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.\" Automatically generated by Pod::Man 4.11 (Pod::Simple 3.35) .\" .\" Standard preamble: .\" ======================================================================== .de Sp \" Vertical space (when we can't use .PP) .if t .sp .5v .if n .sp .. .de Vb \" Begin verbatim text .ft CW .nf .ne \\$1 .. .de Ve \" End verbatim text .ft R .fi .. .\" Set up some character translations and predefined strings. \*(-- will .\" give an unbreakable dash, \*(PI will give pi, \*(L" will give a left .\" double quote, and \*(R" will give a right double quote. \*(C+ will .\" give a nicer C++. Capital omega is used to do unbreakable dashes and .\" therefore won't be available. \*(C` and \*(C' expand to `' in nroff, .\" nothing in troff, for use with C<>. .tr \(*W- .ds C+ C\v'-.1v'\h'-1p'\s-2+\h'-1p'+\s0\v'.1v'\h'-1p' .ie n \{\ . ds -- \(*W- . ds PI pi . if (\n(.H=4u)&(1m=24u) .ds -- \(*W\h'-12u'\(*W\h'-12u'-\" diablo 10 pitch . if (\n(.H=4u)&(1m=20u) .ds -- \(*W\h'-12u'\(*W\h'-8u'-\" diablo 12 pitch . ds L" "" . ds R" "" . ds C` "" . ds C' "" 'br\} .el\{\ . ds -- \|\(em\| . ds PI \(*p . ds L" `` . ds R" '' . ds C` . ds C' 'br\} .\" .\" Escape single quotes in literal strings from groff's Unicode transform. .ie \n(.g .ds Aq \(aq .el .ds Aq ' .\" .\" If the F register is >0, we'll generate index entries on stderr for .\" titles (.TH), headers (.SH), subsections (.SS), items (.Ip), and index .\" entries marked with X<> in POD. Of course, you'll have to process the .\" output yourself in some meaningful fashion. .\" .\" Avoid warning from groff about undefined register 'F'. .de IX .. .nr rF 0 .if \n(.g .if rF .nr rF 1 .if (\n(rF:(\n(.g==0)) \{\ . if \nF \{\ . de IX . tm Index:\\$1\t\\n%\t"\\$2" .. . if !\nF==2 \{\ . nr % 0 . nr F 2 . \} . \} .\} .rr rF .\" .\" Accent mark definitions (@(#)ms.acc 1.5 88/02/08 SMI; from UCB 4.2). .\" Fear. Run. Save yourself. No user-serviceable parts. . \" fudge factors for nroff and troff .if n \{\ . ds #H 0 . ds #V .8m . ds #F .3m . ds #[ \f1 . ds #] \fP .\} .if t \{\ . ds #H ((1u-(\\\\n(.fu%2u))*.13m) . ds #V .6m . ds #F 0 . ds #[ \& . ds #] \& .\} . \" simple accents for nroff and troff .if n \{\ . ds ' \& . ds ` \& . ds ^ \& . ds , \& . ds ~ ~ . ds / .\} .if t \{\ . ds ' \\k:\h'-(\\n(.wu*8/10-\*(#H)'\'\h"|\\n:u" . ds ` \\k:\h'-(\\n(.wu*8/10-\*(#H)'\`\h'|\\n:u' . ds ^ \\k:\h'-(\\n(.wu*10/11-\*(#H)'^\h'|\\n:u' . ds , \\k:\h'-(\\n(.wu*8/10)',\h'|\\n:u' . ds ~ \\k:\h'-(\\n(.wu-\*(#H-.1m)'~\h'|\\n:u' . ds / \\k:\h'-(\\n(.wu*8/10-\*(#H)'\z\(sl\h'|\\n:u' .\} . \" troff and (daisy-wheel) nroff accents .ds : \\k:\h'-(\\n(.wu*8/10-\*(#H+.1m+\*(#F)'\v'-\*(#V'\z.\h'.2m+\*(#F'.