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

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<!DOCTYPE erlref SYSTEM "erlref.dtd">

<erlref>
  <header>
    <copyright>
      <year>1999</year><year>2017</year>
      <holder>Ericsson AB. All Rights Reserved.</holder>
    </copyright>
    <legalnotice>
      Licensed under the Apache License, Version 2.0 (the "License");
      you may not use this file except in compliance with the License.
      You may obtain a copy of the License at

          http://www.apache.org/licenses/LICENSE-2.0

      Unless required by applicable law or agreed to in writing, software
      distributed under the License is distributed on an "AS IS" BASIS,
      WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
      See the License for the specific language governing permissions 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://csrc.nist.gov/publications/fips/fips180-4/fips-180-4.pdf"> Secure Hash Standard</url>,
	<url href="http://www.ietf.org/rfc/rfc1321.txt"> The MD5 Message Digest Algorithm (RFC 1321)</url> and
	<url href="http://www.ietf.org/rfc/rfc1320.txt">The MD4 Message Digest Algorithm (RFC 1320)</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>Cmac functions - <url href="http://www.ietf.org/rfc/rfc4493.txt">The AES-CMAC Algorithm (RFC 4493)</url></p>
      </item>
      <item>
        <p>Block ciphers - <url href="http://csrc.nist.gov/groups/ST/toolkit/block_ciphers.html"> </url> DES and AES in
        Block Cipher Modes - <url href="http://csrc.nist.gov/groups/ST/toolkit/BCM/index.html"> ECB, CBC, CFB, OFB, CTR and GCM </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>
      <item>
        <p>gcm: Dworkin, M., "Recommendation for Block Cipher Modes of
        Operation: Galois/Counter Mode (GCM) and GMAC",
        National Institute of Standards and Technology SP 800-
        38D, November 2007.</p>
      </item>
    </list>
  </description>

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

    <code>key_value() = integer() | binary() </code>
    <p>Always <c>binary()</c> when used as return value</p>

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

    <code>rsa_private() = [key_value()] = [E, N, D] | [E, N, D, P1, P2, E1, E2, C] </code>
    <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 <url href="http://www.ietf.org/rfc/rfc3477.txt"> RFC 3447</url>.</p>

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

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

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

    <code>srp_private() = key_value() </code>
    <p>Where is  <c>a</c> or <c>b</c> from <url href="http://srp.stanford.edu/design.html">SRP design</url></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 generated if not provided) from <url href="http://srp.stanford.edu/design.html">SRP design</url>
    Version = '3' |  '6' |  '6a'
    </p>

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

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

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

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

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

    <code>ecdh_params() = ec_named_curve() | ec_explicit_curve()</code>

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

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

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

    <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|
      brainpoolP160r1| brainpoolP160t1| brainpoolP192r1| brainpoolP192t1| brainpoolP224r1|
      brainpoolP224t1| brainpoolP256r1| brainpoolP256t1| brainpoolP320r1| brainpoolP320t1|
      brainpoolP384r1| brainpoolP384t1| brainpoolP512r1| brainpoolP512t1
    </code>
    <p>Note that the <em>sect</em> curves are GF2m (characteristic two) curves and are only supported if the
    underlying OpenSSL has support for them.
    See also <seealso marker="#supports-0">crypto:supports/0</seealso>
    </p>

    <marker id="engine_key_ref_type"/>
    <code>engine_key_ref() = #{engine   := engine_ref(),
                               key_id   := key_id(),
                               password => password()}</code>

    <code>engine_ref() = term()</code>
    <p>The result of a call to <seealso marker="#engine_load-3">engine_load/3</seealso>.
    </p>

    <code>key_id() = string() | binary()</code>
    <p>Identifies the key to be used. The format depends on the loaded engine. It is passed to
    the <c>ENGINE_load_(private|public)_key</c> functions in libcrypto.
    </p>

    <code>password() = string() | binary()</code>
    <p>The key's password
    </p>

     <code>stream_cipher() = rc4 | aes_ctr </code>

     <code>block_cipher() = aes_cbc | aes_cfb8 | aes_cfb128 | aes_ige256 | blowfish_cbc |
     blowfish_cfb64 | des_cbc | des_cfb | des3_cbc | des3_cfb | des_ede3 | rc2_cbc </code>

