1 /*
2 Name: imath.h
3 Purpose: Arbitrary precision integer arithmetic routines.
4 Author: M. J. Fromberger
5
6 Copyright (C) 2002-2007 Michael J. Fromberger, All Rights Reserved.
7
8 Permission is hereby granted, free of charge, to any person obtaining a copy
9 of this software and associated documentation files (the "Software"), to deal
10 in the Software without restriction, including without limitation the rights
11 to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
12 copies of the Software, and to permit persons to whom the Software is
13 furnished to do so, subject to the following conditions:
14
15 The above copyright notice and this permission notice shall be included in
16 all copies or substantial portions of the Software.
17
18 THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
19 IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
20 FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
21 AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
22 LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
23 OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
24 SOFTWARE.
25 */
26
27 #ifndef IMATH_H_
28 #define IMATH_H_
29
30 #include <limits.h>
31 #include <stdbool.h>
32 #include <stdint.h>
33
34 #ifdef __cplusplus
35 extern "C" {
36 #endif
37
38 typedef unsigned char mp_sign;
39 typedef unsigned int mp_size;
40 typedef int mp_result;
41 typedef long mp_small; /* must be a signed type */
42 typedef unsigned long mp_usmall; /* must be an unsigned type */
43
44
45 /* Build with words as uint64_t by default. */
46 #ifdef USE_32BIT_WORDS
47 typedef uint16_t mp_digit;
48 typedef uint32_t mp_word;
49 # define MP_DIGIT_MAX (UINT16_MAX * 1UL)
50 # define MP_WORD_MAX (UINT32_MAX * 1UL)
51 #else
52 typedef uint32_t mp_digit;
53 typedef uint64_t mp_word;
54 # define MP_DIGIT_MAX (UINT32_MAX * UINT64_C(1))
55 # define MP_WORD_MAX (UINT64_MAX)
56 #endif
57
58 typedef struct {
59 mp_digit single;
60 mp_digit* digits;
61 mp_size alloc;
62 mp_size used;
63 mp_sign sign;
64 } mpz_t, *mp_int;
65
MP_DIGITS(mp_int Z)66 static inline mp_digit* MP_DIGITS(mp_int Z) { return Z->digits; }
MP_ALLOC(mp_int Z)67 static inline mp_size MP_ALLOC(mp_int Z) { return Z->alloc; }
MP_USED(mp_int Z)68 static inline mp_size MP_USED(mp_int Z) { return Z->used; }
MP_SIGN(mp_int Z)69 static inline mp_sign MP_SIGN(mp_int Z) { return Z->sign; }
70
71 extern const mp_result MP_OK;
72 extern const mp_result MP_FALSE;
73 extern const mp_result MP_TRUE;
74 extern const mp_result MP_MEMORY;
75 extern const mp_result MP_RANGE;
76 extern const mp_result MP_UNDEF;
77 extern const mp_result MP_TRUNC;
78 extern const mp_result MP_BADARG;
79 extern const mp_result MP_MINERR;
80
81 #define MP_DIGIT_BIT (sizeof(mp_digit) * CHAR_BIT)
82 #define MP_WORD_BIT (sizeof(mp_word) * CHAR_BIT)
83 #define MP_SMALL_MIN LONG_MIN
84 #define MP_SMALL_MAX LONG_MAX
85 #define MP_USMALL_MAX ULONG_MAX
86
87 #define MP_MIN_RADIX 2
88 #define MP_MAX_RADIX 36
89
90 /** Sets the default number of digits allocated to an `mp_int` constructed by
91 `mp_int_init_size()` with `prec == 0`. Allocations are rounded up to
92 multiples of this value. `MP_DEFAULT_PREC` is the default value. Requires
93 `ndigits > 0`. */
94 void mp_int_default_precision(mp_size ndigits);
95
96 /** Sets the number of digits below which multiplication will use the standard
97 quadratic "schoolbook" multiplication algorithm rather than Karatsuba-Ofman.
