[thirdparty/openssl.git] / crypto / bn / bn_sqrt.c

/*
 * Copyright 2000-2018 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_lcl.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!
 */
{
    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)
                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;
        }

        BNerr(BN_F_BN_MOD_SQRT, 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);
    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, 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 */
            BNerr(BN_F_BN_MOD_SQRT, 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.
         */
        BNerr(BN_F_BN_MOD_SQRT, 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)) {
        BNerr(BN_F_BN_MOD_SQRT, 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 smallest  i  such that  b^(2^i) = 1 */
        i = 1;
        if (!BN_mod_sqr(t, b, p, ctx))
            goto end;
        while (!BN_is_one(t)) {
            i++;
            if (i == e) {
                BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
                goto end;
            }
            if (!BN_mod_mul(t, t, t, p, ctx))
                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)) {
            BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
            err = 1;
        }
    }

 end:
    if (err) {
        if (ret != in)
            BN_clear_free(ret);
        ret = NULL;
    }
    BN_CTX_end(ctx);
    bn_check_top(ret);
    return ret;
}
Commit	Line	Data
0f113f3e	1	/*
c4d3c19b	2	* Copyright 2000-2018 The OpenSSL Project Authors. All Rights Reserved.
cd2eebfd	3	*
367ace68	4	* Licensed under the Apache License 2.0 (the "License"). You may not use
4f22f405 RS	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
cd2eebfd BM	8	*/
cd2eebfd BM	9
b39fc560	10	#include "internal/cryptlib.h"
cd2eebfd BM	11	#include "bn_lcl.h"
cd2eebfd BM	12
0f113f3e MC	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!
cd2eebfd	18	*/
0f113f3e MC	19	{
	20	BIGNUM *ret = in;
	21	int err = 1;
	22	int r;
	23	BIGNUM A, b, q, t, x, y;
	24	int e, i, j;
	25
	26	if (!BN_is_odd(p) \|\| BN_abs_is_word(p, 1)) {
	27	if (BN_abs_is_word(p, 2)) {
	28	if (ret == NULL)
	29	ret = BN_new();
	30	if (ret == NULL)
	31	goto end;
	32	if (!BN_set_word(ret, BN_is_bit_set(a, 0))) {
	33	if (ret != in)
	34	BN_free(ret);
	35	return NULL;
	36	}
	37	bn_check_top(ret);
	38	return ret;
	39	}
	40
	41	BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
26a7d938	42	return NULL;
0f113f3e MC	43	}
	44
	45	if (BN_is_zero(a) \|\| BN_is_one(a)) {
	46	if (ret == NULL)
	47	ret = BN_new();
	48	if (ret == NULL)
	49	goto end;
	50	if (!BN_set_word(ret, BN_is_one(a))) {
	51	if (ret != in)
	52	BN_free(ret);
	53	return NULL;
	54	}
	55	bn_check_top(ret);
	56	return ret;
	57	}
	58
	59	BN_CTX_start(ctx);
	60	A = BN_CTX_get(ctx);
	61	b = BN_CTX_get(ctx);
	62	q = BN_CTX_get(ctx);
	63	t = BN_CTX_get(ctx);
	64	x = BN_CTX_get(ctx);
	65	y = BN_CTX_get(ctx);
	66	if (y == NULL)
	67	goto end;
	68
	69	if (ret == NULL)
	70	ret = BN_new();
	71	if (ret == NULL)
	72	goto end;
	73
	74	/* A = a mod p */
	75	if (!BN_nnmod(A, a, p, ctx))
	76	goto end;
	77
	78	/* now write \|p\| - 1 as 2^eq where q is odd /
	79	e = 1;
	80	while (!BN_is_bit_set(p, e))
	81	e++;
	82	/* we'll set q later (if needed) */
	83
	84	if (e == 1) {
50e735f9 MC	85	/*-
	86	* The easy case: (\|p\|-1)/2 is odd, so 2 has an inverse
	87	* modulo (\|p\|-1)/2, and square roots can be computed
	88	* directly by modular exponentiation.
	89	* We have
	90	* 2 * (\|p\|+1)/4 == 1 (mod (\|p\|-1)/2),
	91	* so we can use exponent (\|p\|+1)/4, i.e. (\|p\|-3)/4 + 1.