     <code>aead_cipher() = aes_gcm | chacha20_poly1305 </code>

     <code>stream_key() = aes_key() | rc4_key() </code>

     <code>block_key() = aes_key() |  blowfish_key() | des_key()| des3_key() </code>

     <code>aes_key() = iodata() </code> <p>Key length is 128, 192 or 256 bits</p>

     <code>rc4_key() = iodata() </code> <p>Variable key length from 8 bits up to 2048 bits (usually between 40 and 256)</p>

     <code>blowfish_key() = iodata() </code> <p>Variable key length from 32 bits up to 448 bits</p>

     <code>des_key() = iodata() </code> <p>Key length is 64 bits (in CBC mode only 8 bits are used)</p>

     <code>des3_key() = [binary(), binary(), binary()] </code> <p>Each key part is 64 bits (in CBC mode only 8 bits are used)</p>

     <code>digest_type() =  md5 | sha | sha224 | sha256 | sha384 | sha512</code>

     <code>rsa_digest_type() = md5 | ripemd160 | sha | sha224 | sha256 | sha384 | sha512</code>

     <code>dss_digest_type() = sha | sha224 | sha256 | sha384 | sha512</code> <p>Note that the actual supported
     dss_digest_type depends on the underlying crypto library. In OpenSSL version >= 1.0.1 the listed digest are supported, while in 1.0.0 only sha, sha224 and sha256 are supported. In version 0.9.8 only sha is supported.</p>

     <code>ecdsa_digest_type() = sha | sha224 | sha256 | sha384 | sha512</code>

     <code>sign_options() = [{rsa_pad, rsa_sign_padding()} | {rsa_pss_saltlen, integer()}]</code>

     <code>rsa_sign_padding() = rsa_pkcs1_padding | rsa_pkcs1_pss_padding</code>

     <code> hash_algorithms() =  md5 | ripemd160 | sha | sha224 | sha256 | sha384 | sha512 </code> <p>md4 is also supported for hash_init/1 and hash/2.
     Note that both md4 and md5 are recommended only for compatibility with existing applications.
     </p>
     <code> cipher_algorithms() = aes_cbc | aes_cfb8 | aes_cfb128 | aes_ctr | aes_gcm |
     aes_ige256 | blowfish_cbc | blowfish_cfb64 | chacha20_poly1305 | des_cbc |
     des_cfb | des3_cbc | des3_cfb | des_ede3 | rc2_cbc | rc4 </code>
     <code> mac_algorithms() = hmac | cmac</code>
     <code> public_key_algorithms() = rsa |dss | ecdsa | dh | ecdh | ec_gf2m</code>
     <p>Note that ec_gf2m is not strictly a public key algorithm, but a restriction on what curves are supported
     with ecdsa and ecdh.
     </p>
     <code>engine_method_type() = engine_method_rsa | engine_method_dsa | engine_method_dh |
     engine_method_rand | engine_method_ecdh | engine_method_ecdsa |
     engine_method_ciphers | engine_method_digests | engine_method_store |
     engine_method_pkey_meths | engine_method_pkey_asn1_meths</code>

