path: root/70/1c3d0e820aaa1d3e524266c008a01c935a7405

Received: from sog-mx-4.v43.ch3.sourceforge.com ([172.29.43.194]
	helo=mx.sourceforge.net)
	by sfs-ml-2.v29.ch3.sourceforge.com with esmtp (Exim 4.76)
	(envelope-from <mark@monetize.io>) id 1VtpM3-0007jq-A5
	for bitcoin-development@lists.sourceforge.net;
	Fri, 20 Dec 2013 01:58:19 +0000
Received: from mail-pb0-f46.google.com ([209.85.160.46])
	by sog-mx-4.v43.ch3.sourceforge.com with esmtps (TLSv1:RC4-SHA:128)
	(Exim 4.76) id 1VtpM0-0003Cp-WC
	for bitcoin-development@lists.sourceforge.net;
	Fri, 20 Dec 2013 01:58:19 +0000
Received: by mail-pb0-f46.google.com with SMTP id md12so1930490pbc.19
	for <bitcoin-development@lists.sourceforge.net>;
	Thu, 19 Dec 2013 17:58:10 -0800 (PST)
X-Google-DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed;
	d=1e100.net; s=20130820;
	h=x-gm-message-state:message-id:date:from:organization:user-agent
	:mime-version:to:subject:content-type:content-transfer-encoding;
	bh=zj2nb2fPFLuFAASGxUTnCHPs8yHMQ9rx2EXnZ8wvLSk=;
	b=Q1PhifIwFZxMRolAIirhdxznQ6MWJlMyerMit4RGXwFU3XLwlgWOV/3S5VtuJQ7jt3
	2GrUmZc27awEBlA6x3hq8WnLeTg3h/Pqg/b16uxeGbZRJnhozYCjdF7E+lDSRI66kKwD
	hqC5a3eFv/AgQcUSfQPOoPItReq7imIkpTCbWRM9zoFwmqEkAW4UJSfy6DSl3OrgSenA
	NT59+xspB1X6wRNuROwNLaC/Lp1PiUkDPADc4ZV6HDtlG7LeHIjCOrDf/TRTtXX4CMmO
	wOgPfXuJmZypbFL9cKqZWJmkusoC+9EptFiFXj1eEjt07LtZgcfCGq6vYXKqFujzO3pk
	mOpQ==
X-Gm-Message-State: ALoCoQkKos/5XsQoKeNgZNXx4ThauCrwG6DxBnItqXlWIdDEbLaQGgOAVZRk19CazcH7ml/T6g7l
X-Received: by 10.66.162.195 with SMTP id yc3mr5480685pab.64.1387504690679;
	Thu, 19 Dec 2013 17:58:10 -0800 (PST)
Received: from [192.168.127.177] (50-0-36-217.dsl.dynamic.sonic.net.
	[50.0.36.217]) by mx.google.com with ESMTPSA id
	pe3sm10332366pbc.23.2013.12.19.17.58.08
	for <bitcoin-development@lists.sourceforge.net>
	(version=TLSv1 cipher=ECDHE-RSA-RC4-SHA bits=128/128);
	Thu, 19 Dec 2013 17:58:09 -0800 (PST)
Message-ID: <52B3A1C8.5000005@monetize.io>
Date: Thu, 19 Dec 2013 17:47:52 -0800
From: Mark Friedenbach <mark@monetize.io>
Organization: Monetize.io Inc.
User-Agent: Mozilla/5.0 (X11; Linux x86_64;
	rv:24.0) Gecko/20100101 Thunderbird/24.2.0
MIME-Version: 1.0
To: Bitcoin Dev <bitcoin-development@lists.sourceforge.net>
X-Enigmail-Version: 1.6
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit
X-Spam-Score: 0.0 (/)
X-Spam-Report: Spam Filtering performed by mx.sourceforge.net.
	See http://spamassassin.org/tag/ for more details.
	0.0 URIBL_BLOCKED ADMINISTRATOR NOTICE: The query to URIBL was blocked.
	See
	http://wiki.apache.org/spamassassin/DnsBlocklists#dnsbl-block
	for more information. [URIs: wikipedia.org]
X-Headers-End: 1VtpM0-0003Cp-WC
Subject: [Bitcoin-development] BIP proposal: Authenticated prefix trees
X-BeenThere: bitcoin-development@lists.sourceforge.net
X-Mailman-Version: 2.1.9
Precedence: list
List-Id: <bitcoin-development.lists.sourceforge.net>
List-Unsubscribe: <https://lists.sourceforge.net/lists/listinfo/bitcoin-development>,
	<mailto:bitcoin-development-request@lists.sourceforge.net?subject=unsubscribe>
List-Archive: <http://sourceforge.net/mailarchive/forum.php?forum_name=bitcoin-development>
List-Post: <mailto:bitcoin-development@lists.sourceforge.net>
List-Help: <mailto:bitcoin-development-request@lists.sourceforge.net?subject=help>
List-Subscribe: <https://lists.sourceforge.net/lists/listinfo/bitcoin-development>,
	<mailto:bitcoin-development-request@lists.sourceforge.net?subject=subscribe>
X-List-Received-Date: Fri, 20 Dec 2013 01:58:19 -0000

