/src/openssl/crypto/bn/bn_sqrt.c

Source
/*
 * Copyright 2000-2022 The OpenSSL Project Authors. All Rights Reserved.
 *
 * Licensed under the Apache License 2.0 (the "License").  You may not use
 * this file except in compliance with the License.  You can obtain a copy
 * in the file LICENSE in the source distribution or at
 * https://www.openssl.org/source/license.html
 */

#include "internal/cryptlib.h"
#include "bn_local.h"

BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
/*
 * Returns 'ret' such that ret^2 == a (mod p), using the Tonelli/Shanks
 * algorithm (cf. Henri Cohen, "A Course in Algebraic Computational Number
 * Theory", algorithm 1.5.1). 'p' must be prime, otherwise an error or
 * an incorrect "result" will be returned.
 */
{
    BIGNUM *ret = in;
    int err = 1;
    int r;
    BIGNUM *A, *b, *q, *t, *x, *y;
    int e, i, j;
    int used_ctx = 0;

    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)
                goto end;
            if (!BN_set_word(ret, BN_is_bit_set(a, 0))) {
                if (ret != in)
                    BN_free(ret);
                return NULL;
            }
            bn_check_top(ret);
            return ret;
        }

        ERR_raise(ERR_LIB_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)
            goto end;
        if (!BN_set_word(ret, BN_is_one(a))) {
            if (ret != in)
                BN_free(ret);
            return NULL;
        }
        bn_check_top(ret);
        return ret;
    }

    BN_CTX_start(ctx);
    used_ctx = 1;
    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))
            goto end;
        if (!BN_mod_exp(ret, A, q, p, ctx))
            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,
         * Subject: Square Roots and Cognate Matters modulo p=8n+5.
         * URL: https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind9211&L=NMBRTHRY&P=4026
         * November 1992.)
         */

        /* t := 2*a */
        if (!BN_mod_lshift1_quick(t, A, p))
            goto end;

        /* b := (2*a)^((|p|-5)/8) */
        if (!BN_rshift(q, p, 3))
            goto end;
        q->neg = 0;
        if (!BN_mod_exp(b, t, q, p, ctx))
            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))
            goto end;
        if (!BN_sub_word(t, 1))
            goto end;

        /* x = a*b*t */
        if (!BN_mod_mul(x, A, b, p, ctx))
            goto end;
        if (!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_priv_rand_ex(y, BN_num_bits(p), 0, 0, 0, ctx))
                goto end;
            if (BN_ucmp(y, p) >= 0) {
                if (!(p->neg ? BN_add : BN_sub)(y, y, p))
                    goto end;
            }
            /* now 0 <= y < |p| */
            if (BN_is_zero(y))
                if (!BN_set_word(y, i))
                    goto end;
        }

        r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */
        if (r < -1)
            goto end;
        if (r == 0) {
            /* m divides p */
            ERR_raise(ERR_LIB_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.
         */
        ERR_raise(ERR_LIB_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(y, y, q, p, ctx))
        goto end;
    if (BN_is_one(y)) {
        ERR_raise(ERR_LIB_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(x, A, t, p, ctx))
            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))
        goto end;
    if (!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 prime. */
        if (i >= e) {
            ERR_raise(ERR_LIB_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))
            goto end;
        if (!BN_mod_mul(x, x, t, p, ctx))
            goto end;
        if (!BN_mod_mul(b, b, y, p, ctx))
            goto end;
        e = i;
    }

vrfy:
    if (!err) {
        /*
         * verify the result -- the input might have been not a square (test
         * added in 0.9.8)
         */

        if (!BN_mod_sqr(x, ret, p, ctx))
            err = 1;

