path: root/lib/crypto/doc/src/crypto.xml

<?xml version="1.0" encoding="iso-8859-1" ?>
<!DOCTYPE erlref SYSTEM "erlref.dtd">

<erlref>
  <header>
    <copyright>
      <year>1999</year><year>2013</year>
      <holder>Ericsson AB. All Rights Reserved.</holder>
    </copyright>
    <legalnotice>
      The contents of this file are subject to the Erlang Public License,
      Version 1.1, (the "License"); you may not use this file except in
      compliance with the License. You should have received a copy of the
      Erlang Public License along with this software. If not, it can be
      retrieved online at http://www.erlang.org/.

      Software distributed under the License is distributed on an "AS IS"
      basis, WITHOUT WARRANTY OF ANY KIND, either express or implied. See
      the License for the specific language governing rights and limitations
      under the License.

    </legalnotice>

    <title>crypto</title>
  </header>
  <module>crypto</module>
  <modulesummary>Crypto Functions</modulesummary>
  <description>
    <p>This module provides a set of cryptographic functions.
      </p>
    <list type="bulleted">
      <item>
	<p>Hash functions - <url href="http://www.ietf.org/rfc/rfc1320.txt">The MD4 Message Digest Algorithm (RFC 1320)</url>,
	<url href="http://www.ietf.org/rfc/rfc1321.txt"> The MD5 Message Digest Algorithm (RFC 1321)</url> and
	<url href="http://csrc.nist.gov/publications/fips/fips180-4/fips-180-4.pdf"> Secure Hash Standard </url>
	</p>
      </item>
      <item>
        <p>Hmac functions - <url href="http://www.ietf.org/rfc/rfc2104.txt"> Keyed-Hashing for Message Authentication (RFC 2104) </url></p>
      </item>
      <item>
        <p>Block ciphers - <url href="http://csrc.nist.gov/groups/ST/toolkit/block_ciphers.html"> </url> DES and AES and
        and Block Cipher Modes - <url href="http://csrc.nist.gov/groups/ST/toolkit/BCM/index.html"> ECB, CBC, CFB, OFB and CTR </url></p>
      </item>
      <item>
        <p><url href="http://www.ietf.org/rfc/rfc1321.txt"> RSA encryption RFC 1321 </url> </p>
      </item>
      <item>
        <p>Digital signatures <url href="http://csrc.nist.gov/publications/drafts/fips186-3/fips_186-3.pdf">Digital Signature Standard (DSS) </url>  and  <url href="http://csrc.nist.gov/groups/STM/cavp/documents/dss2/ecdsa2vs.pdf">Elliptic Curve Digital
	Signature Algorithm (ECDSA) </url> </p>
      </item>
      <item>
        <p><url href="http://www.ietf.org/rfc/rfc2945.txt"> Secure Remote Password Protocol (SRP - RFC 2945) </url></p>
      </item>
    </list>
  </description>

 <section>
    <title>DATA TYPES </title>

    <p><code>byte() = 0 ... 255</code></p>

    <p><code>ioelem() = byte() | binary() | iolist()</code></p>

    <p><code>iolist() = [ioelem()]</code></p>

    <p><code>key_value()  = integer() | binary() </code></p>

    <p><code>rsa_public()  = [key_value()] = [E, N]  </code></p>
    <p> Where E is the public exponent and N is public modulus. </p>

    <p><code>rsa_private() = [key_value()] = [E, N, D] | [E, N, D, P1, P2, E1, E2, C] </code></p>
    <p>Where E is the public exponent, N is public modulus and D is
    the private exponent.The longer key format contains redundant
    information that will make the calculation faster. P1,P2 are first
    and second prime factors. E1,E2 are first and second exponents. C
    is the CRT coefficient. Terminology is taken from RFC 3447. </p>

    <p><code>dss_public() = [key_value()] = [P, Q, G, Y] </code></p>
    <p>Where P, Q and G are the dss parameters and Y is the public key.</p>

    <p><code>dss_private() =  [key_value()] = [P, Q, G, X] </code></p>
    <p>Where P, Q and G are the dss parameters and X is the private key.</p>

    <p><code>dss_public() =  [key_value()] =[P, Q, G, Y] </code></p>

    <p><code>srp_public() =  key_value() </code></p>
    <p>Where is <c>A</c> or <c>B</c> from <url href="http://srp.stanford.edu/design.html">SRP design</url></p>

    <p><code>srp_private() = key_value() </code></p>
    <p>Where is  <c>a</c> or <c>b</c> from <url href="http://srp.stanford.edu/design.html">SRP design</url></p>

    <p><code>srp_params() = {user, [Generator::binary(), Prime::binary(), Version::atom()]} |
    {host, [Verifier::binary(), Generator::binary(), Prime::binary(), Version::atom()]}
    |  {user, [DerivedKey::binary(), Prime::binary(), Generator::binary(), Version::atom() | [Scrambler:binary()]]}
    | {host,[Verifier::binary(), Prime::binary(), Version::atom() | [Scrambler::binary]]} </code></p>

    <p>Where Verifier is <c>v</c>, Generator is <c>g</c> and Prime is<c> N</c>, DerivedKey is <c>X</c>, and Scrambler is
    <c>u</c> (optional will be genrated if not provided) from <url href="http://srp.stanford.edu/design.html">SRP design</url>
    Version = '3' |  '6' |  '6a'
    </p>

    <p><code>dh_public() = key_value() </code></p>

    <p><code>dh_private() = key_value() </code></p>

    <p><code>dh_params() = [key_value()] = [P, G] </code></p>

    <p><code>ecdh_public() = key_value() </code></p>

    <p><code>ecdh_private() = key_value() </code></p>

    <p><code>ecdh_params() =  ec_named_curve() |
    {ec_field(), Prime :: key_value(), Point :: key_value(), Order :: integer(), CoFactor :: none | integer()} </code></p>

    <p><code>ec_field() = {prime_field, Prime :: integer()} |
    {characteristic_two_field, M :: integer(), Basis :: ec_basis()}</code></p>

    <p><code>ec_basis() =  {tpbasis, K :: non_neg_integer()} |
    {ppbasis, K1 :: non_neg_integer(), K2 :: non_neg_integer(), K3 :: non_neg_integer()} |
    onbasis</code></p>