 </section>

  <funcs>
    <func>
      <name>block_encrypt(Type, Key, PlainText) -> CipherText</name>
      <fsummary>Encrypt <c>PlainText</c> according to <c>Type</c> block cipher</fsummary>
      <type>
  <v>Type = des_ecb | blowfish_ecb | aes_ecb </v>
  <v>Key = block_key() </v>
        <v>PlainText = iodata() </v>
      </type>
      <desc>
        <p>Encrypt <c>PlainText</c> according to <c>Type</c> block cipher.</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>block_decrypt(Type, Key, CipherText) -> PlainText</name>
      <fsummary>Decrypt <c>CipherText</c> according to <c>Type</c> block cipher</fsummary>
      <type>
  <v>Type = des_ecb | blowfish_ecb | aes_ecb </v>
  <v>Key = block_key() </v>
        <v>PlainText = iodata() </v>
      </type>
      <desc>
        <p>Decrypt <c>CipherText</c> according to <c>Type</c> block cipher.</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>block_encrypt(Type, Key, Ivec, PlainText) -> CipherText</name>
      <name>block_encrypt(AeadType, Key, Ivec, {AAD, PlainText}) -> {CipherText, CipherTag}</name>
      <name>block_encrypt(aes_gcm, Key, Ivec, {AAD, PlainText, TagLength}) -> {CipherText, CipherTag}</name>
      <fsummary>Encrypt <c>PlainText</c> according to <c>Type</c> block cipher</fsummary>
      <type>
	<v>Type = block_cipher() </v>
	<v>AeadType = aead_cipher() </v>
	<v>Key = block_key() </v>
        <v>PlainText = iodata() </v>
        <v>AAD = IVec = CipherText = CipherTag = binary()</v>
        <v>TagLength = 1..16</v>
      </type>
      <desc>
        <p>Encrypt <c>PlainText</c> according to <c>Type</c> block cipher.
	<c>IVec</c> is an arbitrary initializing vector.</p>
	<p>In AEAD (Authenticated Encryption with Associated Data) mode, encrypt
	<c>PlainText</c>according to <c>Type</c> block cipher and calculate
	<c>CipherTag</c> that also authenticates the <c>AAD</c> (Associated Authenticated Data).</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>block_decrypt(Type, Key, Ivec, CipherText) -> PlainText</name>
      <name>block_decrypt(AeadType, Key, Ivec, {AAD, CipherText, CipherTag}) -> PlainText | error</name>
      <fsummary>Decrypt <c>CipherText</c> according to <c>Type</c> block cipher</fsummary>
      <type>
	<v>Type = block_cipher() </v>
	<v>AeadType = aead_cipher() </v>
	<v>Key = block_key() </v>
        <v>PlainText = iodata() </v>
        <v>AAD = IVec = CipherText = CipherTag = binary()</v>
      </type>
      <desc>
        <p>Decrypt <c>CipherText</c> according to <c>Type</c> block cipher.
	<c>IVec</c> is an arbitrary initializing vector.</p>
        <p>In AEAD (Authenticated Encryption with Associated Data) mode, decrypt
	<c>CipherText</c>according to <c>Type</c> block cipher and check the authenticity
	the <c>PlainText</c> and <c>AAD</c> (Associated Authenticated Data) using the
	<c>CipherTag</c>. May return <c>error</c> if the decryption or validation fail's</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>bytes_to_integer(Bin) -> Integer </name>
      <fsummary>Convert binary representation, of an integer, to an Erlang integer.</fsummary>
      <type>
	<v>Bin = binary() - as returned by crypto functions</v>

        <v>Integer = integer() </v>
      </type>
      <desc>
        <p>Convert binary representation, of an integer, to an Erlang integer.
         </p>
      </desc>
    </func>

    <func>
      <name>compute_key(Type, OthersPublicKey, MyKey, 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>MyKey = dh_private() | ecdh_private() | {srp_public(),srp_private()}</v>
	<v>Params = dh_params() | ecdh_params() | SrpUserParams | SrpHostParams</v>
	<v>SrpUserParams = {user, [DerivedKey::binary(), Prime::binary(), Generator::binary(), Version::atom() | [Scrambler:binary()]]} </v>
	<v>SrpHostParams = {host, [Verifier::binary(), Prime::binary(), Version::atom() | [Scrambler::binary]]} </v>
	<v>SharedSecret = binary()</v>
      </type>
      <desc>
	<p>Computes the shared secret from the private key and the other party's public key.
	 See also <seealso marker="public_key:public_key#compute_key-2">public_key:compute_key/2</seealso>
	</p>
      </desc>
    </func>