98 Requires `ndigits >= sizeof(mp_word)`. */
99 void mp_int_multiply_threshold(mp_size ndigits);
100
101 /** A sign indicating a (strictly) negative value. */
102 extern const mp_sign MP_NEG;
103
104 /** A sign indicating a zero or positive value. */
105 extern const mp_sign MP_ZPOS;
106
107 /** Reports whether `z` is odd, having remainder 1 when divided by 2. */
mp_int_is_odd(mp_int z)108 static inline bool mp_int_is_odd(mp_int z) { return (z->digits[0] & 1) != 0; }
109
110 /** Reports whether `z` is even, having remainder 0 when divided by 2. */
mp_int_is_even(mp_int z)111 static inline bool mp_int_is_even(mp_int z) { return (z->digits[0] & 1) == 0; }
112
113 /** Initializes `z` with 1-digit precision and sets it to zero. This function
114 cannot fail unless `z == NULL`. */
115 mp_result mp_int_init(mp_int z);
116
117 /** Allocates a fresh zero-valued `mpz_t` on the heap, returning NULL in case
118 of error. The only possible error is out-of-memory. */
119 mp_int mp_int_alloc(void);
120
121 /** Initializes `z` with at least `prec` digits of storage, and sets it to
122 zero. If `prec` is zero, the default precision is used. In either case the
123 size is rounded up to the nearest multiple of the word size. */
124 mp_result mp_int_init_size(mp_int z, mp_size prec);
125
126 /** Initializes `z` to be a copy of an already-initialized value in `old`. The
127 new copy does not share storage with the original. */
128 mp_result mp_int_init_copy(mp_int z, mp_int old);
129
130 /** Initializes `z` to the specified signed `value` at default precision. */
131 mp_result mp_int_init_value(mp_int z, mp_small value);
132
133 /** Initializes `z` to the specified unsigned `value` at default precision. */
134 mp_result mp_int_init_uvalue(mp_int z, mp_usmall uvalue);
135
136 /** Sets `z` to the value of the specified signed `value`. */
137 mp_result mp_int_set_value(mp_int z, mp_small value);
138
139 /** Sets `z` to the value of the specified unsigned `value`. */
140 mp_result mp_int_set_uvalue(mp_int z, mp_usmall uvalue);
141
142 /** Releases the storage used by `z`. */
143 void mp_int_clear(mp_int z);
144
145 /** Releases the storage used by `z` and also `z` itself.
146 This should only be used for `z` allocated by `mp_int_alloc()`. */
147 void mp_int_free(mp_int z);
148
149 /** Replaces the value of `c` with a copy of the value of `a`. No new memory is
150 allocated unless `a` has more significant digits than `c` has allocated. */
151 mp_result mp_int_copy(mp_int a, mp_int c);
152
153 /** Swaps the values and storage between `a` and `c`. */
154 void mp_int_swap(mp_int a, mp_int c);
155
156 /** Sets `z` to zero. The allocated storage of `z` is not changed. */
157 void mp_int_zero(mp_int z);
158
159 /** Sets `c` to the absolute value of `a`. */
160 mp_result mp_int_abs(mp_int a, mp_int c);
161
162 /** Sets `c` to the additive inverse (negation) of `a`. */
163 mp_result mp_int_neg(mp_int a, mp_int c);
164
165 /** Sets `c` to the sum of `a` and `b`. */
166 mp_result mp_int_add(mp_int a, mp_int b, mp_int c);
167
168 /** Sets `c` to the sum of `a` and `value`. */
169 mp_result mp_int_add_value(mp_int a, mp_small value, mp_int c);
170
171 /** Sets `c` to the difference of `a` less `b`. */
172 mp_result mp_int_sub(mp_int a, mp_int b, mp_int c);
173
174 /** Sets `c` to the difference of `a` less `value`. */
175 mp_result mp_int_sub_value(mp_int a, mp_small value, mp_int c);
176
177 /** Sets `c` to the product of `a` and `b`. */
178 mp_result mp_int_mul(mp_int a, mp_int b, mp_int c);
179
180 /** Sets `c` to the product of `a` and `value`. */
181 mp_result mp_int_mul_value(mp_int a, mp_small value, mp_int c);
182
183 /** Sets `c` to the product of `a` and `2^p2`. Requires `p2 >= 0`. */
184 mp_result mp_int_mul_pow2(mp_int a, mp_small p2, mp_int c);
185
186 /** Sets `c` to the square of `a`. */
187 mp_result mp_int_sqr(mp_int a, mp_int c);
188
189 /** Sets `q` and `r` to the quotent and remainder of `a / b`. Division by
190 powers of 2 is detected and handled efficiently. The remainder is pinned
191 to `0 <= r < b`.