    <p><code>ec_named_curve() ->
      sect571r1| sect571k1| sect409r1| sect409k1| secp521r1| secp384r1| secp224r1| secp224k1|
      secp192k1| secp160r2| secp128r2| secp128r1| sect233r1| sect233k1| sect193r2| sect193r1|
      sect131r2| sect131r1| sect283r1| sect283k1| sect163r2| secp256k1| secp160k1| secp160r1|
      secp112r2| secp112r1| sect113r2| sect113r1| sect239k1| sect163r1| sect163k1| secp256r1|
      secp192r1 </code></p>

 </section>

  <funcs>
      <func>
      <name>algorithms() -> [atom()]</name>
      <fsummary>Provide a list of available crypto algorithms.</fsummary>
      <desc>
        <p>Provides the available crypto algorithms in terms of a list
          of atoms.</p>
      </desc>
    </func>

    <func>
      <name>compute_key(Type, OthersPublicKey, MyPrivateKey, Params) -> SharedSecret</name>
      <fsummary>Computes the shared secret</fsummary>
      <type>
	<v> Type = dh | ecdh | srp </v>
	<v>OthersPublicKey =  dh_public() | ecdh_public() | srp_public() </v>
	<v>MyPrivate =  dh_private() | ecdh_private() | srp_private()  </v>
	<v>Params = dh_params() | edhc_params() | srp_params() </v>
	<v>SharedSecret = binary()</v>
      </type>
      <desc>
	<p>Computes the shared secret from the private key and the other party's public key.
	</p>
      </desc>
    </func>

    <func>
      <name>exor(Data1, Data2) -> Result</name>
      <fsummary>XOR data</fsummary>
      <type>
	<v>Data1, Data2 = iolist() | binary()</v>
	<v>Result = binary()</v>
      </type>
      <desc>
	<p>Performs bit-wise XOR (exclusive or) on the data supplied.</p>
      </desc>
    </func>

   <func>
      <name>generate_key(Type, Params) -> {PublicKey, PrivateKey} </name>
      <name>generate_key(Type, Params, PrivateKey) -> {PublicKey, PrivateKey} </name>
      <fsummary>Generates a public keys of type <c>Type</c></fsummary>
      <type>
	<v> Type = dh | ecdh | srp </v>
	<v>Params = dh_params() | edhc_params() | srp_params() </v>
	<v>PublicKey =  dh_public() | ecdh_public() | srp_public() </v>
	<v>PrivateKey =  dh_private() | ecdh_private() | srp_private() </v>
      </type>
      <desc>
	<p>Generates public keys of type <c>Type</c>.
	</p>
      </desc>
    </func>

     <func>
      <name>hash(Type, Data) -> Digest</name>
      <fsummary></fsummary>
      <type>
	<v>Type = md4 | md5 | ripemd160 | sha | sha224 | sha256 | sha384 | sha512</v>
	<v>Data = iodata()</v>
	<v>Digest = binary()</v>
      </type>
      <desc>
        <p>Computes a message digest of type <c>Type</c> from <c>Data</c>.</p>
	<p>May throw exception <c>notsup</c> in case the chosen <c>Type</c>
	is not supported by the underlying OpenSSL implementation.</p>
      </desc>
    </func>

    <func>
      <name>hash_init(Type) -> Context</name>
      <fsummary></fsummary>
      <type>
	<v>Type = md4 | md5 | ripemd160 | sha | sha224 | sha256 | sha384 | sha512</v>
      </type>
      <desc>
        <p>Initializes the context for streaming hash operations. <c>Type</c> determines
        which digest to use. The returned context should be used as argument
	to <seealso marker="#hash_update/2">hash_update</seealso>.</p>
	<p>May throw exception <c>notsup</c> in case the chosen <c>Type</c>
	is not supported by the underlying OpenSSL implementation.</p>
      </desc>
    </func>

    <func>
      <name>hash_update(Context, Data) -> NewContext</name>
      <fsummary></fsummary>
      <type>
	<v>Data = iodata()</v>
      </type>
      <desc>
        <p>Updates the digest represented by <c>Context</c> using the given <c>Data</c>. <c>Context</c>
        must have been generated using <seealso marker="#hash_init/1">hash_init</seealso>
	or a previous call to this function. <c>Data</c> can be any length. <c>NewContext</c>
        must be passed into the next call to <c>hash_update</c>
	or <seealso marker="#hash_final/1">hash_final</seealso>.</p>
      </desc>
    </func>
    <func>
      <name>hash_final(Context) -> Digest</name>
      <fsummary></fsummary>
      <type>
	<v>Digest = binary()</v>
      </type>
      <desc>
        <p>Finalizes the hash operation referenced by <c>Context</c> returned
	from a previous call to <seealso marker="#hash_update/2">hash_update</seealso>.
	The size of <c>Digest</c> is determined by the type of hash
	function used to generate it.</p>
      </desc>
    </func>

    <func>
      <name>hmac(Type, Key, Data) -> Mac</name>
      <name>hmac(Type, Key, Data, MacLength) -> Mac</name>
      <fsummary></fsummary>
      <type>
	<v>Type = md5 | sha | sha224 | sha256 | sha384 | sha512</v>
        <v>Key = iodata()</v>
	<v>Data = iodata()</v>
        <v>MacLength = integer()</v>
	<v>Mac = binary()</v>
      </type>
      <desc>
        <p>Computes a HMAC of type <c>Type</c> from <c>Data</c> using
        <c>Key</c> as the authentication key.</p> <c>MacLength</c>
        will limit the size of the resultant <c>Mac</c>.
      </desc>
    </func>

    <func>
      <name>hmac_init(Type, Key) -> Context</name>
      <fsummary></fsummary>
      <type>
        <v>Type = md5 | ripemd160 | sha | sha224 | sha256 | sha384 | sha512</v>
        <v>Key = iolist() | binary()</v>
        <v>Context = binary()</v>
      </type>
      <desc>
        <p>Initializes the context for streaming HMAC operations. <c>Type</c> determines
        which hash function to use in the HMAC operation. <c>Key</c> is the authentication
        key. The key can be any length.</p>
      </desc>
    </func>

    <func>
      <name>hmac_update(Context, Data) -> NewContext</name>
      <fsummary></fsummary>
      <type>
        <v>Context = NewContext = binary()</v>
        <v>Data = iolist() | binary()</v>
      </type>
      <desc>
        <p>Updates the HMAC represented by <c>Context</c> using the given <c>Data</c>. <c>Context</c>
        must have been generated using an HMAC init function (such as 
        <seealso marker="#hmac_init/2">hmac_init</seealso>). <c>Data</c> can be any length. <c>NewContext</c>
        must be passed into the next call to <c>hmac_update</c>.</p>
      </desc>
    </func>

