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1.\" $OpenBSD: BN_add.3,v 1.20 2023/04/27 09:47:03 tb Exp $
2.\" full merge up to: OpenSSL e9b77246 Jan 20 19:58:49 2017 +0100
3.\"
4.\" This file is a derived work.
5.\" The changes are covered by the following Copyright and license:
6.\"
7.\" Copyright (c) 2021 Ingo Schwarze <schwarze@openbsd.org>
8.\"
9.\" Permission to use, copy, modify, and distribute this software for any
10.\" purpose with or without fee is hereby granted, provided that the above
11.\" copyright notice and this permission notice appear in all copies.
12.\"
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14.\" WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
15.\" MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
16.\" ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
17.\" WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
18.\" ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
19.\" OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
20.\"
21.\" The original file was written by Ulf Moeller <ulf@openssl.org>
22.\" and Bodo Moeller <bodo@openssl.org>.
23.\" Copyright (c) 2000, 2015 The OpenSSL Project. All rights reserved.
24.\"
25.\" Redistribution and use in source and binary forms, with or without
26.\" modification, are permitted provided that the following conditions
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28.\"
29.\" 1. Redistributions of source code must retain the above copyright
30.\" notice, this list of conditions and the following disclaimer.
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32.\" 2. Redistributions in binary form must reproduce the above copyright
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34.\" the documentation and/or other materials provided with the
35.\" distribution.
36.\"
37.\" 3. All advertising materials mentioning features or use of this
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39.\" "This product includes software developed by the OpenSSL Project
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66.\" ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
67.\" OF THE POSSIBILITY OF SUCH DAMAGE.
68.\"
69.Dd $Mdocdate: April 27 2023 $
70.Dt BN_ADD 3
71.Os
72.Sh NAME
73.Nm BN_add ,
74.Nm BN_uadd ,
75.Nm BN_sub ,
76.Nm BN_usub ,
77.Nm BN_mul ,
78.Nm BN_sqr ,
79.Nm BN_div ,
80.Nm BN_mod ,
81.Nm BN_nnmod ,
82.Nm BN_mod_add ,
83.Nm BN_mod_add_quick ,
84.Nm BN_mod_sub ,
85.Nm BN_mod_sub_quick ,
86.Nm BN_mod_mul ,
87.Nm BN_mod_sqr ,
88.Nm BN_mod_lshift ,
89.Nm BN_mod_lshift_quick ,
90.Nm BN_mod_lshift1 ,
91.Nm BN_mod_lshift1_quick ,
92.Nm BN_exp ,
93.Nm BN_mod_exp ,
94.\" The following are public, but intentionally undocumented for now:
95.\" .Nm BN_mod_exp_mont , r \(== a ^ p (mod m)
96.\" .Nm BN_mod_exp_mont_consttime ,
97.\" .Nm BN_mod_exp_mont_word ,
98.\" .Nm BN_mod_exp_simple ,
99.\" .Nm BN_mod_exp2_mont r \(== (a1 ^ p1) * (a2 ^ p2) (mod m)
100.\" Maybe they should be deleted from <openssl/bn.h>.
101.Nm BN_gcd
102.Nd arithmetic operations on BIGNUMs
103.Sh SYNOPSIS
104.In openssl/bn.h
105.Ft int
106.Fo BN_add
