Skip to content
GitLab
Menu
Projects
Groups
Snippets
Loading...
Help
Help
Support
Community forum
Keyboard shortcuts
?
Submit feedback
Contribute to GitLab
Sign in / Register
Toggle navigation
Menu
Open sidebar
LSH
lsh
Commits
8e2381c5
Commit
8e2381c5
authored
Jan 22, 2002
by
Niels Möller
Browse files
(Public-key algorithms): New chapter.
Rev: src/nettle/nettle.texinfo:1.11
parent
b4721b22
Changes
1
Hide whitespace changes
Inline
Side-by-side
src/nettle/nettle.texinfo
View file @
8e2381c5
...
...
@@ -5,6 +5,7 @@
@settitle The Nettle low-level cryptographic library.
@c
%**end of header
@footnotestyle end
@syncodeindex fn cp
@dircategory GNU Libraries
...
...
@@ -305,6 +306,8 @@ This chapter describes all the Nettle functions, grouped by family.
* Cipher functions::
* Cipher Block Chaining::
* Keyed hash functions::
* Public-key algorithms::
* Randomness::
* Miscellaneous functions::
* Compatibility functions::
@end menu
...
...
@@ -531,7 +534,7 @@ changes to the message.
It is recommended to @emph
{
always
}
use an authentication mechanism in
addition to encrypting the messages. Popular choices are Message
Authetication Codes like @acronym
{
HMAC-SHA1
}
@pxref
{
Keyed hash
functions
}
, or digital signatures like @acronym
{
rsa
}
.
functions
}
, or digital signatures like @acronym
{
RSA
}
.
Some ciphers have so called "weak keys", keys that results in
undesirable structure after the key setup processing, and should be
...
...
@@ -606,8 +609,8 @@ key before feeding it to ARCFOUR. For example
@example
/* A more robust key setup function for ARCFOUR */
void
my
_
arcfour
_
set
_
key(struct arcfour
_
ctx *ctx,
unsigned length, const uint8
_
t *key)
arcfour
_
set
_
key
_
hashed
(struct arcfour
_
ctx *ctx,
unsigned length, const uint8
_
t *key)
@
{
struct sha1
_
ctx hash;
uint8
_
t digest[SHA1
_
DIGEST
_
SIZE];
...
...
@@ -1058,7 +1061,7 @@ 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.
@node Keyed hash functions,
Miscellaneous function
s, Cipher Block Chaining, Reference
@node Keyed hash functions,
Public-key algorithm
s, Cipher Block Chaining, Reference
@comment node-name, next, previous, up
@section Keyed Hash Functions
...
...
@@ -1258,9 +1261,263 @@ This function also resets the context for processing new messages, with
the same key.
@end deftypefun
@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
paper 1976. In traditional, "symmetric", cryptography, sender and
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
@code
{
y = f(x)
}
. Two examples are cryptographic hash functions, and
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
@dfn
{
keypair
}
: A public key and a corresponding private key. The private
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
signature operation on that very message, and that the "someone" also
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
the configuration of the server can be understood as "grant access to
whoever knows the private key corresponding to this particular public
key, and to no others".
@subsection @acronym
{
RSA
}
The @acronym
{
RSA
}
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.
It's remarkably simple to describe trapdoor function behind
@acronym
{
RSA
}
. The "one-way"-function used is
@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.
@code
{
e
}
can be quite small, even @code
{
e = 3
}
have been used, although
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?
Let's first look at how @acronym
{
RSA
}
keypairs are generated. First
@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
the @acronym
{
RSA
}
keypair.