    <func>
      <name>hmac_final(Context) -> Mac</name>
      <fsummary></fsummary>
      <type>
        <v>Context = Mac = binary()</v>
      </type>
      <desc>
        <p>Finalizes the HMAC operation referenced by <c>Context</c>. The size of the resultant MAC is
        determined by the type of hash function used to generate it.</p>
      </desc>
    </func>

    <func>
      <name>hmac_final_n(Context, HashLen) -> Mac</name>
      <fsummary></fsummary>
      <type>
        <v>Context = Mac = binary()</v>
        <v>HashLen = non_neg_integer()</v>
      </type>
      <desc>
        <p>Finalizes the HMAC operation referenced by <c>Context</c>. <c>HashLen</c> must be greater than
        zero. <c>Mac</c> will be a binary with at most <c>HashLen</c> bytes. Note that if HashLen is greater than the actual number of bytes returned from the underlying hash, the returned hash will have fewer than <c>HashLen</c> bytes.</p>
      </desc>
    </func>

      <func>
      <name>info() -> [atom()]</name>
      <fsummary>Provide a list of available crypto functions.</fsummary>
      <desc>
        <p>Provides the available crypto functions in terms of a list
          of atoms.</p>
      </desc>
    </func>

    <func>
      <name>info_lib() -> [{Name,VerNum,VerStr}]</name>
      <fsummary>Provides information about the libraries used by crypto.</fsummary>
      <type>
        <v>Name = binary()</v>
        <v>VerNum = integer()</v>
        <v>VerStr = binary()</v>
      </type>
      <desc>
        <p>Provides the name and version of the libraries used by crypto.</p>
        <p><c>Name</c> is the name of the library. <c>VerNum</c> is
        the numeric version according to the library's own versioning
        scheme. <c>VerStr</c> contains a text variant of the version.</p>
        <pre>
> <input>info_lib().</input>
[{&lt;&lt;"OpenSSL"&gt;&gt;,9469983,&lt;&lt;"OpenSSL 0.9.8a 11 Oct 2005"&gt;&gt;}]
        </pre>
	<note><p>
	  From OTP R16 the <em>numeric version</em> represents the version of the OpenSSL
	  <em>header files</em> (<c>openssl/opensslv.h</c>) used when crypto was compiled.
	  The text variant represents the OpenSSL library used at runtime.
	  In earlier OTP versions both numeric and text was taken from the library.
	</p></note>
      </desc>
    </func>

    <func>
      <name>mod_exp_prime(N, P, M) -> Result</name>
      <fsummary>Computes the function: N^P mod M</fsummary>
      <type>
	<v>N, P, M = binary()</v>
	<v>Result = binary() | error</v>
      </type>
      <desc>
	<p>Computes the function <c>N^P mod M</c>.</p>
      </desc>
    </func>

    <func>
      <name>rand_bytes(N) -> binary()</name>
      <fsummary>Generate a binary of random bytes</fsummary>
      <type>
        <v>N = integer()</v>
      </type>
      <desc>
        <p>Generates N bytes randomly uniform 0..255, and returns the
          result in a binary. Uses the <c>crypto</c> library pseudo-random
          number generator.</p>
      </desc>
    </func>

     <func>
      <name>rand_uniform(Lo, Hi) -> N</name>
      <fsummary>Generate a random number</fsummary>
      <type>
        <v>Lo, Hi, N = integer()</v>
      </type>
      <desc>
        <p>Generate a random number <c><![CDATA[N, Lo =< N < Hi.]]></c> Uses the
	<c>crypto</c> library pseudo-random number generator.
	<c>Hi</c> must be larger than <c>Lo</c>.</p>
      </desc>
    </func>

    <func>
      <name>sign(Algorithm, DigestType, Msg, Key) -> binary()</name>
      <fsummary> Create digital signature.</fsummary>
      <type>
	<v>Algorithm = rsa | dss | ecdsa </v>
	<v>Msg = binary() | {digest,binary()}</v>
	<d>The msg is either the binary "plain text" data to be
	signed or it is the hashed value of "plain text" i.e. the
	digest.</d>
	<v>DigestType = digest_type()</v>
	<v>Key = rsa_private_key() | dsa_private_key() | ec_private_key()</v>
      </type>
      <desc>
	<p> Creates a digital signature.</p>
      </desc>
    </func>

    <func>
      <name>start() -> ok</name>
      <fsummary> Equivalent to application:start(crypto). </fsummary>
      <desc>
        <p> Equivalent to application:start(crypto).</p>
      </desc>
    </func>
    <func>
      <name>stop() -> ok</name>
      <fsummary> Equivalent to application:stop(crypto).</fsummary>
      <desc>
        <p> Equivalent to application:stop(crypto).</p>
      </desc>
    </func>

    <func>
      <name>strong_rand_bytes(N) -> binary()</name>
      <fsummary>Generate a binary of random bytes</fsummary>
      <type>
        <v>N = integer()</v>
      </type>
      <desc>
        <p>Generates N bytes randomly uniform 0..255, and returns the
        result in a binary. Uses a cryptographically secure prng seeded and
        periodically mixed with operating system provided entropy. By default
        this is the <c>RAND_bytes</c> method from OpenSSL.</p>
	<p>May throw exception <c>low_entropy</c> in case the random generator
	failed due to lack of secure "randomness".</p>
      </desc>
    </func>
    <func>
      <name>verify(Algorithm, DigestType, Msg, Signature, Key) -> boolean()</name>
      <fsummary>Verifies a digital signature.</fsummary>
      <type>
	<v> Algorithm = rsa | dss | ecdsa </v>
	<v>Msg = binary() | {digest,binary()}</v>
	<d>The msg is either the binary "plain text" data
        or it is the hashed value of "plain text" i.e. the digest.</d>
	<v>DigestType = digest_type()</v>
	<v>Signature = binary()</v>
	<v>Key = rsa_public_key() | dsa_public_key() | ec_public_key()</v>
      </type>
      <desc>
	<p>Verifies a digital signature</p>
      </desc>
 </func>

