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.\" Automatically generated by Pod::Man 2.27 (Pod::Simple 3.28) .\" .\" 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 turned on, we'll generate index entries on stderr for .\" titles (.TH), headers (.SH), subsections (.SS), items (.Ip), and index .\" entries marked with X<> in POD. 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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 "des 3" .TH des 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" DES_random_key, DES_set_key, DES_key_sched, DES_set_key_checked, DES_set_key_unchecked, DES_set_odd_parity, DES_is_weak_key, DES_ecb_encrypt, DES_ecb2_encrypt, DES_ecb3_encrypt, DES_ncbc_encrypt, DES_cfb_encrypt, DES_ofb_encrypt, DES_pcbc_encrypt, DES_cfb64_encrypt, DES_ofb64_encrypt, DES_xcbc_encrypt, DES_ede2_cbc_encrypt, DES_ede2_cfb64_encrypt, DES_ede2_ofb64_encrypt, DES_ede3_cbc_encrypt, DES_ede3_cbcm_encrypt, DES_ede3_cfb64_encrypt, DES_ede3_ofb64_encrypt, DES_cbc_cksum, DES_quad_cksum, DES_string_to_key, DES_string_to_2keys, DES_fcrypt, DES_crypt, DES_enc_read, DES_enc_write \- DES encryption .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 1 \& #include <openssl/des.h> \& \& void DES_random_key(DES_cblock *ret); \& \& int DES_set_key(const_DES_cblock *key, DES_key_schedule *schedule); \& int DES_key_sched(const_DES_cblock *key, DES_key_schedule *schedule); \& int DES_set_key_checked(const_DES_cblock *key, \& DES_key_schedule *schedule); \& void DES_set_key_unchecked(const_DES_cblock *key, \& DES_key_schedule *schedule); \& \& void DES_set_odd_parity(DES_cblock *key); \& int DES_is_weak_key(const_DES_cblock *key); \& \& void DES_ecb_encrypt(const_DES_cblock *input, DES_cblock *output, \& DES_key_schedule *ks, int enc); \& void DES_ecb2_encrypt(const_DES_cblock *input, DES_cblock *output, \& DES_key_schedule *ks1, DES_key_schedule *ks2, int enc); \& void DES_ecb3_encrypt(const_DES_cblock *input, DES_cblock *output, \& DES_key_schedule *ks1, DES_key_schedule *ks2, \& DES_key_schedule *ks3, int enc); \& \& void DES_ncbc_encrypt(const unsigned char *input, unsigned char *output, \& long length, DES_key_schedule *schedule, DES_cblock *ivec, \& int enc); \& void DES_cfb_encrypt(const unsigned char *in, unsigned char *out, \& int numbits, long length, DES_key_schedule *schedule, \& DES_cblock *ivec, int enc); \& void DES_ofb_encrypt(const unsigned char *in, unsigned char *out, \& int numbits, long length, DES_key_schedule *schedule, \& DES_cblock *ivec); \& void DES_pcbc_encrypt(const unsigned char *input, unsigned char *output, \& long length, DES_key_schedule *schedule, DES_cblock *ivec, \& int enc); \& void DES_cfb64_encrypt(const unsigned char *in, unsigned char *out, \& long length, DES_key_schedule *schedule, DES_cblock *ivec, \& int *num, int enc); \& void DES_ofb64_encrypt(const unsigned char *in, unsigned char *out, \& long length, DES_key_schedule *schedule, DES_cblock *ivec, \& int *num); \& \& void DES_xcbc_encrypt(const unsigned char *input, unsigned char *output, \& long length, DES_key_schedule *schedule, DES_cblock *ivec, \& const_DES_cblock *inw, const_DES_cblock *outw, int enc); \& \& void DES_ede2_cbc_encrypt(const unsigned char *input, \& unsigned char *output, long length, DES_key_schedule *ks1, \& DES_key_schedule *ks2, DES_cblock *ivec, int enc); \& void DES_ede2_cfb64_encrypt(const unsigned char *in, \& unsigned char *out, long length, DES_key_schedule *ks1, \& DES_key_schedule *ks2, DES_cblock *ivec, int *num, int enc); \& void DES_ede2_ofb64_encrypt(const unsigned char *in, \& unsigned char *out, long length, DES_key_schedule *ks1, \& DES_key_schedule *ks2, DES_cblock *ivec, int *num); \& \& void DES_ede3_cbc_encrypt(const unsigned char *input, \& unsigned char *output, long length, DES_key_schedule *ks1, \& DES_key_schedule *ks2, DES_key_schedule *ks3, DES_cblock *ivec, \& int enc); \& void DES_ede3_cbcm_encrypt(const unsigned char *in, unsigned char *out, \& long length, DES_key_schedule *ks1, DES_key_schedule *ks2, \& DES_key_schedule *ks3, DES_cblock *ivec1, DES_cblock *ivec2, \& int enc); \& void DES_ede3_cfb64_encrypt(const unsigned char *in, unsigned char *out, \& long length, DES_key_schedule *ks1, DES_key_schedule *ks2, \& DES_key_schedule *ks3, DES_cblock *ivec, int *num, int enc); \& void DES_ede3_ofb64_encrypt(const unsigned char *in, unsigned char *out, \& long length, DES_key_schedule *ks1, \& DES_key_schedule *ks2, DES_key_schedule *ks3, \& DES_cblock *ivec, int *num); \& \& DES_LONG DES_cbc_cksum(const unsigned char *input, DES_cblock *output, \& long length, DES_key_schedule *schedule, \& const_DES_cblock *ivec); \& DES_LONG DES_quad_cksum(const unsigned char *input, DES_cblock output[], \& long length, int out_count, DES_cblock *seed); \& void DES_string_to_key(const char *str, DES_cblock *key); \& void DES_string_to_2keys(const char *str, DES_cblock *key1, \& DES_cblock *key2); \& \& char *DES_fcrypt(const char *buf, const char *salt, char *ret); \& char *DES_crypt(const char *buf, const char *salt); \& \& int DES_enc_read(int fd, void *buf, int len, DES_key_schedule *sched, \& DES_cblock *iv); \& int DES_enc_write(int fd, const void *buf, int len, \& DES_key_schedule *sched, DES_cblock *iv); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This library contains a fast implementation of the \s-1DES\s0 encryption algorithm. .PP There are two phases to the use of \s-1DES\s0 encryption. The first is the generation of a \fIDES_key_schedule\fR from a key, the second is the actual encryption. A \s-1DES\s0 key is of type \fIDES_cblock\fR. This type is consists of 8 bytes with odd parity. The least significant bit in each byte is the parity bit. The key schedule is an expanded form of the key; it is used to speed the encryption process. .PP \&\fIDES_random_key()\fR generates a random key. The \s-1PRNG\s0 must be seeded prior to using this function (see \fIrand\fR\|(3)). If the \s-1PRNG\s0 could not generate a secure key, 0 is returned. .PP Before a \s-1DES\s0 key can be used, it must be converted into the architecture dependent \fIDES_key_schedule\fR via the \&\fIDES_set_key_checked()\fR or \fIDES_set_key_unchecked()\fR function. .PP \&\fIDES_set_key_checked()\fR will check that the key passed is of odd parity and is not a weak or semi-weak key. If the parity is wrong, then \-1 is returned. If the key is a weak key, then \-2 is returned. If an error is returned, the key schedule is not generated. .PP \&\fIDES_set_key()\fR works like \&\fIDES_set_key_checked()\fR if the \fIDES_check_key\fR flag is non-zero, otherwise like \fIDES_set_key_unchecked()\fR. These functions are available for compatibility; it is recommended to use a function that does not depend on a global variable. .PP \&\fIDES_set_odd_parity()\fR sets the parity of the passed \fIkey\fR to odd. .PP \&\fIDES_is_weak_key()\fR returns 1 if the passed key is a weak key, 0 if it is ok. .PP The following routines mostly operate on an input and output stream of \&\fIDES_cblock\fRs. .PP \&\fIDES_ecb_encrypt()\fR is the basic \s-1DES\s0 encryption routine that encrypts or decrypts a single 8\-byte \fIDES_cblock\fR in \fIelectronic code book\fR (\s-1ECB\s0) mode. It always transforms the input data, pointed to by \&\fIinput\fR, into the output data, pointed to by the \fIoutput\fR argument. If the \fIencrypt\fR argument is non-zero (\s-1DES_ENCRYPT\s0), the \fIinput\fR (cleartext) is encrypted in to the \fIoutput\fR (ciphertext) using the key_schedule specified by the \fIschedule\fR argument, previously set via \&\fIDES_set_key\fR. If \fIencrypt\fR is zero (\s-1DES_DECRYPT\s0), the \fIinput\fR (now ciphertext) is decrypted into the \fIoutput\fR (now cleartext). Input and output may overlap. \fIDES_ecb_encrypt()\fR does not return a value. .PP \&\fIDES_ecb3_encrypt()\fR encrypts/decrypts the \fIinput\fR block by using three-key Triple-DES encryption in \s-1ECB\s0 mode. This involves encrypting the input with \fIks1\fR, decrypting with the key schedule \fIks2\fR, and then encrypting with \fIks3\fR. This routine greatly reduces the chances of brute force breaking of \s-1DES\s0 and has the advantage of if \fIks1\fR, \&\fIks2\fR and \fIks3\fR are the same, it is equivalent to just encryption using \s-1ECB\s0 mode and \fIks1\fR as the key. .PP The macro \fIDES_ecb2_encrypt()\fR is provided to perform two-key Triple-DES encryption by using \fIks1\fR for the final encryption. .PP \&\fIDES_ncbc_encrypt()\fR encrypts/decrypts using the \fIcipher-block-chaining\fR (\s-1CBC\s0) mode of \s-1DES. \s0 If the \fIencrypt\fR argument is non-zero, the routine cipher-block-chain encrypts the cleartext data pointed to by the \fIinput\fR argument into the ciphertext pointed to by the \fIoutput\fR argument, using the key schedule provided by the \fIschedule\fR argument, and initialization vector provided by the \fIivec\fR argument. If the \&\fIlength\fR argument is not an integral multiple of eight bytes, the last block is copied to a temporary area and zero filled. The output is always an integral multiple of eight bytes. .PP \&\fIDES_xcbc_encrypt()\fR is \s-1RSA\s0's \s-1DESX\s0 mode of \s-1DES. \s0 It uses \fIinw\fR and \&\fIoutw\fR to 'whiten' the encryption. \fIinw\fR and \fIoutw\fR are secret (unlike the iv) and are as such, part of the key. So the key is sort of 24 bytes. This is much better than \s-1CBC DES.\s0 .PP \&\fIDES_ede3_cbc_encrypt()\fR implements outer triple \s-1CBC DES\s0 encryption with three keys. This means that each \s-1DES\s0 operation inside the \s-1CBC\s0 mode is an \f(CW\*(C`C=E(ks3,D(ks2,E(ks1,M)))\*(C'\fR. This mode is used by \s-1SSL.\s0 .PP The \fIDES_ede2_cbc_encrypt()\fR macro implements two-key Triple-DES by reusing \fIks1\fR for the final encryption. \f(CW\*(C`C=E(ks1,D(ks2,E(ks1,M)))\*(C'\fR. This form of Triple-DES is used by the \s-1RSAREF\s0 library. .PP \&\fIDES_pcbc_encrypt()\fR encrypt/decrypts using the propagating cipher block chaining mode used by Kerberos v4. Its parameters are the same as \&\fIDES_ncbc_encrypt()\fR. .PP \&\fIDES_cfb_encrypt()\fR encrypt/decrypts using cipher feedback mode. This method takes an array of characters as input and outputs and array of characters. It does not require any padding to 8 character groups. Note: the \fIivec\fR variable is changed and the new changed value needs to be passed to the next call to this function. Since this function runs a complete \s-1DES ECB\s0 encryption per \fInumbits\fR, this function is only suggested for use when sending small numbers of characters. .PP \&\fIDES_cfb64_encrypt()\fR implements \s-1CFB\s0 mode of \s-1DES\s0 with 64bit feedback. Why is this useful you ask? Because this routine will allow you to encrypt an arbitrary number of bytes, no 8 byte padding. Each call to this routine will encrypt the