• Home
  • Line#
  • Scopes#
  • Navigate#
  • Raw
  • Download
1 /**
2  *  Core bignum functions
3  *
4  *  This interface should only be used by the legacy bignum module (bignum.h)
5  *  and the modular bignum modules (bignum_mod.c, bignum_mod_raw.c). All other
6  *  modules should use the high-level modular bignum interface (bignum_mod.h)
7  *  or the legacy bignum interface (bignum.h).
8  *
9  * This module is about processing non-negative integers with a fixed upper
10  * bound that's of the form 2^n-1 where n is a multiple of #biL.
11  * These can be thought of integers written in base 2^#biL with a fixed
12  * number of digits. Digits in this base are called *limbs*.
13  * Many operations treat these numbers as the principal representation of
14  * a number modulo 2^n or a smaller bound.
15  *
16  * The functions in this module obey the following conventions unless
17  * explicitly indicated otherwise:
18  *
19  * - **Overflow**: some functions indicate overflow from the range
20  *   [0, 2^n-1] by returning carry parameters, while others operate
21  *   modulo and so cannot overflow. This should be clear from the function
22  *   documentation.
23  * - **Bignum parameters**: Bignums are passed as pointers to an array of
24  *   limbs. A limb has the type #mbedtls_mpi_uint. Unless otherwise specified:
25  *     - Bignum parameters called \p A, \p B, ... are inputs, and are
26  *       not modified by the function.
27  *     - For operations modulo some number, the modulus is called \p N
28  *       and is input-only.
29  *     - Bignum parameters called \p X, \p Y are outputs or input-output.
30  *       The initial content of output-only parameters is ignored.
31  *     - Some functions use different names that reflect traditional
32  *       naming of operands of certain operations (e.g.
33  *       divisor/dividend/quotient/remainder).
34  *     - \p T is a temporary storage area. The initial content of such
35  *       parameter is ignored and the final content is unspecified.
36  * - **Bignum sizes**: bignum sizes are always expressed in limbs.
37  *   Most functions work on bignums of a given size and take a single
38  *   \p limbs parameter that applies to all parameters that are limb arrays.
39  *   All bignum sizes must be at least 1 and must be significantly less than
40  *   #SIZE_MAX. The behavior if a size is 0 is undefined. The behavior if the
41  *   total size of all parameters overflows #SIZE_MAX is undefined.
42  * - **Parameter ordering**: for bignum parameters, outputs come before inputs.
43  *   Temporaries come last.
44  * - **Aliasing**: in general, output bignums may be aliased to one or more
45  *   inputs. As an exception, parameters that are documented as a modulus value
46  *   may not be aliased to an output. Outputs may not be aliased to one another.
47  *   Temporaries may not be aliased to any other parameter.
48  * - **Overlap**: apart from aliasing of limb array pointers (where two
49  *   arguments are equal pointers), overlap is not supported and may result
50  *   in undefined behavior.
51  * - **Error handling**: This is a low-level module. Functions generally do not
52  *   try to protect against invalid arguments such as nonsensical sizes or
53  *   null pointers. Note that some functions that operate on bignums of
54  *   different sizes have constraints about their size, and violating those
55  *   constraints may lead to buffer overflows.
56  * - **Modular representatives**: functions that operate modulo \p N expect
57  *   all modular inputs to be in the range [0, \p N - 1] and guarantee outputs
58  *   in the range [0, \p N - 1]. If an input is out of range, outputs are
59  *   fully unspecified, though bignum values out of range should not cause
60  *   buffer overflows (beware that this is not extensively tested).
61  */
62 
63 /*
64  *  Copyright The Mbed TLS Contributors
65  *  SPDX-License-Identifier: Apache-2.0
66  *
67  *  Licensed under the Apache License, Version 2.0 (the "License"); you may
68  *  not use this file except in compliance with the License.