    <func>
      <name>aes_cfb_128_encrypt(Key, IVec, Text) -> Cipher</name>
      <fsummary>Encrypt <c>Text</c>according to AES in Cipher Feedback mode</fsummary>
      <type>
        <v>Key = Text = iolist() | binary()</v>
        <v>IVec = Cipher = binary()</v>
      </type>
      <desc>
        <p>Encrypts <c>Text</c> according to AES in Cipher Feedback 
          mode (CFB). <c>Key</c> is the
          AES key, and <c>IVec</c> is an arbitrary initializing vector.
          The lengths of <c>Key</c> and <c>IVec</c> must be 128 bits
          (16 bytes).</p>
      </desc>
    </func>

    <func>
      <name>aes_cfb_128_decrypt(Key, IVec, Cipher) -> Text</name>
      <fsummary>Decrypt <c>Cipher</c>according to AES in Cipher Feedback mode</fsummary>
      <type>
        <v>Key = Cipher = iolist() | binary()</v>
        <v>IVec = Text = binary()</v>
      </type>
      <desc>
        <p>Decrypts <c>Cipher</c> according to AES in Cipher Feedback Mode (CFB).
          <c>Key</c> is the AES key, and <c>IVec</c> is an arbitrary
          initializing vector. <c>Key</c> and <c>IVec</c> must have
          the same values as those used when encrypting. The lengths of
          <c>Key</c> and <c>IVec</c> must be 128 bits (16 bytes).</p>
      </desc>
    </func>

    <func>
      <name>aes_cbc_128_encrypt(Key, IVec, Text) -> Cipher</name>
      <fsummary>Encrypt <c>Text</c>according to AES in Cipher Block Chaining mode</fsummary>
      <type>
        <v>Key = Text = iolist() | binary()</v>
        <v>IVec = Cipher = binary()</v>
      </type>
      <desc>
          <p>Encrypts <c>Text</c> according to AES in Cipher Block Chaining
          mode (CBC). <c>Text</c>
          must be a multiple of 128 bits (16 bytes). <c>Key</c> is the
          AES key, and <c>IVec</c> is an arbitrary initializing vector.
          The lengths of <c>Key</c> and <c>IVec</c> must be 128 bits
          (16 bytes).</p>
      </desc>
    </func>

    <func>
      <name>aes_cbc_128_decrypt(Key, IVec, Cipher) -> Text</name>
      <fsummary>Decrypt <c>Cipher</c>according to AES in Cipher Block Chaining mode</fsummary>
      <type>
        <v>Key = Cipher = iolist() | binary()</v>
        <v>IVec = Text = binary()</v>
      </type>
      <desc>
          <p>Decrypts <c>Cipher</c> according to AES in Cipher Block
          Chaining mode (CBC).
          <c>Key</c> is the AES key, and <c>IVec</c> is an arbitrary
          initializing vector. <c>Key</c> and <c>IVec</c> must have
          the same values as those used when encrypting. <c>Cipher</c>
          must be a multiple of 128 bits (16 bytes).  The lengths of
          <c>Key</c> and <c>IVec</c> must be 128 bits (16 bytes).</p>
      </desc>
    </func>

    <func>
      <name>aes_cbc_ivec(Data) -> IVec</name>
      <fsummary>Get <c>IVec</c> to be used in next iteration of
                <c>aes_cbc_*_[ecrypt|decrypt]</c></fsummary>
      <type>
        <v>Data = iolist() | binary()</v>
        <v>IVec = binary()</v>
      </type>
      <desc>
        <p>Returns the <c>IVec</c> to be used in a next iteration of 
          <c>aes_cbc_*_[encrypt|decrypt]</c>. <c>Data</c> is the encrypted
          data from the previous iteration step.</p>
      </desc>
    </func>

    <func>
      <name>aes_ctr_encrypt(Key, IVec, Text) -> Cipher</name>
      <fsummary>Encrypt <c>Text</c>according to AES in Counter mode</fsummary>
      <type>
        <v>Key = Text = iolist() | binary()</v>
        <v>IVec = Cipher = binary()</v>
      </type>
      <desc>
      <p>Encrypts <c>Text</c> according to AES in Counter mode (CTR). <c>Text</c>
        can be any number of bytes. <c>Key</c> is the AES key and must be either
        128, 192 or 256 bits long. <c>IVec</c> is an arbitrary initializing vector of 128 bits
        (16 bytes).</p>
    </desc>
    </func>

    <func>
      <name>aes_ctr_decrypt(Key, IVec, Cipher) -> Text</name>
      <fsummary>Decrypt <c>Cipher</c>according to AES in Counter mode</fsummary>
      <type>
        <v>Key = Cipher = iolist() | binary()</v>
        <v>IVec = Text = binary()</v>
      </type>
      <desc>
      <p>Decrypts <c>Cipher</c> according to AES in Counter mode (CTR). <c>Cipher</c>
        can be any number of bytes. <c>Key</c> is the AES key and must be either
        128, 192 or 256 bits long. <c>IVec</c> is an arbitrary initializing vector of 128 bits
        (16 bytes).</p>
    </desc>
    </func>

    <func>
      <name>aes_ctr_stream_init(Key, IVec) -> State</name>
      <fsummary></fsummary>
      <type>
        <v>State = { K, I, E, C }</v>
        <v>Key = K = iolist()</v>
        <v>IVec = I = E = binary()</v>
        <v>C = integer()</v>
      </type>
      <desc>
        <p>Initializes the state for use in streaming AES encryption using Counter mode (CTR).
        <c>Key</c> is the AES key and must be either 128, 192, or 256 bts long. <c>IVec</c> is
        an arbitrary initializing vector of 128 bits (16 bytes). This state is for use with
        <seealso marker="#aes_ctr_stream_encrypt/2">aes_ctr_stream_encrypt</seealso> and 
        <seealso marker="#aes_ctr_stream_decrypt/2">aes_ctr_stream_decrypt</seealso>.</p>
      </desc>
    </func>

    <func>
      <name>aes_ctr_stream_encrypt(State, Text) -> { NewState, Cipher}</name>
      <fsummary></fsummary>
      <type>
        <v>Text = iolist() | binary()</v>
        <v>Cipher = binary()</v>
      </type>
      <desc>
      <p>Encrypts <c>Text</c> according to AES in Counter mode (CTR).  This function can be
        used to encrypt a stream of text using a series of calls instead of requiring all
        text to be in memory. <c>Text</c> can be any number of bytes. State is initialized using 
        <seealso marker="#aes_ctr_stream_init/2">aes_ctr_stream_init</seealso>. <c>NewState</c> is the new streaming
        encryption state that must be passed to the next call to <c>aes_ctr_stream_encrypt</c>.
        <c>Cipher</c> is the encrypted cipher text.</p>
      </desc>
    </func>