input bytes to output and then update ivec and num. num contains 'how far' we are though ivec. If this does not make much sense, read more about cfb mode of \s-1DES :\-\s0). .PP \&\fIDES_ede3_cfb64_encrypt()\fR and \fIDES_ede2_cfb64_encrypt()\fR is the same as \&\fIDES_cfb64_encrypt()\fR except that Triple-DES is used. .PP \&\fIDES_ofb_encrypt()\fR encrypts using output feedback mode. This method takes an array of characters as input and outputs and array of characters. It does not require any padding to 8 character groups. Note: the \fIivec\fR variable is changed and the new changed value needs to be passed to the next call to this function. Since this function runs a complete \s-1DES ECB\s0 encryption per numbits, this function is only suggested for use when sending small numbers of characters. .PP \&\fIDES_ofb64_encrypt()\fR is the same as \fIDES_cfb64_encrypt()\fR using Output Feed Back mode. .PP \&\fIDES_ede3_ofb64_encrypt()\fR and \fIDES_ede2_ofb64_encrypt()\fR is the same as \&\fIDES_ofb64_encrypt()\fR, using Triple-DES. .PP The following functions are included in the \s-1DES\s0 library for compatibility with the \s-1MIT\s0 Kerberos library. .PP \&\fIDES_cbc_cksum()\fR produces an 8 byte checksum based on the input stream (via \s-1CBC\s0 encryption). The last 4 bytes of the checksum are returned and the complete 8 bytes are placed in \fIoutput\fR. This function is used by Kerberos v4. Other applications should use \&\fIEVP_DigestInit\fR\|(3) etc. instead. .PP \&\fIDES_quad_cksum()\fR is a Kerberos v4 function. It returns a 4 byte checksum from the input bytes. The algorithm can be iterated over the input, depending on \fIout_count\fR, 1, 2, 3 or 4 times. If \fIoutput\fR is non-NULL, the 8 bytes generated by each pass are written into \&\fIoutput\fR. .PP The following are DES-based transformations: .PP \&\fIDES_fcrypt()\fR is a fast version of the Unix \fIcrypt\fR\|(3) function. This version takes only a small amount of space relative to other fast \&\fIcrypt()\fR implementations. This is different to the normal crypt in that the third parameter is the buffer that the return value is written into. It needs to be at least 14 bytes long. This function is thread safe, unlike the normal crypt. .PP \&\fIDES_crypt()\fR is a faster replacement for the normal system \fIcrypt()\fR. This function calls \fIDES_fcrypt()\fR with a static array passed as the third parameter. This emulates the normal non-thread safe semantics of \fIcrypt\fR\|(3). .PP \&\fIDES_enc_write()\fR writes \fIlen\fR bytes to file descriptor \fIfd\fR from buffer \fIbuf\fR. The data is encrypted via \fIpcbc_encrypt\fR (default) using \fIsched\fR for the key and \fIiv\fR as a starting vector. The actual data send down \fIfd\fR consists of 4 bytes (in network byte order) containing the length of the following encrypted data. The encrypted data then follows, padded with random data out to a multiple of 8 bytes. .PP \&\fIDES_enc_read()\fR is used to read \fIlen\fR bytes from file descriptor \&\fIfd\fR into buffer \fIbuf\fR. The data being read from \fIfd\fR is assumed to have come from \fIDES_enc_write()\fR and is decrypted using \fIsched\fR for the key schedule and \fIiv\fR for the initial vector. .PP \&\fBWarning:\fR The data format used by \fIDES_enc_write()\fR and \fIDES_enc_read()\fR has a cryptographic weakness: When asked to write more than \s-1MAXWRITE\s0 bytes, \fIDES_enc_write()\fR will split the data into several chunks that are all encrypted using the same \s-1IV. \s0 So don't use these functions unless you are sure you know what you do (in which case you might