3 * Jumping around a BBS sequence
5 * (c) 1999 Straylight/Edgeware
8 /*----- Licensing notice --------------------------------------------------*
10 * This file is part of Catacomb.
12 * Catacomb is free software; you can redistribute it and/or modify
13 * it under the terms of the GNU Library General Public License as
14 * published by the Free Software Foundation; either version 2 of the
15 * License, or (at your option) any later version.
17 * Catacomb is distributed in the hope that it will be useful,
18 * but WITHOUT ANY WARRANTY; without even the implied warranty of
19 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
20 * GNU Library General Public License for more details.
22 * You should have received a copy of the GNU Library General Public
23 * License along with Catacomb; if not, write to the Free
24 * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
28 /*----- Header files ------------------------------------------------------*/
32 #include "mpbarrett.h"
36 /*----- Main code ---------------------------------------------------------*/
40 * Arguments: @bbs *b@ = pointer to BBS generator context
41 * @const bbs_priv *bp@ = pointer to BBS modulus factors
42 * @mp *n@ = number of steps to move
43 * @mp *px@ = exponent mod @p@ for a one-step jump
44 * @mp *qx@ = exponent mod @q@ for a one-step jump
48 * Use: Jumps a BBS context a certain number of places (assuming the
49 * arguments are right).
51 * Let the BBS modulus be %$n = pq$% and the current residue be
52 * %$x$%. Then the computations performed are:
54 * * Calculate %$x_p = x \bmod p$% and %$x_q = x \bmod q$%.
56 * * Determine %$e_p = px^n \bmod (p - 1)$% and similarly
57 * %$e_q = qx^n \bmod (p - 1)$%.
59 * * Calculate %$x_p' = x_p^{e_p} \bmod p$% and
60 * %$x_q' = x_q^{e_q} \bmod q$%.
62 * * Combine %$x_p'$% and %$x_q'$% using the Chinese Remainder
65 * If you want to step the generator forwards, simply set
66 * %$px = qx = 2$%. If you want to step backwards, make
67 * %$px = (p + 1)/4$% and %$qx = (q + 1)/4$%. Note that, if
68 * %$x$% is a quadratic residue mod $%p$%, then
70 * %$(x^2) ^ {(p + 1)/4}$%
71 * %${} = x^{(p + 1)/2}$%
72 * %${} = x \cdot x^{(p - 1)/2}$%
75 * Simple, no? (Note that the division works because
76 * %$p \equiv 3 \pmod 4$%.)
79 static void jump(bbs
*b
, const bbs_priv
*bp
, mp
*n
,
83 mp
*v
[2] = { MP_NEW
, MP_NEW
};
85 /* --- First work out the exponents --- */
91 m
= mp_sub(MP_NEW
, bp
->p
, MP_ONE
);
92 mpbarrett_create(&mb
, m
);
93 ep
= mpbarrett_exp(&mb
, MP_NEW
, px
, n
);
94 mpbarrett_destroy(&mb
);
98 m
= mp_sub(m
, bp
->q
, MP_ONE
);
99 mpbarrett_create(&mb
, m
);
100 eq
= mpbarrett_exp(&mb
, MP_NEW
, qx
, n
);
101 mpbarrett_destroy(&mb
);
107 /* --- Now calculate the residues of @x@ --- */
109 mp_div(0, &v
[0], b
->x
, bp
->p
);
110 mp_div(0, &v
[1], b
->x
, bp
->q
);
112 /* --- Exponentiate --- */
117 mpbarrett_create(&mb
, bp
->p
);
118 v
[0] = mpbarrett_exp(&mb
, v
[0], v
[0], ep
);
119 mpbarrett_destroy(&mb
);
121 mpbarrett_create(&mb
, bp
->q
);
122 v
[1] = mpbarrett_exp(&mb
, v
[1], v
[1], eq
);
123 mpbarrett_destroy(&mb
);
129 /* --- Sort out the result using the Chinese Remainder Theorem --- */
136 mv
[0].m
= MP_COPY(bp
->p
);
137 mv
[1].m
= MP_COPY(bp
->q
);
138 for (i
= 0; i
< 2; i
++)
139 mv
[i
].n
= mv
[i
].ni
= mv
[i
].nni
= MP_NEW
;
140 mpcrt_create(&c
, mv
, 2, b
->mb