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
}
"Algorithm Identifier" for the hash function, and the message digest
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
mod n
}
@footnote
{
Actuelly, the computation is not done like this, it is
done more efficiently using @code
{
p
}
, @code
{
q
}
and the chinese remainder
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
@deftypefun void rsa
_
init
_
public
_
key (struct rsa
_
public
_
key *@var
{
pub
}
)
@deftypefunx void rsa
_
init
_
private
_
key (struct rsa
_
private
_
key *@var
{
key
}
)
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
@deftypefun void rsa
_
clear
_
public
_
key (struct rsa
_
public
_
key *@var
{
pub
}
)
@deftypefunx void rsa
_
clear
_
private
_
key (struct rsa
_
private
_
key *@var
{
key
}
)
Calls @code
{
mpz
_
clear
}
on all numbers in the key struct.
@end deftypefun
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
above, and temporaries, are allocated dynamically. For informationon how
to customize allocation, see @xref
{
Custom Allocation,,GMP Allocation,gmp
}
.
When you have assigned values to the attributes of a key, you must call
@node Miscellaneous functions, Compatibility functions, Keyed hash functions, Reference
@deftypefun int rsa
_
prepare
_
public
_
key (struct rsa
_
public
_
key *@var
{
pub
}
)
@deftypefunx int rsa
_
prepare
_
private
_
key (struct rsa
_
private
_
key *@var
{
key
}
)
Computes the octet size of the key (stored in the @code
{
size
}
attribute,
and may also do other basig sanity checks. Returns one if successful, or
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
to the rsa function, and it will extract the message digest and do the
rest of the work.
Creation and verification of signatures is done with the following functions:
@deftypefun void rsa
_
md5
_
sign (struct rsa
_
private
_
key *@var
{
key
}
, struct md5
_
ctx *@var
{
hash
}
, mpz
_
t @var
{
signature
}
)
@deftypefunx void rsa
_
sha1
_
sign (struct rsa
_
private
_
key *@var
{
key
}
, struct sha1
_
ctx *@var
{
hash
}
, mpz
_
t @var
{
signature
}
)
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
@deftypefun int rsa
_
md5
_
verify (struct rsa
_
public
_
key *@var
{
key
}
, struct md5
_
ctx *@var
{
hash
}
, const mpz
_
t @var
{
signature
}
)
@deftypefunx int rsa
_
sha1
_
verify (struct rsa
_
public
_
key *@var
{
key
}
, struct sha1
_
ctx *@var
{
hash
}
, const mpz
_
t @var
{
signature
}
)
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
If you need to use the @acronym
{
RSA
}
trapdoor, the private key, in a way
that isn't support by the above functions Nettle also includes a
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
don't need to call @code
{
rsa
_
prepare
_
public
_
key
}
or
@code
{
rsa
_
prepare
_
private
_
key
}
after key generation.
@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
@node Randomness, Miscellaneous functions, Public-key algorithms, Reference
@comment node-name, next, previous, up
@section Randomness
XXX Not yet written.
@node Miscellaneous functions, Compatibility functions, Randomness, Reference
@comment node-name, next, previous, up
@section Miscellaneous functions
...
...
@@ -1306,13 +1563,22 @@ For the serious nettle hacker, here is a recipe for nettlesoup.
4 servings
@itemize @w
@item
1 litre fresh nettles (urtica dioica)
@item
2 tablespoons butter
@item
3 tablespoons flour
@item
1 litre buillon (meat or vegetable)
@item
1/2 teaspoon salt
@item
a tad white pepper
@item
some cream or milk
@end itemize
Gather 1 ltre fresh nettles. Use gloves! Small, tender shoots are
preferable but the tops of larger nettles can also be used.
...
...
@@ -1329,6 +1595,7 @@ Season with salt and pepper.
Serve with boiled egghalves.
@c And the original Swedish version.
@ignore
Recept på nässelsoppa
4 portioner
...
...
@@ -1351,8 +1618,6 @@ sp
smaksätt med salt och peppar.
Servera med kokta ägghalvor.
@ignore
@end ignore
@node Installation, Index, Nettle soup, Top
...
...
Write
Preview
Markdown
is supported
0%
Try again
or
attach a new file
.
Attach a file
Cancel
You are about to add
0
people
to the discussion. Proceed with caution.
Finish editing this message first!
Cancel
Please
register
or
sign in
to comment