107.Fa "BIGNUM *r"
108.Fa "const BIGNUM *a"
109.Fa "const BIGNUM *b"
110.Fc
111.Ft int
112.Fo BN_uadd
113.Fa "BIGNUM *r"
114.Fa "const BIGNUM *a"
115.Fa "const BIGNUM *b"
116.Fc
117.Ft int
118.Fo BN_sub
119.Fa "BIGNUM *r"
120.Fa "const BIGNUM *a"
121.Fa "const BIGNUM *b"
122.Fc
123.Ft int
124.Fo BN_usub
125.Fa "BIGNUM *r"
126.Fa "const BIGNUM *a"
127.Fa "const BIGNUM *b"
128.Fc
129.Ft int
130.Fo BN_mul
131.Fa "BIGNUM *r"
132.Fa "const BIGNUM *a"
133.Fa "const BIGNUM *b"
134.Fa "BN_CTX *ctx"
135.Fc
136.Ft int
137.Fo BN_sqr
138.Fa "BIGNUM *r"
139.Fa "const BIGNUM *a"
140.Fa "BN_CTX *ctx"
141.Fc
142.Ft int
143.Fo BN_div
144.Fa "BIGNUM *dv"
145.Fa "BIGNUM *rem"
146.Fa "const BIGNUM *a"
147.Fa "const BIGNUM *d"
148.Fa "BN_CTX *ctx"
149.Fc
150.Ft int
151.Fo BN_mod
152.Fa "BIGNUM *rem"
153.Fa "const BIGNUM *a"
154.Fa "const BIGNUM *m"
155.Fa "BN_CTX *ctx"
156.Fc
157.Ft int
158.Fo BN_nnmod
159.Fa "BIGNUM *r"
160.Fa "const BIGNUM *a"
161.Fa "const BIGNUM *m"
162.Fa "BN_CTX *ctx"
163.Fc
164.Ft int
165.Fo BN_mod_add
166.Fa "BIGNUM *r"
167.Fa "const BIGNUM *a"
168.Fa "const BIGNUM *b"
169.Fa "const BIGNUM *m"
170.Fa "BN_CTX *ctx"
171.Fc
172.Ft int
173.Fo BN_mod_add_quick
174.Fa "BIGNUM *r"
175.Fa "const BIGNUM *a"
176.Fa "const BIGNUM *b"
177.Fa "const BIGNUM *m"
178.Fc
179.Ft int
180.Fo BN_mod_sub
181.Fa "BIGNUM *r"
182.Fa "const BIGNUM *a"
183.Fa "const BIGNUM *b"
184.Fa "const BIGNUM *m"
185.Fa "BN_CTX *ctx"
186.Fc
187.Ft int
188.Fo BN_mod_sub_quick
189.Fa "BIGNUM *r"
190.Fa "const BIGNUM *a"
191.Fa "const BIGNUM *b"
192.Fa "const BIGNUM *m"
193.Fc
194.Ft int
195.Fo BN_mod_mul
196.Fa "BIGNUM *r"
197.Fa "const BIGNUM *a"
198.Fa "const BIGNUM *b"
199.Fa "const BIGNUM *m"
200.Fa "BN_CTX *ctx"
201.Fc
202.Ft int
203.Fo BN_mod_sqr
204.Fa "BIGNUM *r"
205.Fa "const BIGNUM *a"
206.Fa "const BIGNUM *m"
207.Fa "BN_CTX *ctx"
208.Fc
209.Ft int
210.Fo BN_mod_lshift
211.Fa "BIGNUM *r"
212.Fa "const BIGNUM *a"
213.Fa "int n"
214.Fa "const BIGNUM *m"
215.Fa "BN_CTX *ctx"
216.Fc
217.Ft int
218.Fo BN_mod_lshift_quick
219.Fa "BIGNUM *r"
220.Fa "const BIGNUM *a"
221.Fa "int n"
222.Fa "const BIGNUM *m"
223.Fc
224.Ft int
225.Fo BN_mod_lshift1
226.Fa "BIGNUM *r"
227.Fa "const BIGNUM *a"
228.Fa "const BIGNUM *m"
229.Fa "BN_CTX *ctx"
230.Fc
231.Ft int
232.Fo BN_mod_lshift1_quick
233.Fa "BIGNUM *r"
234.Fa "const BIGNUM *a"
235.Fa "const BIGNUM *m"
236.Fc
237.Ft int
238.Fo BN_exp
239.Fa "BIGNUM *r"
240.Fa "const BIGNUM *a"
241.Fa "const BIGNUM *p"
242.Fa "BN_CTX *ctx"
243.Fc
244.Ft int
245.Fo BN_mod_exp
246.Fa "BIGNUM *r"
247.Fa "const BIGNUM *a"
248.Fa "const BIGNUM *p"
249.Fa "const BIGNUM *m"
250.Fa "BN_CTX *ctx"
251.Fc
252.Ft int
253.Fo BN_gcd
254.Fa "BIGNUM *r"
255.Fa "const BIGNUM *a"
256.Fa "const BIGNUM *b"
257.Fa "BN_CTX *ctx"
258.Fc
259.Sh DESCRIPTION
260.Fn BN_add
261adds
262.Fa a
263and
264.Fa b
265and places the result in
266.Fa r
267.Pq Li r=a+b .
268.Fa r
269may be the same
270.Vt BIGNUM
271as
272.Fa a
273or
274.Fa b .
275.Pp
276.Fn BN_uadd
277adds the absolute values of
278.Fa a
279and
280.Fa b
281and places the result in
282.Fa r
283.Pq Li r=|a|+|b|\& .
