/src/boringssl/crypto/fipsmodule/bn/sqrt.cc.inc

Source
// Copyright 2000-2016 The OpenSSL Project Authors. All Rights Reserved.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     https://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

#include <openssl/bn.h>

#include <openssl/err.h>

#include "internal.h"


using namespace bssl;

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 == nullptr) {
        ret = BN_new();
      }
      if (ret == nullptr || !BN_set_word(ret, BN_is_bit_set(a, 0))) {
        if (ret != in) {
          BN_free(ret);
        }
        return nullptr;
      }
      return ret;
    }

    OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
    return nullptr;
  }

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

  BN_CTXScope scope(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 == nullptr) {
    goto end;
  }

  if (ret == nullptr) {
    ret = BN_new();
  }
  if (ret == nullptr) {
    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, nullptr)) {
      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, nullptr)) {
      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_rand_range_ex(y, 22, p)) {
        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, nullptr)) {
    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, nullptr)) {
      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 = nullptr;
  }
  return ret;
}

Coverage Report

Created: 2026-06-15 07:04

Line	Count	Source
1		// Copyright 2000-2016 The OpenSSL Project Authors. All Rights Reserved.
2		//
3		// Licensed under the Apache License, Version 2.0 (the "License");
4		// you may not use this file except in compliance with the License.
5		// You may obtain a copy of the License at
6		//
7		// https://www.apache.org/licenses/LICENSE-2.0
8		//
9		// Unless required by applicable law or agreed to in writing, software
10		// distributed under the License is distributed on an "AS IS" BASIS,
11		// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
12		// See the License for the specific language governing permissions and
13		// limitations under the License.
14
15		#include <openssl/bn.h>
16
17		#include <openssl/err.h>
18
19		#include "internal.h"
20
21
22		using namespace bssl;
23
24	4.12k	BIGNUM BN_mod_sqrt(BIGNUM in, const BIGNUM a, const BIGNUM p, BN_CTX *ctx) {
25		// Compute a square root of \|a\| mod \|p\| using the Tonelli/Shanks algorithm
26		// (cf. Henri Cohen, "A Course in Algebraic Computational Number Theory",
27		// algorithm 1.5.1). \|p\| is assumed to be a prime.
28
29	4.12k	BIGNUM *ret = in;
30	4.12k	int err = 1;
31	4.12k	int r;
32	4.12k	BIGNUM A, b, q, t, x, y;
33	4.12k	int e, i, j;
34
35	4.12k	if (!BN_is_odd(p) \|\| BN_abs_is_word(p, 1)) {
36	0	if (BN_abs_is_word(p, 2)) {
37	0	if (ret == nullptr) {
38	0	ret = BN_new();
39	0	}
40	0	if (ret == nullptr \|\| !BN_set_word(ret, BN_is_bit_set(a, 0))) {
41	0	if (ret != in) {
42	0	BN_free(ret);
43	0	}
44	0	return nullptr;
45	0	}
46	0	return ret;
47	0	}
48
49	0	OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
50	0	return nullptr;
51	0	}
52
53	4.12k	if (BN_is_zero(a) \|\| BN_is_one(a)) {
54	0	if (ret == nullptr) {
55	0	ret = BN_new();
56	0	}
57	0	if (ret == nullptr \|\| !BN_set_word(ret, BN_is_one(a))) {
58	0	if (ret != in) {
59	0	BN_free(ret);
60	0	}
61	0	return nullptr;
62	0	}
63	0	return ret;
64	0	}
65
66	4.12k	BN_CTXScope scope(ctx);
67	4.12k	A = BN_CTX_get(ctx);
68	4.12k	b = BN_CTX_get(ctx);
69	4.12k	q = BN_CTX_get(ctx);
70	4.12k	t = BN_CTX_get(ctx);
71	4.12k	x = BN_CTX_get(ctx);
72	4.12k	y = BN_CTX_get(ctx);
73	4.12k	if (y == nullptr) {
74	0	goto end;
