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\input texinfo          @c -*-texinfo-*-

@c %**start of header
@setfilename nettle.info
@settitle The Nettle low-level cryptographic library.
@c %**end of header

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@footnotestyle end
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@syncodeindex fn cp

@dircategory GNU Libraries
@direntry
* Nettle: (nettle).           A low-level cryptographics library.
@end direntry

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@set UPDATED-FOR 1.12
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@c Latin-1 doesn't work with TeX output.
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@c Also lookout for é characters.

@set AUTHOR Niels Möller
@ifinfo
Draft manual for the Nettle library. This manual corresponds to version
@value{UPDATED-FOR}.

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Copyright 2001, 2004 @value{AUTHOR}
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Permission is granted to make and distribute verbatim
copies of this manual provided the copyright notice and
this permission notice are preserved on all copies.

@ignore
Permission is granted to process this file through TeX
and print the results, provided the printed document
carries a copying permission notice identical to this
one except for the removal of this paragraph (this
paragraph not being relevant to the printed manual).

@end ignore
Permission is granted to copy and distribute modified
versions of this manual under the conditions for
verbatim copying, provided also that the sections
entitled ``Copying'' and ``GNU General Public License''
are included exactly as in the original, and provided
that the entire resulting derived work is distributed
under the terms of a permission notice identical to this
one.

Permission is granted to copy and distribute
translations of this manual into another language,
under the above conditions for modified versions,
except that this permission notice may be stated in a
translation approved by the Free Software Foundation.

@end ifinfo

@titlepage
@sp 10
@c @center @titlefont{Nettle Manual}

@title Nettle Manual
@subtitle For the Nettle Library version @value{UPDATED-FOR}

@author @value{AUTHOR}

@c The following two commands start the copyright page.
@page
@vskip 0pt plus 1filll
Copyright @copyright{} 2001 @value{AUTHOR}

Permission is granted to make and distribute verbatim
copies of this manual provided the copyright notice and
this permission notice are preserved on all copies.

Permission is granted to copy and distribute modified
versions of this manual under the conditions for
verbatim copying, provided also that the sections
entitled ``Copying'' and ``GNU General Public License''
are included exactly as in the original, and provided
that the entire resulting derived work is distributed
under the terms of a permission notice identical to this
one.

Permission is granted to copy and distribute
translations of this manual into another language,
under the above conditions for modified versions,
except that this permission notice may be stated in a
translation approved by the Free Software Foundation.

@end titlepage

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@contents

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@ifnottex
@node     Top, Introduction, (dir), (dir)
@comment  node-name,  next,  previous,  up
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@top Nettle
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This document describes the nettle low-level cryptographic library. You
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can use the library directly from your C programs, or (recommended)
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write or use an object-oriented wrapper for your favorite language or
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application.

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This manual corresponds to version @value{UPDATED-FOR} of the library.
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@menu
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* Introduction::                What is Nettle?
* Copyright::                   Your rights.
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* Conventions::                 
* Example::                     
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* Reference::                   All Nettle functions and features.
* Nettle soup::                 For the serious nettle hacker.
* Installation::                How to install Nettle.
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* Index::                       
@end menu

@end ifnottex

@node Introduction, Copyright, Top, Top
@comment  node-name,  next,  previous,  up
@chapter Introduction

Nettle is a cryptographic library that is designed to fit easily in more
or less any context: In crypto toolkits for object-oriented languages
(C++, Python, Pike, ...), in applications like LSH or GNUPG, or even in
kernel space. In most contexts, you need more than the basic
cryptographic algorithms, you also need some way to keep track of available
algorithms, their properties and variants. You often have some algorithm
selection process, often dictated by a protocol you want to implement.

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And as the requirements of applications differ in subtle and not so
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subtle ways, an API that fits one application well can be a pain to use
in a different context. And that is why there are so many different
cryptographic libraries around.

Nettle tries to avoid this problem by doing one thing, the low-level
crypto stuff, and providing a @emph{simple} but general interface to it.
In particular, Nettle doesn't do algorithm selection. It doesn't do
memory allocation. It doesn't do any I/O.

The idea is that one can build several application and context specific
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interfaces on top of Nettle, and share the code, test cases, benchmarks,
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documentation, etc. For this first version, the only application using
Nettle is LSH, and it uses an object-oriented abstraction on top of the
library. 

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This manual explains how to use the Nettle library. It also tries to
provide some background on the cryptography, and advice on how to best
put it to use.

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@node Copyright, Conventions, Introduction, Top
@comment  node-name,  next,  previous,  up
@chapter Copyright

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Nettle is distributed under the GNU General Public License (GPL) (see
the file COPYING for details). However, most of the individual files
are dual licensed under less restrictive licenses like the GNU Lesser
General Public License (LGPL), or are in the public domain. This means
that if you don't use the parts of nettle that are GPL-only, you have
the option to use the Nettle library just as if it were licensed under
the LGPL. To find the current status of particular files, you have to
read the copyright notices at the top of the files.
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A list of the supported algorithms, their origins and licenses:

@table @emph
@item AES
The implementation of the AES cipher (also known as rijndael) is written
by Rafael Sevilla. Released under the LGPL.

@item ARCFOUR
The implementation of the ARCFOUR (also known as RC4) cipher is written
by Niels Möller. Released under the LGPL.

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@item ARCTWO
The implementation of the ARCTWO (also known as RC2) cipher is written
by Nikos Mavroyanopoulos and modified by Werner Koch and Simon
Josefsson. Released under the LGPL.

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@item BLOWFISH
The implementation of the BLOWFISH cipher is written by Werner Koch,
copyright owned by the Free Software Foundation. Also hacked by Ray
Dassen and Niels Möller. Released under the GPL.

@item CAST128
The implementation of the CAST128 cipher is written by Steve Reid.
Released into the public domain.

@item DES
The implementation of the DES cipher is written by Dana L. How, and
released under the LGPL.

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@item MD2
The implementation of MD2 is written by Andrew Kuchling, and hacked
some by Andreas Sigfridsson and Niels Möller. Python Cryptography
Toolkit license (essentially public domain).

@item MD4
This is almost the same code as for MD5 below, with modifications by
Marcus Comstedt. Released into the public domain.

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@item MD5
The implementation of the MD5 message digest is written by Colin Plumb.
It has been hacked some more by Andrew Kuchling and Niels Möller.
Released into the public domain.

@item SERPENT
The implementation of the SERPENT cipher is written by Ross Anderson,
Eli Biham, and Lars Knudsen, adapted to LSH by Rafael Sevilla, and to
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Nettle by Niels Möller. Released under the GPL.
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@item SHA1
The implementation of the SHA1 message digest is written by Peter
Gutmann, and hacked some more by Andrew Kuchling and Niels Möller.
Released into the public domain.

@item TWOFISH
The implementation of the TWOFISH cipher is written by Ruud de Rooij.
Released under the LGPL.
@end table

@node Conventions, Example, Copyright, Top
@comment  node-name,  next,  previous,  up
@chapter Conventions

For each supported algorithm, there is an include file that defines a
@emph{context struct}, a few constants, and declares functions for
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operating on the context. The context struct encapsulates all information
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needed by the algorithm, and it can be copied or moved in memory with no
unexpected effects.

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For consistency, functions for different algorithms are very similar,
but there are some differences, for instance reflecting if the key setup
or encryption function differ for encryption and encryption, and whether
or not key setup can fail. There are also differences between algorithms
that don't show in function prototypes, but which the application must
nevertheless be aware of. There is no big difference between the
functions for stream ciphers and for block ciphers, although they should
be used quite differently by the application.
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If your application uses more than one algorithm, you should probably
create an interface that is tailor-made for your needs, and then write a
few lines of glue code on top of Nettle.

By convention, for an algorithm named @code{foo}, the struct tag for the
context struct is @code{foo_ctx}, constants and functions uses prefixes
like @code{FOO_BLOCK_SIZE} (a constant) and @code{foo_set_key} (a
function).

In all functions, strings are represented with an explicit length, of
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type @code{unsigned}, and a pointer of type @code{uint8_t *} or
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@code{const uint8_t *}. For functions that transform one string to
another, the argument order is length, destination pointer and source
pointer. Source and destination areas are of the same length. Source and
destination may be the same, so that you can process strings in place,
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but they @emph{must not} overlap in any other way.
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@c FIXME: Say something about the name mangling.

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@node Example, Reference, Conventions, Top
@comment  node-name,  next,  previous,  up
@chapter Example

A simple example program that reads a file from standard in and writes
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its SHA1 checksum on standard output should give the flavor of Nettle.
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@example
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@verbatiminclude sha-example.c
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@end example

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@node Reference, Nettle soup, Example, Top
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@comment  node-name,  next,  previous,  up
@chapter Reference

This chapter describes all the Nettle functions, grouped by family.

@menu
* Hash functions::              
* Cipher functions::            
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* Cipher Block Chaining::       
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* Keyed hash functions::        
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* Public-key algorithms::       
* Randomness::                  
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* Miscellaneous functions::     
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* Compatibility functions::     
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@end menu

@node Hash functions, Cipher functions, Reference, Reference
@comment  node-name,  next,  previous,  up
@section Hash functions

A cryptographic @dfn{hash function} is a function that takes variable
size strings, and maps them to strings of fixed, short, length. There
are naturally lots of collisions, as there are more possible 1MB files
than 20 byte strings. But the function is constructed such that is hard
to find the collisions. More precisely, a cryptographic hash function
@code{H} should have the following properties:

@table @emph

@item One-way
Given a hash value @code{H(x)} it is hard to find a string @code{x}
that hashes to that value.