.m
);
141 b
->x
= mpcrt_solve(&c
, b
->x
, v
);
145 /* --- Tidy away --- */
153 /* --- @bbs_{ff,rew}{,n}@ --- *
155 * Arguments: @bbs *b@ = pointer to a BBS generator state
156 * @const bbs_priv *bp@ = pointer to BBS modulus factors
157 * @mp *n@, @unsigned long n@ = number of steps to make
161 * Use: `Fast-forwards' or rewinds a Blum-Blum-Shub generator by @n@
162 * steps. The @...n@ versions take an @unsigned long@ argument;
163 * the non-@...n@ versions a multiprecision integer. If @n@ is
164 * negative then the generator is stepped in the reverse
168 static void ff(bbs
*b
, const bbs_priv
*bp
, mp
*n
)
169 { jump(b
, bp
, n
, MP_TWO
, MP_TWO
); }
171 static void rew(bbs
*b
, const bbs_priv
*bp
, mp
*n
)
173 mp
*px
= mp_lsr(MP_NEW
, bp
->p
, 2);
174 mp
*qx
= mp_lsr(MP_NEW
, bp
->q
, 2);
175 px
= mp_add(px
, px
, MP_ONE
);
176 qx
= mp_add(qx
, qx
, MP_ONE
);
177 jump(b
, bp
, n
, px
, qx
);
182 void bbs_ff(bbs
*b
, const bbs_priv
*bp
, mp
*n
)
187 n
= mp_neg(MP_NEW
, n
);
193 void bbs_ffn(bbs
*b
, const bbs_priv
*bp
, unsigned long n
)
194 { mp
*nn
= mp_fromulong(MP_NEW
, n
); ff(b
, bp
, nn
); mp_drop(nn
); }
196 void bbs_rew(bbs
*b
, const bbs_priv
*bp
, mp
*n
)
201 n
= mp_neg(MP_NEW
, n
);
207 void bbs_rewn(bbs
*b
, const bbs_priv
*bp
, unsigned long n
)
208 { mp
*nn
= mp_fromulong(MP_NEW
, n
); bbs_rew(b
, bp
, nn
); mp_drop(nn
); }
210 /*----- Test rig ----------------------------------------------------------*/
214 static int verify(dstr
*v
)
224 bp
.p
= *(mp
**)v
[0].buf
;
225 bp
.q
= *(mp
**)v
[1].buf
;
226 bp
.n
= mp_mul(MP_NEW
, bp
.p
, bp
.q
);
227 x
= *(mp
**)v
[2].buf
;
228 n
= *(unsigned long *)v
[3].buf
;
230 bbs_create(&b
, bp
.n
, x
);
231 p
= bbs_bits(&b
, 32);
234 for (i
= 0; i
< n
; i
++)
236 q
= bbs_bits(&b
, 32);
239 bbs_rewn(&b
, &bp
, n
+ (32 + b
.k
- 1) / b
.k
);
240 r
= bbs_bits(&b
, 32);
243 fputs("\n*** bbs rewind failure\n", stderr
);
244 fputs("p = ", stderr
); mp_writefile(bp
.p
, stderr
, 10); fputc('\n', stderr
);
245 fputs("q = ", stderr
); mp_writefile(bp
.q
, stderr
, 10); fputc('\n', stderr
);
246 fputs("n = ", stderr
); mp_writefile(bp
.n
, stderr
, 10); fputc('\n', stderr
);
247 fputs("x = ", stderr
); mp_writefile(x
, stderr
, 10); fputc('\n', stderr
);
248 fprintf(stderr
, "stepped %lu back\n", n
+ (32 + b
.k
- 1) / b
.k
);
249 fprintf(stderr
, "expected output = %08lx, found %08lx\n",
250 (unsigned long)p
, (unsigned long)r
);
256 r
= bbs_bits(&b
, 32);
259 fputs("\n*** bbs fastforward failure\n", stderr
);
260 fputs("p = ", stderr
); mp_writefile(bp
.p
, stderr
, 10); fputc('\n', stderr
);
261 fputs("q = ", stderr
); mp_writefile(bp
.q
, stderr
, 10); fputc('\n', stderr
);
262 fputs("n = ", stderr
); mp_writefile(bp
.n
, stderr
, 10); fputc('\n', stderr
);
263 fputs("x = ", stderr
); mp_writefile(x
, stderr
, 10); fputc('\n', stderr
);
264 fprintf(stderr
, "stepped %lu back\n", n
+ (32 + b
.k
- 1) / b
.k
);
265 fprintf(stderr
, "expected output = %08lx, found %08lx\n",
266 (unsigned long)q
, (unsigned long)r
);
276 assert(mparena_count(MPARENA_GLOBAL
) == 0);
280 static test_chunk tests
[] = {
281 { "bbs-jump", verify
, { &type_mp
, &type_mp
, &type_mp
, &type_ulong
, 0 } },
285 int main(int argc
, char *argv
[])
288 test_run(argc
, argv
, tests
, SRCDIR
"/t/bbs");
294 /*----- That's all, folks -------------------------------------------------*/