284.Fa r
285may be the same
286.Vt BIGNUM
287as
288.Fa a
289or
290.Fa b .
291.Pp
292.Fn BN_sub
293subtracts
294.Fa b
295from
296.Fa a
297and places the result in
298.Fa r
299.Pq Li r=a-b .
300.Fa r
301may be the same
302.Vt BIGNUM
303as
304.Fa a
305or
306.Fa b .
307.Pp
308.Fn BN_usub
309subtracts the absolute value of
310.Fa b
311from the absolute value of
312.Fa a
313and places the result in
314.Fa r
315.Pq Li r=|a|-|b|\& .
316It requires the absolute value of
317.Fa a
318to be greater than the absolute value of
319.Fa b ;
320otherwise it will fail.
321.Fa r
322may be the same
323.Vt BIGNUM
324as
325.Fa a
326or
327.Fa b .
328.Pp
329.Fn BN_mul
330multiplies
331.Fa a
332and
333.Fa b
334and places the result in
335.Fa r
336.Pq Li r=a*b .
337.Fa r
338may be the same
339.Vt BIGNUM
340as
341.Fa a
342or
343.Fa b .
344For multiplication by powers of 2, use
345.Xr BN_lshift 3 .
346.Pp
347.Fn BN_sqr
348takes the square of
349.Fa a
350and places the result in
351.Fa r
352.Pq Li r=a^2 .
353.Fa r
354and
355.Fa a
356may be the same
357.Vt BIGNUM .
358This function is faster than
359.Fn BN_mul r a a .
360.Pp
361.Fn BN_div
362divides
363.Fa a
364by
365.Fa d
366and places the result in
367.Fa dv
368and the remainder in
369.Fa rem
370.Pq Li dv=a/d , rem=a%d .
371If the flag
372.Dv BN_FLG_CONSTTIME
373is set on
374.Fa a
375or
376.Fa d ,
377it operates in constant time.
378Either of
379.Fa dv
380and
381.Fa rem
382may be
383.Dv NULL ,
384in which case the respective value is not returned.
385The result is rounded towards zero; thus if
386.Fa a
387is negative, the remainder will be zero or negative.
388For division by powers of 2, use
389.Fn BN_rshift 3 .
390.Pp
391.Fn BN_mod
392corresponds to
393.Fn BN_div
394with
395.Fa dv
396set to
397.Dv NULL .
398It is implemented as a macro.
399.Pp
400.Fn BN_nnmod
401reduces
402.Fa a
403modulo
404.Fa m
405and places the non-negative remainder in
406.Fa r .
407.Pp
408.Fn BN_mod_add
409adds
410.Fa a
411to
412.Fa b
413modulo
414.Fa m
415and places the non-negative result in
416.Fa r .
417.Pp
418.Fn BN_mod_add_quick
419is a variant of
420.Fn BN_mod_add
421that requires
422.Fa a
423and
424.Fa b
425to both be non-negative and smaller than
426.Fa m .
427If any of these constraints are violated,
428it silently produces wrong results.
429.Pp
430.Fn BN_mod_sub
431subtracts
432.Fa b
433from
434.Fa a
435modulo
436.Fa m
437and places the non-negative result in
438.Fa r .
439.Pp
440.Fn BN_mod_sub_quick
441is a variant of
442.Fn BN_mod_sub
443that requires
444.Fa a
445and
446.Fa b
447to both be non-negative and smaller than
448.Fa m .
449If any of these constraints are violated,
450it silently produces wrong results.
451.Pp
452.Fn BN_mod_mul
453multiplies
454.Fa a
455by
456.Fa b
457and finds the non-negative remainder respective to modulus
458.Fa m
459.Pq Li r=(a*b)%m .
460.Fa r
461may be the same
462.Vt BIGNUM
463as
464.Fa a
465or
466.Fa b .
467For a more efficient algorithm for repeated computations using the same
468modulus, see
469.Xr BN_mod_mul_montgomery 3 .
470.Pp
471.Fn BN_mod_sqr
472takes the square of
473.Fa a
474modulo
475.Fa m
476and places the result in
477.Fa r .
478.Pp
479.Fn BN_mod_lshift
480shifts
481.Fa a
482left by
483.Fa n
484bits, reduces the result modulo
485.Fa m ,
486and places the non-negative remainder in
487.Fa r
488.Pq Li r=a*2^n mod m .