69  *  You may obtain a copy of the License at
70  *
71  *  http://www.apache.org/licenses/LICENSE-2.0
72  *
73  *  Unless required by applicable law or agreed to in writing, software
74  *  distributed under the License is distributed on an "AS IS" BASIS, WITHOUT
75  *  WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
76  *  See the License for the specific language governing permissions and
77  *  limitations under the License.
78  */
79 
80 #ifndef MBEDTLS_BIGNUM_CORE_H
81 #define MBEDTLS_BIGNUM_CORE_H
82 
83 #include "common.h"
84 
85 #if defined(MBEDTLS_BIGNUM_C)
86 #include "mbedtls/bignum.h"
87 #endif
88 
89 #define ciL    ( sizeof(mbedtls_mpi_uint) )   /** chars in limb  */
90 #define biL    ( ciL << 3 )                   /** bits  in limb  */
91 #define biH    ( ciL << 2 )                   /** half limb size */
92 
93 /*
94  * Convert between bits/chars and number of limbs
95  * Divide first in order to avoid potential overflows
96  */
97 #define BITS_TO_LIMBS(i)  ( (i) / biL + ( (i) % biL != 0 ) )
98 #define CHARS_TO_LIMBS(i) ( (i) / ciL + ( (i) % ciL != 0 ) )
99 /* Get a specific byte, without range checks. */
100 #define GET_BYTE( X, i )                                \
101     ( ( (X)[(i) / ciL] >> ( ( (i) % ciL ) * 8 ) ) & 0xff )
102 
103 /** Count leading zero bits in a given integer.
104  *
105  * \param a     Integer to count leading zero bits.
106  *
107  * \return      The number of leading zero bits in \p a.
108  */
109 size_t mbedtls_mpi_core_clz( mbedtls_mpi_uint a );
110 
111 /** Return the minimum number of bits required to represent the value held
112  * in the MPI.
113  *
114  * \note This function returns 0 if all the limbs of \p A are 0.
115  *
116  * \param[in] A     The address of the MPI.
117  * \param A_limbs   The number of limbs of \p A.
118  *
119  * \return      The number of bits in \p A.
120  */
121 size_t mbedtls_mpi_core_bitlen( const mbedtls_mpi_uint *A, size_t A_limbs );
122 
123 /** Convert a big-endian byte array aligned to the size of mbedtls_mpi_uint
124  * into the storage form used by mbedtls_mpi.
125  *
126  * \param[in,out] A     The address of the MPI.
127  * \param A_limbs       The number of limbs of \p A.
128  */
129 void mbedtls_mpi_core_bigendian_to_host( mbedtls_mpi_uint *A,
130                                          size_t A_limbs );
131 
132 /**
133  * \brief   Perform a safe conditional copy of an MPI which doesn't reveal
134  *          whether assignment was done or not.
135  *
136  * \param[out] X        The address of the destination MPI.
137  *                      This must be initialized. Must have enough limbs to
138  *                      store the full value of \p A.
139  * \param[in]  A        The address of the source MPI. This must be initialized.
140  * \param      limbs    The number of limbs of \p A.
141  * \param      assign   The condition deciding whether to perform the
142  *                      assignment or not. Must be either 0 or 1:
143  *                      * \c 1: Perform the assignment `X = A`.
144  *                      * \c 0: Keep the original value of \p X.
145  *
146  * \note           This function avoids leaking any information about whether
147  *                 the assignment was done or not.
148  *
149  * \warning        If \p assign is neither 0 nor 1, the result of this function
150  *                 is indeterminate, and the resulting value in \p X might be
151  *                 neither its original value nor the value in \p A.
152  */
153 void mbedtls_mpi_core_cond_assign( mbedtls_mpi_uint *X,
154                                    const mbedtls_mpi_uint *A,
155                                    size_t limbs,
156                                    unsigned char assign );
157 
158 /**
159  * \brief   Perform a safe conditional swap of two MPIs which doesn't reveal
160  *          whether the swap was done or not.