75	0	}
76
77	4.12k	if (ret == nullptr) {
78	0	ret = BN_new();
79	0	}
80	4.12k	if (ret == nullptr) {
81	0	goto end;
82	0	}
83
84		// A = a mod p
85	4.12k	if (!BN_nnmod(A, a, p, ctx)) {
86	0	goto end;
87	0	}
88
89		// now write \|p\| - 1 as 2^e*q where q is odd
90	4.12k	e = 1;
91	281k	while (!BN_is_bit_set(p, e)) {
92	277k	e++;
93	277k	}
94		// we'll set q later (if needed)
95
96	4.12k	if (e == 1) {
97		// The easy case: (\|p\|-1)/2 is odd, so 2 has an inverse
98		// modulo (\|p\|-1)/2, and square roots can be computed
99		// directly by modular exponentiation.
100		// We have
101		// 2 * (\|p\|+1)/4 == 1 (mod (\|p\|-1)/2),
102		// so we can use exponent (\|p\|+1)/4, i.e. (\|p\|-3)/4 + 1.
103	1.20k	if (!BN_rshift(q, p, 2)) {
104	0	goto end;
105	0	}
106	1.20k	q->neg = 0;
107	1.20k	if (!BN_add_word(q, 1) \|\| !BN_mod_exp_mont(ret, A, q, p, ctx, nullptr)) {
108	0	goto end;
109	0	}
110	1.20k	err = 0;
111	1.20k	goto vrfy;
112	1.20k	}
113
114	2.92k	if (e == 2) {
115		// \|p\| == 5 (mod 8)
116		//
117		// In this case 2 is always a non-square since
118		// Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime.
119		// So if a really is a square, then 2*a is a non-square.
120		// Thus for
121		// b := (2*a)^((\|p\|-5)/8),
122		// i := (2a)b^2
123		// we have
124		// i^2 = (2a)^((1 + (\|p\|-5)/4)2)
125		// = (2*a)^((p-1)/2)
126		// = -1;
127		// so if we set
128		// x := ab(i-1),
129		// then
130		// x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
131		// = a^2 * b^2 * (-2*i)
132		// = a(-i)(2ab^2)
133		// = a(-i)i
134		// = a.
135		//
136		// (This is due to A.O.L. Atkin,
137		// <URL:
138		//http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
139		// November 1992.)
140
141		// t := 2*a
142	0	if (!bn_mod_lshift1_consttime(t, A, p, ctx)) {
143	0	goto end;
144	0	}
145
146		// b := (2*a)^((\|p\|-5)/8)
147	0	if (!BN_rshift(q, p, 3)) {
148	0	goto end;
149	0	}
150	0	q->neg = 0;
151	0	if (!BN_mod_exp_mont(b, t, q, p, ctx, nullptr)) {
152	0	goto end;
153	0	}
154
155		// y := b^2
156	0	if (!BN_mod_sqr(y, b, p, ctx)) {
157	0	goto end;
158	0	}
159
160		// t := (2a)b^2 - 1
161	0	if (!BN_mod_mul(t, t, y, p, ctx) \|\|
162	0	!BN_sub_word(t, 1)) {
163	0	goto end;
164	0	}
165
166		// x = abt
167	0	if (!BN_mod_mul(x, A, b, p, ctx) \|\|
168	0	!BN_mod_mul(x, x, t, p, ctx)) {
169	0	goto end;
170	0	}
171
172	0	if (!BN_copy(ret, x)) {
173	0	goto end;
174	0	}
175	0	err = 0;
176	0	goto vrfy;
177	0	}
178
179		// e > 2, so we really have to use the Tonelli/Shanks algorithm.
180		// First, find some y that is not a square.
181	2.92k	if (!BN_copy(q, p)) {
182	0	goto end; // use 'q' as temp
183	0	}
184	2.92k	q->neg = 0;
185	2.92k	i = 2;
186	29.2k	do {
187		// For efficiency, try small numbers first;
188		// if this fails, try random numbers.
189	29.2k	if (i < 22) {
190	29.2k	if (!BN_set_word(y, i)) {
191	0	goto end;
192	0	}
193	29.2k	} else {
194	0	if (!BN_rand_range_ex(y, 22, p)) {
195	0	goto end;
196	0	}
197	0	}
198
199	29.2k	r = bn_jacobi(y, q, ctx); // here 'q' is \|p\|
200	29.2k	if (r < -1) {
201	0	goto end;
202	0	}
203	29.2k	if (r == 0) {
204		// m divides p
205	0	OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
206	0	goto end;
207	0	}
208	29.2k	} while (r == 1 && ++i < 82);
209
210	2.92k	if (r != -1) {
211		// Many rounds and still no non-square -- this is more likely
212		// a bug than just bad luck.
213		// Even if p is not prime, we should have found some y
214		// such that r == -1.
215	0	OPENSSL_PUT_ERROR(BN, BN_R_TOO_MANY_ITERATIONS);
216	0	goto end;
217	0	}
218
219		// Here's our actual 'q':
220	2.92k	if (!BN_rshift(q, q, e)) {
221	0	goto end;
222	0	}
223
