditions
	* are met:
	*
	* 1. Redistributions of source code must retain the above copyright
	* notice, this list of conditions and the following disclaimer.
	*
	* 2. Redistributions in binary form must reproduce the above copyright
	* notice, this list of conditions and the following disclaimer in
	* the documentation and/or other materials provided with the
	* distribution.
	*
	* 3. All advertising materials mentioning features or use of this
	* software must display the following acknowledgment:
	* "This product includes software developed by the OpenSSL Project
	* for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
	*
	* 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
	* endorse or promote products derived from this software without
	* prior written permission. For written permission, please contact
	* openssl-core@openssl.org.
	*
	* 5. Products derived from this software may not be called "OpenSSL"
	* nor may "OpenSSL" appear in their names without prior written
	* permission of the OpenSSL Project.
	*
	* 6. Redistributions of any form whatsoever must retain the following
	* acknowledgment:
	* "This product includes software developed by the OpenSSL Project
	* for use in the OpenSSL Toolkit (http://www.openssl.org/)"
	*
	* THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
	* EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
	* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
	* PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
	* ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
	* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
	* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
	* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
	* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
	* STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
	* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
	* OF THE POSSIBILITY OF SUCH DAMAGE.
	* ====================================================================
	*
	* This product includes cryptographic software written by Eric Young
	* (eay@cryptsoft.com). This product includes software written by Tim
	* Hudson (tjh@cryptsoft.com). */

	#include <openssl/bn.h>

	#include <openssl/err.h>

	#include "internal.h"


	BIGNUM BN_mod_sqrt(BIGNUM in, const BIGNUM a, const BIGNUM p, BN_CTX *ctx) {
	// Compute a square root of \|a\| mod \|p\| using the Tonelli/Shanks algorithm
	// (cf. Henri Cohen, "A Course in Algebraic Computational Number Theory",
	// algorithm 1.5.1). \|p\| is assumed to be a prime.

	BIGNUM *ret = in;
	int err = 1;
	int r;
	BIGNUM A, b, q, t, x, y;
	int e, i, j;

	if (!BN_is_odd(p) \|\| BN_abs_is_word(p, 1)) {
	if (BN_abs_is_word(p, 2)) {
	if (ret == NULL) {
	ret = BN_new();
	}
	if (ret == NULL \|\|
	!BN_set_word(ret, BN_is_bit_set(a, 0))) {
	if (ret != in) {
	BN_free(ret);
	}
	return NULL;
	}
	return ret;
	}

	OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
	return NULL;
	}

	if (BN_is_zero(a) \|\| BN_is_one(a)) {
	if (ret == NULL) {
	ret = BN_new();
	}
	if (ret == NULL \|\|
	!BN_set_word(ret, BN_is_one(a))) {
	if (ret != in) {
	BN_free(ret);
	}
	return NULL;
	}
	return ret;
	}

	BN_CTX_start(ctx);
	A = BN_CTX_get(ctx);
	b = BN_CTX_get(ctx);
	q = BN_CTX_get(ctx);
	t = BN_CTX_get(ctx);
	x = BN_CTX_get(ctx);
	y = BN_CTX_get(ctx);
	if (y == NULL) {
	goto end;
	}

	if (ret == NULL) {
	ret = BN_new();
	}
	if (ret == NULL) {
	goto end;
	}

	// A = a mod p
	if (!BN_nnmod(A, a, p, ctx)) {
	goto end;
	}

	// now write \|p\| - 1 as 2^e*q where q is odd
	e = 1;
	while (!BN_is_bit_set(p, e)) {
	e++;
	}
	// we'll set q later (if needed)

	if (e == 1) {
	// The easy case: (\|p\|-1)/2 is odd, so 2 has an inverse
	// modulo (\|p\|-1)/2, and square roots can be computed
	// directly by modular exponentiation.
	// We have
	// 2 * (\|p\|+1)/4 == 1 (mod (\|p\|-1)/2),
	// so we can use exponent (\|p\|+1)/4, i.e. (\|p\|-3)/4 + 1.
	if (!BN_rshift(q, p, 2)) {
	goto end;
	}
	q->neg = 0;
	if (!BN_add_word(q, 1) \|\|
	!BN_mod_exp_mont(ret, A, q, p, ctx, NULL)) {
	goto end;
	}
	err = 0;
	goto vrfy;
	}

	if (e == 2) {
	// \|p\| == 5 (mod 8)
	//
	// In this case 2 is always a non-square since
	// Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime.
	// So if a really is a square, then 2*a is a non-square.
	// Thus for
	// b := (2*a)^((\|p\|-5)/8),
	// i := (2a)b^2
	// we have
	// i^2 = (2a)^((1 + (\|p\|-5)/4)2)
	// = (2*a)^((p-1)/2)
	// = -1;
	// so if we set
	// x := ab(i-1),
	// then
	// x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
	// = a^2 * b^2 * (-2*i)
	// = a(-i)(2ab^2)
	// = a(-i)i
	// = a.
	//
	// (This is due to A.O.L. Atkin,
	// <URL:
	//http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
	// November 1992.)