161  *
162  * \param[in,out] X         The address of the first MPI.
163  *                          This must be initialized.
164  * \param[in,out] Y         The address of the second MPI.
165  *                          This must be initialized.
166  * \param         limbs     The number of limbs of \p X and \p Y.
167  * \param         swap      The condition deciding whether to perform
168  *                          the swap or not. Must be either 0 or 1:
169  *                          * \c 1: Swap the values of \p X and \p Y.
170  *                          * \c 0: Keep the original values of \p X and \p Y.
171  *
172  * \note           This function avoids leaking any information about whether
173  *                 the swap was done or not.
174  *
175  * \warning        If \p swap is neither 0 nor 1, the result of this function
176  *                 is indeterminate, and both \p X and \p Y might end up with
177  *                 values different to either of the original ones.
178  */
179 void mbedtls_mpi_core_cond_swap( mbedtls_mpi_uint *X,
180                                  mbedtls_mpi_uint *Y,
181                                  size_t limbs,
182                                  unsigned char swap );
183 
184 /** Import X from unsigned binary data, little-endian.
185  *
186  * The MPI needs to have enough limbs to store the full value (including any
187  * most significant zero bytes in the input).
188  *
189  * \param[out] X         The address of the MPI.
190  * \param X_limbs        The number of limbs of \p X.
191  * \param[in] input      The input buffer to import from.
192  * \param input_length   The length bytes of \p input.
193  *
194  * \return       \c 0 if successful.
195  * \return       #MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL if \p X isn't
196  *               large enough to hold the value in \p input.
197  */
198 int mbedtls_mpi_core_read_le( mbedtls_mpi_uint *X,
199                               size_t X_limbs,
200                               const unsigned char *input,
201                               size_t input_length );
202 
203 /** Import X from unsigned binary data, big-endian.
204  *
205  * The MPI needs to have enough limbs to store the full value (including any
206  * most significant zero bytes in the input).
207  *
208  * \param[out] X        The address of the MPI.
209  *                      May only be #NULL if \X_limbs is 0 and \p input_length
210  *                      is 0.
211  * \param X_limbs       The number of limbs of \p X.
212  * \param[in] input     The input buffer to import from.
213  *                      May only be #NULL if \p input_length is 0.
214  * \param input_length  The length in bytes of \p input.
215  *
216  * \return       \c 0 if successful.
217  * \return       #MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL if \p X isn't
218  *               large enough to hold the value in \p input.
219  */
220 int mbedtls_mpi_core_read_be( mbedtls_mpi_uint *X,
221                               size_t X_limbs,
222                               const unsigned char *input,
223                               size_t input_length );
224 
225 /** Export A into unsigned binary data, little-endian.
226  *
227  * \note If \p output is shorter than \p A the export is still successful if the
228  *       value held in \p A fits in the buffer (that is, if enough of the most
229  *       significant bytes of \p A are 0).
230  *
231  * \param[in] A         The address of the MPI.
232  * \param A_limbs       The number of limbs of \p A.
233  * \param[out] output   The output buffer to export to.
234  * \param output_length The length in bytes of \p output.
235  *
236  * \return       \c 0 if successful.
237  * \return       #MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL if \p output isn't
238  *               large enough to hold the value of \p A.
239  */
240 int mbedtls_mpi_core_write_le( const mbedtls_mpi_uint *A,
241                                size_t A_limbs,
242                                unsigned char *output,
243                                size_t output_length );
244 
245 /** Export A into unsigned binary data, big-endian.
246  *
247  * \note If \p output is shorter than \p A the export is still successful if the
248  *       value held in \p A fits in the buffer (that is, if enough of the most
249  *       significant bytes of \p A are 0).
250  *
251  * \param[in] A         The address of the MPI.
252  * \param A_limbs       The number of limbs of \p A.
253  * \param[out] output   The output buffer to export to.
254  * \param output_length The length in bytes of \p output.
255  *
256  * \return       \c 0 if successful.