@item Collision-resistant
It is hard to find two different strings, @code{x} and @code{y}, such
that @code{H(x)} = @code{H(y)}.

@end table

Hash functions are useful as building blocks for digital signatures,
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message authentication codes, pseudo random generators, association of
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unique id:s to documents, and many other things.

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There are several fairly popular hash functions. Collisions have been
found the the compression function of MD4, and in variants of MD5.
These functions are therefore not recommended for new applications.
The recommended hash function for new applications is SHA1.

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@subsection @acronym{MD5}

MD5 is a message digest function constructed by Ronald Rivest, and
described in @cite{RFC 1321}. It outputs message digests of 128 bits, or
16 octets. Nettle defines MD5 in @file{<nettle/md5.h>}.

@deftp {Context struct} {struct md5_ctx}
@end deftp

@defvr Constant MD5_DIGEST_SIZE
The size of an MD5 digest, i.e. 16.
@end defvr

@defvr Constant MD5_DATA_SIZE
The internal block size of MD5. Useful for some special constructions,
in particular HMAC-MD5.
@end defvr

@deftypefun void md5_init (struct md5_ctx *@var{ctx})
Initialize the MD5 state.
@end deftypefun

@deftypefun void md5_update (struct md5_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{data})
Hash some more data.
@end deftypefun

@deftypefun void md5_digest (struct md5_ctx *@var{ctx}, unsigned @var{length}, uint8_t *@var{digest})
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Performs final processing and extracts the message digest, writing it
to @var{digest}. @var{length} may be smaller than
@code{MD5_DIGEST_SIZE}, in which case only the first @var{length}
octets of the digest are written.
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This function also resets the context in the same way as
@code{md5_init}.
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@end deftypefun

The normal way to use MD5 is to call the functions in order: First
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@code{md5_init}, then @code{md5_update} zero or more times, and finally
@code{md5_digest}. After @code{md5_digest}, the context is reset to
its initial state, so you can start over calling @code{md5_update} to
hash new data.
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To start over, you can call @code{md5_init} at any time.

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@subsection @acronym{MD2}

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MD2 is another hash function of Ronald Rivest's, described in
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@cite{RFC 1319}. It outputs message digests of 128 bits, or 16 octets.
Nettle defines MD2 in @file{<nettle/md2.h>}.

@deftp {Context struct} {struct md2_ctx}
@end deftp

@defvr Constant MD2_DIGEST_SIZE
The size of an MD2 digest, i.e. 16.
@end defvr

@defvr Constant MD2_DATA_SIZE
The internal block size of MD2.
@end defvr

@deftypefun void md2_init (struct md2_ctx *@var{ctx})
Initialize the MD2 state.
@end deftypefun

@deftypefun void md2_update (struct md2_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{data})
Hash some more data.
@end deftypefun

@deftypefun void md2_digest (struct md2_ctx *@var{ctx}, unsigned @var{length}, uint8_t *@var{digest})
Performs final processing and extracts the message digest, writing it
to @var{digest}. @var{length} may be smaller than
@code{MD2_DIGEST_SIZE}, in which case only the first @var{length}
octets of the digest are written.

This function also resets the context in the same way as
@code{md2_init}.
@end deftypefun

@subsection @acronym{MD4}

MD4 is a predecessor of MD5, described in @cite{RFC 1320}. Like MD5,
it is constructed by Ronald Rivest. It outputs message digests of 128
bits, or 16 octets. Nettle defines MD4 in @file{<nettle/md4.h>}.
Because of discovered weaknesses, use of MD4 is not recommended, but
it is sometimes needed for compatibility with existing applications
and protocols.

@deftp {Context struct} {struct md4_ctx}
@end deftp

@defvr Constant MD4_DIGEST_SIZE
The size of an MD4 digest, i.e. 16.
@end defvr

@defvr Constant MD4_DATA_SIZE
The internal block size of MD4.
@end defvr

@deftypefun void md4_init (struct md4_ctx *@var{ctx})
Initialize the MD4 state.
@end deftypefun

@deftypefun void md4_update (struct md4_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{data})
Hash some more data.
@end deftypefun

@deftypefun void md4_digest (struct md4_ctx *@var{ctx}, unsigned @var{length}, uint8_t *@var{digest})
Performs final processing and extracts the message digest, writing it
to @var{digest}. @var{length} may be smaller than
@code{MD4_DIGEST_SIZE}, in which case only the first @var{length}
octets of the digest are written.

This function also resets the context in the same way as
@code{md4_init}.
@end deftypefun

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@subsection @acronym{SHA1}

SHA1 is a hash function specified by @dfn{NIST} (The U.S. National Institute
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for Standards and Technology). It outputs hash values of 160 bits, or 20
octets. Nettle defines SHA1 in @file{<nettle/sha.h>}.
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The functions are analogous to the MD5 ones.

@deftp {Context struct} {struct sha1_ctx}
@end deftp

@defvr Constant SHA1_DIGEST_SIZE
The size of an SHA1 digest, i.e. 20.
@end defvr

@defvr Constant SHA1_DATA_SIZE
The internal block size of SHA1. Useful for some special constructions,
in particular HMAC-SHA1.
@end defvr

@deftypefun void sha1_init (struct sha1_ctx *@var{ctx})
Initialize the SHA1 state.
@end deftypefun

@deftypefun void sha1_update (struct sha1_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{data})
Hash some more data.
@end deftypefun

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@deftypefun void sha1_digest (struct sha1_ctx *@var{ctx}, unsigned @var{length}, uint8_t *@var{digest})
Performs final processing and extracts the message digest, writing it
to @var{digest}. @var{length} may be smaller than
@code{SHA1_DIGEST_SIZE}, in which case only the first @var{length}
octets of the digest are written.

This function also resets the context in the same way as
@code{sha1_init}.
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@end deftypefun

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@subsection @acronym{SHA256}

SHA256 is another hash function specified by @dfn{NIST}, intended as a
replacement for @acronym{SHA1}, generating larger digests. It outputs
hash values of 256 bits, or 32 octets. Nettle defines SHA256 in
@file{<nettle/sha.h>}.

The functions are analogous to the MD5 ones.

@deftp {Context struct} {struct sha256_ctx}
@end deftp

@defvr Constant SHA256_DIGEST_SIZE
The size of an SHA256 digest, i.e. 20.
@end defvr

@defvr Constant SHA256_DATA_SIZE
The internal block size of SHA256. Useful for some special constructions,
in particular HMAC-SHA256.
@end defvr

@deftypefun void sha256_init (struct sha256_ctx *@var{ctx})
Initialize the SHA256 state.
@end deftypefun

@deftypefun void sha256_update (struct sha256_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{data})
Hash some more data.
@end deftypefun

@deftypefun void sha256_digest (struct sha256_ctx *@var{ctx}, unsigned @var{length}, uint8_t *@var{digest})
Performs final processing and extracts the message digest, writing it
to @var{digest}. @var{length} may be smaller than
@code{SHA256_DIGEST_SIZE}, in which case only the first @var{length}
octets of the digest are written.
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This function also resets the context in the same way as
@code{sha256_init}.
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@end deftypefun

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@subsection @code{struct nettle_hash}

Nettle includes a struct including information about the supported hash
functions. It is defined in @file{<nettle/nettle-meta.h>}, and is used
by Nettle's implementation of @acronym{HMAC} @pxref{Keyed hash
functions}.

@deftp {Meta struct} @code{struct nettle_hash} name context_size digest_size block_size init update digest
The last three attributes are function pointers, of types
@code{nettle_hash_init_func}, @code{nettle_hash_update_func}, and
@code{nettle_hash_digest_func}. The first argument to these functions is
@code{void *} pointer so a context struct, which is of size
@code{context_size}. 
@end deftp

@deftypevr {Constant Struct} {struct nettle_cipher} nettle_md5
@deftypevrx {Constant Struct} {struct nettle_cipher} nettle_sha1
@deftypevrx {Constant Struct} {struct nettle_cipher} nettle_sha256

These are all the hash functions that Nettle implements.
@end deftypevr

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@node Cipher functions, Cipher Block Chaining, Hash functions, Reference
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@comment  node-name,  next,  previous,  up
@section Cipher functions

A @dfn{cipher} is a function that takes a message or @dfn{plaintext}
and a secret @dfn{key} and transforms it to a @dfn{ciphertext}. Given
only the ciphertext, but not the key, it should be hard to find the
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plaintext. Given matching pairs of plaintext and ciphertext, it should
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be hard to find the key.

There are two main classes of ciphers: Block ciphers and stream ciphers.

A block cipher can process data only in fixed size chunks, called
@dfn{blocks}. Typical block sizes are 8 or 16 octets. To encrypt
arbitrary messages, you usually have to pad it to an integral number of
blocks, split it into blocks, and then process each block. The simplest
way is to process one block at a time, independent of each other. That
mode of operation is called @dfn{ECB}, Electronic Code Book mode.
However, using ECB is usually a bad idea. For a start, plaintext blocks
that are equal are transformed to ciphertext blocks that are equal; that
leaks information about the plaintext. Usually you should apply the
cipher is some feedback mode, @dfn{CBC} (Cipher Block Chaining) being one
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of the most popular. @xref{Cipher Block Chaining}, for information on
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how to apply @acronym{CBC} with Nettle.
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A stream cipher can be used for messages of arbitrary length; a typical
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stream cipher is a keyed pseudo-random generator. To encrypt a plaintext
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message of @var{n} octets, you key the generator, generate @var{n}
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octets of pseudo-random data, and XOR it with the plaintext. To decrypt,
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regenerate the same stream using the key, XOR it to the ciphertext, and
the plaintext is recovered.