\h'|\\n:u'\v'\*(#V' .ds 8 \h'\*(#H'\(*b\h'-\*(#H' .ds o \\k:\h'-(\\n(.wu+\w'\(de'u-\*(#H)/2u'\v'-.3n'\*(#[\z\(de\v'.3n'\h'|\\n:u'\*(#] .ds d- \h'\*(#H'\(pd\h'-\w'~'u'\v'-.25m'\f2\(hy\fP\v'.25m'\h'-\*(#H' .ds D- D\\k:\h'-\w'D'u'\v'-.11m'\z\(hy\v'.11m'\h'|\\n:u' .ds th \*(#[\v'.3m'\s+1I\s-1\v'-.3m'\h'-(\w'I'u*2/3)'\s-1o\s+1\*(#] .ds Th \*(#[\s+2I\s-2\h'-\w'I'u*3/5'\v'-.3m'o\v'.3m'\*(#] .ds ae a\h'-(\w'a'u*4/10)'e .ds Ae A\h'-(\w'A'u*4/10)'E . \" corrections for vroff .if v .ds ~ \\k:\h'-(\\n(.wu*9/10-\*(#H)'\s-2\u~\d\s+2\h'|\\n:u' .if v .ds ^ \\k:\h'-(\\n(.wu*10/11-\*(#H)'\v'-.4m'^\v'.4m'\h'|\\n:u' . \" for low resolution devices (crt and lpr) .if \n(.H>23 .if \n(.V>19 \ \{\ . ds : e . ds 8 ss . ds o a . ds d- d\h'-1'\(ga . ds D- D\h'-1'\(hy . ds th \o'bp' . ds Th \o'LP' . ds ae ae . ds Ae AE .\} .rm #[ #] #H #V #F C .\" ======================================================================== .\" .IX Title "bn_internal 3" .TH bn_internal 3 "2019-12-20" "1.0.2u" "OpenSSL" .\" For nroff, turn off justification. Always turn off hyphenation; it makes .\" way too many mistakes in technical documents. .if n .ad l .nh .SH "NAME" bn_mul_words, bn_mul_add_words, bn_sqr_words, bn_div_words, bn_add_words, bn_sub_words, bn_mul_comba4, bn_mul_comba8, bn_sqr_comba4, bn_sqr_comba8, bn_cmp_words, bn_mul_normal, bn_mul_low_normal, bn_mul_recursive, bn_mul_part_recursive, bn_mul_low_recursive, bn_mul_high, bn_sqr_normal, bn_sqr_recursive, bn_expand, bn_wexpand, bn_expand2, bn_fix_top, bn_check_top, bn_print, bn_dump, bn_set_max, bn_set_high, bn_set_low \- BIGNUM library internal functions .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 1 \& #include <openssl/bn.h> \& \& BN_ULONG bn_mul_words(BN_ULONG *rp, BN_ULONG *ap, int num, BN_ULONG w); \& BN_ULONG bn_mul_add_words(BN_ULONG *rp, BN_ULONG *ap, int num, \& BN_ULONG w); \& void bn_sqr_words(BN_ULONG *rp, BN_ULONG *ap, int num); \& BN_ULONG bn_div_words(BN_ULONG h, BN_ULONG l, BN_ULONG d); \& BN_ULONG bn_add_words(BN_ULONG *rp, BN_ULONG *ap, BN_ULONG *bp, \& int num); \& BN_ULONG bn_sub_words(BN_ULONG *rp, BN_ULONG *ap, BN_ULONG *bp, \& int num); \& \& void bn_mul_comba4(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b); \& void bn_mul_comba8(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b); \& void bn_sqr_comba4(BN_ULONG *r, BN_ULONG *a); \& void bn_sqr_comba8(BN_ULONG *r, BN_ULONG *a); \& \& int bn_cmp_words(BN_ULONG *a, BN_ULONG *b, int n); \& \& void bn_mul_normal(BN_ULONG *r, BN_ULONG *a, int na, BN_ULONG *b, \& int nb); \& void bn_mul_low_normal(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b, int n); \& void bn_mul_recursive(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b, int n2, \& int dna,int dnb,BN_ULONG *tmp); \& void bn_mul_part_recursive(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b, \& int n, int tna,int tnb, BN_ULONG *tmp); \& void bn_mul_low_recursive(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b, \& int n2, BN_ULONG *tmp); \& void bn_mul_high(BN_ULONG *r, BN_ULONG *a, BN_ULONG *b, BN_ULONG *l, \& int n2, BN_ULONG *tmp); \& \& void bn_sqr_normal(BN_ULONG *r, BN_ULONG *a, int n, BN_ULONG *tmp); \& void bn_sqr_recursive(BN_ULONG *r, BN_ULONG *a, int n2, BN_ULONG *tmp); \& \& void mul(BN_ULONG r, BN_ULONG a, BN_ULONG w, BN_ULONG