257  * \return       #MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL if \p output isn't
258  *               large enough to hold the value of \p A.
259  */
260 int mbedtls_mpi_core_write_be( const mbedtls_mpi_uint *A,
261                                size_t A_limbs,
262                                unsigned char *output,
263                                size_t output_length );
264 
265 /** \brief              Shift an MPI right in place by a number of bits.
266  *
267  *                      Shifting by more bits than there are bit positions
268  *                      in \p X is valid and results in setting \p X to 0.
269  *
270  *                      This function's execution time depends on the value
271  *                      of \p count (and of course \p limbs).
272  *
273  * \param[in,out] X     The number to shift.
274  * \param limbs         The number of limbs of \p X. This must be at least 1.
275  * \param count         The number of bits to shift by.
276  */
277 void mbedtls_mpi_core_shift_r( mbedtls_mpi_uint *X, size_t limbs,
278                                size_t count );
279 
280 /**
281  * \brief Add two fixed-size large unsigned integers, returning the carry.
282  *
283  * Calculates `A + B` where `A` and `B` have the same size.
284  *
285  * This function operates modulo `2^(biL*limbs)` and returns the carry
286  * (1 if there was a wraparound, and 0 otherwise).
287  *
288  * \p X may be aliased to \p A or \p B.
289  *
290  * \param[out] X    The result of the addition.
291  * \param[in] A     Little-endian presentation of the left operand.
292  * \param[in] B     Little-endian presentation of the right operand.
293  * \param limbs     Number of limbs of \p X, \p A and \p B.
294  *
295  * \return          1 if `A + B >= 2^(biL*limbs)`, 0 otherwise.
296  */
297 mbedtls_mpi_uint mbedtls_mpi_core_add( mbedtls_mpi_uint *X,
298                                        const mbedtls_mpi_uint *A,
299                                        const mbedtls_mpi_uint *B,
300                                        size_t limbs );
301 
302 /**
303  * \brief Conditional addition of two fixed-size large unsigned integers,
304  *        returning the carry.
305  *
306  * Functionally equivalent to
307  *
308  * ```
309  * if( cond )
310  *    X += A;
311  * return carry;
312  * ```
313  *
314  * This function operates modulo `2^(biL*limbs)`.
315  *
316  * \param[in,out] X  The pointer to the (little-endian) array
317  *                   representing the bignum to accumulate onto.
318  * \param[in] A      The pointer to the (little-endian) array
319  *                   representing the bignum to conditionally add
320  *                   to \p X. This may be aliased to \p X but may not
321  *                   overlap otherwise.
322  * \param limbs      Number of limbs of \p X and \p A.
323  * \param cond       Condition bit dictating whether addition should
324  *                   happen or not. This must be \c 0 or \c 1.
325  *
326  * \warning          If \p cond is neither 0 nor 1, the result of this function
327  *                   is unspecified, and the resulting value in \p X might be
328  *                   neither its original value nor \p X + \p A.
329  *
330  * \return           1 if `X + cond * A >= 2^(biL*limbs)`, 0 otherwise.
331  */
332 mbedtls_mpi_uint mbedtls_mpi_core_add_if( mbedtls_mpi_uint *X,
333                                           const mbedtls_mpi_uint *A,
334                                           size_t limbs,
335                                           unsigned cond );
336 
337 /**
338  * \brief Subtract two fixed-size large unsigned integers, returning the borrow.
339  *
340  * Calculate `A - B` where \p A and \p B have the same size.
341  * This function operates modulo `2^(biL*limbs)` and returns the carry
342  * (1 if there was a wraparound, i.e. if `A < B`, and 0 otherwise).
343  *
344  * \p X may be aliased to \p A or \p B, or even both, but may not overlap
345  * either otherwise.
346  *
347  * \param[out] X    The result of the subtraction.
348  * \param[in] A     Little-endian presentation of left operand.
349  * \param[in] B     Little-endian presentation of right operand.