    <func>
      <name>aes_ctr_stream_decrypt(State, Cipher) -> { NewState, Text }</name>
      <fsummary></fsummary>
      <type>
        <v>Cipher = iolist() | binary()</v>
        <v>Text = binary()</v>
      </type>
      <desc>
      <p>Decrypts <c>Cipher</c> according to AES in Counter mode (CTR). This function can be
        used to decrypt a stream of ciphertext using a series of calls instead of requiring all
        ciphertext to be in memory.  <c>Cipher</c> can be any number of bytes. State is initialized using 
        <seealso marker="#aes_ctr_stream_init/2">aes_ctr_stream_init</seealso>. <c>NewState</c> is the new streaming
        encryption state that must be passed to the next call to <c>aes_ctr_stream_encrypt</c>.
        <c>Text</c> is the decrypted data.</p>
      </desc>
    </func>

      <func>
	<name>blowfish_ecb_encrypt(Key, Text) -> Cipher</name>
	<fsummary>Encrypt the first 64 bits of <c>Text</c> using Blowfish in ECB mode</fsummary>
	<type>
        <v>Key = Text = iolist() | binary()</v>
        <v>Cipher = binary()</v>
      </type>
      <desc>
        <p>Encrypts the first 64 bits of <c>Text</c> using Blowfish in ECB mode. <c>Key</c> is the Blowfish key. The length of <c>Text</c> must be at least 64 bits (8 bytes).</p>
      </desc>
    </func>

    <func>
      <name>blowfish_ecb_decrypt(Key, Text) -> Cipher</name>
      <fsummary>Decrypt the first 64 bits of <c>Text</c> using Blowfish in ECB mode</fsummary>
      <type>
        <v>Key = Text = iolist() | binary()</v>
        <v>Cipher = binary()</v>
      </type>
      <desc>
        <p>Decrypts the first 64 bits of <c>Text</c> using Blowfish in ECB mode. <c>Key</c> is the Blowfish key. The length of <c>Text</c> must be at least 64 bits (8 bytes).</p>
      </desc>
    </func>

    <func>
      <name>blowfish_cbc_encrypt(Key, IVec, Text) -> Cipher</name>
      <fsummary>Encrypt <c>Text</c> using Blowfish in CBC mode</fsummary>
      <type>
        <v>Key = Text = iolist() | binary()</v>
        <v>IVec = Cipher = binary()</v>
      </type>
      <desc>
        <p>Encrypts <c>Text</c> using Blowfish in CBC mode. <c>Key</c> is the Blowfish key, and <c>IVec</c> is an
	arbitrary initializing vector. The length of <c>IVec</c>
          must be 64 bits (8 bytes). The length of <c>Text</c> must be a multiple of 64 bits (8 bytes).</p>
      </desc>
    </func>
    <func>
      <name>blowfish_cbc_decrypt(Key, IVec, Text) -> Cipher</name>
      <fsummary>Decrypt <c>Text</c> using Blowfish in CBC mode</fsummary>
      <type>
        <v>Key = Text = iolist() | binary()</v>
        <v>IVec = Cipher = binary()</v>
      </type>
      <desc>
        <p>Decrypts <c>Text</c> using Blowfish in CBC mode. <c>Key</c> is the Blowfish key, and <c>IVec</c> is an
          arbitrary initializing vector. The length of <c>IVec</c>
          must be 64 bits (8 bytes). The length of <c>Text</c> must be a multiple 64 bits (8 bytes).</p>
      </desc>
    </func>

    <func>
      <name>blowfish_cfb64_encrypt(Key, IVec, Text) -> Cipher</name>
      <fsummary>Encrypt <c>Text</c>using Blowfish in CFB mode with 64
        bit feedback</fsummary>
      <type>
        <v>Key = Text = iolist() | binary()</v>
        <v>IVec = Cipher = binary()</v>
      </type>
      <desc>
        <p>Encrypts <c>Text</c> using Blowfish in CFB mode with 64 bit
          feedback. <c>Key</c> is the Blowfish key, and <c>IVec</c> is an
          arbitrary initializing vector. The length of <c>IVec</c>
          must be 64 bits (8 bytes).</p>
      </desc>
    </func>

    <func>
      <name>blowfish_cfb64_decrypt(Key, IVec, Text) -> Cipher</name>
      <fsummary>Decrypt <c>Text</c>using Blowfish in CFB mode with 64
        bit feedback</fsummary>
      <type>
        <v>Key = Text = iolist() | binary()</v>
        <v>IVec = Cipher = binary()</v>
      </type>
      <desc>
        <p>Decrypts <c>Text</c> using Blowfish in CFB mode with 64 bit
          feedback. <c>Key</c> is the Blowfish key, and <c>IVec</c> is an
          arbitrary initializing vector. The length of <c>IVec</c>
          must be 64 bits (8 bytes).</p>
      </desc>
    </func>

    <func>
      <name>blowfish_ofb64_encrypt(Key, IVec, Text) -> Cipher</name>
      <fsummary>Encrypt <c>Text</c>using Blowfish in OFB mode with 64
        bit feedback</fsummary>
      <type>
        <v>Key = Text = iolist() | binary()</v>
        <v>IVec = Cipher = binary()</v>
      </type>
      <desc>
        <p>Encrypts <c>Text</c> using Blowfish in OFB mode with 64 bit
          feedback. <c>Key</c> is the Blowfish key, and <c>IVec</c> is an
          arbitrary initializing vector. The length of <c>IVec</c>
          must be 64 bits (8 bytes).</p>
      </desc>
    </func>

    <func>
      <name>des_cbc_encrypt(Key, IVec, Text) -> Cipher</name>
      <fsummary>Encrypt <c>Text</c>according to DES in CBC mode</fsummary>
      <type>
        <v>Key = Text = iolist() | binary()</v>
        <v>IVec = Cipher = binary()</v>
      </type>
      <desc>
        <p>Encrypts <c>Text</c> according to DES in CBC
          mode. <c>Text</c> must be a multiple of 64 bits (8
          bytes). <c>Key</c> is the DES key, and <c>IVec</c> is an
          arbitrary initializing vector. The lengths of <c>Key</c> and
          <c>IVec</c> must be 64 bits (8 bytes).</p>
      </desc>
    </func>