c); \& void mul_add(BN_ULONG r, BN_ULONG a, BN_ULONG w, BN_ULONG c); \& void sqr(BN_ULONG r0, BN_ULONG r1, BN_ULONG a); \& \& BIGNUM *bn_expand(BIGNUM *a, int bits); \& BIGNUM *bn_wexpand(BIGNUM *a, int n); \& BIGNUM *bn_expand2(BIGNUM *a, int n); \& void bn_fix_top(BIGNUM *a); \& \& void bn_check_top(BIGNUM *a); \& void bn_print(BIGNUM *a); \& void bn_dump(BN_ULONG *d, int n); \& void bn_set_max(BIGNUM *a); \& void bn_set_high(BIGNUM *r, BIGNUM *a, int n); \& void bn_set_low(BIGNUM *r, BIGNUM *a, int n); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This page documents the internal functions used by the OpenSSL \&\fB\s-1BIGNUM\s0\fR implementation. They are described here to facilitate debugging and extending the library. They are \fInot\fR to be used by applications. .SS "The \s-1BIGNUM\s0 structure" .IX Subsection "The BIGNUM structure" .Vb 1 \& typedef struct bignum_st BIGNUM; \& \& struct bignum_st \& { \& BN_ULONG *d; /* Pointer to an array of \*(AqBN_BITS2\*(Aq bit chunks. */ \& int top; /* Index of last used d +1. */ \& /* The next are internal book keeping for bn_expand. */ \& int dmax; /* Size of the d array. */ \& int neg; /* one if the number is negative */ \& int flags; \& }; .Ve .PP The integer value is stored in \fBd\fR, a \fBmalloc()\fRed array of words (\fB\s-1BN_ULONG\s0\fR), least significant word first. A \fB\s-1BN_ULONG\s0\fR can be either 16, 32 or 64 bits in size, depending on the 'number of bits' (\fB\s-1BITS2\s0\fR) specified in \&\f(CW\*(C`openssl/bn.h\*(C'\fR. .PP \&\fBdmax\fR is the size of the \fBd\fR array that has been allocated. \fBtop\fR is the number of words being used, so for a value of 4, bn.d[0]=4 and bn.top=1. \fBneg\fR is 1 if the number is negative. When a \fB\s-1BIGNUM\s0\fR is \&\fB0\fR, the \fBd\fR field can be \fB\s-1NULL\s0\fR and \fBtop\fR == \fB0\fR. .PP \&\fBflags\fR is a bit field of flags which are defined in \f(CW\*(C`openssl/bn.h\*(C'\fR. The flags begin with \fB\s-1BN_FLG_\s0\fR. The macros BN_set_flags(b,n) and BN_get_flags(b,n) exist to enable or fetch flag(s) \fBn\fR from \fB\s-1BIGNUM\s0\fR structure \fBb\fR. .PP Various routines in this library require the use of temporary \&\fB\s-1BIGNUM\s0\fR variables during their execution. Since dynamic memory allocation to create \fB\s-1BIGNUM\s0\fRs is rather expensive when used in conjunction with repeated subroutine calls, the \fB\s-1BN_CTX\s0\fR structure is used. This structure contains \fB\s-1BN_CTX_NUM\s0\fR \fB\s-1BIGNUM\s0\fRs, see \&\fBBN_CTX_start\fR\|(3). .SS "Low-level arithmetic operations" .IX Subsection "Low-level arithmetic operations" These functions are implemented in C and for several platforms in assembly language: .PP bn_mul_words(\fBrp\fR, \fBap\fR, \fBnum\fR, \fBw\fR) operates on the \fBnum\fR word arrays \fBrp\fR and \fBap\fR. It computes \fBap\fR * \fBw\fR, places the result in \fBrp\fR, and returns the high word (carry). .PP bn_mul_add_words(\fBrp\fR, \fBap\fR, \fBnum\fR, \fBw\fR) operates on