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

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

Coverage Report

Created: 2025-12-10 06:24

Line	Count	Source
1		/*
2		* Copyright 2000-2022 The OpenSSL Project Authors. All Rights Reserved.
3		*
4		* Licensed under the Apache License 2.0 (the "License"). You may not use
5		* this file except in compliance with the License. You can obtain a copy
6		* in the file LICENSE in the source distribution or at
7		* https://www.openssl.org/source/license.html
8		*/
9
10		#include "internal/cryptlib.h"
11		#include "bn_local.h"
12
13		BIGNUM BN_mod_sqrt(BIGNUM in, const BIGNUM a, const BIGNUM p, BN_CTX *ctx)
14		/*
15		* Returns 'ret' such that ret^2 == a (mod p), using the Tonelli/Shanks
16		* algorithm (cf. Henri Cohen, "A Course in Algebraic Computational Number
17		* Theory", algorithm 1.5.1). 'p' must be prime, otherwise an error or
18		* an incorrect "result" will be returned.
19		*/
20	0	{
21	0	BIGNUM *ret = in;
22	0	int err = 1;
23	0	int r;
24	0	BIGNUM A, b, q, t, x, y;
25	0	int e, i, j;
26	0	int used_ctx = 0;
27
28	0	if (!BN_is_odd(p) \|\| BN_abs_is_word(p, 1)) {
29	0	if (BN_abs_is_word(p, 2)) {
30	0	if (ret == NULL)
31	0	ret = BN_new();
32	0	if (ret == NULL)
33	0	goto end;
34	0	if (!BN_set_word(ret, BN_is_bit_set(a, 0))) {
35	0	if (ret != in)
36	0	BN_free(ret);
37	0	return NULL;
38	0	}
39	0	bn_check_top(ret);
40	0	return ret;
41	0	}
42
43	0	ERR_raise(ERR_LIB_BN, BN_R_P_IS_NOT_PRIME);
44	0	return NULL;
45	0	}
46
47	0	if (BN_is_zero(a) \|\| BN_is_one(a)) {
48	0	if (ret == NULL)
49	0	ret = BN_new();
50	0	if (ret == NULL)
51	0	goto end;
52	0	if (!BN_set_word(ret, BN_is_one(a))) {
53	0	if (ret != in)
54	0	BN_free(ret);
55	0	return NULL;
56	0	}
57	0	bn_check_top(ret);
58	0	return ret;
59	0	}
60
61	0	BN_CTX_start(ctx);
62	0	used_ctx = 1;
63	0	A = BN_CTX_get(ctx);
64	0	b = BN_CTX_get(ctx);
65	0	q = BN_CTX_get(ctx);
66	0	t = BN_CTX_get(ctx);
67	0	x = BN_CTX_get(ctx);
68	0	y = BN_CTX_get(ctx);
69	0	if (y == NULL)
70	0	goto end;
71
72	0	if (ret == NULL)
73	0	ret = BN_new();
74	0	if (ret == NULL)
75	0	goto end;
76
77		/* A = a mod p */
78	0	if (!BN_nnmod(A, a, p, ctx))
79	0	goto end;
80
81		/* now write \|p\| - 1 as 2^eq where q is odd /
82	0	e = 1;
83	0	while (!BN_is_bit_set(p, e))
84	0	e++;
85		/* we'll set q later (if needed) */
86
87	0	if (e == 1) {
88		/*-
89		* The easy case: (\|p\|-1)/2 is odd, so 2 has an inverse
90		* modulo (\|p\|-1)/2, and square roots can be computed
91		* directly by modular exponentiation.
92		* We have
93		* 2 * (\|p\|+1)/4 == 1 (mod (\|p\|-1)/2),
94		* so we can use exponent (\|p\|+1)/4, i.e. (\|p\|-3)/4 + 1.
95		*/
96	0	if (!BN_rshift(q, p, 2))
97	0	goto end;
98	0	q->neg = 0;
99	0	if (!BN_add_word(q, 1))
100	0	goto end;
101	0	if (!BN_mod_exp(ret, A, q, p, ctx))
102	0	goto end;
103	0	err = 0;
104	0	goto vrfy;
105	0	}
106