    <func>
      <name>exor(Data1, Data2) -> Result</name>
      <fsummary>XOR data</fsummary>
      <type>
	<v>Data1, Data2 = iodata()</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, PrivKeyOut} </name>
      <name>generate_key(Type, Params, PrivKeyIn) -> {PublicKey, PrivKeyOut} </name>
      <fsummary>Generates a public key of type <c>Type</c></fsummary>
      <type>
	<v> Type = dh | ecdh | rsa | srp </v>
	<v>Params = dh_params() | ecdh_params() | RsaParams | SrpUserParams | SrpHostParams </v>
	<v>RsaParams = {ModulusSizeInBits::integer(), PublicExponent::key_value()}</v>
	<v>SrpUserParams = {user, [Generator::binary(), Prime::binary(), Version::atom()]}</v>
	<v>SrpHostParams = {host, [Verifier::binary(), Generator::binary(), Prime::binary(), Version::atom()]}</v>
	<v>PublicKey =  dh_public() | ecdh_public() | rsa_public() | srp_public() </v>
	<v>PrivKeyIn = undefined | dh_private() | ecdh_private() | srp_private() </v>
	<v>PrivKeyOut =  dh_private() | ecdh_private() | rsa_private() | srp_private() </v>
      </type>
      <desc>
	<p>Generates a public key of type <c>Type</c>.
	See also <seealso marker="public_key:public_key#generate_key-1">public_key:generate_key/1</seealso>.
	May throw exception an exception of class <c>error</c>:
        </p>
        <list type="bulleted">
          <item><c>badarg</c>: an argument is of wrong type or has an illegal value,</item>
	  <item><c>low_entropy</c>: the random generator failed due to lack of secure "randomness",</item>
          <item><c>computation_failed</c>: the computation fails of another reason than <c>low_entropy</c>.</item>
        </list>
	<note>
	  <p>RSA key generation is only available if the runtime was
	  built with dirty scheduler support. Otherwise, attempting to
	  generate an RSA key will throw exception <c>error:notsup</c>.</p>
	</note>
      </desc>
    </func>

     <func>
      <name>hash(Type, Data) -> Digest</name>
      <fsummary></fsummary>
      <type>
	<v>Type = md4 | hash_algorithms()</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 | hash_algorithms()</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 = hash_algorithms() - except ripemd160</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> <p><c>MacLength</c>
        will limit the size of the resultant <c>Mac</c>.</p>
      </desc>
    </func>

    <func>
      <name>hmac_init(Type, Key) -> Context</name>
      <fsummary></fsummary>
      <type>
	<v>Type = hash_algorithms() - except ripemd160</v>
        <v>Key = iodata()</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 = iodata()</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>
	or to one of the functions <seealso marker="#hmac_final-1">hmac_final</seealso> and
	<seealso marker="#hmac_final_n-2">hmac_final_n</seealso>
	</p>
	<warning><p>Do not use a <c>Context</c> as argument in more than one
	call to	hmac_update or hmac_final. The semantics of reusing old contexts
	in any way is undefined and could even crash the VM in earlier releases.
	The reason for this limitation is a lack of support in the underlying
	OpenSSL	API.</p></warning>
      </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>cmac(Type, Key, Data) -> Mac</name>
      <name>cmac(Type, Key, Data, MacLength) -> Mac</name>
      <fsummary>Calculates the Cipher-based Message Authentication Code.</fsummary>
      <type>
        <v>Type = block_cipher()</v>
        <v>Key = iodata()</v>
        <v>Data = iodata()</v>
        <v>MacLength = integer()</v>
        <v>Mac = binary()</v>
      </type>
      <desc>
        <p>Computes a CMAC of type <c>Type</c> from <c>Data</c> using
        <c>Key</c> as the authentication key.</p> <p><c>MacLength</c>
        will limit the size of the resultant <c>Mac</c>.</p>
      </desc>
    </func>

    <func>
      <name>info_fips() -> Status</name>
      <fsummary>Provides information about the FIPS operating status.</fsummary>
      <type>
        <v>Status = enabled | not_enabled | not_supported</v>
      </type>
      <desc>
        <p>Provides information about the FIPS operating status of
        crypto and the underlying OpenSSL library. If crypto was built
        with FIPS support this can be either <c>enabled</c> (when
        running in FIPS mode) or <c>not_enabled</c>. For other builds
        this value is always <c>not_supported</c>.</p>
        <warning>
          <p>In FIPS mode all non-FIPS compliant algorithms are
          disabled and throw exception <c>not_supported</c>. Check
          <seealso marker="#supports-0">supports</seealso> that in
          FIPS mode returns the restricted list of available
          algorithms.</p>
        </warning>
      </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;,269484095,&lt;&lt;"OpenSSL 1.1.0c  10 Nov 2016""&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_pow(N, P, M) -> Result</name>
      <fsummary>Computes the function: N^P mod M</fsummary>
      <type>
	<v>N, P, M = binary() | integer()</v>
	<v>Result = binary() | error</v>
      </type>
      <desc>
	<p>Computes the function <c>N^P mod M</c>.</p>
      </desc>
    </func>