the \fBnum\fR word arrays \fBrp\fR and \fBap\fR. It computes \fBap\fR * \fBw\fR + \fBrp\fR, places the result in \fBrp\fR, and returns the high word (carry). .PP bn_sqr_words(\fBrp\fR, \fBap\fR, \fBn\fR) operates on the \fBnum\fR word array \&\fBap\fR and the 2*\fBnum\fR word array \fBap\fR. It computes \fBap\fR * \fBap\fR word-wise, and places the low and high bytes of the result in \fBrp\fR. .PP bn_div_words(\fBh\fR, \fBl\fR, \fBd\fR) divides the two word number (\fBh\fR,\fBl\fR) by \fBd\fR and returns the result. .PP bn_add_words(\fBrp\fR, \fBap\fR, \fBbp\fR, \fBnum\fR) operates on the \fBnum\fR word arrays \fBap\fR, \fBbp\fR and \fBrp\fR. It computes \fBap\fR + \fBbp\fR, places the result in \fBrp\fR, and returns the high word (carry). .PP bn_sub_words(\fBrp\fR, \fBap\fR, \fBbp\fR, \fBnum\fR) operates on the \fBnum\fR word arrays \fBap\fR, \fBbp\fR and \fBrp\fR. It computes \fBap\fR \- \fBbp\fR, places the result in \fBrp\fR, and returns the carry (1 if \fBbp\fR > \fBap\fR, 0 otherwise). .PP bn_mul_comba4(\fBr\fR, \fBa\fR, \fBb\fR) operates on the 4 word arrays \fBa\fR and \&\fBb\fR and the 8 word array \fBr\fR. It computes \fBa\fR*\fBb\fR and places the result in \fBr\fR. .PP bn_mul_comba8(\fBr\fR, \fBa\fR, \fBb\fR) operates on the 8 word arrays \fBa\fR and \&\fBb\fR and the 16 word array \fBr\fR. It computes \fBa\fR*\fBb\fR and places the result in \fBr\fR. .PP bn_sqr_comba4(\fBr\fR, \fBa\fR, \fBb\fR) operates on the 4 word arrays \fBa\fR and \&\fBb\fR and the 8 word array \fBr\fR. .PP bn_sqr_comba8(\fBr\fR, \fBa\fR, \fBb\fR) operates on the 8 word arrays \fBa\fR and \&\fBb\fR and the 16 word array \fBr\fR. .PP The following functions are implemented in C: .PP bn_cmp_words(\fBa\fR, \fBb\fR, \fBn\fR) operates on the \fBn\fR word arrays \fBa\fR and \fBb\fR. It returns 1, 0 and \-1 if \fBa\fR is greater than, equal and less than \fBb\fR. .PP bn_mul_normal(\fBr\fR, \fBa\fR, \fBna\fR, \fBb\fR, \fBnb\fR) operates on the \fBna\fR word array \fBa\fR, the \fBnb\fR word array \fBb\fR and the \fBna\fR+\fBnb\fR word array \fBr\fR. It computes \fBa\fR*\fBb\fR and places the result in \fBr\fR. .PP bn_mul_low_normal(\fBr\fR, \fBa\fR, \fBb\fR, \fBn\fR) operates on the \fBn\fR word arrays \fBr\fR, \fBa\fR and \fBb\fR. It computes the \fBn\fR low words of \&\fBa\fR*\fBb\fR and places the result in \fBr\fR. .PP bn_mul_recursive(\fBr\fR, \fBa\fR, \fBb\fR, \fBn2\fR, \fBdna\fR, \fBdnb\fR, \fBt\fR) operates on the word arrays \fBa\fR and \fBb\fR of length \fBn2\fR+\fBdna\fR and \fBn2\fR+\fBdnb\fR (\fBdna\fR and \fBdnb\fR are currently allowed to be 0 or negative) and the 2*\fBn2\fR word arrays \fBr\fR and \fBt\fR. \fBn2\fR must be a power of 2. It computes \&\fBa\fR*\fBb\fR and places the result in \fBr\fR. .PP bn_mul_part_recursive(\fBr\fR, \fBa\fR, \fBb\fR, \fBn\fR, \fBtna\fR, \fBtnb\fR, \fBtmp\fR) operates on the word arrays \fBa\fR and \fBb\fR of length \fBn\fR+\fBtna\fR and \&\fBn\fR+\fBtnb\fR and the 4*\fBn\fR word arrays \fBr\fR and \fBtmp\fR. .PP bn_mul_low_recursive(\fBr\fR, \fBa\fR, \fBb\fR, \fBn2\fR, \fBtmp\fR) operates on the \&\fBn2\fR word arrays \fBr\fR and \fBtmp\fR and the \fBn2\fR/2 word arrays \fBa\fR and \fBb\fR. .PP bn_mul_high(\fBr\fR, \fBa\fR, \fBb\fR, \fBl\fR, \fBn2\fR, \fBtmp\fR) operates on the \&\fBn2\fR word arrays \fBr\fR, \fBa\fR, \fBb\fR and \fBl\fR (?) and the 3*\fBn2\fR word array \fBtmp\fR. .PP \&\fBBN_mul()\fR calls \fBbn_mul_normal()\fR, or an optimized implementation if the factors have the same size: \fBbn_mul_comba8()\fR is used if they are 8 words long, \fBbn_mul_recursive()\fR if they are larger than \&\fB\s-1BN_MULL_SIZE_NORMAL\s0\fR and the size is an exact multiple of the word size, and \fBbn_mul_part_recursive()\fR for others that are larger than \&\fB\s-1BN_MULL_SIZE_NORMAL\s0\fR. .PP bn_sqr_normal(\fBr\fR, \fBa\fR, \fBn\fR, \fBtmp\fR) operates on the \fBn\fR word array \&\fBa\fR and the 2*\fBn\fR word arrays \fBtmp\fR and \fBr\fR. .PP The implementations use the following macros which, depending on the architecture, may use \*(L"long long\*(R" C operations or inline assembler. They are defined in \f(CW\*(C`bn_lcl.h\*(C'\fR. .PP mul(\fBr\fR, \fBa\fR, \fBw\fR, \fBc\fR) computes \fBw\fR*\fBa\fR+\fBc\fR and places the low word of the result in \fBr\fR and the high word in \fBc\fR. .PP mul_add(\fBr\fR, \fBa\fR, \fBw\fR, \fBc\fR) computes \fBw\fR*\fBa\fR+\fBr\fR+\fBc\fR and places the low word of the result in \fBr\fR and the high word in \fBc\fR. .PP sqr(\fBr0\fR, \fBr1\fR, \fBa\fR) computes \fBa\fR*\fBa\fR and places the low word of the result in \fBr0\fR and the high word in \fBr1\fR. .SS "Size changes" .IX Subsection "Size changes" \&\fBbn_expand()\fR ensures that \fBb\fR has enough space for a \fBbits\fR bit number. \fBbn_wexpand()\fR ensures that \fBb\fR has enough space for an \&\fBn\fR word number. If the number has to be expanded, both macros call \fBbn_expand2()\fR, which allocates a new \fBd\fR array and copies the data. They return \fB\s-1NULL\s0\fR on error, \fBb\fR otherwise. .PP The \fBbn_fix_top()\fR macro reduces \fBa\->top\fR to point to the most significant non-zero word plus one when \fBa\fR has shrunk. .SS "Debugging" .IX Subsection "Debugging" \&\fBbn_check_top()\fR verifies that \f(CW\*(C`((a)\->top >= 0 && (a)\->top <= (a)\->dmax)\*(C'\fR. A violation will cause the program to abort. .PP \&\fBbn_print()\fR prints \fBa\fR to stderr. \fBbn_dump()\fR prints \fBn\fR words at \fBd\fR (in reverse order, i.e. most significant word first) to stderr. .PP \&\fBbn_set_max()\fR makes \fBa\fR a static number with a \fBdmax\fR of its current size. This is used by \fBbn_set_low()\fR and \fBbn_set_high()\fR to make \fBr\fR a read-only \&\fB\s-1BIGNUM\s0\fR that contains the \fBn\fR low or high words of \fBa\fR. .PP If \fB\s-1BN_DEBUG\s0\fR is not defined, \fBbn_check_top()\fR, \fBbn_print()\fR, \fBbn_dump()\fR and \fBbn_set_max()\fR are defined as empty macros. .SH "SEE ALSO" .IX Header "SEE ALSO" \&\fBbn\fR\|(3)