350  * \param limbs     Number of limbs of \p X, \p A and \p B.
351  *
352  * \return          1 if `A < B`.
353  *                  0 if `A >= B`.
354  */
355 mbedtls_mpi_uint mbedtls_mpi_core_sub( mbedtls_mpi_uint *X,
356                                        const mbedtls_mpi_uint *A,
357                                        const mbedtls_mpi_uint *B,
358                                        size_t limbs );
359 
360 /**
361  * \brief Perform a fixed-size multiply accumulate operation: X += b * A
362  *
363  * \p X may be aliased to \p A (when \p X_limbs == \p A_limbs), but may not
364  * otherwise overlap.
365  *
366  * This function operates modulo `2^(biL*X_limbs)`.
367  *
368  * \param[in,out] X  The pointer to the (little-endian) array
369  *                   representing the bignum to accumulate onto.
370  * \param X_limbs    The number of limbs of \p X. This must be
371  *                   at least \p A_limbs.
372  * \param[in] A      The pointer to the (little-endian) array
373  *                   representing the bignum to multiply with.
374  *                   This may be aliased to \p X but may not overlap
375  *                   otherwise.
376  * \param A_limbs    The number of limbs of \p A.
377  * \param b          X scalar to multiply with.
378  *
379  * \return           The carry at the end of the operation.
380  */
381 mbedtls_mpi_uint mbedtls_mpi_core_mla( mbedtls_mpi_uint *X, size_t X_limbs,
382                                        const mbedtls_mpi_uint *A, size_t A_limbs,
383                                        mbedtls_mpi_uint b );
384 
385 /**
386  * \brief Calculate initialisation value for fast Montgomery modular
387  *        multiplication
388  *
389  * \param[in] N  Little-endian presentation of the modulus. This must have
390  *               at least one limb.
391  *
392  * \return       The initialisation value for fast Montgomery modular multiplication
393  */
394 mbedtls_mpi_uint mbedtls_mpi_core_montmul_init( const mbedtls_mpi_uint *N );
395 
396 /**
397  * \brief Montgomery multiplication: X = A * B * R^-1 mod N (HAC 14.36)
398  *
399  * \p A and \p B must be in canonical form. That is, < \p N.
400  *
401  * \p X may be aliased to \p A or \p N, or even \p B (if \p AN_limbs ==
402  * \p B_limbs) but may not overlap any parameters otherwise.
403  *
404  * \p A and \p B may alias each other, if \p AN_limbs == \p B_limbs. They may
405  * not alias \p N (since they must be in canonical form, they cannot == \p N).
406  *
407  * \param[out]    X         The destination MPI, as a little-endian array of
408  *                          length \p AN_limbs.
409  *                          On successful completion, X contains the result of
410  *                          the multiplication `A * B * R^-1` mod N where
411  *                          `R = 2^(biL*AN_limbs)`.
412  * \param[in]     A         Little-endian presentation of first operand.
413  *                          Must have the same number of limbs as \p N.
414  * \param[in]     B         Little-endian presentation of second operand.
415  * \param[in]     B_limbs   The number of limbs in \p B.
416  *                          Must be <= \p AN_limbs.
417  * \param[in]     N         Little-endian presentation of the modulus.
418  *                          This must be odd, and have exactly the same number
419  *                          of limbs as \p A.
420  *                          It may alias \p X, but must not alias or otherwise
421  *                          overlap any of the other parameters.
422  * \param[in]     AN_limbs  The number of limbs in \p X, \p A and \p N.
423  * \param         mm        The Montgomery constant for \p N: -N^-1 mod 2^biL.
424  *                          This can be calculated by `mbedtls_mpi_core_montmul_init()`.
425  * \param[in,out] T         Temporary storage of size at least 2*AN_limbs+1 limbs.
426  *                          Its initial content is unused and
427  *                          its final content is indeterminate.
428  *                          It must not alias or otherwise overlap any of the
429  *                          other parameters.