    <func>
   <name>next_iv(Type, Data) -> NextIVec</name>
   <name>next_iv(Type, Data, IVec) -> NextIVec</name>
   <fsummary></fsummary>
   <type>
     <v>Type = des_cbc | des3_cbc | aes_cbc | des_cfb</v>
     <v>Data = iodata()</v>
     <v>IVec = NextIVec = binary()</v>
   </type>
   <desc>
     <p>Returns the initialization vector to be used in the next
     iteration of encrypt/decrypt of type <c>Type</c>. <c>Data</c> is the
     encrypted data from the previous iteration step. The <c>IVec</c>
     argument is only needed for <c>des_cfb</c> as the vector used
     in the previous iteration step.</p>
   </desc>
    </func>

    <func>
      <name>private_decrypt(Type, CipherText, PrivateKey, Padding) -> PlainText</name>
      <fsummary>Decrypts CipherText using the private Key.</fsummary>
      <type>
	<v>Type = rsa</v>
	<v>CipherText = binary()</v>
	<v>PrivateKey = rsa_private() | engine_key_ref()</v>
	<v>Padding = rsa_pkcs1_padding | rsa_pkcs1_oaep_padding | rsa_no_padding</v>
        <v>PlainText = binary()</v>
      </type>
      <desc>
	<p>Decrypts the <c>CipherText</c>, encrypted with
	<seealso marker="#public_encrypt-4">public_encrypt/4</seealso> (or equivalent function)
	  using the <c>PrivateKey</c>, and returns the
	  plaintext (message digest). This is a low level signature  verification operation
	used for instance by older versions of the SSL protocol.
	  See also <seealso marker="public_key:public_key#decrypt_private-2">public_key:decrypt_private/[2,3]</seealso>
	</p>
      </desc>
    </func>

    <func>
      <name>privkey_to_pubkey(Type, EnginePrivateKeyRef) -> PublicKey</name>
      <fsummary>Fetches a public key from an Engine stored private key.</fsummary>
      <type>
	<v>Type = rsa | dss</v>
	<v>EnginePrivateKeyRef = engine_key_ref()</v>
	<v>PublicKey =  rsa_public() | dss_public()</v>
      </type>
      <desc>
	<p>Fetches the corresponding public key from a private key stored in an Engine.
	The key must be of the type indicated by the Type parameter.
	</p>	
	<p>	  
	  May throw exception notsup in case there is 
	  no engine support in the underlying OpenSSL implementation.
	</p>
      </desc>
    </func>

    <func>
      <name>private_encrypt(Type, PlainText, PrivateKey, Padding) -> CipherText</name>
      <fsummary>Encrypts PlainText using the private Key.</fsummary>
      <type>
	<v>Type = rsa</v>
	<v>PlainText = binary()</v>
	<d> The size of the <c>PlainText</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, where N is public modulus of the RSA key.</d>
	<v>PrivateKey = rsa_private() | engine_key_ref()</v>
	<v>Padding = rsa_pkcs1_padding | rsa_no_padding</v>
        <v>CipherText = binary()</v>
      </type>
      <desc>
	<p>Encrypts the <c>PlainText</c> using the <c>PrivateKey</c>
	and returns the ciphertext. This is a low level signature operation
	used for instance by older versions of the SSL protocol. See
	also <seealso
	marker="public_key:public_key#encrypt_private-2">public_key:encrypt_private/[2,3]</seealso>
	</p>
      </desc>
    </func>
    <func>
      <name>public_decrypt(Type, CipherText, PublicKey, Padding) -> PlainText</name>
      <fsummary>Decrypts CipherText using the public Key.</fsummary>
      <type>
	<v>Type = rsa</v>
	<v>CipherText = binary()</v>
	<v>PublicKey =  rsa_public() | engine_key_ref()</v>
	<v>Padding = rsa_pkcs1_padding | rsa_no_padding</v>
        <v>PlainText = binary()</v>
      </type>
      <desc>
	<p>Decrypts the <c>CipherText</c>, encrypted with
	<seealso marker="#private_encrypt-4">private_encrypt/4</seealso>(or equivalent function)
	  using the <c>PrivateKey</c>, and returns the
	  plaintext (message digest). This is a low level signature verification operation
	  used for instance by older versions of the SSL protocol.
	  See also <seealso marker="public_key:public_key#decrypt_public-2">public_key:decrypt_public/[2,3]</seealso>
	</p>
      </desc>
    </func>