192
193 Either of `q` or `r` may be NULL, but not both, and `q` and `r` may not
194 point to the same value. */
195 mp_result mp_int_div(mp_int a, mp_int b, mp_int q, mp_int r);
196
197 /** Sets `q` and `*r` to the quotent and remainder of `a / value`. Division by
198 powers of 2 is detected and handled efficiently. The remainder is pinned to
199 `0 <= *r < b`. Either of `q` or `r` may be NULL. */
200 mp_result mp_int_div_value(mp_int a, mp_small value, mp_int q, mp_small *r);
201
202 /** Sets `q` and `r` to the quotient and remainder of `a / 2^p2`. This is a
203 special case for division by powers of two that is more efficient than
204 using ordinary division. Note that `mp_int_div()` will automatically handle
205 this case, this function is for cases where you have only the exponent. */
206 mp_result mp_int_div_pow2(mp_int a, mp_small p2, mp_int q, mp_int r);
207
208 /** Sets `c` to the remainder of `a / m`.
209 The remainder is pinned to `0 <= c < m`. */
210 mp_result mp_int_mod(mp_int a, mp_int m, mp_int c);
211
212 /** Sets `c` to the value of `a` raised to the `b` power.
213 It returns `MP_RANGE` if `b < 0`. */
214 mp_result mp_int_expt(mp_int a, mp_small b, mp_int c);
215
216 /** Sets `c` to the value of `a` raised to the `b` power.
217 It returns `MP_RANGE` if `b < 0`. */
218 mp_result mp_int_expt_value(mp_small a, mp_small b, mp_int c);
219
220 /** Sets `c` to the value of `a` raised to the `b` power.
221 It returns `MP_RANGE`) if `b < 0`. */
222 mp_result mp_int_expt_full(mp_int a, mp_int b, mp_int c);
223
224 /** Sets `*r` to the remainder of `a / value`.
225 The remainder is pinned to `0 <= r < value`. */
226 static inline
mp_int_mod_value(mp_int a,mp_small value,mp_small * r)227 mp_result mp_int_mod_value(mp_int a, mp_small value, mp_small* r) {
228 return mp_int_div_value(a, value, 0, r);
229 }
230
231 /** Returns the comparator of `a` and `b`. */
232 int mp_int_compare(mp_int a, mp_int b);
233
234 /** Returns the comparator of the magnitudes of `a` and `b`, disregarding their
235 signs. Neither `a` nor `b` is modified by the comparison. */
236 int mp_int_compare_unsigned(mp_int a, mp_int b);
237
238 /** Returns the comparator of `z` and zero. */
239 int mp_int_compare_zero(mp_int z);
240
241 /** Returns the comparator of `z` and the signed value `v`. */
242 int mp_int_compare_value(mp_int z, mp_small v);
243
244 /** Returns the comparator of `z` and the unsigned value `uv`. */
245 int mp_int_compare_uvalue(mp_int z, mp_usmall uv);
246
247 /** Reports whether `a` is divisible by `v`. */
248 bool mp_int_divisible_value(mp_int a, mp_small v);
249
250 /** Returns `k >= 0` such that `z` is `2^k`, if such a `k` exists. If no such
251 `k` exists, the function returns -1. */
252 int mp_int_is_pow2(mp_int z);
253
254 /** Sets `c` to the value of `a` raised to the `b` power, reduced modulo `m`.
255 It returns `MP_RANGE` if `b < 0` or `MP_UNDEF` if `m == 0`. */
256 mp_result mp_int_exptmod(mp_int a, mp_int b, mp_int m, mp_int c);
257
258 /** Sets `c` to the value of `a` raised to the `value` power, modulo `m`.
259 It returns `MP_RANGE` if `value < 0` or `MP_UNDEF` if `m == 0`. */
260 mp_result mp_int_exptmod_evalue(mp_int a, mp_small value, mp_int m, mp_int c);
261
262 /** Sets `c` to the value of `value` raised to the `b` power, modulo `m`.