not want to use them anyway). They cannot handle non-blocking sockets. \&\fIDES_enc_read()\fR uses an internal state and thus cannot be used on multiple files. .PP \&\fIDES_rw_mode\fR is used to specify the encryption mode to use with \&\fIDES_enc_read()\fR and \fIDES_end_write()\fR. If set to \fI\s-1DES_PCBC_MODE\s0\fR (the default), DES_pcbc_encrypt is used. If set to \fI\s-1DES_CBC_MODE\s0\fR DES_cbc_encrypt is used. .SH "NOTES" .IX Header "NOTES" Single-key \s-1DES\s0 is insecure due to its short key size. \s-1ECB\s0 mode is not suitable for most applications; see \fIdes_modes\fR\|(7). .PP The \fIevp\fR\|(3) library provides higher-level encryption functions. .SH "BUGS" .IX Header "BUGS" \&\fIDES_3cbc_encrypt()\fR is flawed and must not be used in applications. .PP \&\fIDES_cbc_encrypt()\fR does not modify \fBivec\fR; use \fIDES_ncbc_encrypt()\fR instead. .PP \&\fIDES_cfb_encrypt()\fR and \fIDES_ofb_encrypt()\fR operates on input of 8 bits. What this means is that if you set numbits to 12, and length to 2, the first 12 bits will come from the 1st input byte and the low half of the second input byte. The second 12 bits will have the low 8 bits taken from the 3rd input byte and the top 4 bits taken from the 4th input byte. The same holds for output. This function has been implemented this way because most people will be using a multiple of 8 and because once you get into pulling bytes input bytes apart things get ugly! .PP \&\fIDES_string_to_key()\fR is available for backward compatibility with the \&\s-1MIT\s0 library. New applications should use a cryptographic hash function. The same applies for \fIDES_string_to_2key()\fR. .SH "CONFORMING TO" .IX Header "CONFORMING TO" \&\s-1ANSI X3.106\s0 .PP The \fBdes\fR library was written to be source code compatible with the \s-1MIT\s0 Kerberos library. .SH "SEE ALSO" .IX Header "SEE ALSO" \&\fIcrypt\fR\|(3), \fIdes_modes\fR\|(7), \fIevp\fR\|(3), \fIrand\fR\|(3) .SH "HISTORY" .IX Header "HISTORY" In OpenSSL 0.9.7, all des_ functions were renamed to \s-1DES_\s0 to avoid clashes with older versions of libdes. Compatibility des_ functions are provided for a short while, as well as \fIcrypt()\fR. Declarations for these are in <openssl/des_old.h>. There is no \s-1DES_\s0 variant for \fIdes_random_seed()\fR. This will happen to other functions as well if they are deemed redundant (\fIdes_random_seed()\fR just calls \&\fIRAND_seed()\fR and is present for backward compatibility only), buggy or already scheduled for removal. .PP \&\fIdes_cbc_cksum()\fR, \fIdes_cbc_encrypt()\fR, \fIdes_ecb_encrypt()\fR, \&\fIdes_is_weak_key()\fR, \fIdes_key_sched()\fR, \fIdes_pcbc_encrypt()\fR, \&\fIdes_quad_cksum()\fR, \fIdes_random_key()\fR and \fIdes_string_to_key()\fR are available in the \s-1MIT\s0 Kerberos library; \&\fIdes_check_key_parity()\fR, \fIdes_fixup_key_parity()\fR and \fIdes_is_weak_key()\fR are available in newer versions of that library. .PP \&\fIdes_set_key_checked()\fR and \fIdes_set_key_unchecked()\fR were added in OpenSSL 0.9.5. .PP \&\fIdes_generate_random_block()\fR, \fIdes_init_random_number_generator()\fR, \&\fIdes_new_random_key()\fR, \fIdes_set_random_generator_seed()\fR and \&\fIdes_set_sequence_number()\fR and \fIdes_rand_data()\fR are used in newer versions of Kerberos but are not implemented here. .PP \&\fIdes_random_key()\fR generated cryptographically weak random data in SSLeay and in OpenSSL prior version 0.9.5, as well as in the original \&\s-1MIT\s0 library. .SH "AUTHOR" .IX Header "AUTHOR" Eric Young (eay@cryptsoft.com). Modified for the OpenSSL project (http://www.openssl.org).