@strong{Caution:} The first rule for this kind of cipher is the
same as for a One Time Pad: @emph{never} ever use the same key twice.

A common misconception is that encryption, by itself, implies
authentication. Say that you and a friend share a secret key, and you
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receive an encrypted message. You apply the key, and get a plaintext
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message that makes sense to you. Can you then be sure that it really was
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your friend that wrote the message you're reading? The answer is no. For
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example, if you were using a block cipher in ECB mode, an attacker may
pick up the message on its way, and reorder, delete or repeat some of
the blocks. Even if the attacker can't decrypt the message, he can
change it so that you are not reading the same message as your friend
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wrote. If you are using a block cipher in @acronym{CBC} mode rather than
ECB, or are using a stream cipher, the possibilities for this sort of
attack are different, but the attacker can still make predictable
changes to the message.
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It is recommended to @emph{always} use an authentication mechanism in
addition to encrypting the messages. Popular choices are Message
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Authentication Codes like @acronym{HMAC-SHA1} @pxref{Keyed hash
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functions}, or digital signatures like @acronym{RSA}.
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Some ciphers have so called ``weak keys'', keys that results in
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undesirable structure after the key setup processing, and should be
avoided. In Nettle, the presence of weak keys for a cipher mean that the
key setup function can fail, so you have to check its return value. In
addition, the context struct has a field @code{status}, that is set to a
non-zero value if key setup fails. When possible, avoid algorithm that
have weak keys. There are several good ciphers that don't have any weak
keys.

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To encrypt a message, you first initialize a cipher context for
encryption or decryption with a particular key. You then use the context
to process plaintext or ciphertext messages. The initialization is known
as called @dfn{key setup}. With Nettle, it is recommended to use each
context struct for only one direction, even if some of the ciphers use a
single key setup function that can be used for both encryption and
decryption.

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@subsection AES
AES is a quite new block cipher, specified by NIST as a replacement for
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the older DES standard. The standard is the result of a competition
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between cipher designers. The winning design, also known as RIJNDAEL,
was constructed by Joan Daemen and Vincent Rijnmen.
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Like all the AES candidates, the winning design uses a block size of 128
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bits, or 16 octets, and variable key-size, 128, 192 and 256 bits (16, 24
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and 32 octets) being the allowed key sizes. It does not have any weak
keys. Nettle defines AES in @file{<nettle/aes.h>}.
 
@deftp {Context struct} {struct aes_ctx}
@end deftp

@defvr Constant AES_BLOCK_SIZE
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The AES block-size, 16
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@end defvr

@defvr Constant AES_MIN_KEY_SIZE
@end defvr

@defvr Constant AES_MAX_KEY_SIZE
@end defvr

@defvr Constant AES_KEY_SIZE
Default AES key size, 32
@end defvr

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@deftypefun void aes_set_encrypt_key (struct aes_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{key})
@deftypefunx void aes_set_decrypt_key (struct aes_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{key})
Initialize the cipher, for encryption or decryption, respectively.
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@end deftypefun

@deftypefun void aes_encrypt (struct aes_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{dst}, uint8_t *@var{src})
Encryption function. @var{length} must be an integral multiple of the
block size. If it is more than one block, the data is processed in ECB
mode. @code{src} and @code{dst} may be equal, but they must not overlap
in any other way.
@end deftypefun

@deftypefun void aes_decrypt (struct aes_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{dst}, uint8_t *@var{src})
Analogous to @code{aes_encrypt}
@end deftypefun

@subsection ARCFOUR
ARCFOUR is a stream cipher, also known under the trade marked name RC4,
and it is one of the fastest ciphers around. A problem is that the key
setup of ARCFOUR is quite weak, you should never use keys with
structure, keys that are ordinary passwords, or sequences of keys like
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``secret:1'', ``secret:2'', @enddots{}. If you have keys that don't look
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like random bit strings, and you want to use ARCFOUR, always hash the
key before feeding it to ARCFOUR. For example

@example
/* A more robust key setup function for ARCFOUR */
void
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arcfour_set_key_hashed(struct arcfour_ctx *ctx,
                       unsigned length, const uint8_t *key)
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@{
  struct sha1_ctx hash;
  uint8_t digest[SHA1_DIGEST_SIZE];

  sha1_init(&hash);
  sha1_update(&hash, length, key);
  sha1_digest(&hash, SHA1_DIGEST_SIZE, digest);

  arcfour_set_key(ctx, SHA1_DIGEST_SIZE, digest);
@}
@end example

Nettle defines ARCFOUR in @file{<nettle/arcfour.h>}.

@deftp {Context struct} {struct arcfour_ctx}
@end deftp

@defvr Constant ARCFOUR_MIN_KEY_SIZE
Minimum key size, 1
@end defvr

@defvr Constant ARCFOUR_MAX_KEY_SIZE
Maximum key size, 256
@end defvr

@defvr Constant ARCFOUR_KEY_SIZE
Default ARCFOUR key size, 16
@end defvr

@deftypefun void arcfour_set_key (struct arcfour_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{key})
Initialize the cipher. The same function is used for both encryption and
decryption. 
@end deftypefun

@deftypefun void arcfour_crypt (struct arcfour_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{key})
Encrypt some data. The same function is used for both encryption and
decryption. Unlike the block ciphers, this function modifies the
context, so you can split the data into arbitrary chunks and encrypt
them one after another. The result is the same as if you had called
@code{arcfour_crypt} only once with all the data.
@end deftypefun

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@subsection ARCTWO
ARCTWO (also known as the trade marked name RC2) is a block cipher
specified in RFC 2268. Nettle also include a variation of the ARCTWO
set key operation that lack one step, to be compatible with the
reverse engineered RRC2 cipher description, as described in a Usenet
post to @code{sci.crypt} by Peter Gutmann.

ARCTWO uses a block size of 64 bits, and variable key-size ranging
from 1 to 128 octets. Besides the key, ARCTWO also has a second
parameter to key setup, the number of effective key bits, @code{ekb}.
This parameter can be used to artificially reduce the key size. In
practice, @code{ekb} is usually set equal to the input key size.
Nettle defines ARCTWO in @file{<nettle/arctwo.h>}.

We do not recommend the use of ARCTWO; the Nettle implementation is
provided primarily for interoperability with existing applications and
standards.

@deftp {Context struct} {struct arctwo_ctx}
@end deftp

@defvr Constant ARCTWO_BLOCK_SIZE
The AES block-size, 8
@end defvr

@defvr Constant ARCTWO_MIN_KEY_SIZE
@end defvr

@defvr Constant ARCTWO_MAX_KEY_SIZE
@end defvr

@defvr Constant ARCTWO_KEY_SIZE
Default ARCTWO key size, 8
@end defvr

@deftypefun void arctwo_set_key_ekb (struct arctwo_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{key}, unsigned @var{ekb})
@deftypefunx void arctwo_set_key (struct arctwo_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{key})
@deftypefunx void arctwo_set_key_gutmann (struct arctwo_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{key})
Initialize the cipher. The same function is used for both encryption
and decryption. The first function is the most general one, which lets
you provide both the variable size key, and the desired effective key
size (in bits). The maximum value for @var{ekb} is 1024, and for
convenience, @code{ekb = 0} has the same effect as @code{ekb = 1024}.

@code{arctwo_set_key(ctx, length, key)} is equivalent to
@code{arctwo_set_key_ekb(ctx, length, key, 8*length)}, and
@code{arctwo_set_key_gutmann(ctx, length, key)} is equivalent to
@code{arctwo_set_key_ekb(ctx, length, key, 1024)}
@end deftypefun

@deftypefun void arctwo_encrypt (struct arctwo_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{dst}, uint8_t *@var{src})
Encryption function. @var{length} must be an integral multiple of the
block size. If it is more than one block, the data is processed in ECB
mode. @code{src} and @code{dst} may be equal, but they must not
overlap in any other way.
@end deftypefun

@deftypefun void arctwo_decrypt (struct arctwo_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{dst}, uint8_t *@var{src})
Analogous to @code{arctwo_encrypt}
@end deftypefun

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@subsection CAST128

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CAST-128 is a block cipher, specified in @cite{RFC 2144}. It uses a 64
bit (8 octets) block size, and a variable key size of up to 128 bits.
Nettle defines cast128 in @file{<nettle/cast128.h>}.

@deftp {Context struct} {struct cast128_ctx}
@end deftp

@defvr Constant CAST128_BLOCK_SIZE
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The CAST128 block-size, 8
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@end defvr

@defvr Constant CAST128_MIN_KEY_SIZE
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Minimum CAST128 key size, 5
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@end defvr

@defvr Constant CAST128_MAX_KEY_SIZE
Maximum CAST128 key size, 16
@end defvr

@defvr Constant CAST128_KEY_SIZE
Default CAST128 key size, 16
@end defvr

@deftypefun void cast128_set_key (struct cast128_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{key})
Initialize the cipher. The same function is used for both encryption and
decryption. 
@end deftypefun

@deftypefun void cast128_encrypt (struct cast128_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{dst}, uint8_t *@var{src})
Encryption function. @var{length} must be an integral multiple of the
block size. If it is more than one block, the data is processed in ECB
mode. @code{src} and @code{dst} may be equal, but they must not overlap
in any other way.
@end deftypefun

@deftypefun void cast128_decrypt (struct cast128_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{dst}, uint8_t *@var{src})
Analogous to @code{cast128_encrypt}
@end deftypefun

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@subsection BLOWFISH

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BLOWFISH is a block cipher designed by Bruce Schneier. It uses a block
size of 64 bits (8 octets), and a variable key size, up to 448 bits. It
has some weak keys. Nettle defines BLOWFISH in @file{<nettle/blowfish.h>}.