    <func>
      <name>public_encrypt(Type, PlainText, PublicKey, Padding) -> CipherText</name>
      <fsummary>Encrypts PlainText using the public Key.</fsummary>
      <type>
	<v>Type = rsa</v>
	<v>PlainText = binary()</v>
	<d> The size of the <c>PlainText</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, where N is public modulus of the RSA key.</d>
	<v>PublicKey = rsa_public() | engine_key_ref()</v>
	<v>Padding = rsa_pkcs1_padding | rsa_pkcs1_oaep_padding | rsa_no_padding</v>
        <v>CipherText = binary()</v>
      </type>
      <desc>
	<p>Encrypts the <c>PlainText</c> (message digest) using the <c>PublicKey</c>
	and returns the <c>CipherText</c>. This is a low level signature operation
	used for instance by older versions of the SSL protocol. See also <seealso
	marker="public_key:public_key#encrypt_public-2">public_key:encrypt_public/[2,3]</seealso>
	</p>
      </desc>
    </func>

    <func>
      <name>rand_seed(Seed) -> ok</name>
      <fsummary>Set the seed for random bytes generation</fsummary>
      <type>
        <v>Seed = binary()</v>
      </type>
      <desc>
        <p>Set the seed for PRNG to the given binary. This calls the
        RAND_seed function from openssl. Only use this if the system
        you are running on does not have enough "randomness" built in.
        Normally this is when
        <seealso marker="#strong_rand_bytes/1">strong_rand_bytes/1</seealso>
        throws <c>low_entropy</c></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>
      <name>sign(Algorithm, DigestType, Msg, Key, Options) -> 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 "cleartext" data to be
	signed or it is the hashed value of "cleartext" i.e. the
	digest (plaintext).</d>
	<v>DigestType = rsa_digest_type() | dss_digest_type() | ecdsa_digest_type()</v>
	<v>Key = rsa_private() | dss_private() | [ecdh_private(),ecdh_params()] | engine_key_ref()</v>
	<v>Options = sign_options()</v>
      </type>
      <desc>
	<p>Creates a digital signature.</p>
	<p>Algorithm <c>dss</c> can only be used together with digest type
	<c>sha</c>.</p>
	<p>See also <seealso marker="public_key:public_key#sign-3">public_key:sign/3</seealso>.</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>rand_seed() -> rand:state()</name>
      <fsummary>Strong random number generation plugin state</fsummary>
      <desc>
	<p>
          Creates state object for
	  <seealso marker="stdlib:rand">random number generation</seealso>,
          in order to generate cryptographically strong random numbers
          (based on OpenSSL's <c>BN_rand_range</c>),
          and saves it on process dictionary before returning it as well.
          See also
	  <seealso marker="stdlib:rand#seed-1">rand:seed/1</seealso>.
	</p>
        <p><em>Example</em></p>
        <pre>
_ = crypto:rand_seed(),
_IntegerValue = rand:uniform(42), % [1; 42]
_FloatValue = rand:uniform().     % [0.0; 1.0[</pre>
      </desc>
    </func>

    <func>
      <name>rand_seed_s() -> rand:state()</name>
      <fsummary>Strong random number generation plugin state</fsummary>
      <desc>
	<p>
          Creates state object for
	  <seealso marker="stdlib:rand">random number generation</seealso>,
          in order to generate cryptographically strongly random numbers
          (based on OpenSSL's <c>BN_rand_range</c>).
          See also
	  <seealso marker="stdlib:rand#seed_s-1">rand:seed_s/1</seealso>.
	</p>
      </desc>
    </func>

    <func>
      <name>stream_init(Type, Key) -> State</name>
      <fsummary></fsummary>
      <type>
	<v>Type = rc4 </v>
        <v>State = opaque() </v>
        <v>Key = iodata()</v>
      </type>
      <desc>
        <p>Initializes the state for use in RC4 stream encryption
        <seealso marker="#stream_encrypt-2">stream_encrypt</seealso> and
        <seealso marker="#stream_decrypt-2">stream_decrypt</seealso></p>
      </desc>
    </func>