	// t := 2*a
	if (!bn_mod_lshift1_consttime(t, A, p, ctx)) {
	goto end;
	}

	// b := (2*a)^((\|p\|-5)/8)
	if (!BN_rshift(q, p, 3)) {
	goto end;
	}
	q->neg = 0;
	if (!BN_mod_exp_mont(b, t, q, p, ctx, NULL)) {
	goto end;
	}

	// y := b^2
	if (!BN_mod_sqr(y, b, p, ctx)) {
	goto end;
	}

	// t := (2a)b^2 - 1
	if (!BN_mod_mul(t, t, y, p, ctx) \|\|
	!BN_sub_word(t, 1)) {
	goto end;
	}

	// x = abt
	if (!BN_mod_mul(x, A, b, p, ctx) \|\|
	!BN_mod_mul(x, x, t, p, ctx)) {
	goto end;
	}

	if (!BN_copy(ret, x)) {
	goto end;
	}
	err = 0;
	goto vrfy;
	}

	// e > 2, so we really have to use the Tonelli/Shanks algorithm.
	// First, find some y that is not a square.
	if (!BN_copy(q, p)) {
	goto end; // use 'q' as temp
	}
	q->neg = 0;
	i = 2;
	do {
	// For efficiency, try small numbers first;
	// if this fails, try random numbers.
	if (i < 22) {
	if (!BN_set_word(y, i)) {
	goto end;
	}
	} else {
	if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) {
	goto end;
	}
	if (BN_ucmp(y, p) >= 0) {
	if (BN_usub(y, y, p)) {
	goto end;
	}
	}
	// now 0 <= y < \|p\|
	if (BN_is_zero(y)) {
	if (!BN_set_word(y, i)) {
	goto end;
	}
	}
	}

	r = bn_jacobi(y, q, ctx); // here 'q' is \|p\|
	if (r < -1) {
	goto end;
	}
	if (r == 0) {
	// m divides p
	OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
	goto end;
	}
	} while (r == 1 && ++i < 82);

	if (r != -1) {
	// Many rounds and still no non-square -- this is more likely
	// a bug than just bad luck.
	// Even if p is not prime, we should have found some y
	// such that r == -1.
	OPENSSL_PUT_ERROR(BN, BN_R_TOO_MANY_ITERATIONS);
	goto end;
	}

	// Here's our actual 'q':
	if (!BN_rshift(q, q, e)) {
	goto end;
	}

	// Now that we have some non-square, we can find an element
	// of order 2^e by computing its q'th power.
	if (!BN_mod_exp_mont(y, y, q, p, ctx, NULL)) {
	goto end;
	}
	if (BN_is_one(y)) {
	OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
	goto end;
	}

	// Now we know that (if p is indeed prime) there is an integer
	// k, 0 <= k < 2^e, such that
	//
	// a^q * y^k == 1 (mod p).
	//
	// As a^q is a square and y is not, k must be even.
	// q+1 is even, too, so there is an element
	//
	// X := a^((q+1)/2) * y^(k/2),
	//
	// and it satisfies
	//
	// X^2 = a^q * a * y^k
	// = a,
	//
	// so it is the square root that we are looking for.

	// t := (q-1)/2 (note that q is odd)
	if (!BN_rshift1(t, q)) {
	goto end;
	}

	// x := a^((q-1)/2)
	if (BN_is_zero(t)) { // special case: p = 2^e + 1
	if (!BN_nnmod(t, A, p, ctx)) {
	goto end;
	}
	if (BN_is_zero(t)) {
	// special case: a == 0 (mod p)
	BN_zero(ret);
	err = 0;
	goto end;
	} else if (!BN_one(x)) {
	goto end;
	}
	} else {
	if (!BN_mod_exp_mont(x, A, t, p, ctx, NULL)) {
	goto end;
	}
	if (BN_is_zero(x)) {
	// special case: a == 0 (mod p)
	BN_zero(ret);
	err = 0;
	goto end;
	}
	}

	// b := a*x^2 (= a^q)
	if (!BN_mod_sqr(b, x, p, ctx) \|\|
	!BN_mod_mul(b, b, A, p, ctx)) {
	goto end;
	}