@deftp {Context struct} {struct blowfish_ctx}
@end deftp

@defvr Constant BLOWFISH_BLOCK_SIZE
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The BLOWFISH block-size, 8
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@end defvr

@defvr Constant BLOWFISH_MIN_KEY_SIZE
Minimum BLOWFISH key size, 8
@end defvr

@defvr Constant BLOWFISH_MAX_KEY_SIZE
Maximum BLOWFISH key size, 56
@end defvr

@defvr Constant BLOWFISH_KEY_SIZE
Default BLOWFISH key size, 16
@end defvr

@deftypefun int blowfish_set_key (struct blowfish_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{key})
Initialize the cipher. The same function is used for both encryption and
decryption. Returns 1 on success, and 0 if the key was weak. Calling
@code{blowfish_encrypt} or @code{blowfish_decrypt} with a weak key will
crash with an assert violation.
@end deftypefun

@deftypefun void blowfish_encrypt (struct blowfish_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{dst}, uint8_t *@var{src})
Encryption function. @var{length} must be an integral multiple of the
block size. If it is more than one block, the data is processed in ECB
mode. @code{src} and @code{dst} may be equal, but they must not overlap
in any other way.
@end deftypefun

@deftypefun void blowfish_decrypt (struct blowfish_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{dst}, uint8_t *@var{src})
Analogous to @code{blowfish_encrypt}
@end deftypefun

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@subsection DES
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DES is the old Data Encryption Standard, specified by NIST. It uses a
block size of 64 bits (8 octets), and a key size of 56 bits. However,
the key bits are distributed over 8 octets, where the least significant
bit of each octet is used for parity. A common way to use DES is to
generate 8 random octets in some way, then set the least significant bit
of each octet to get odd parity, and initialize DES with the resulting
key.

The key size of DES is so small that keys can be found by brute force,
using specialized hardware or lots of ordinary work stations in
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parallel. One shouldn't be using plain DES at all today, if one uses
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DES at all one should be using DES3 or ``triple DES'', see below.
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DES also has some weak keys. Nettle defines DES in @file{<nettle/des.h>}.

@deftp {Context struct} {struct des_ctx}
@end deftp

@defvr Constant DES_BLOCK_SIZE
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The DES block-size, 8
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@end defvr

@defvr Constant DES_KEY_SIZE
DES key size, 8
@end defvr

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@deftypefun int des_set_key (struct des_ctx *@var{ctx}, const uint8_t *@var{key})
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Initialize the cipher. The same function is used for both encryption and
decryption. Returns 1 on success, and 0 if the key was weak or had bad
parity. Calling @code{des_encrypt} or @code{des_decrypt} with a bad key
will crash with an assert violation.
@end deftypefun

@deftypefun void des_encrypt (struct des_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{dst}, uint8_t *@var{src})
Encryption function. @var{length} must be an integral multiple of the
block size. If it is more than one block, the data is processed in ECB
mode. @code{src} and @code{dst} may be equal, but they must not overlap
in any other way.
@end deftypefun

@deftypefun void des_decrypt (struct des_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{dst}, uint8_t *@var{src})
Analogous to @code{des_encrypt}
@end deftypefun
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@deftypefun void des_fix_parity (unsigned @var{length}, uint8_t *@var{dst}, const uint8_t *@var{src})
Adjusts the parity bits to match DES's requirements. You need this
function if you have created a random-looking string by a key agreement
protocol, and want to use it as a DES key. @var{dst} and @var{src} may
be equal.
@end deftypefun

@subsection DES3
The inadequate key size of DES has already been mentioned. One way to
increase the key size is to pipe together several DES boxes with
independent keys. It turns out that using two DES ciphers is not as
secure as one might think, even if the key size of the combination is a
respectable 112 bits.

The standard way to increase DES's key size is to use three DES boxes.
The mode of operation is a little peculiar: the middle DES box is wired
in the reverse direction. To encrypt a block with DES3, you encrypt it
using the first 56 bits of the key, then @emph{decrypt} it using the
middle 56 bits of the key, and finally encrypt it again using the last
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56 bits of the key. This is known as ``ede'' triple-DES, for
``encrypt-decrypt-encrypt''.
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The ``ede'' construction provides some backward compatibility, as you get
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plain single DES simply by feeding the same key to all three boxes. That
should help keeping down the gate count, and the price, of hardware
circuits implementing both plain DES and DES3.

DES3 has a key size of 168 bits, but just like plain DES, useless parity
bits are inserted, so that keys are represented as 24 octets (192 bits).
As a 112 bit key is large enough to make brute force attacks
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impractical, some applications uses a ``two-key'' variant of triple-DES.
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In this mode, the same key bits are used for the first and the last DES
box in the pipe, while the middle box is keyed independently. The
two-key variant is believed to be secure, i.e. there are no known
attacks significantly better than brute force.

Naturally, it's simple to implement triple-DES on top of Nettle's DES
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functions. Nettle includes an implementation of three-key ``ede''
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triple-DES, it is defined in the same place as plain DES,
@file{<nettle/des.h>}.

@deftp {Context struct} {struct des3_ctx}
@end deftp

@defvr Constant DES3_BLOCK_SIZE
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The DES3 block-size is the same as DES_BLOCK_SIZE, 8
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@end defvr

@defvr Constant DES3_KEY_SIZE
DES key size, 24
@end defvr

@deftypefun int des3_set_key (struct des3_ctx *@var{ctx}, const uint8_t *@var{key})
Initialize the cipher. The same function is used for both encryption and
decryption. Returns 1 on success, and 0 if the key was weak or had bad
parity. Calling @code{des_encrypt} or @code{des_decrypt} with a bad key
will crash with an assert violation.
@end deftypefun

For random-looking strings, you can use @code{des_fix_parity} to adjust
the parity bits before calling @code{des3_set_key}.

@deftypefun void des3_encrypt (struct des3_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{dst}, uint8_t *@var{src})
Encryption function. @var{length} must be an integral multiple of the
block size. If it is more than one block, the data is processed in ECB
mode. @code{src} and @code{dst} may be equal, but they must not overlap
in any other way.
@end deftypefun

@deftypefun void des3_decrypt (struct des3_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{dst}, uint8_t *@var{src})
Analogous to @code{des_encrypt}
@end deftypefun

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@subsection SERPENT
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SERPENT is one of the AES finalists, designed by Ross Anderson, Eli
Biham and Lars Knudsen. Thus, the interface and properties are similar
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to AES'. One peculiarity is that it is quite pointless to use it with
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anything but the maximum key size, smaller keys are just padded to
larger ones. Nettle defines SERPENT in @file{<nettle/serpent.h>}.

@deftp {Context struct} {struct serpent_ctx}
@end deftp

@defvr Constant SERPENT_BLOCK_SIZE
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The SERPENT block-size, 16
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@end defvr

@defvr Constant SERPENT_MIN_KEY_SIZE
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Minimum SERPENT key size, 16
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@end defvr

@defvr Constant SERPENT_MAX_KEY_SIZE
Maximum SERPENT key size, 32
@end defvr

@defvr Constant SERPENT_KEY_SIZE
Default SERPENT key size, 32
@end defvr

@deftypefun void serpent_set_key (struct serpent_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{key})
Initialize the cipher. The same function is used for both encryption and
decryption. 
@end deftypefun

@deftypefun void serpent_encrypt (struct serpent_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{dst}, uint8_t *@var{src})
Encryption function. @var{length} must be an integral multiple of the
block size. If it is more than one block, the data is processed in ECB
mode. @code{src} and @code{dst} may be equal, but they must not overlap
in any other way.
@end deftypefun

@deftypefun void serpent_decrypt (struct serpent_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{dst}, uint8_t *@var{src})
Analogous to @code{serpent_encrypt}
@end deftypefun

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@subsection TWOFISH
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Another AES finalist, this one designed by Bruce Schneier and others.
Nettle defines it in @file{<nettle/twofish.h>}.

@deftp {Context struct} {struct twofish_ctx}
@end deftp

@defvr Constant TWOFISH_BLOCK_SIZE
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The TWOFISH block-size, 16
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1024
@end defvr

@defvr Constant TWOFISH_MIN_KEY_SIZE
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Minimum TWOFISH key size, 16
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@end defvr

@defvr Constant TWOFISH_MAX_KEY_SIZE
Maximum TWOFISH key size, 32
@end defvr

@defvr Constant TWOFISH_KEY_SIZE
Default TWOFISH key size, 32
@end defvr

@deftypefun void twofish_set_key (struct twofish_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{key})
Initialize the cipher. The same function is used for both encryption and
decryption. 
@end deftypefun

@deftypefun void twofish_encrypt (struct twofish_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{dst}, uint8_t *@var{src})
Encryption function. @var{length} must be an integral multiple of the
block size. If it is more than one block, the data is processed in ECB
mode. @code{src} and @code{dst} may be equal, but they must not overlap
in any other way.
@end deftypefun

@deftypefun void twofish_decrypt (struct twofish_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{dst}, uint8_t *@var{src})
Analogous to @code{twofish_encrypt}
@end deftypefun

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@c @node nettle_cipher, Cipher Block Chaining, Cipher functions, Reference
@c @comment  node-name,  next,  previous,  up
@subsection @code{struct nettle_cipher}

Nettle includes a struct including information about some of the more
regular cipher functions. It should be considered a little experimental,
but can be useful for applications that need a simple way to handle
various algorithms. Nettle defines these structs in
@file{<nettle/nettle-meta.h>}. 