430  */
431 void mbedtls_mpi_core_montmul( mbedtls_mpi_uint *X,
432                                const mbedtls_mpi_uint *A,
433                                const mbedtls_mpi_uint *B, size_t B_limbs,
434                                const mbedtls_mpi_uint *N, size_t AN_limbs,
435                                mbedtls_mpi_uint mm, mbedtls_mpi_uint *T );
436 
437 /**
438  * \brief Calculate the square of the Montgomery constant. (Needed
439  *        for conversion and operations in Montgomery form.)
440  *
441  * \param[out] X  A pointer to the result of the calculation of
442  *                the square of the Montgomery constant:
443  *                2^{2*n*biL} mod N.
444  * \param[in]  N  Little-endian presentation of the modulus, which must be odd.
445  *
446  * \return        0 if successful.
447  * \return        #MBEDTLS_ERR_MPI_ALLOC_FAILED if there is not enough space
448  *                to store the value of Montgomery constant squared.
449  * \return        #MBEDTLS_ERR_MPI_DIVISION_BY_ZERO if \p N modulus is zero.
450  * \return        #MBEDTLS_ERR_MPI_NEGATIVE_VALUE if \p N modulus is negative.
451  */
452 int mbedtls_mpi_core_get_mont_r2_unsafe( mbedtls_mpi *X,
453                                          const mbedtls_mpi *N );
454 
455 #if defined(MBEDTLS_TEST_HOOKS)
456 /**
457  * Copy an MPI from a table without leaking the index.
458  *
459  * \param dest              The destination buffer. This must point to a writable
460  *                          buffer of at least \p limbs limbs.
461  * \param table             The address of the table. This must point to a readable
462  *                          array of \p count elements of \p limbs limbs each.
463  * \param limbs             The number of limbs in each table entry.
464  * \param count             The number of entries in \p table.
465  * \param index             The (secret) table index to look up. This must be in the
466  *                          range `0 .. count-1`.
467  */
468 void mbedtls_mpi_core_ct_uint_table_lookup( mbedtls_mpi_uint *dest,
469                                             const mbedtls_mpi_uint *table,
470                                             size_t limbs,
471                                             size_t count,
472                                             size_t index );
473 #endif /* MBEDTLS_TEST_HOOKS */
474 
475 /**
476  * \brief          Fill an integer with a number of random bytes.
477  *
478  * \param X        The destination MPI.
479  * \param X_limbs  The number of limbs of \p X.
480  * \param bytes    The number of random bytes to generate.
481  * \param f_rng    The RNG function to use. This must not be \c NULL.
482  * \param p_rng    The RNG parameter to be passed to \p f_rng. This may be
483  *                 \c NULL if \p f_rng doesn't need a context argument.
484  *
485  * \return         \c 0 if successful.
486  * \return         #MBEDTLS_ERR_MPI_BAD_INPUT_DATA if \p X does not have
487  *                 enough room for \p bytes bytes.
488  * \return         A negative error code on RNG failure.
489  *
490  * \note           The bytes obtained from the RNG are interpreted
491  *                 as a big-endian representation of an MPI; this can
492  *                 be relevant in applications like deterministic ECDSA.
493  */
494 int mbedtls_mpi_core_fill_random( mbedtls_mpi_uint *X, size_t X_limbs,
495                                   size_t bytes,
496                                   int (*f_rng)(void *, unsigned char *, size_t),
497                                   void *p_rng );
498 
499 /* BEGIN MERGE SLOT 1 */
500 
501 /**
502  * \brief          Returns the number of limbs of working memory required for
503  *                 a call to `mbedtls_mpi_core_exp_mod()`.
504  *
505  * \param AN_limbs The number of limbs in the input `A` and the modulus `N`
506  *                 (they must be the same size) that will be given to
507  *                 `mbedtls_mpi_core_exp_mod()`.
508  * \param E_limbs  The number of limbs in the exponent `E` that will be given
509  *                 to `mbedtls_mpi_core_exp_mod()`.