263 It returns `MP_RANGE` if `b < 0` or `MP_UNDEF` if `m == 0`. */
264 mp_result mp_int_exptmod_bvalue(mp_small value, mp_int b, mp_int m, mp_int c);
265
266 /** Sets `c` to the value of `a` raised to the `b` power, reduced modulo `m`,
267 given a precomputed reduction constant `mu` defined for Barrett's modular
268 reduction algorithm.
269
270 It returns `MP_RANGE` if `b < 0` or `MP_UNDEF` if `m == 0`. */
271 mp_result mp_int_exptmod_known(mp_int a, mp_int b, mp_int m, mp_int mu, mp_int c);
272
273 /** Sets `c` to the reduction constant for Barrett reduction by modulus `m`.
274 Requires that `c` and `m` point to distinct locations. */
275 mp_result mp_int_redux_const(mp_int m, mp_int c);
276
277 /** Sets `c` to the multiplicative inverse of `a` modulo `m`, if it exists.
278 The least non-negative representative of the congruence class is computed.
279
280 It returns `MP_UNDEF` if the inverse does not exist, or `MP_RANGE` if `a ==
281 0` or `m <= 0`. */
282 mp_result mp_int_invmod(mp_int a, mp_int m, mp_int c);
283
284 /** Sets `c` to the greatest common divisor of `a` and `b`.
285
286 It returns `MP_UNDEF` if the GCD is undefined, such as for example if `a`
287 and `b` are both zero. */
288 mp_result mp_int_gcd(mp_int a, mp_int b, mp_int c);
289
290 /** Sets `c` to the greatest common divisor of `a` and `b`, and sets `x` and
291 `y` to values satisfying Bezout's identity `gcd(a, b) = ax + by`.
292
293 It returns `MP_UNDEF` if the GCD is undefined, such as for example if `a`
294 and `b` are both zero. */
295 mp_result mp_int_egcd(mp_int a, mp_int b, mp_int c, mp_int x, mp_int y);
296
297 /** Sets `c` to the least common multiple of `a` and `b`.
298
299 It returns `MP_UNDEF` if the LCM is undefined, such as for example if `a`
300 and `b` are both zero. */
301 mp_result mp_int_lcm(mp_int a, mp_int b, mp_int c);
302
303 /** Sets `c` to the greatest integer not less than the `b`th root of `a`,
304 using Newton's root-finding algorithm.
305 It returns `MP_UNDEF` if `a < 0` and `b` is even. */
306 mp_result mp_int_root(mp_int a, mp_small b, mp_int c);
307
308 /** Sets `c` to the greatest integer not less than the square root of `a`.
309 This is a special case of `mp_int_root()`. */
310 static inline
mp_int_sqrt(mp_int a,mp_int c)311 mp_result mp_int_sqrt(mp_int a, mp_int c) { return mp_int_root(a, 2, c); }
312
313 /** Returns `MP_OK` if `z` is representable as `mp_small`, else `MP_RANGE`.
314 If `out` is not NULL, `*out` is set to the value of `z` when `MP_OK`. */
315 mp_result mp_int_to_int(mp_int z, mp_small *out);
316
317 /** Returns `MP_OK` if `z` is representable as `mp_usmall`, or `MP_RANGE`.
318 If `out` is not NULL, `*out` is set to the value of `z` when `MP_OK`. */
319 mp_result mp_int_to_uint(mp_int z, mp_usmall *out);
320
321 /** Converts `z` to a zero-terminated string of characters in the specified
322 `radix`, writing at most `limit` characters to `str` including the
323 terminating NUL value. A leading `-` is used to indicate a negative value.
324
325 Returns `MP_TRUNC` if `limit` was to small to write all of `z`.
326 Requires `MP_MIN_RADIX <= radix <= MP_MAX_RADIX`. */
327 mp_result mp_int_to_string(mp_int z, mp_size radix, char *str, int limit);
328
329 /** Reports the minimum number of characters required to represent `z` as a
330 zero-terminated string in the given `radix`.
331 Requires `MP_MIN_RADIX <= radix <= MP_MAX_RADIX`. */
332 mp_result mp_int_string_len(mp_int z, mp_size radix);
333
334 /** Reads a string of ASCII digits in the specified `radix` from the zero
335 terminated `str` provided into `z`. For values of `radix > 10`, the letters
336 `A`..`Z` or `a`..`z` are accepted. Letters are interpreted without respect
337 to case.