@deftp {Meta struct} @code{struct nettle_cipher} name context_size block_size key_size set_encrypt_key set_decrypt_key encrypt decrypt
The last four attributes are function pointers, of types
@code{nettle_set_key_func} and @code{nettle_crypt_func}. The first
argument to these functions is a @code{void *} pointer to a context
struct, which is of size @code{context_size}.
@end deftp

@deftypevr {Constant Struct} {struct nettle_cipher} nettle_aes128
@deftypevrx {Constant Struct} {struct nettle_cipher} nettle_aes192
@deftypevrx {Constant Struct} {struct nettle_cipher} nettle_aes256

@deftypevrx {Constant Struct} {struct nettle_cipher} nettle_arcfour128
@deftypevrx {Constant Struct} {struct nettle_cipher} nettle_cast128

@deftypevrx {Constant Struct} {struct nettle_cipher} nettle_serpent128
@deftypevrx {Constant Struct} {struct nettle_cipher} nettle_serpent192
@deftypevrx {Constant Struct} {struct nettle_cipher} nettle_serpent256

@deftypevrx {Constant Struct} {struct nettle_cipher} nettle_twofish128
@deftypevrx {Constant Struct} {struct nettle_cipher} nettle_twofish192
@deftypevrx {Constant Struct} {struct nettle_cipher} nettle_twofish256

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@deftypevrx {Constant Struct} {struct nettle_cipher} nettle_arctwo40;
@deftypevrx {Constant Struct} {struct nettle_cipher} nettle_arctwo64;
@deftypevrx {Constant Struct} {struct nettle_cipher} nettle_arctwo128;
@deftypevrx {Constant Struct} {struct nettle_cipher} nettle_arctwo_gutmann128;

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Nettle includes such structs for all the @emph{regular} ciphers, i.e.
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ones without weak keys or other oddities.
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@end deftypevr

@node Cipher Block Chaining, Keyed hash functions, Cipher functions, Reference
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@comment  node-name,  next,  previous,  up
@section Cipher Block Chaining

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When using @acronym{CBC} mode, plaintext blocks are not encrypted
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independently of each other, like in Electronic Cook Book mode. Instead,
when encrypting a block in @acronym{CBC} mode, the previous ciphertext
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block is XOR:ed with the plaintext before it is fed to the block cipher.
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When encrypting the first block, a random block called an @dfn{IV}, or
Initialization Vector, is used as the ``previous ciphertext block''. The
IV should be chosen randomly, but it need not be kept secret, and can
even be transmitted in the clear together with the encrypted data.
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In symbols, if @code{E_k} is the encryption function of a block cipher,
and @code{IV} is the initialization vector, then @code{n} plaintext blocks
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@code{M_1},@dots{} @code{M_n} are transformed into @code{n} ciphertext blocks
@code{C_1},@dots{} @code{C_n} as follows:

@example
C_1 = E_k(IV  XOR M_1)
C_2 = E_k(C_1 XOR M_2)

@dots{}

C_n = E_k(C_(n-1) XOR M_n)
@end example

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Nettle includes a few utility functions for applying a block cipher in
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Cipher Block Chaining (@acronym{CBC}) mode. The functions uses @code{void *} to
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pass cipher contexts around.

@deftypefun {void} cbc_encrypt (void *@var{ctx}, void (*@var{f})(), unsigned @var{block_size}, uint8_t *@var{iv}, unsigned @var{length}, uint8_t *@var{dst}, const uint8_t *@var{src})
@deftypefunx {void} cbc_decrypt (void *@var{ctx}, void (*@var{f})(), unsigned @var{block_size}, uint8_t *@var{iv}, unsigned @var{length}, uint8_t *@var{dst}, const uint8_t *@var{src})

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Applies the encryption or decryption function @var{f} in @acronym{CBC}
mode. The function @var{f} is really typed as
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@code{void f (void *@var{ctx}, unsigned @var{length}, uint8_t @var{dst},
const uint8_t *@var{src})},

@noindent and the @code{cbc_encrypt} and @code{cbc_decrypt} functions pass their
argument @var{ctx} on to @var{f}.
@end deftypefun
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There are also some macros to help use these functions correctly.
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@deffn Macro CBC_CTX (@var{context_type}, @var{block_size})
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Expands into
@example
@{
   context_type ctx;
   uint8_t iv[block_size];
@}
@end example
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@end deffn

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It can be used to define a @acronym{CBC} context struct, either directly,
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@example
struct CBC_CTX(struct aes_ctx, AES_BLOCK_SIZE) ctx;
@end example

or to give it a struct tag,

@example
struct aes_cbc_ctx CBC_CTX (struct aes_ctx, AES_BLOCK_SIZE);
@end example

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@deffn Macro CBC_SET_IV (@var{ctx}, @var{iv})
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First argument is a pointer to a context struct as defined by @code{CBC_CTX},
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and the second is a pointer to an Initialization Vector (IV) that is
copied into that context.
@end deffn
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@deffn Macro CBC_ENCRYPT (@var{ctx}, @var{f}, @var{length}, @var{dst}, @var{src})
@deffnx Macro CBC_DECRYPT (@var{ctx}, @var{f}, @var{length}, @var{dst}, @var{src})
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A simpler way to invoke @code{cbc_encrypt} and @code{cbc_decrypt}. The
first argument is a pointer to a context struct as defined by
@code{CBC_CTX}, and the second argument is an encryption or decryption
function following Nettle's conventions. The last three arguments define
the source and destination area for the operation.
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@end deffn
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These macros use some tricks to make the compiler display a warning if
the types of @var{f} and @var{ctx} don't match, e.g. if you try to use
an @code{struct aes_ctx} context with the @code{des_encrypt} function.
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@node Keyed hash functions, Public-key algorithms, Cipher Block Chaining, Reference
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@comment  node-name,  next,  previous,  up
@section Keyed Hash Functions

A @dfn{keyed hash function}, or @dfn{Message Authentication Code}
(@acronym{MAC}) is a function that takes a key and a message, and
produces fixed size @acronym{MAC}. It should be hard to compute a
message and a matching @acronym{MAC} without knowledge of the key. It
should also be hard to compute the key given only messages and
corresponding @acronym{MAC}s.

Keyed hash functions are useful primarily for message authentication,
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when Alice and Bob shares a secret: The sender, Alice, computes the
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@acronym{MAC} and attaches it to the message. The receiver, Bob, also computes
the @acronym{MAC} of the message, using the same key, and compares that
to Alice's value. If they match, Bob can be assured that
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the message has not been modified on its way from Alice.
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However, unlike digital signatures, this assurance is not transferable.
Bob can't show the message and the @acronym{MAC} to a third party and
prove that Alice sent that message. Not even if he gives away the key to
the third party. The reason is that the @emph{same} key is used on both
sides, and anyone knowing the key can create a correct @acronym{MAC} for
any message. If Bob believes that only he and Alice knows the key, and
he knows that he didn't attach a @acronym{MAC} to a particular message,
he knows it must be Alice who did it. However, the third party can't
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distinguish between a @acronym{MAC} created by Alice and one created by
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Bob.

Keyed hash functions are typically a lot faster than digital signatures
as well.

@subsection @acronym{HMAC}

One can build keyed hash functions from ordinary hash functions. Older
constructions simply concatenate secret key and message and hashes that, but
such constructions have weaknesses. A better construction is
@acronym{HMAC}, described in @cite{RFC 2104}.

For an underlying hash function @code{H}, with digest size @code{l} and
internal block size @code{b}, @acronym{HMAC-H} is constructed as
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follows: From a given key @code{k}, two distinct subkeys @code{k_i} and
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@code{k_o} are constructed, both of length @code{b}. The
@acronym{HMAC-H} of a message @code{m} is then computed as @code{H(k_o |
H(k_i | m))}, where @code{|} denotes string concatenation.

@acronym{HMAC} keys can be of any length, but it is recommended to use
keys of length @code{l}, the digest size of the underlying hash function
@code{H}. Keys that are longer than @code{b} are shortened to length
@code{l} by hashing with @code{H}, so arbitrarily long keys aren't
very useful. 

Nettle's @acronym{HMAC} functions are defined in @file{<nettle/hmac.h>}.
There are abstract functions that use a pointer to a @code{struct
nettle_hash} to represent the underlying hash function and @code{void
*} pointers that point to three different context structs for that hash
function. There are also concrete functions for @acronym{HMAC-MD5},
@acronym{HMAC-SHA1}, and @acronym{HMAC-SHA256}. First, the abstract
functions:

@deftypefun void hmac_set_key (void *@var{outer}, void *@var{inner}, void *@var{state}, const struct nettle_hash *@var{H}, unsigned @var{length}, const uint8_t *@var{key})
Initializes the three context structs from the key. The @var{outer} and
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@var{inner} contexts corresponds to the subkeys @code{k_o} and
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@code{k_i}. @var{state} is used for hashing the message, and is
initialized as a copy of the @var{inner} context.
@end deftypefun

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@deftypefun void hmac_update (void *@var{state}, const struct nettle_hash *@var{H}, unsigned @var{length}, const uint8_t *@var{data})
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This function is called zero or more times to process the message.
Actually, @code{hmac_update(state, H, length, data)} is equivalent to
@code{H->update(state, length, data)}, so if you wish you can use the
ordinary update function of the underlying hash function instead.
@end deftypefun

@deftypefun void hmac_digest (const void *@var{outer}, const void *@var{inner}, void *@var{state}, const struct nettle_hash *@var{H}, unsigned @var{length}, uint8_t *@var{digest})
Extracts the @acronym{MAC} of the message, writing it to @var{digest}.
@var{outer} and @var{inner} are not modified. @var{length} is usually
equal to @code{H->digest_size}, but if you provide a smaller value,
only the first @var{length} octets of the @acronym{MAC} are written.