107	0	if (e == 2) {
108		/*-
109		* \|p\| == 5 (mod 8)
110		*
111		* In this case 2 is always a non-square since
112		* Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime.
113		* So if a really is a square, then 2*a is a non-square.
114		* Thus for
115		* b := (2*a)^((\|p\|-5)/8),
116		* i := (2a)b^2
117		* we have
118		* i^2 = (2a)^((1 + (\|p\|-5)/4)2)
119		* = (2*a)^((p-1)/2)
120		* = -1;
121		* so if we set
122		* x := ab(i-1),
123		* then
124		* x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
125		* = a^2 * b^2 * (-2*i)
126		* = a(-i)(2ab^2)
127		* = a(-i)i
128		* = a.
129		*
130		* (This is due to A.O.L. Atkin,
131		* Subject: Square Roots and Cognate Matters modulo p=8n+5.
132		* URL: https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind9211&L=NMBRTHRY&P=4026
133		* November 1992.)
134		*/
135
136		/* t := 2a /
137	0	if (!BN_mod_lshift1_quick(t, A, p))
138	0	goto end;
139
140		/* b := (2a)^((\|p\|-5)/8) /
141	0	if (!BN_rshift(q, p, 3))
142	0	goto end;
143	0	q->neg = 0;
144	0	if (!BN_mod_exp(b, t, q, p, ctx))
145	0	goto end;
146
147		/* y := b^2 */
148	0	if (!BN_mod_sqr(y, b, p, ctx))
149	0	goto end;
150
151		/* t := (2a)b^2 - 1 */
152	0	if (!BN_mod_mul(t, t, y, p, ctx))
153	0	goto end;
154	0	if (!BN_sub_word(t, 1))
155	0	goto end;
156
157		/* x = abt */
158	0	if (!BN_mod_mul(x, A, b, p, ctx))
159	0	goto end;
160	0	if (!BN_mod_mul(x, x, t, p, ctx))
161	0	goto end;
162
163	0	if (!BN_copy(ret, x))
164	0	goto end;
165	0	err = 0;
166	0	goto vrfy;
167	0	}
168
169		/*
170		* e > 2, so we really have to use the Tonelli/Shanks algorithm. First,
171		* find some y that is not a square.
172		*/
173	0	if (!BN_copy(q, p))
174	0	goto end; /* use 'q' as temp */
175	0	q->neg = 0;
176	0	i = 2;
177	0	do {
178		/*
179		* For efficiency, try small numbers first; if this fails, try random
180		* numbers.
181		*/
182	0	if (i < 22) {
183	0	if (!BN_set_word(y, i))
184	0	goto end;
185	0	} else {
186	0	if (!BN_priv_rand_ex(y, BN_num_bits(p), 0, 0, 0, ctx))
187	0	goto end;
188	0	if (BN_ucmp(y, p) >= 0) {
189	0	if (!(p->neg ? BN_add : BN_sub)(y, y, p))
190	0	goto end;
191	0	}
192		/* now 0 <= y < \|p\| */
193	0	if (BN_is_zero(y))
194	0	if (!BN_set_word(y, i))
195	0	goto end;
196	0	}
197
198	0	r = BN_kronecker(y, q, ctx); /* here 'q' is \|p\| */
199	0	if (r < -1)
200	0	goto end;
201	0	if (r == 0) {
202		/* m divides p */
203	0	ERR_raise(ERR_LIB_BN, BN_R_P_IS_NOT_PRIME);
204	0	goto end;
205	0	}
206	0	} while (r == 1 && ++i < 82);
207
208	0	if (r != -1) {
209		/*
210		* Many rounds and still no non-square -- this is more likely a bug
211		* than just bad luck. Even if p is not prime, we should have found
212		* some y such that r == -1.
213		*/
214	0	ERR_raise(ERR_LIB_BN, BN_R_TOO_MANY_ITERATIONS);
215	0	goto end;
216	0	}
217
218		/* Here's our actual 'q': */
219	0	if (!BN_rshift(q, q, e))
220	0	goto end;
221
222		/*