	92	*/
0f113f3e MC	93	if (!BN_rshift(q, p, 2))
	94	goto end;
	95	q->neg = 0;
	96	if (!BN_add_word(q, 1))
	97	goto end;
	98	if (!BN_mod_exp(ret, A, q, p, ctx))
	99	goto end;
	100	err = 0;
	101	goto vrfy;
	102	}
	103
	104	if (e == 2) {
35a1cc90 MC	105	/*-
	106	* \|p\| == 5 (mod 8)
	107	*
	108	* In this case 2 is always a non-square since
	109	* Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime.
	110	* So if a really is a square, then 2*a is a non-square.
	111	* Thus for
	112	* b := (2*a)^((\|p\|-5)/8),
	113	* i := (2a)b^2
	114	* we have
	115	* i^2 = (2a)^((1 + (\|p\|-5)/4)2)
	116	* = (2*a)^((p-1)/2)
	117	* = -1;
	118	* so if we set
	119	* x := ab(i-1),
	120	* then
	121	* x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
	122	* = a^2 * b^2 * (-2*i)
	123	* = a(-i)(2ab^2)
	124	* = a(-i)i
	125	* = a.
	126	*
	127	* (This is due to A.O.L. Atkin,
e98e586b BE	128	* Subject: Square Roots and Cognate Matters modulo p=8n+5.
e98e586b BE	129	* URL: https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind9211&L=NMBRTHRY&P=4026
35a1cc90 MC	130	* November 1992.)
35a1cc90 MC	131	*/
0f113f3e MC	132
	133	/* t := 2a /
	134	if (!BN_mod_lshift1_quick(t, A, p))
	135	goto end;
	136
	137	/* b := (2a)^((\|p\|-5)/8) /
	138	if (!BN_rshift(q, p, 3))
	139	goto end;
	140	q->neg = 0;
	141	if (!BN_mod_exp(b, t, q, p, ctx))
	142	goto end;
	143
	144	/* y := b^2 */
	145	if (!BN_mod_sqr(y, b, p, ctx))
	146	goto end;
	147
	148	/* t := (2a)b^2 - 1 */
	149	if (!BN_mod_mul(t, t, y, p, ctx))
	150	goto end;
	151	if (!BN_sub_word(t, 1))
	152	goto end;
	153
	154	/* x = abt */
	155	if (!BN_mod_mul(x, A, b, p, ctx))
	156	goto end;
	157	if (!BN_mod_mul(x, x, t, p, ctx))
	158	goto end;
	159
	160	if (!BN_copy(ret, x))
	161	goto end;
	162	err = 0;
	163	goto vrfy;
	164	}
	165
	166	/*
	167	* e > 2, so we really have to use the Tonelli/Shanks algorithm. First,
	168	* find some y that is not a square.
	169	*/
	170	if (!BN_copy(q, p))
	171	goto end; /* use 'q' as temp */
	172	q->neg = 0;
	173	i = 2;
	174	do {
	175	/*
	176	* For efficiency, try small numbers first; if this fails, try random
	177	* numbers.
	178	*/
	179	if (i < 22) {
	180	if (!BN_set_word(y, i))
	181	goto end;
	182	} else {
2934be91	183	if (!BN_priv_rand_ex(y, BN_num_bits(p), 0, 0, ctx))
0f113f3e MC	184	goto end;
	185	if (BN_ucmp(y, p) >= 0) {
	186	if (!(p->neg ? BN_add : BN_sub) (y, y, p))
	187	goto end;
	188	}
	189	/* now 0 <= y < \|p\| */
	190	if (BN_is_zero(y))
	191	if (!BN_set_word(y, i))
	192	goto end;
	193	}
	194
	195	r = BN_kronecker(y, q, ctx); /* here 'q' is \|p\| */
	196	if (r < -1)
	197	goto end;
	198	if (r == 0) {
	199	/* m divides p */
	200	BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
	201	goto end;
	202	}
	203	}
	204	while (r == 1 && ++i < 82);
	205
	206	if (r != -1) {
	207	/*
	208	* Many rounds and still no non-square -- this is more likely a bug
	209	* than just bad luck. Even if p is not prime, we should have found
	210	* some y such that r == -1.
	211	*/
	212	BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS);
	213	goto end;
	214	}
	215
	216	/* Here's our actual 'q': */
	217	if (!BN_rshift(q, q, e))
	218	goto end;
	219
	220	/*
	221	* Now that we have some non-square, we can find an element of order 2^e
	222	* by computing its q'th power.