    <func>
      <name>des_cbc_decrypt(Key, IVec, Cipher) -> Text</name>
      <fsummary>Decrypt <c>Cipher</c>according to DES in CBC mode</fsummary>
      <type>
        <v>Key = Cipher = iolist() | binary()</v>
        <v>IVec = Text = binary()</v>
      </type>
      <desc>
        <p>Decrypts <c>Cipher</c> according to DES in CBC mode.
          <c>Key</c> is the DES key, and <c>IVec</c> is an arbitrary
          initializing vector.  <c>Key</c> and <c>IVec</c> must have
          the same values as those used when encrypting. <c>Cipher</c>
          must be a multiple of 64 bits (8 bytes).  The lengths of
          <c>Key</c> and <c>IVec</c> must be 64 bits (8 bytes).</p>
      </desc>
    </func>

    <func>
      <name>des_cbc_ivec(Data) -> IVec</name>
      <fsummary>Get <c>IVec</c> to be used in next iteration of
                <c>des_cbc_[ecrypt|decrypt]</c></fsummary>
      <type>
        <v>Data = iolist() | binary()</v>
        <v>IVec = binary()</v>
      </type>
      <desc>
        <p>Returns the <c>IVec</c> to be used in a next iteration of
          <c>des_cbc_[encrypt|decrypt]</c>. <c>Data</c> is the encrypted
          data from the previous iteration step.</p>
      </desc>
    </func>

    <func>
      <name>des_cfb_encrypt(Key, IVec, Text) -> Cipher</name>
      <fsummary>Encrypt <c>Text</c>according to DES in CFB mode</fsummary>
      <type>
        <v>Key = Text = iolist() | binary()</v>
        <v>IVec = Cipher = binary()</v>
      </type>
      <desc>
        <p>Encrypts <c>Text</c> according to DES in 8-bit CFB
          mode. <c>Key</c> is the DES key, and <c>IVec</c> is an
          arbitrary initializing vector. The lengths of <c>Key</c> and
          <c>IVec</c> must be 64 bits (8 bytes).</p>
      </desc>
    </func>

    <func>
      <name>des_cfb_decrypt(Key, IVec, Cipher) -> Text</name>
      <fsummary>Decrypt <c>Cipher</c>according to DES in CFB mode</fsummary>
      <type>
        <v>Key = Cipher = iolist() | binary()</v>
        <v>IVec = Text = binary()</v>
      </type>
      <desc>
        <p>Decrypts <c>Cipher</c> according to DES in 8-bit CFB mode.
          <c>Key</c> is the DES key, and <c>IVec</c> is an arbitrary
          initializing vector.  <c>Key</c> and <c>IVec</c> must have
          the same values as those used when encrypting.  The lengths of
          <c>Key</c> and <c>IVec</c> must be 64 bits (8 bytes).</p>
      </desc>
    </func>

    <func>
      <name>des_cfb_ivec(IVec, Data) -> NextIVec</name>
      <fsummary>Get <c>IVec</c> to be used in next iteration of
                <c>des_cfb_[ecrypt|decrypt]</c></fsummary>
      <type>
        <v>IVec = iolist() | binary()</v>
        <v>Data = iolist() | binary()</v>
        <v>NextIVec = binary()</v>
      </type>
      <desc>
        <p>Returns the <c>IVec</c> to be used in a next iteration of
          <c>des_cfb_[encrypt|decrypt]</c>. <c>IVec</c> is the vector
          used in the previous iteration step. <c>Data</c> is the encrypted
          data from the previous iteration step.</p>
      </desc>
    </func>

    <func>
      <name>des3_cbc_encrypt(Key1, Key2, Key3, IVec, Text) -> Cipher</name>
      <fsummary>Encrypt <c>Text</c>according to DES3 in CBC mode</fsummary>
      <type>
        <v>Key1 =Key2 = Key3 Text = iolist() | binary()</v>
        <v>IVec = Cipher = binary()</v>
      </type>
      <desc>
        <p>Encrypts <c>Text</c> according to DES3 in CBC
          mode. <c>Text</c> must be a multiple of 64 bits (8
          bytes). <c>Key1</c>, <c>Key2</c>, <c>Key3</c>, are the DES
          keys, and <c>IVec</c> is an arbitrary initializing
          vector. The lengths of each of <c>Key1</c>, <c>Key2</c>,
          <c>Key3</c> and <c>IVec</c> must be 64 bits (8 bytes).</p>
      </desc>
    </func>

    <func>
      <name>des3_cbc_decrypt(Key1, Key2, Key3, IVec, Cipher) -> Text</name>
      <fsummary>Decrypt <c>Cipher</c>according to DES3 in CBC mode</fsummary>
      <type>
        <v>Key1 = Key2 = Key3 = Cipher = iolist() | binary()</v>
        <v>IVec = Text = binary()</v>
      </type>
      <desc>
        <p>Decrypts <c>Cipher</c> according to DES3 in CBC mode.
          <c>Key1</c>, <c>Key2</c>, <c>Key3</c> are the DES key, and
          <c>IVec</c> is an arbitrary initializing vector.
          <c>Key1</c>, <c>Key2</c>, <c>Key3</c> and <c>IVec</c> must
          and <c>IVec</c> must have the same values as those used when
          encrypting. <c>Cipher</c> must be a multiple of 64 bits (8
          bytes).  The lengths of <c>Key1</c>, <c>Key2</c>,
          <c>Key3</c>, and <c>IVec</c> must be 64 bits (8 bytes).</p>
      </desc>
    </func>

    <func>
      <name>des3_cfb_encrypt(Key1, Key2, Key3, IVec, Text) -> Cipher</name>
      <fsummary>Encrypt <c>Text</c>according to DES3 in CFB mode</fsummary>
      <type>
        <v>Key1 =Key2 = Key3 Text = iolist() | binary()</v>
        <v>IVec = Cipher = binary()</v>
      </type>
      <desc>
        <p>Encrypts <c>Text</c> according to DES3 in 8-bit CFB
          mode. <c>Key1</c>, <c>Key2</c>, <c>Key3</c>, are the DES
          keys, and <c>IVec</c> is an arbitrary initializing
          vector. The lengths of each of <c>Key1</c>, <c>Key2</c>,
          <c>Key3</c> and <c>IVec</c> must be 64 bits (8 bytes).</p>
	<p>May throw exception <c>notsup</c> for old OpenSSL
	  versions (0.9.7) that does not support this encryption mode.</p>
      </desc>
    </func>