510  *
511  * \return         The number of limbs of working memory required by
512  *                 `mbedtls_mpi_core_exp_mod()`.
513  */
514 size_t mbedtls_mpi_core_exp_mod_working_limbs( size_t AN_limbs, size_t E_limbs );
515 
516 /**
517  * \brief            Perform a modular exponentiation with secret exponent:
518  *                   X = A^E mod N, where \p A is already in Montgomery form.
519  *
520  * \param[out] X     The destination MPI, as a little endian array of length
521  *                   \p AN_limbs.
522  * \param[in] A      The base MPI, as a little endian array of length \p AN_limbs.
523  *                   Must be in Montgomery form.
524  * \param[in] N      The modulus, as a little endian array of length \p AN_limbs.
525  * \param AN_limbs   The number of limbs in \p X, \p A, \p N, \p RR.
526  * \param[in] E      The exponent, as a little endian array of length \p E_limbs.
527  * \param E_limbs    The number of limbs in \p E.
528  * \param[in] RR     The precomputed residue of 2^{2*biL} modulo N, as a little
529  *                   endian array of length \p AN_limbs.
530  * \param[in,out] T  Temporary storage of at least the number of limbs returned
531  *                   by `mbedtls_mpi_core_exp_mod_working_limbs()`.
532  *                   Its initial content is unused and its final content is
533  *                   indeterminate.
534  *                   It must not alias or otherwise overlap any of the other
535  *                   parameters.
536  *                   It is up to the caller to zeroize \p T when it is no
537  *                   longer needed, and before freeing it if it was dynamically
538  *                   allocated.
539  */
540 void mbedtls_mpi_core_exp_mod( mbedtls_mpi_uint *X,
541                                const mbedtls_mpi_uint *A,
542                                const mbedtls_mpi_uint *N, size_t AN_limbs,
543                                const mbedtls_mpi_uint *E, size_t E_limbs,
544                                const mbedtls_mpi_uint *RR,
545                                mbedtls_mpi_uint *T );
546 
547 /* END MERGE SLOT 1 */
548 
549 /* BEGIN MERGE SLOT 2 */
550 
551 /* END MERGE SLOT 2 */
552 
553 /* BEGIN MERGE SLOT 3 */
554 
555 /**
556  * \brief Subtract unsigned integer from known-size large unsigned integers.
557  *        Return the borrow.
558  *
559  * \param[out] X    The result of the subtraction.
560  * \param[in] A     The left operand.
561  * \param b         The unsigned scalar to subtract.
562  * \param limbs     Number of limbs of \p X and \p A.
563  *
564  * \return          1 if `A < b`.
565  *                  0 if `A >= b`.
566  */
567 mbedtls_mpi_uint mbedtls_mpi_core_sub_int( mbedtls_mpi_uint *X,
568                                            const mbedtls_mpi_uint *A,
569                                            mbedtls_mpi_uint b,
570                                            size_t limbs );
571 
572 /* END MERGE SLOT 3 */
573 
574 /* BEGIN MERGE SLOT 4 */
575 
576 /* END MERGE SLOT 4 */
577 
578 /* BEGIN MERGE SLOT 5 */
579 
580 /* END MERGE SLOT 5 */
581 
582 /* BEGIN MERGE SLOT 6 */
583 
584 /* END MERGE SLOT 6 */
585 
586 /* BEGIN MERGE SLOT 7 */
587 
588 /* END MERGE SLOT 7 */
589 
590 /* BEGIN MERGE SLOT 8 */
591 
592 /* END MERGE SLOT 8 */
593 
594 /* BEGIN MERGE SLOT 9 */
595 
596 /* END MERGE SLOT 9 */
597 
598 /* BEGIN MERGE SLOT 10 */
599 
600 /* END MERGE SLOT 10 */
601 
602 #endif /* MBEDTLS_BIGNUM_CORE_H */
603