338
339 Leading whitespace is ignored, and a leading `+` or `-` is interpreted as a
340 sign flag. Processing stops when a NUL or any other character out of range
341 for a digit in the given radix is encountered.
342
343 If the whole string was consumed, `MP_OK` is returned; otherwise
344 `MP_TRUNC`. is returned.
345
346 Requires `MP_MIN_RADIX <= radix <= MP_MAX_RADIX`. */
347 mp_result mp_int_read_string(mp_int z, mp_size radix, const char *str);
348
349 /** Reads a string of ASCII digits in the specified `radix` from the zero
350 terminated `str` provided into `z`. For values of `radix > 10`, the letters
351 `A`..`Z` or `a`..`z` are accepted. Letters are interpreted without respect
352 to case.
353
354 Leading whitespace is ignored, and a leading `+` or `-` is interpreted as a
355 sign flag. Processing stops when a NUL or any other character out of range
356 for a digit in the given radix is encountered.
357
358 If the whole string was consumed, `MP_OK` is returned; otherwise
359 `MP_TRUNC`. is returned. If `end` is not NULL, `*end` is set to point to
360 the first unconsumed byte of the input string (the NUL byte if the whole
361 string was consumed). This emulates the behavior of the standard C
362 `strtol()` function.
363
364 Requires `MP_MIN_RADIX <= radix <= MP_MAX_RADIX`. */
365 mp_result mp_int_read_cstring(mp_int z, mp_size radix, const char *str, char **end);
366
367 /** Returns the number of significant bits in `z`. */
368 mp_result mp_int_count_bits(mp_int z);
369
370 /** Converts `z` to 2's complement binary, writing at most `limit` bytes into
371 the given `buf`. Returns `MP_TRUNC` if the buffer limit was too small to
372 contain the whole value. If this occurs, the contents of buf will be
373 effectively garbage, as the function uses the buffer as scratch space.
374
375 The binary representation of `z` is in base-256 with digits ordered from
376 most significant to least significant (network byte ordering). The
377 high-order bit of the first byte is set for negative values, clear for
378 non-negative values.
379
380 As a result, non-negative values will be padded with a leading zero byte if
381 the high-order byte of the base-256 magnitude is set. This extra byte is
382 accounted for by the `mp_int_binary_len()` function. */
383 mp_result mp_int_to_binary(mp_int z, unsigned char *buf, int limit);
384
385 /** Reads a 2's complement binary value from `buf` into `z`, where `len` is the
386 length of the buffer. The contents of `buf` may be overwritten during
387 processing, although they will be restored when the function returns. */
388 mp_result mp_int_read_binary(mp_int z, unsigned char *buf, int len);
389
390 /** Returns the number of bytes to represent `z` in 2's complement binary. */
391 mp_result mp_int_binary_len(mp_int z);
392
393 /** Converts the magnitude of `z` to unsigned binary, writing at most `limit`
394 bytes into the given `buf`. The sign of `z` is ignored, but `z` is not
395 modified. Returns `MP_TRUNC` if the buffer limit was too small to contain
396 the whole value. If this occurs, the contents of `buf` will be effectively
397 garbage, as the function uses the buffer as scratch space during
398 conversion.
399
400 The binary representation of `z` is in base-256 with digits ordered from
401 most significant to least significant (network byte ordering). */
402 mp_result mp_int_to_unsigned(mp_int z, unsigned char *buf, int limit);
403
404 /** Reads an unsigned binary value from `buf` into `z`, where `len` is the
405 length of the buffer. The contents of `buf` are not modified during
406 processing. */
407 mp_result mp_int_read_unsigned(mp_int z, unsigned char *buf, int len);
408
409 /** Returns the number of bytes required to represent `z` as an unsigned binary
410 value in base 256. */
411 mp_result mp_int_unsigned_len(mp_int z);
412
413 /** Returns a pointer to a brief, human-readable, zero-terminated string
414 describing `res`. The returned string is statically allocated and must not
415 be freed by the caller. */
416 const char *mp_error_string(mp_result res);
417
418 #ifdef __cplusplus
419 }
420 #endif
421 #endif /* end IMATH_H_ */
422