    <func>
      <name>des3_cfb_decrypt(Key1, Key2, Key3, IVec, Cipher) -> Text</name>
      <fsummary>Decrypt <c>Cipher</c>according to DES3 in CFB mode</fsummary>
      <type>
        <v>Key1 = Key2 = Key3 = Cipher = iolist() | binary()</v>
        <v>IVec = Text = binary()</v>
      </type>
      <desc>
        <p>Decrypts <c>Cipher</c> according to DES3 in 8-bit CFB mode.
          <c>Key1</c>, <c>Key2</c>, <c>Key3</c> are the DES key, and
          <c>IVec</c> is an arbitrary initializing vector.
          <c>Key1</c>, <c>Key2</c>, <c>Key3</c> and <c>IVec</c> must
          and <c>IVec</c> must have the same values as those used when
          encrypting.  The lengths of <c>Key1</c>, <c>Key2</c>,
          <c>Key3</c>, and <c>IVec</c> must be 64 bits (8 bytes).</p>
	<p>May throw exception <c>notsup</c> for old OpenSSL
	  versions (0.9.7) that does not support this encryption mode.</p>
      </desc>
    </func>

    <func>
      <name>des_ecb_encrypt(Key, Text) -> Cipher</name>
      <fsummary>Encrypt <c>Text</c>according to DES in ECB mode</fsummary>
      <type>
        <v>Key = Text = iolist() | binary()</v>
        <v>Cipher = binary()</v>
      </type>
      <desc>
        <p>Encrypts <c>Text</c> according to DES in ECB mode.
          <c>Key</c> is the DES key. The lengths of <c>Key</c> and
          <c>Text</c> must be 64 bits (8 bytes).</p>
      </desc>
    </func>
    <func>
      <name>des_ecb_decrypt(Key, Cipher) -> Text</name>
      <fsummary>Decrypt <c>Cipher</c>according to DES in ECB mode</fsummary>
      <type>
        <v>Key = Cipher = iolist() | binary()</v>
        <v>Text = binary()</v>
      </type>
      <desc>
        <p>Decrypts <c>Cipher</c> according to DES in ECB mode.
          <c>Key</c> is the DES key. The lengths of <c>Key</c> and
          <c>Cipher</c> must be 64 bits (8 bytes).</p>
      </desc>
    </func>
      <func>
      <name>rc2_cbc_encrypt(Key, IVec, Text) -> Cipher</name>
      <fsummary>Encrypt <c>Text</c>according to RC2 in CBC mode</fsummary>
      <type>
        <v>Key = Text = iolist() | binary()</v>
        <v>Ivec = Cipher = binary()</v>
      </type>
      <desc>
        <p>Encrypts <c>Text</c> according to RC2 in CBC mode.</p>
      </desc>
    </func>

    <func>
      <name>rc2_cbc_decrypt(Key, IVec, Cipher) -> Text</name>
      <fsummary>Decrypts <c>Cipher</c>according to RC2 in CBC mode</fsummary>
      <type>
        <v>Key = Text = iolist() | binary()</v>
        <v>Ivec = Cipher = binary()</v>
      </type>
      <desc>
        <p>Decrypts <c>Cipher</c> according to RC2 in CBC mode.</p>
      </desc>
    </func>
    
    <func>
      <name>rc4_encrypt(Key, Data) -> Result</name>
      <fsummary>Encrypt data using RC4</fsummary>
      <type>
	<v>Key, Data = iolist() | binary()</v>
	<v>Result = binary()</v>
      </type>
      <desc>
	<p>Encrypts the data with RC4 symmetric stream encryption.
	  Since it is symmetric, the same function is used for
	  decryption.</p>
      </desc>
    </func>


    <func>
      <name>rsa_public_encrypt(PlainText, PublicKey, Padding) -> ChipherText</name>
      <fsummary>Encrypts Msg using the public Key.</fsummary>
      <type>
	<v>PlainText = binary()</v>
	<v>PublicKey = [E, N]</v>
	<v>E, N = integer()</v>
	<d>Where <c>E</c> is the public exponent and <c>N</c> is public modulus.</d>
	<v>Padding = rsa_pkcs1_padding | rsa_pkcs1_oaep_padding | rsa_no_padding</v>
        <v>ChipherText = binary()</v>
      </type>
      <desc>
	<p>Encrypts the <c>PlainText</c> (usually a session key) using the <c>PublicKey</c>
	  and returns the cipher. The <c>Padding</c> decides what padding mode is used,
	  <c>rsa_pkcs1_padding</c> is PKCS #1 v1.5 currently the most
	  used mode and <c>rsa_pkcs1_oaep_padding</c> is EME-OAEP as
	  defined in PKCS #1 v2.0 with SHA-1, MGF1 and an empty encoding
	  parameter. This mode is recommended for all new applications.
	  The size of the <c>Msg</c> must be less
	  than <c>byte_size(N)-11</c> if
	  <c>rsa_pkcs1_padding</c> is used, <c>byte_size(N)-41</c> if
	  <c>rsa_pkcs1_oaep_padding</c> is used and <c>byte_size(N)</c> if <c>rsa_no_padding</c>
	  is used.
	</p>
      </desc>
    </func>

    <func>
      <name>rsa_private_decrypt(ChipherText, PrivateKey, Padding) -> PlainText</name>
      <fsummary>Decrypts ChipherText using the private Key.</fsummary>
      <type>
	<v>ChipherText = binary()</v>
	<v>PrivateKey = [E, N, D] | [E, N, D, P1, P2, E1, E2, C]</v>
	<v>E, N, D = integer()</v>
	<d>Where <c>E</c> is the public exponent, <c>N</c> is public modulus  and
	  <c>D</c> is the private exponent.</d>
	<v>P1, P2, E1, E2, C = integer()</v>
	<d>The longer key format contains redundant information that will make
	   the calculation faster. <c>P1,P2</c> are first and second prime factors.
           <c>E1,E2</c> are first and second exponents. <c>C</c> is the CRT coefficient.
           Terminology is taken from RFC 3447.</d>
	<v>Padding = rsa_pkcs1_padding | rsa_pkcs1_oaep_padding | rsa_no_padding</v>
        <v>PlainText = binary()</v>
      </type>
      <desc>
	<p>Decrypts the <c>ChipherText</c> (usually a session key encrypted with
	  <seealso marker="#rsa_public_encrypt/3">rsa_public_encrypt/3</seealso>)
	  using the <c>PrivateKey</c> and returns the
	  message. The <c>Padding</c> is the padding mode that was
	  used to encrypt the data,
	  see <seealso marker="#rsa_public_encrypt/3">rsa_public_encrypt/3</seealso>.
	</p>
      </desc>
    </func>