   <func>
      <name>stream_init(Type, Key, IVec) -> State</name>
      <fsummary></fsummary>
      <type>
	<v>Type = aes_ctr </v>
        <v>State = opaque() </v>
        <v>Key = iodata()</v>
        <v>IVec = binary()</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 bits long. <c>IVec</c> is
        an arbitrary initializing vector of 128 bits (16 bytes). This state is for use with
        <seealso marker="#stream_encrypt-2">stream_encrypt</seealso> and
        <seealso marker="#stream_decrypt-2">stream_decrypt</seealso>.</p>
      </desc>
    </func>

    <func>
      <name>stream_encrypt(State, PlainText) -> { NewState, CipherText}</name>
      <fsummary></fsummary>
      <type>
        <v>Text = iodata()</v>
        <v>CipherText = binary()</v>
      </type>
      <desc>
	<p>Encrypts <c>PlainText</c> according to the stream cipher <c>Type</c> specified in stream_init/3.
	<c>Text</c> can be any number of bytes. The initial <c>State</c> is created using
        <seealso marker="#stream_init-2">stream_init</seealso>.
	<c>NewState</c> must be passed into the next call to <c>stream_encrypt</c>.</p>
      </desc>
    </func>

    <func>
      <name>stream_decrypt(State, CipherText) -> { NewState, PlainText }</name>
      <fsummary></fsummary>
      <type>
        <v>CipherText = iodata()</v>
        <v>PlainText = binary()</v>
      </type>
      <desc>
	<p>Decrypts <c>CipherText</c> according to the stream cipher <c>Type</c> specified in stream_init/3.
	<c>PlainText</c> can be any number of bytes. The initial <c>State</c> is created using
        <seealso marker="#stream_init-2">stream_init</seealso>.
	<c>NewState</c> must be passed into the next call to <c>stream_decrypt</c>.</p>
      </desc>
    </func>

   <func>
      <name>supports() -> AlgorithmList </name>
      <fsummary>Provide a list of available crypto algorithms.</fsummary>
      <type>
      <v> AlgorithmList =  [{hashs, [hash_algorithms()]},
                            {ciphers, [cipher_algorithms()]},
			    {public_keys, [public_key_algorithms()]},
			    {macs, [mac_algorithms()]}]
      </v>
      </type>
      <desc>
        <p> Can be used to determine which crypto algorithms that are supported
	by the underlying OpenSSL library</p>
      </desc>
    </func>

    <func>
      <name>ec_curves() -> EllipticCurveList </name>
      <fsummary>Provide a list of available named elliptic curves.</fsummary>
      <type>
      <v>EllipticCurveList = [ec_named_curve()]</v>
      </type>
      <desc>
        <p>Can be used to determine which named elliptic curves are supported.</p>
      </desc>
    </func>

    <func>
      <name>ec_curve(NamedCurve) -> EllipticCurve </name>
      <fsummary>Get the defining parameters of a elliptic curve.</fsummary>
      <type>
      <v>NamedCurve = ec_named_curve()</v>
      <v>EllipticCurve = ec_explicit_curve()</v>
      </type>
      <desc>
        <p>Return the defining parameters of a elliptic curve.</p>
      </desc>
    </func>

  <func>
      <name>verify(Algorithm, DigestType, Msg, Signature, Key) -> boolean()</name>
      <name>verify(Algorithm, DigestType, Msg, Signature, Key, Options) -> 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 "cleartext" data
        or it is the hashed value of "cleartext" i.e. the digest (plaintext).</d>
	<v>DigestType = rsa_digest_type() | dss_digest_type() | ecdsa_digest_type()</v>
	<v>Signature = binary()</v>
	<v>Key = rsa_public() | dss_public() | [ecdh_public(),ecdh_params()] | engine_key_ref()</v>
	<v>Options = sign_options()</v>
      </type>
      <desc>
	<p>Verifies a digital signature</p>
	<p>Algorithm <c>dss</c> can only be used together with digest type
	<c>sha</c>.</p>

	<p>See also <seealso marker="public_key:public_key#verify-4">public_key:verify/4</seealso>.</p>
      </desc>
    </func>