This function also resets the @var{state} context so that you can start
over processing a new message (with the same key).
@end deftypefun

Like for @acronym{CBC}, there are some macros to help use these
functions correctly.

@deffn Macro HMAC_CTX (@var{type})
Expands into
@example
@{
   type outer;
   type inner;
   type state;
@}
@end example
@end deffn

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It can be used to define a @acronym{HMAC} context struct, either
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directly,

@example
struct HMAC_CTX(struct md5_ctx) ctx;
@end example

or to give it a struct tag,

@example
struct hmac_md5_ctx HMAC_CTX (struct md5_ctx);
@end example

@deffn Macro HMAC_SET_KEY (@var{ctx}, @var{H}, @var{length}, @var{key})
@var{ctx} is a pointer to a context struct as defined by
@code{HMAC_CTX}, @var{H} is a pointer to a @code{const struct
nettle_hash} describing the underlying hash function (so it must match
the type of the components of @var{ctx}). The last two arguments specify
the secret key.
@end deffn

@deffn Macro HMAC_DIGEST (@var{ctx}, @var{H}, @var{length}, @var{digest})
@var{ctx} is a pointer to a context struct as defined by
@code{HMAC_CTX}, @var{H} is a pointer to a @code{const struct
nettle_hash} describing the underlying hash function. The last two
arguments specify where the digest is written.
@end deffn

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Note that there is no @code{HMAC_UPDATE} macro; simply call
@code{hmac_update} function directly, or the update function of the
underlying hash function.
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@subsection Concrete @acronym{HMAC} functions
Now we come to the specialized @acronym{HMAC} functions, which are
easier to use than the general @acronym{HMAC} functions.

@subsubsection @acronym{HMAC-MD5}

@deftp {Context struct} {struct hmac_md5_ctx}
@end deftp

@deftypefun void hmac_md5_set_key (struct hmac_md5_ctx *@var{ctx}, unsigned @var{key_length}, const uint8_t *@var{key})
Initializes the context with the key.
@end deftypefun

@deftypefun void hmac_md5_update (struct hmac_md5_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{data})
Process some more data.
@end deftypefun

@deftypefun void hmac_md5_digest (struct hmac_md5_ctx *@var{ctx}, unsigned @var{length}, uint8_t *@var{digest})
Extracts the @acronym{MAC}, writing it to @var{digest}. @var{length} may be smaller than
@code{MD5_DIGEST_SIZE}, in which case only the first @var{length}
octets of the @acronym{MAC} are written.

This function also resets the context for processing new messages, with
the same key.
@end deftypefun

@subsubsection @acronym{HMAC-SHA1}

@deftp {Context struct} {struct hmac_sha1_ctx}
@end deftp

@deftypefun void hmac_sha1_set_key (struct hmac_sha1_ctx *@var{ctx}, unsigned @var{key_length}, const uint8_t *@var{key})
Initializes the context with the key.
@end deftypefun

@deftypefun void hmac_sha1_update (struct hmac_sha1_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{data})
Process some more data.
@end deftypefun

@deftypefun void hmac_sha1_digest (struct hmac_sha1_ctx *@var{ctx}, unsigned @var{length}, uint8_t *@var{digest})
Extracts the @acronym{MAC}, writing it to @var{digest}. @var{length} may be smaller than
@code{SHA1_DIGEST_SIZE}, in which case only the first @var{length}
octets of the @acronym{MAC} are written.

This function also resets the context for processing new messages, with
the same key.
@end deftypefun


@subsubsection @acronym{HMAC-SHA256}

@deftp {Context struct} {struct hmac_sha256_ctx}
@end deftp

@deftypefun void hmac_sha256_set_key (struct hmac_sha256_ctx *@var{ctx}, unsigned @var{key_length}, const uint8_t *@var{key})
Initializes the context with the key.
@end deftypefun

@deftypefun void hmac_sha256_update (struct hmac_sha256_ctx *@var{ctx}, unsigned @var{length}, const uint8_t *@var{data})
Process some more data.
@end deftypefun

@deftypefun void hmac_sha256_digest (struct hmac_sha256_ctx *@var{ctx}, unsigned @var{length}, uint8_t *@var{digest})
Extracts the @acronym{MAC}, writing it to @var{digest}. @var{length} may be smaller than
@code{SHA256_DIGEST_SIZE}, in which case only the first @var{length}
octets of the @acronym{MAC} are written.

This function also resets the context for processing new messages, with
the same key.
@end deftypefun

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@node Public-key algorithms, Randomness, Keyed hash functions, Reference
@comment  node-name,  next,  previous,  up
@section Public-key algorithms

Nettle uses @acronym{GMP}, the GNU bignum library, for all calculations
with large numbers. In order to use the public-key features of Nettle,
you must install @acronym{GMP}, at least version 3.0, before compiling
Nettle, and you need to link your programs with @code{-lgmp}.

The concept of @dfn{Public-key} encryption and digital signatures was
discovered by Whitfield Diffie and Martin E. Hellman and described in a
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paper 1976. In traditional, ``symmetric'', cryptography, sender and
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receiver share the same keys, and these keys must be distributed in a
secure way. And if there are many users or entities that need to
communicate, each @emph{pair} needs a shared secret key known by nobody
else.

Public-key cryptography uses trapdoor one-way functions. A
@dfn{one-way function} is a function @code{F} such that it is easy to
compute the value @code{F(x)} for any @code{x}, but given a value
@code{y}, it is hard to compute a corresponding @code{x} such that
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@code{y = F(x)}. Two examples are cryptographic hash functions, and
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exponentiation in certain groups.

A @dfn{trapdoor one-way function} is a function @code{F} that is
one-way, unless one knows some secret information about @code{F}. If one
knows the secret, it is easy to compute both @code{F} and it's inverse.
If this sounds strange, look at the @acronym{RSA} example below.

Two important uses for one-way functions with trapdoors are public-key
encryption, and digital signatures. Of these, I won't say more about
public-key encryption, as that isn't yet supported by Nettle. So the
rest of this chapter is about digital signatures.

To use a digital signature algorithm, one must first create a
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@dfn{key-pair}: A public key and a corresponding private key. The private
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key is used to sign messages, while the public key is used for verifying
that that signatures and messages match. Some care must be taken when
distributing the public key; it need not be kept secret, but if a bad
guy is able to replace it (in transit, or in some user's list of known
public keys), bad things may happen.

There are two operations one can do with the keys. The signature
operation takes a message and a private key, and creates a signature for
the message. A signature is some string of bits, usually at most a few
thousand bits or a few hundred octets. Unlike paper-and-ink signatures,
the digital signature depends on the message, so one can't cut it out of
context and glue it to a different message.

The verification operation takes a public key, a message, and a string
that is claimed to be a signature on the message, and returns true or
false. If it returns true, that means that the three input values
matched, and the verifier can be sure that someone went through with the
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signature operation on that very message, and that the ``someone'' also
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knows the private key corresponding to the public key.

The desired properties of a digital signature algorithm are as follows:
Given the public key and pairs of messages and valid signatures on them,
it should be hard to compute the private key, and it should also be hard
to create a new message and signature that is accepted by the
verification operation.

Besides signing meaningful messages, digital signatures can be used for
authorization. A server can be configured with a public key, such that
any client that connects to the service is given a random nonce message.
If the server gets a reply with a correct signature matching the nonce
message and the configured public key, the client is granted access. So
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the configuration of the server can be understood as ``grant access to
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whoever knows the private key corresponding to this particular public
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key, and to no others''.
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@menu
* RSA::                         The RSA public key algorithm.
* DSA::                         The DSA digital signature algorithm.
@end menu

@node RSA, DSA, Public-key algorithms, Public-key algorithms
@comment  node-name,  next,  previous,  up
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@subsection @acronym{RSA}

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The @acronym{RSA} algorithm was the first practical digital signature
algorithm that was constructed. It was described 1978 in a paper by
Ronald Rivest, Adi Shamir and L.M. Adleman, and the technique was also
patented in 1983. The patent expired on September 20, 2000, and since
that day, @acronym{RSA} can be used freely.
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It's remarkably simple to describe the trapdoor function behind
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@acronym{RSA}. The ``one-way''-function used is
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@example
F(x) = x^e mod n
@end example

I.e. raise x to the @code{e}:th power, while discarding all multiples of
@code{n}. The pair of numbers @code{n} and @code{e} is the public key.
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@code{e} can be quite small, even @code{e = 3} has been used, although
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slightly larger numbers are recommended. @code{n} should be about 1000
bits or larger.

If @code{n} is large enough, and properly chosen, the inverse of F,
the computation of @code{e}:th roots modulo @code{n}, is very difficult.
But, where's the trapdoor?