224		// Now that we have some non-square, we can find an element
225		// of order 2^e by computing its q'th power.
226	2.92k	if (!BN_mod_exp_mont(y, y, q, p, ctx, nullptr)) {
227	0	goto end;
228	0	}
229	2.92k	if (BN_is_one(y)) {
230	0	OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
231	0	goto end;
232	0	}
233
234		// Now we know that (if p is indeed prime) there is an integer
235		// k, 0 <= k < 2^e, such that
236		//
237		// a^q * y^k == 1 (mod p).
238		//
239		// As a^q is a square and y is not, k must be even.
240		// q+1 is even, too, so there is an element
241		//
242		// X := a^((q+1)/2) * y^(k/2),
243		//
244		// and it satisfies
245		//
246		// X^2 = a^q * a * y^k
247		// = a,
248		//
249		// so it is the square root that we are looking for.
250
251		// t := (q-1)/2 (note that q is odd)
252	2.92k	if (!BN_rshift1(t, q)) {
253	0	goto end;
254	0	}
255
256		// x := a^((q-1)/2)
257	2.92k	if (BN_is_zero(t)) { // special case: p = 2^e + 1
258	0	if (!BN_nnmod(t, A, p, ctx)) {
259	0	goto end;
260	0	}
261	0	if (BN_is_zero(t)) {
262		// special case: a == 0 (mod p)
263	0	BN_zero(ret);
264	0	err = 0;
265	0	goto end;
266	0	} else if (!BN_one(x)) {
267	0	goto end;
268	0	}
269	2.92k	} else {
270	2.92k	if (!BN_mod_exp_mont(x, A, t, p, ctx, nullptr)) {
271	0	goto end;
272	0	}
273	2.92k	if (BN_is_zero(x)) {
274		// special case: a == 0 (mod p)
275	0	BN_zero(ret);
276	0	err = 0;
277	0	goto end;
278	0	}
279	2.92k	}
280
281		// b := a*x^2 (= a^q)
282	2.92k	if (!BN_mod_sqr(b, x, p, ctx) \|\|
283	2.92k	!BN_mod_mul(b, b, A, p, ctx)) {
284	0	goto end;
285	0	}
286
287		// x := a*x (= a^((q+1)/2))
288	2.92k	if (!BN_mod_mul(x, x, A, p, ctx)) {
289	0	goto end;
290	0	}
291
292	112k	while (1) {
293		// Now b is a^q * y^k for some even k (0 <= k < 2^E
294		// where E refers to the original value of e, which we
295		// don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
296		//
297		// We have a*b = x^2,
298		// y^2^(e-1) = -1,
299		// b^2^(e-1) = 1.
300	112k	if (BN_is_one(b)) {
301	2.31k	if (!BN_copy(ret, x)) {
302	0	goto end;
303	0	}
304	2.31k	err = 0;
305	2.31k	goto vrfy;
306	2.31k	}
307
308		// Find the smallest i, 0 < i < e, such that b^(2^i) = 1
309	5.34M	for (i = 1; i < e; i++) {
310	5.34M	if (i == 1) {
311	110k	if (!BN_mod_sqr(t, b, p, ctx)) {
312	0	goto end;
313	0	}
314	5.23M	} else {
315	5.23M	if (!BN_mod_mul(t, t, t, p, ctx)) {
316	0	goto end;
317	0	}
318	5.23M	}
319	5.34M	if (BN_is_one(t)) {
320	109k	break;
321	109k	}
322	5.34M	}
323		// If not found, a is not a square or p is not a prime.
324	110k	if (i >= e) {
325	603	OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
326	603	goto end;
327	603	}
328
329		// t := y^2^(e - i - 1)
330	109k	if (!BN_copy(t, y)) {
331	0	goto end;
332	0	}
333	217k	for (j = e - i - 1; j > 0; j--) {
334	107k	if (!BN_mod_sqr(t, t, p, ctx)) {
335	0	goto end;
336	0	}
337	107k	}
338	109k	if (!BN_mod_mul(y, t, t, p, ctx) \|\|
339	109k	!BN_mod_mul(x, x, t, p, ctx) \|\|
340	109k	!BN_mod_mul(b, b, y, p, ctx)) {
341	0	goto end;
342	0	}
343
344		// e decreases each iteration, so this loop will terminate.
345	109k	assert(i < e);
346	109k	e = i;
347	109k	}
348
349	3.52k	vrfy:
350	3.52k	if (!err) {
351		// Verify the result. The input might have been not a square.
352	3.52k	if (!BN_mod_sqr(x, ret, p, ctx)) {
353	0	err = 1;
354	0	}
355
356	3.52k	if (!err && 0 != BN_cmp(x, A)) {
357	372	OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
358	372	err = 1;
359	372	}
360	3.52k	}
361
362	4.12k	end:
363	4.12k	if (err) {
364	975	if (ret != in) {
365	0	BN_clear_free(ret);
366	0	}
367	975	ret = nullptr;
368	975	}
369	4.12k	return ret;
370	3.52k	}