	223	*/
	224	if (!BN_mod_exp(y, y, q, p, ctx))
	225	goto end;
	226	if (BN_is_one(y)) {
	227	BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
	228	goto end;
	229	}
	230
50e735f9 MC	231	/*-
	232	* Now we know that (if p is indeed prime) there is an integer
	233	* k, 0 <= k < 2^e, such that
	234	*
	235	* a^q * y^k == 1 (mod p).
	236	*
	237	* As a^q is a square and y is not, k must be even.
	238	* q+1 is even, too, so there is an element
	239	*
	240	* X := a^((q+1)/2) * y^(k/2),
	241	*
	242	* and it satisfies
	243	*
	244	* X^2 = a^q * a * y^k
	245	* = a,
	246	*
	247	* so it is the square root that we are looking for.
	248	*/
0f113f3e MC	249
	250	/* t := (q-1)/2 (note that q is odd) */
	251	if (!BN_rshift1(t, q))
	252	goto end;
	253
	254	/* x := a^((q-1)/2) */
	255	if (BN_is_zero(t)) { /* special case: p = 2^e + 1 */
	256	if (!BN_nnmod(t, A, p, ctx))
	257	goto end;
	258	if (BN_is_zero(t)) {
	259	/* special case: a == 0 (mod p) */
	260	BN_zero(ret);
	261	err = 0;
	262	goto end;
	263	} else if (!BN_one(x))
	264	goto end;
	265	} else {
	266	if (!BN_mod_exp(x, A, t, p, ctx))
	267	goto end;
	268	if (BN_is_zero(x)) {
	269	/* special case: a == 0 (mod p) */
	270	BN_zero(ret);
	271	err = 0;
	272	goto end;
	273	}
	274	}
	275
	276	/* b := ax^2 (= a^q) /
	277	if (!BN_mod_sqr(b, x, p, ctx))
	278	goto end;
	279	if (!BN_mod_mul(b, b, A, p, ctx))
	280	goto end;
	281
	282	/* x := ax (= a^((q+1)/2)) /
	283	if (!BN_mod_mul(x, x, A, p, ctx))
	284	goto end;
	285
	286	while (1) {
50e735f9 MC	287	/*-
	288	* Now b is a^q * y^k for some even k (0 <= k < 2^E
	289	* where E refers to the original value of e, which we
	290	* don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
	291	*
	292	* We have a*b = x^2,
	293	* y^2^(e-1) = -1,
	294	* b^2^(e-1) = 1.
	295	*/
0f113f3e MC	296
	297	if (BN_is_one(b)) {
	298	if (!BN_copy(ret, x))
	299	goto end;
	300	err = 0;
	301	goto vrfy;
	302	}
	303
	304	/* find smallest i such that b^(2^i) = 1 */
	305	i = 1;
	306	if (!BN_mod_sqr(t, b, p, ctx))
	307	goto end;
	308	while (!BN_is_one(t)) {
	309	i++;
	310	if (i == e) {
	311	BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
	312	goto end;
	313	}
	314	if (!BN_mod_mul(t, t, t, p, ctx))
	315	goto end;
	316	}
	317
	318	/* t := y^2^(e - i - 1) */
	319	if (!BN_copy(t, y))
	320	goto end;
	321	for (j = e - i - 1; j > 0; j--) {
	322	if (!BN_mod_sqr(t, t, p, ctx))
	323	goto end;
	324	}
	325	if (!BN_mod_mul(y, t, t, p, ctx))
	326	goto end;
	327	if (!BN_mod_mul(x, x, t, p, ctx))
	328	goto end;
	329	if (!BN_mod_mul(b, b, y, p, ctx))
	330	goto end;
	331	e = i;
	332	}
cd2eebfd	333
6fb60a84	334	vrfy:
0f113f3e MC	335	if (!err) {
	336	/*
	337	* verify the result -- the input might have been not a square (test
	338	* added in 0.9.8)
	339	*/
	340
	341	if (!BN_mod_sqr(x, ret, p, ctx))
	342	err = 1;
	343
	344	if (!err && 0 != BN_cmp(x, A)) {
	345	BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
	346	err = 1;
	347	}
	348	}
6fb60a84	349
cd2eebfd	350	end:
0f113f3e	351	if (err) {
23a1d5e9	352	if (ret != in)
0f113f3e	353	BN_clear_free(ret);
0f113f3e MC	354	ret = NULL;
	355	}
	356	BN_CTX_end(ctx);
	357	bn_check_top(ret);
	358	return ret;
	359	}