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Let's first look at how @acronym{RSA} key-pairs are generated. First
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@code{n} is chosen as the product of two large prime numbers @code{p}
and @code{q} of roughly the same size (so if @code{n} is 1000 bits,
@code{p} and @code{q} are about 500 bits each). One also computes the
number @code{phi = (p-1)(q-1)}, in mathematical speak, phi is the order
of the multiplicative group of integers modulo n.

Next, @code{e} is chosen. It must have no factors in common with phi (in
particular, it must be odd), but can otherwise be chosen more or less
randomly. @code{e = 65537} is a popular choice, because it makes raising
to the @code{e}:th power particularly efficient, and being prime, it
usually has no factors common with @code{phi}.

Finally, a number @code{d}, @code{d < n} is computed such that @code{e d
mod phi = 1}. It can be shown that such a number exists (this is why
@code{e} and @code{phi} must have no common factors), and that for all x,

@example
(x^e)^d mod n = x^(ed) mod n = (x^d)^e mod n = x
@end example

Using Euclid's algorithm, @code{d} can be computed quite easily from
@code{phi} and @code{e}. But it is still hard to get @code{d} without
knowing @code{phi}, which depends on the factorization of @code{n}.

So @code{d} is the trapdoor, if we know @code{d} and @code{y = F(x)}, we can
recover x as @code{y^d mod n}. @code{d} is also the private half of
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the @acronym{RSA} key-pair.
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The most common signature operation for @acronym{RSA} is defined in
@cite{PKCS#1}, a specification by RSA Laboratories. The message to be
signed is first hashed using a cryptographic hash function, e.g.
@acronym{MD5} or @acronym{SHA1}. Next, some padding, the @acronym{ASN.1}
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``Algorithm Identifier'' for the hash function, and the message digest
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itself, are concatenated and converted to a number @code{x}. The
signature is computed from @code{x} and the private key as @code{s = x^d
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mod n}@footnote{Actually, the computation is not done like this, it is
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done more efficiently using @code{p}, @code{q} and the Chinese remainder
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theorem (@acronym{CRT}). But the result is the same.}. The signature, @code{s} is a
number of about the same size of @code{n}, and it usually encoded as a
sequence of octets, most significant octet first.

The verification operation is straight-forward, @code{x} is computed
from the message in the same way as above. Then @code{s^e mod n} is
computed, the operation returns true if and only if the result equals
@code{x}.

@subsection Nettle's @acronym{RSA} support

Nettle represents @acronym{RSA} keys using two structures that contain
large numbers (of type @code{mpz_t}).

@deftp {Context struct} {rsa_public_key} size n e
@code{size} is the size, in octets, of the modulo, and is used internally.
@code{n} and @code{e} is the public key.
@end deftp

@deftp {Context struct} {rsa_private_key} size d p q a b c
@code{size} is the size, in octets, of the modulo, and is used internally.
@code{d} is the secret exponent, but it is not actually used when
signing. Instead, the factors @code{p} and @code{q}, and the parameters
@code{a}, @code{b} and @code{c} are used. They are computed from @code{p},
@code{q} and @code{d} such that @code{a e mod (p - 1) = 1, b e mod (q -
1) = 1, c q mod p= 1}.
@end deftp

Before use, these structs must be initialized by calling one of

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@deftypefun void rsa_public_key_init (struct rsa_public_key *@var{pub})
@deftypefunx void rsa_private_key_init (struct rsa_private_key *@var{key})
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Calls @code{mpz_init} on all numbers in the key struct.
@end deftypefun

and when finished with them, the space for the numbers must be
deallocated by calling one of

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@deftypefun void rsa_public_key_clear (struct rsa_public_key *@var{pub})
@deftypefunx void rsa_private_key_clear (struct rsa_private_key *@var{key})
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Calls @code{mpz_clear} on all numbers in the key struct.
@end deftypefun

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In general, Nettle's @acronym{RSA} functions deviates from Nettle's ``no
memory allocation''-policy. Space for all the numbers, both in the key structs
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above, and temporaries, are allocated dynamically. For information on how
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to customize allocation, see
@xref{Custom Allocation,,GMP Allocation,gmp, GMP Manual}.
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When you have assigned values to the attributes of a key, you must call
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@deftypefun int rsa_public_key_prepare (struct rsa_public_key *@var{pub})
@deftypefunx int rsa_private_key_prepare (struct rsa_private_key *@var{key})
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Computes the octet size of the key (stored in the @code{size} attribute,
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and may also do other basic sanity checks. Returns one if successful, or
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zero if the key can't be used, for instance if the modulo is smaller
than the minimum size specified by PKCS#1.
@end deftypefun

Before signing or verifying a message, you first hash it with the
appropriate hash function. You pass the hash function's context struct
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to the @acronym{RSA} signature function, and it will extract the message
digest and do the rest of the work. There are also alternative functions
that take the @acronym{MD5} or @acronym{SHA1} hash digest as argument.
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Creation and verification of signatures is done with the following functions:

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@deftypefun void rsa_md5_sign (const struct rsa_private_key *@var{key}, struct md5_ctx *@var{hash}, mpz_t @var{signature})
@deftypefunx void rsa_sha1_sign (const struct rsa_private_key *@var{key}, struct sha1_ctx *@var{hash}, mpz_t @var{signature})
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The signature is stored in @var{signature} (which must have been
@code{mpz_init}:ed earlier). The hash context is reset so that it can be
used for new messages.
@end deftypefun

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@deftypefun void rsa_md5_sign_digest (const struct rsa_private_key *@var{key}, const uint8_t *@var{digest}, mpz_t @var{signature})
@deftypefunx void rsa_sha1_sign_digest (const struct rsa_private_key *@var{key}, const uint8_t *@var{digest}, mpz_t @var{signature});
Creates a signature from the given hash digest. @var{digest} should
point to a digest of size @code{MD5_DIGEST_SIZE} or
@code{SHA1_DIGEST_SIZE}, respectively. The signature is stored in
@var{signature} (which must have been @code{mpz_init}:ed earlier)
@end deftypefun

@deftypefun int rsa_md5_verify (const struct rsa_public_key *@var{key}, struct md5_ctx *@var{hash}, const mpz_t @var{signature})
@deftypefunx int rsa_sha1_verify (const struct rsa_public_key *@var{key}, struct sha1_ctx *@var{hash}, const mpz_t @var{signature})
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Returns 1 if the signature is valid, or 0 if it isn't. In either case,
the hash context is reset so that it can be used for new messages.
@end deftypefun

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@deftypefun int rsa_md5_verify_digest (const struct rsa_public_key *@var{key}, const uint8_t *@var{digest}, const mpz_t @var{signature})
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@deftypefunx int rsa_sha1_verify_digest (const struct rsa_public_key *@var{key}, const uint8_t *@var{digest}, const mpz_t @var{signature})
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Returns 1 if the signature is valid, or 0 if it isn't. @var{digest} should
point to a digest of size @code{MD5_DIGEST_SIZE} or
@code{SHA1_DIGEST_SIZE}, respectively.
@end deftypefun

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If you need to use the @acronym{RSA} trapdoor, the private key, in a way
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that isn't supported by the above functions Nettle also includes a
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function that computes @code{x^d mod n} and nothing more, using the
@acronym{CRT} optimization.

@deftypefun void rsa_compute_root (struct rsa_private_key *@var{key}, mpz_t @var{x}, const mpz_t @var{m})
Computes @code{x = m^d}, efficiently.
@end deftypefun

At last, how do you create new keys?

@deftypefun int rsa_generate_keypair (struct rsa_public_key *@var{pub}, struct rsa_private_key *@var{key}, void *@var{random_ctx}, nettle_random_func @var{random}, void *@var{progress_ctx}, nettle_progress_func @var{progress}, unsigned @var{n_size}, unsigned @var{e_size});
There are lots of parameters. @var{pub} and @var{key} is where the
resulting key pair is stored. The structs should be initialized, but you
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don't need to call @code{rsa_public_key_prepare} or
@code{rsa_private_key_prepare} after key generation.
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@var{random_ctx} and @var{random} is a randomness generator.
@code{random(random_ctx, length, dst)} should generate @code{length}
random octets and store them at @code{dst}. For advice, see
@xref{Randomness}.

@var{progress} and @var{progress_ctx} can be used to get callbacks
during the key generation process, in order to uphold an illusion of
progress. @var{progress} can be NULL, in that case there are no
callbacks.

@var{size_n} is the desired size of the modulo, in bits. If @var{size_e}
is non-zero, it is the desired size of the public exponent and a random
exponent of that size is selected. But if @var{e_size} is zero, it is
assumed that the caller has already chosen a value for @code{e}, and
stored it in @var{pub}.
Returns 1 on success, and 0 on failure. The function can fail for
example if if @var{n_size} is too small, or if @var{e_size} is zero and
@code{pub->e} is an even number.
@end deftypefun

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@node DSA,  , RSA, Public-key algorithms
@comment  node-name,  next,  previous,  up
@subsection Nettle's @acronym{DSA} support

The @acronym{DSA} digital signature algorithm is more complex than
@acronym{RSA}. It was specified during the early 1990s, and in 1994 NIST
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published @acronym{FIPS} 186 which is the authoritative specification.
Sometimes @acronym{DSA} is referred to using the acronym @acronym{DSS},
for Digital Signature Standard.
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For @acronym{DSA}, the underlying mathematical problem is the
computation of discreet logarithms. The public key consists of a large
prime @code{p}, a small prime @code{q} which is a factor of @code{p-1},
a number @code{g} which generates a subgroup of order @code{q} modulo
@code{p}, and an element @code{y} in that subgroup.