    <func>
      <name>rsa_private_encrypt(PlainText, PrivateKey, Padding) -> ChipherText</name>
      <fsummary>Encrypts Msg using the private Key.</fsummary>
      <type>
	<v>PlainText = binary()</v>
	<v>PrivateKey = [E, N, D] | [E, N, D, P1, P2, E1, E2, C]</v>
	<v>E, N, D = integer()</v>
	<d>Where <c>E</c> is the public exponent, <c>N</c> is public modulus and
	  <c>D</c> is the private exponent.</d>
	<v>P1, P2, E1, E2, C = integer()</v>
	<d>The longer key format contains redundant information that will make
	   the calculation faster. <c>P1,P2</c> are first and second prime factors.
           <c>E1,E2</c> are first and second exponents. <c>C</c> is the CRT coefficient.
           Terminology is taken from RFC 3447.</d>
	<v>Padding = rsa_pkcs1_padding | rsa_no_padding</v>
        <v>ChipherText = binary()</v>
      </type>
      <desc>
	<p>Encrypts the <c>PlainText</c> using the <c>PrivateKey</c>
	  and returns the cipher. The <c>Padding</c> decides what padding mode is used,
	  <c>rsa_pkcs1_padding</c> is PKCS #1 v1.5 currently the most
	  used mode.
	  The size of the <c>Msg</c> must be less than <c>byte_size(N)-11</c> if
	  <c>rsa_pkcs1_padding</c> is used, and <c>byte_size(N)</c> if <c>rsa_no_padding</c>
	  is used.
	</p>
      </desc>
    </func>
    <func>
      <name>rsa_public_decrypt(ChipherText, PublicKey, Padding) -> PlainText</name>
      <fsummary>Decrypts ChipherText using the public Key.</fsummary>
      <type>
	<v>ChipherText = binary()</v>
	<v>PublicKey = [E, N]</v>
	<v>E, N = integer() </v>
	<d>Where <c>E</c> is the public exponent and <c>N</c> is public modulus</d>
	<v>Padding = rsa_pkcs1_padding | rsa_no_padding</v>
        <v>PlainText = binary()</v>
      </type>
      <desc>
	<p>Decrypts the <c>ChipherText</c> (encrypted with
	  <seealso marker="#rsa_private_encrypt/3">rsa_private_encrypt/3</seealso>)
	  using the <c>PrivateKey</c> and returns the
	  message. The <c>Padding</c> is the padding mode that was
	  used to encrypt the data,
	  see <seealso marker="#rsa_private_encrypt/3">rsa_private_encrypt/3</seealso>.
	</p>
      </desc>
    </func>
 </funcs>

  <section>
    <title>DES in CBC mode</title>
    <p>The Data Encryption Standard (DES) defines an algorithm for
      encrypting and decrypting an 8 byte quantity using an 8 byte key
      (actually only 56 bits of the key is used).
      </p>
    <p>When it comes to encrypting and decrypting blocks that are
      multiples of 8 bytes various modes are defined (NIST SP
      800-38A). One of those modes is the Cipher Block Chaining (CBC)
      mode, where the encryption of an 8 byte segment depend not only
      of the contents of the segment itself, but also on the result of
      encrypting the previous segment: the encryption of the previous
      segment becomes the initializing vector of the encryption of the
      current segment.
      </p>
    <p>Thus the encryption of every segment depends on the encryption
      key (which is secret) and the encryption of the previous
      segment, except the first segment which has to be provided with
      an initial initializing vector. That vector could be chosen at
      random, or be a counter of some kind. It does not have to be
      secret.
      </p>
    <p>The following example is drawn from the old FIPS 81 standard
      (replaced by NIST SP 800-38A), where both the plain text and the
      resulting cipher text is settled. The following code fragment
      returns `true'.
      </p>
    <pre><![CDATA[

      Key = <<16#01,16#23,16#45,16#67,16#89,16#ab,16#cd,16#ef>>,
      IVec = <<16#12,16#34,16#56,16#78,16#90,16#ab,16#cd,16#ef>>,
      P = "Now is the time for all ",
      C = crypto:des_cbc_encrypt(Key, IVec, P),
         % Which is the same as 
      P1 = "Now is t", P2 = "he time ", P3 = "for all ",
      C1 = crypto:des_cbc_encrypt(Key, IVec, P1),
      C2 = crypto:des_cbc_encrypt(Key, C1, P2),
      C3 = crypto:des_cbc_encrypt(Key, C2, P3),

      C = <<C1/binary, C2/binary, C3/binary>>,
      C = <<16#e5,16#c7,16#cd,16#de,16#87,16#2b,16#f2,16#7c,
             16#43,16#e9,16#34,16#00,16#8c,16#38,16#9c,16#0f,
             16#68,16#37,16#88,16#49,16#9a,16#7c,16#05,16#f6>>,
      <<"Now is the time for all ">> == 
                        crypto:des_cbc_decrypt(Key, IVec, C).
    ]]></pre>
    <p>The following is true for the DES CBC mode. For all
      decompositions <c>P1 ++ P2 = P</c> of a plain text message
      <c>P</c> (where the length of all quantities are multiples of 8
      bytes), the encryption <c>C</c> of <c>P</c> is equal to <c>C1 ++
      C2</c>, where <c>C1</c> is obtained by encrypting <c>P1</c> with
      <c>Key</c> and the initializing vector <c>IVec</c>, and where
      <c>C2</c> is obtained by encrypting <c>P2</c> with <c>Key</c>
      and the initializing vector <c>last8(C1)</c>,
      where <c>last(Binary)</c> denotes the last 8 bytes of the
      binary <c>Binary</c>.
      </p>
    <p>Similarly, for all decompositions <c>C1 ++ C2 = C</c> of a
      cipher text message <c>C</c> (where the length of all quantities
      are multiples of 8 bytes), the decryption <c>P</c> of <c>C</c>
      is equal to <c>P1 ++ P2</c>, where <c>P1</c> is obtained by
      decrypting <c>C1</c> with <c>Key</c> and the initializing vector
      <c>IVec</c>, and where <c>P2</c> is obtained by decrypting
      <c>C2</c> with <c>Key</c> and the initializing vector
      <c>last8(C1)</c>, where <c>last8(Binary)</c> is as above.
      </p>
    <p>For DES3 (which uses three 64 bit keys) the situation is the
      same.
    </p>
  </section>
</erlref>