    <!-- Engine functions -->
    <func>
      <name>engine_get_all_methods() -> Result</name>
      <fsummary>Return list of all possible engine methods</fsummary>
      <type>
	<v>Result = [EngineMethod::atom()]</v>
      </type>
      <desc>
	<p>
	  Returns a list of all possible engine methods.
	</p>
	<p>	  
	  May throw exception notsup in case there is 
	  no engine support in the underlying OpenSSL implementation.
	</p>
	<p>
	  See also the chapter <seealso marker="crypto:engine_load#engine_load">Engine Load</seealso>
	  in the User's Guide.
	</p>
      </desc>
    </func>

    <func>
      <name>engine_load(EngineId, PreCmds, PostCmds) -> Result</name>
      <fsummary>Dynamical load an encryption engine</fsummary>
      <type>
	<v>EngineId = unicode:chardata()</v>
	<v>PreCmds, PostCmds = [{unicode:chardata(), unicode:chardata()}]</v>
	<v>Result = {ok, Engine::term()} | {error, Reason::term()}</v>
      </type>
      <desc>
	<p>
	  Loads the OpenSSL engine given by <c>EngineId</c> if it is available and then returns ok and
	  an engine handle. This function is the same as calling <c>engine_load/4</c> with 
	  <c>EngineMethods</c> set to a list of all the possible methods.  An error tuple is 
	  returned if the engine can't be loaded.
	</p>
	<p>
	  The function throws a badarg if the parameters are in wrong format.
	  It may also throw the exception notsup in case there is 
	  no engine support in the underlying OpenSSL implementation.
	</p>
	<p>
	  See also the chapter <seealso marker="crypto:engine_load#engine_load">Engine Load</seealso>
	  in the User's Guide.
	</p>
      </desc>
    </func>

    <func>
      <name>engine_load(EngineId, PreCmds, PostCmds, EngineMethods) -> Result</name>
      <fsummary>Dynamical load an encryption engine</fsummary>
      <type>
	<v>EngineId = unicode:chardata()</v>
	<v>PreCmds, PostCmds = [{unicode:chardata(), unicode:chardata()}]</v>
	<v>EngineMethods = [engine_method_type()]</v>
	<v>Result = {ok, Engine::term()} | {error, Reason::term()}</v>
      </type>
      <desc>
	<p>
	  Loads the OpenSSL engine given by <c>EngineId</c> if it is available and then returns ok and
	  an engine handle. An error tuple is returned if the engine can't be loaded.
	</p>
	<p>
	  The function throws a badarg if the parameters are in wrong format.
	  It may also throw the exception notsup in case there is 
	  no engine support in the underlying OpenSSL implementation.
	</p>
	<p>
	  See also the chapter <seealso marker="crypto:engine_load#engine_load">Engine Load</seealso>
	  in the User's Guide.
	</p>
      </desc>
    </func>

    <func>
      <name>engine_unload(Engine) -> Result</name>
      <fsummary>Dynamical load an encryption engine</fsummary>
      <type>
	<v>Engine = term()</v>
	<v>Result = ok | {error, Reason::term()}</v>
      </type>
      <desc>
	<p>
	  Unloads the OpenSSL engine given by <c>EngineId</c>.
	  An error tuple is returned if the engine can't be unloaded.
	</p>
	<p>
	  The function throws a badarg if the parameter is in wrong format.
	  It may also throw the exception notsup in case there is 
	  no engine support in the underlying OpenSSL implementation.
	</p>
	<p>
	  See also the chapter <seealso marker="crypto:engine_load#engine_load">Engine Load</seealso>
	  in the User's Guide.
	</p>
      </desc>
    </func>

    <func>
      <name>engine_list() -> Result</name>
      <fsummary>List the known engine ids</fsummary>
      <type>
	<v>Result = [EngineId::unicode:chardata()]</v>
      </type>
      <desc>
	<p>List the id's of all engines in OpenSSL's internal list.</p>
	<p>
	  It may also throw the exception notsup in case there is 
	  no engine support in the underlying OpenSSL implementation.
	</p>
	<p>
	  See also the chapter <seealso marker="crypto:engine_load#engine_load">Engine Load</seealso> 
	  in the User's Guide.
	</p>
      </desc>
    </func>

 </funcs>

  <!-- Maybe put this in the users guide -->
  <!-- <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>