The size of @code{q} is fixed to 160 bits, to match with the
@acronym{SHA1} hash algorithm which is used in @acronym{DSA}. The size
of @code{q} is in principle unlimited, but the standard specifies only
nine specific sizes: @code{512 + l*64}, where @code{l} is between 0 and
8. Thus, the maximum size of @code{p} is 1024 bits, at that is also the
recommended size.

The subgroup requirement means that if you compute 

@example
g^t mod p
@end example

for all possible integers @code{t}, you will get precisely @code{q}
distinct values.

The private key is a secret exponent @code{x}, such that

@example
g^x = y mod p
@end example

In mathematical speak, @code{x} is the @dfn{discrete logarithm} of
@code{y} mod @code{p}, with respect to the generator @code{d}. The size
of @code{x} will also be about 160 bits.

The signature generation algorithm is randomized; in order to create a
@acronym{DSA} signature, you need a good source for random numbers
(@pxref{Randomness}).

To create a signature, one starts with the hash digest of the message,
@code{h}, which is a 160 bit number, and a random number @code{k,
0<k<q}, also 160 bits. Next, one computes 

@example
r = (g^k mod p) mod q
s = k^-1 (h + x r) mod q
@end example

The signature is the pair @code{(r, s)}, two 160 bit numbers. Note the
two different mod operations when computing @code{r}, and the use of the
secret exponent @code{x}.

To verify a signature, one first checks that @code{0 < r,s < q}, and
then one computes backwards,

@example
w = s^-1 mod q
v = (g^(w h) y^(w r) mod p) mod q
@end example

The signature is valid if @code{v = r}. This works out because @code{w =
s^-1 mod q = k (h + x r)^-1 mod q}, so that

@example
g^(w h) y^(w r) = g^(w h) (g^x)^(w r) = g^(w (h + x r)) = g^k 
@end example

When reducing mod @code{q} this yields @code{r}. Note that when
verifying a signature, we don't know either @code{k} or @code{x}: those
numbers are secret.

If you can choose between @acronym{RSA} and @acronym{DSA}, which one is
best? Both are believed to be secure. @acronym{DSA} gained popularity
in the late 1990s, as a patent free alternative to @acronym{RSA}. Now
that the @acronym{RSA} patents have expired, there's no compelling
reason to want to use @acronym{DSA}.

@acronym{DSA} signatures are smaller than @acronym{RSA} signatures,
which is important for some specialized applications.

From a practical point of view, @acronym{DSA}'s need for a good
randomness source is a serious disadvantage. If you ever use the same
@code{k} (and @code{r}) for two different message, you leak your private
key.

@subsection Nettle's @acronym{DSA} support

Like for @acronym{RSA}, Nettle represents @acronym{DSA} keys using two
structures, containing values of type @code{mpz_t}. For information on
how to customize allocation, see @xref{Custom Allocation,,GMP
Allocation,gmp, GMP Manual}.

Most of the @acronym{DSA} functions are very similar to the
corresponding @acronym{RSA} functions, but there are a few differences
pointed out below. For a start, there are no functions corresponding to
@code{rsa_public_key_prepare} and @code{rsa_private_key_prepare}.

@deftp {Context struct} {dsa_public_key} p q g y
The public parameters described above.
@end deftp

@deftp {Context struct} {dsa_private_key} x
The private key @code{x}.
@end deftp

Before use, these structs must be initialized by calling one of

@deftypefun void dsa_public_key_init (struct dsa_public_key *@var{pub})
@deftypefunx void dsa_private_key_init (struct dsa_private_key *@var{key})
Calls @code{mpz_init} on all numbers in the key struct.
@end deftypefun

When finished with them, the space for the numbers must be
deallocated by calling one of

@deftypefun void dsa_public_key_clear (struct dsa_public_key *@var{pub})
@deftypefunx void dsa_private_key_clear (struct dsa_private_key *@var{key})
Calls @code{mpz_clear} on all numbers in the key struct.
@end deftypefun

Signatures are represented using the structure below, and need to be
initialized and cleared in the same way as the key structs.

@deftp {Context struct} {dsa_signature} r s
@end deftp

@deftypefun void dsa_signature_init (struct dsa_signature *@var{signature})
@deftypefunx void dsa_signature_clear (struct dsa_signature *@var{signature})
You must call @code{dsa_signature_init} before creating or using a
signature, and call @code{dsa_signature_clear} when you are finished
with it.
@end deftypefun

For signing, you need to provide both the public and the private key
(unlike @acronym{RSA}, where the private key struct includes all
information needed for signing), and a source for random numbers.
Signatures always use the @acronym{SHA1} hash function.

@deftypefun void dsa_sign (const struct dsa_public_key *@var{pub}, const struct dsa_private_key *@var{key}, void *@var{random_ctx}, nettle_random_func @var{random}, struct sha1_ctx *@var{hash}, struct dsa_signature *@var{signature})
@deftypefunx void dsa_sign_digest (const struct dsa_public_key *@var{pub}, const struct dsa_private_key *@var{key}, void *@var{random_ctx}, nettle_random_func @var{random}, const uint8_t *@var{digest}, struct dsa_signature *@var{signature})
Creates a signature from the given hash context or digest.
@var{random_ctx} and @var{random} is a randomness generator.
@code{random(random_ctx, length, dst)} should generate @code{length}
random octets and store them at @code{dst}. For advice, see
@xref{Randomness}.
@end deftypefun

Verifying signatures is a little easier, since no randomness generator is
needed. The functions are

@deftypefun int dsa_verify (const struct dsa_public_key *@var{key}, struct sha1_ctx *@var{hash}, const struct dsa_signature *@var{signature})
@deftypefunx int dsa_verify_digest (const struct dsa_public_key *@var{key}, const uint8_t *@var{digest}, const struct dsa_signature *@var{signature})
Verifies a signature. Returns 1 if the signature is valid, otherwise 0.
@end deftypefun

Key generation uses mostly the same parameters as the corresponding
@acronym{RSA} function.

@deftypefun int dsa_generate_keypair (struct dsa_public_key *@var{pub}, struct dsa_private_key *@var{key}, void *@var{random_ctx}, nettle_random_func @var{random}, void *@var{progress_ctx}, nettle_progress_func @var{progress}, unsigned @var{bits})
@var{pub} and @var{key} is where the resulting key pair is stored. The
structs should be initialized before you call this function.

@var{random_ctx} and @var{random} is a randomness generator.
@code{random(random_ctx, length, dst)} should generate @code{length}
random octets and store them at @code{dst}. For advice, see
@xref{Randomness}.

@var{progress} and @var{progress_ctx} can be used to get callbacks
during the key generation process, in order to uphold an illusion of
progress. @var{progress} can be NULL, in that case there are no
callbacks.

@var{bits} is the desired size of @code{p}, in bits. To generate keys
that conform to the standard, you must use a value of the form @code{512
+ l*64}, for @code{0 <= l <= 8}. Keys smaller than 768 bits are not
considered secure, so you should probably stick to 1024. Non-standard
sizes are possible, in particular sizes larger than 1024 bits, although
@acronym{DSA} implementations can not in general be expected to support
such keys. Also note that using very large keys doesn't make much sense,
because the security is also limited by the size of the smaller prime
@code{q}, which is always 160 bits.

Returns 1 on success, and 0 on failure. The function will fail if
@var{bits} is too small.
@end deftypefun

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@node Randomness, Miscellaneous functions, Public-key algorithms, Reference
@comment  node-name,  next,  previous,  up
@section Randomness

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A crucial ingredient in many cryptographic contexts is randomness: Let
@code{p} be a random prime, choose a random initialization vector
@code{iv}, a random key @code{k} and a random exponent @code{e}, etc. In
the theories, it is assumed that you have plenty of randomness around.
If this assumption is not true in practice, systems that are otherwise
perfectly secure, can be broken. Randomness has often turned out to be
the weakest link in the chain.

In non-cryptographic applications, such as games as well as scientific
simulation, a good randomness generator usually means a generator that
has good statistical properties, and is seeded by some simple function
of things like the current time, process id, and host name.

However, such a generator is inadequate for cryptography, for at least
two reasons:

@itemize

@item
It's too easy for an attacker to guess the initial seed. Even if it will
take some 2^32 tries before he guesses right, that's far too easy. For
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example, if the process id is 16 bits, the resolution of ``current time''
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is one second, and the attacker knows what day the generator was seeded,
there are only about 2^32 possibilities to try if all possible values
for the process id and time-of-day are tried.

@item
The generator output reveals too much. By observing only a small segment
of the generator's output, its internal state can be recovered, and from
there, all previous output and all future output can be computed by the
attacker. 
@end itemize

A randomness generator that is used for cryptographic purposes must have
better properties. Let's first look at the seeding, as the issues here
are mostly independent of the rest of the generator. The initial state
of the generator (its seed) must be unguessable by the attacker. So
what's unguessable? It depends on what the attacker already knows. The
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concept used in information theory to reason about such things is called
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``entropy'', or ``conditional entropy'' (not to be confused with the
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thermodynamic concept with the same name). A reasonable requirement is
that the seed contains a conditional entropy of at least some 80-100
bits. This property can be explained as follows: Allow the attacker to
ask @code{n} yes-no-questions, of his own choice, about the seed. If
the attacker, using this question-and-answer session, as well as any
other information he knows about the seeding process, still can't gue