223		* Now that we have some non-square, we can find an element of order 2^e
224		* by computing its q'th power.
225		*/
226	0	if (!BN_mod_exp(y, y, q, p, ctx))
227	0	goto end;
228	0	if (BN_is_one(y)) {
229	0	ERR_raise(ERR_LIB_BN, BN_R_P_IS_NOT_PRIME);
230	0	goto end;
231	0	}
232
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
252		/* t := (q-1)/2 (note that q is odd) */
253	0	if (!BN_rshift1(t, q))
254	0	goto end;
255
256		/* x := a^((q-1)/2) */
257	0	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	if (BN_is_zero(t)) {
261		/* special case: a == 0 (mod p) */
262	0	BN_zero(ret);
263	0	err = 0;
264	0	goto end;
265	0	} else if (!BN_one(x))
266	0	goto end;
267	0	} else {
268	0	if (!BN_mod_exp(x, A, t, p, ctx))
269	0	goto end;
270	0	if (BN_is_zero(x)) {
271		/* special case: a == 0 (mod p) */
272	0	BN_zero(ret);
273	0	err = 0;
274	0	goto end;
275	0	}
276	0	}
277
278		/* b := ax^2 (= a^q) /
279	0	if (!BN_mod_sqr(b, x, p, ctx))
280	0	goto end;
281	0	if (!BN_mod_mul(b, b, A, p, ctx))
282	0	goto end;
283
284		/* x := ax (= a^((q+1)/2)) /
285	0	if (!BN_mod_mul(x, x, A, p, ctx))
286	0	goto end;
287
288	0	while (1) {
289		/*-
290		* Now b is a^q * y^k for some even k (0 <= k < 2^E
291		* where E refers to the original value of e, which we
292		* don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
293		*
294		* We have a*b = x^2,
295		* y^2^(e-1) = -1,
296		* b^2^(e-1) = 1.
297		*/
298
299	0	if (BN_is_one(b)) {
300	0	if (!BN_copy(ret, x))
301	0	goto end;
302	0	err = 0;
303	0	goto vrfy;
304	0	}
305
306		/* Find the smallest i, 0 < i < e, such that b^(2^i) = 1. */
307	0	for (i = 1; i < e; i++) {
308	0	if (i == 1) {
309	0	if (!BN_mod_sqr(t, b, p, ctx))
310	0	goto end;
311
312	0	} else {
313	0	if (!BN_mod_mul(t, t, t, p, ctx))
314	0	goto end;
315	0	}
316	0	if (BN_is_one(t))
317	0	break;
318	0	}
319		/* If not found, a is not a square or p is not prime. */
320	0	if (i >= e) {
321	0	ERR_raise(ERR_LIB_BN, BN_R_NOT_A_SQUARE);
322	0	goto end;
323	0	}
324
325		/* t := y^2^(e - i - 1) */
326	0	if (!BN_copy(t, y))
327	0	goto end;
328	0	for (j = e - i - 1; j > 0; j--) {
329	0	if (!BN_mod_sqr(t, t, p, ctx))
330	0	goto end;
331	0	}
332	0	if (!BN_mod_mul(y, t, t, p, ctx))
333	0	goto end;
334	0	if (!BN_mod_mul(x, x, t, p, ctx))
335	0	goto end;
336	0	if (!BN_mod_mul(b, b, y, p, ctx))
337	0	goto end;
338	0	e = i;
339	0	}
340
341	0	vrfy:
342	0	if (!err) {
343		/*
344		* verify the result -- the input might have been not a square (test
345		* added in 0.9.8)
346		*/
347
348	0	if (!BN_mod_sqr(x, ret, p, ctx))
349	0	err = 1;
350
351	0	if (!err && 0 != BN_cmp(x, A)) {
352	0	ERR_raise(ERR_LIB_BN, BN_R_NOT_A_SQUARE);
353	0	err = 1;
354	0	}
355	0	}
356
357	0	end:
358	0	if (err) {
359	0	if (ret != in)
360	0	BN_clear_free(ret);
361	0	ret = NULL;
362	0	}
363	0	if (used_ctx)
364	0	BN_CTX_end(ctx);
365	0	bn_check_top(ret);
366	0	return ret;
367	0	}