489.Pp
490.Fn BN_mod_lshift1
491shifts
492.Fa a
493left by one bit, reduces the result modulo
494.Fa m ,
495and places the non-negative remainder in
496.Fa r
497.Pq Li r=a*2 mod m .
498.Pp
499.Fn BN_mod_lshift_quick
500and
501.Fn BN_mod_lshift1_quick
502are variants of
503.Fn BN_mod_lshift
504and
505.Fn BN_mod_lshift1 ,
506respectively, that require
507.Fa a
508to be non-negative and less than
509.Fa m .
510If either of these constraints is violated, they sometimes fail
511and sometimes silently produce wrong results.
512.Pp
513.Fn BN_exp
514raises
515.Fa a
516to the
517.Fa p Ns -th
518power and places the result in
519.Fa r
520.Pq Li r=a^p .
521This function is faster than repeated applications of
522.Fn BN_mul .
523.Pp
524.Fn BN_mod_exp
525computes
526.Fa a
527to the
528.Fa p Ns -th
529power modulo
530.Fa m
531.Pq Li r=(a^p)%m .
532If the flag
533.Dv BN_FLG_CONSTTIME
534is set on
535.Fa p ,
536it operates in constant time.
537This function uses less time and space than
538.Fn BN_exp .
539.Pp
540.Fn BN_gcd
541computes the greatest common divisor of
542.Fa a
543and
544.Fa b
545and places the result in
546.Fa r .
547.Fa r
548may be the same
549.Vt BIGNUM
550as
551.Fa a
552or
553.Fa b .
554.Pp
555For all functions,
556.Fa ctx
557is a previously allocated
558.Vt BN_CTX
559used for temporary variables; see
560.Xr BN_CTX_new 3 .
561.Pp
562Unless noted otherwise, the result
563.Vt BIGNUM
564must be different from the arguments.
565.Sh RETURN VALUES
566For all functions, 1 is returned for success, 0 on error.
567The return value should always be checked, for example:
568.Pp
569.Dl if (!BN_add(r,a,b)) goto err;
570.Pp
571The error codes can be obtained by
572.Xr ERR_get_error 3 .
573.Sh SEE ALSO
574.Xr BN_add_word 3 ,
575.Xr BN_CTX_new 3 ,
576.Xr BN_new 3 ,
577.Xr BN_set_bit 3 ,
578.Xr BN_set_flags 3 ,
579.Xr BN_set_negative 3
580.Sh HISTORY
581.Fn BN_add ,
582.Fn BN_sub ,
583.Fn BN_mul ,
584.Fn BN_sqr ,
585.Fn BN_div ,
586.Fn BN_mod ,
587.Fn BN_mod_mul ,
588.Fn BN_mod_exp ,
589and
590.Fn BN_gcd
591first appeared in SSLeay 0.5.1.
592.Fn BN_exp
593first appeared in SSLeay 0.9.0.
594All these functions have been available since
595.Ox 2.4 .
596.Pp
597.Fn BN_uadd ,
598.Fn BN_usub ,
599and the
600.Fa ctx
601argument to
602.Fn BN_mul
603first appeared in SSLeay 0.9.1 and have been available since
604.Ox 2.6 .
605.Pp
606.Fn BN_nnmod ,
607.Fn BN_mod_add ,
608.Fn BN_mod_add_quick ,
609.Fn BN_mod_sub ,
610.Fn BN_mod_sub_quick ,
611.Fn BN_mod_sqr ,
612.Fn BN_mod_lshift ,
613.Fn BN_mod_lshift_quick ,
614.Fn BN_mod_lshift1 ,
615and
616.Fn BN_mod_lshift1_quick
617first appeared in OpenSSL 0.9.7 and have been available since
618.Ox 3.2 .
619.Sh BUGS
620Even if the
621.Dv BN_FLG_CONSTTIME
622flag is set on
623.Fa a
624or
625.Fa b ,
626.Fn BN_gcd
627neither fails nor operates in constant time, potentially allowing
628timing side-channel attacks.
629.Pp
630Even if the
631.Dv BN_FLG_CONSTTIME
632flag is set on
633.Fa p ,
634if the modulus
635.Fa m
636is even,
637.Fn BN_mod_exp
638does not operate in constant time, potentially allowing
639timing side-channel attacks.
640.Pp
641If
642.Dv BN_FLG_CONSTTIME
643is set on
644.Fa p ,
645.Fn BN_exp
646fails instead of operating in constant time.
generated by cgit v1.2.3-55-g6feb (git 2.39.1) at 2025-05-06 16:43:54 +0000