	// x := a*x (= a^((q+1)/2))
	if (!BN_mod_mul(x, x, A, p, ctx)) {
	goto end;
	}

	while (1) {
	// Now b is a^q * y^k for some even k (0 <= k < 2^E
	// where E refers to the original value of e, which we
	// don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
	//
	// We have a*b = x^2,
	// y^2^(e-1) = -1,
	// b^2^(e-1) = 1.
	if (BN_is_one(b)) {
	if (!BN_copy(ret, x)) {
	goto end;
	}
	err = 0;
	goto vrfy;
	}

	// Find the smallest i, 0 < i < e, such that b^(2^i) = 1
	for (i = 1; i < e; i++) {
	if (i == 1) {
	if (!BN_mod_sqr(t, b, p, ctx)) {
	goto end;
	}
	} else {
	if (!BN_mod_mul(t, t, t, p, ctx)) {
	goto end;
	}
	}
	if (BN_is_one(t)) {
	break;
	}
	}
	// If not found, a is not a square or p is not a prime.
	if (i >= e) {
	OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
	goto end;
	}

	// t := y^2^(e - i - 1)
	if (!BN_copy(t, y)) {
	goto end;
	}
	for (j = e - i - 1; j > 0; j--) {
	if (!BN_mod_sqr(t, t, p, ctx)) {
	goto end;
	}
	}
	if (!BN_mod_mul(y, t, t, p, ctx) \|\|
	!BN_mod_mul(x, x, t, p, ctx) \|\|
	!BN_mod_mul(b, b, y, p, ctx)) {
	goto end;
	}

	// e decreases each iteration, so this loop will terminate.
	assert(i < e);
	e = i;
	}

	vrfy:
	if (!err) {
	// Verify the result. The input might have been not a square.
	if (!BN_mod_sqr(x, ret, p, ctx)) {
	err = 1;
	}

	if (!err && 0 != BN_cmp(x, A)) {
	OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
	err = 1;
	}
	}

	end:
	if (err) {
	if (ret != in) {
	BN_clear_free(ret);
	}
	ret = NULL;
	}
	BN_CTX_end(ctx);
	return ret;
	}

	int BN_sqrt(BIGNUM out_sqrt, const BIGNUM in, BN_CTX *ctx) {
	BIGNUM estimate, tmp, delta, last_delta, *tmp2;
	int ok = 0, last_delta_valid = 0;

	if (in->neg) {
	OPENSSL_PUT_ERROR(BN, BN_R_NEGATIVE_NUMBER);
	return 0;
	}
	if (BN_is_zero(in)) {
	BN_zero(out_sqrt);
	return 1;
	}

	BN_CTX_start(ctx);
	if (out_sqrt == in) {
	estimate = BN_CTX_get(ctx);
	} else {
	estimate = out_sqrt;
	}
	tmp = BN_CTX_get(ctx);
	last_delta = BN_CTX_get(ctx);
	delta = BN_CTX_get(ctx);
	if (estimate == NULL \|\| tmp == NULL \|\| last_delta == NULL \|\| delta == NULL) {
	goto err;
	}

	// We estimate that the square root of an n-bit number is 2^{n/2}.
	if (!BN_lshift(estimate, BN_value_one(), BN_num_bits(in)/2)) {
	goto err;
	}

	// This is Newton's method for finding a root of the equation \|estimate\|^2 -
	// \|in\| = 0.
	for (;;) {
	// \|estimate\| = 1/2 * (\|estimate\| + \|in\|/\|estimate\|)
	if (!BN_div(tmp, NULL, in, estimate, ctx) \|\|
	!BN_add(tmp, tmp, estimate) \|\|
	!BN_rshift1(estimate, tmp) \|\|
	// \|tmp\| = \|estimate\|^2
	!BN_sqr(tmp, estimate, ctx) \|\|
	// \|delta\| = \|in\| - \|tmp\|
	!BN_sub(delta, in, tmp)) {
	OPENSSL_PUT_ERROR(BN, ERR_R_BN_LIB);
	goto err;
	}

	delta->neg = 0;
	// The difference between \|in\| and \|estimate\| squared is required to always
	// decrease. This ensures that the loop always terminates, but I don't have
	// a proof that it always finds the square root for a given square.
	if (last_delta_valid && BN_cmp(delta, last_delta) >= 0) {
	break;
	}

	last_delta_valid = 1;

	tmp2 = last_delta;
	last_delta = delta;
	delta = tmp2;
	}

	if (BN_cmp(tmp, in) != 0) {
	OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
	goto err;
	}

	ok = 1;

	err:
	if (ok && out_sqrt == in && !BN_copy(out_sqrt, estimate)) {
	ok = 0;
	}
	BN_CTX_end(ctx);
	return ok;
	}