-----BEGIN PGP SIGNED MESSAGE-----
Hash: SHA1

Hello fellow bitcoin developers. Included below is the first draft of
a BIP for a new Merkle-compressed data structure. The need for this
data structure arose out of the misnamed "Ultimate blockchain
compression" project, but it has since been recognized to have many
other applications.

In addition to this BIP I am preparing three additional BIPs
describing the use of this data structure in stateless validation &
mining, the UBC address index for "SPV+" operating modes, document
timestamping and merged mining.

A Python implementation of this data structure is available here:

https://github.com/monetizeio/python-bitcoin

A C++ implementation is being worked on.

As per the BIP-1 procedure, I am submitting this rough draft to the
community for discussion. I welcome all comments and criticisms of
both form and content.

- -Mark


==Abstract==

This BIP describes a [http://en.wikipedia.org/wiki/Hash_tree Merkle
hash tree] variant of the [http://en.wikipedia.org/wiki/Trie
prefix-tree data structure], ideally suited for encoding key-value
indices which support memory-efficient proofs.

==Motivation==

There are a number of applications which would benefit from having a
data structure with the following properties:

* '''Arbitrary mapping of keys to values.''' A ''key'' can be any
bytestring, and its ''value'' any other bytestring.
* '''Duplicate keys disallowed.''' Every key has one, and only one
value associated with it. Some applications demand assurance that no
key value is reused, and that this constraint can be checked without
requiring access to the entire data structure.
* '''Efficient look-up by key.''' The data structure should support
sub-linear lookup operations with respect to the number of keys in the
mapping. Logarithmic time or linear with respect to the length of the
key should be achievable and would be sufficient for realistic
applications.
* '''Merkle compression of mapping structure.''' It should be possible
to produce a reduced description of the tree consisting of a single
root hash value which is deterministically calculated from the mapping
structure.
* '''Efficient proofs of inclusion.''' It should be possible to
extract a proof of key/value mapping which is limited in size and
verification time by the length of the key in the worst case.
* '''Computation of updates using local information.''' Given a set of
inclusion proofs, it should be possible to calculate adjustments to
the local mapping structure (update or deletion of included mappings,
or insertion between two included mappings which are adjacent in the
global structure).

Such applications include committed validation indices which enable
stateless mining nodes, committed wallet indices which enable
trust-less querying of the unspent transaction output set by
<code>scriptPubKey</code>, efficient document time-stamping, and
secure & efficient merged mining. This BIP describes an authenticated
prefix tree which has the above properties, but leaves the myriad
applications to be formalized in future BIPs.

==Data structure==

This BIP defines a binary prefix tree. Such a structure provides a
mapping of bitstrings (the ''keys'') to bytestrings (the ''values'').
It is an acyclic binary tree which implicitly encodes keys within the
traversal path -- a "left" branch is a 0, and a "right" branch is a 1.
Each node is reachable by only one unique path, and reading off the
branches taken (0 for each left, 1 for each right) as one follows the
path from root to target yields the node's key.

The particular binary prefix tree defined by this BIP is a hybrid
PATRICIA / de la Brandais tree structure.
[http://en.wikipedia.org/wiki/Radix_tree PATRICIA trees] compress a
long sequence of non-branching nodes into a single interior node with
a per-branch ''skip prefix''. This achieves significant savings in
storage space, root hash calculation, and traversal time.

A de la Brandais trie achieves compression by only storing branches
actually taken in a node. The space savings are minimal for a binary
tree, but place the serialized size of a non-branching interior node
under the SHA-256 block size, thereby reducing the number of hash
operations required to perform updates and validate proofs.

This BIP describes the authenticated prefix tree and its many
variations in terms of its serialized representation. Additional BIPs
describe the application of authenticated prefix trees to such
applications as committed indices, document time-stamping, and merged
mining.

==Serialization format==

As a hierarchical structure, the serialization of an entire tree is
the serialization of its root node. A serialized node is the
concatenation of five structures:

    node := flags || VARCHAR(extra) || value || left || right

The <code>flags</code> is a single byte field whose composite values
determine the bytes that follow.

    flags = (left_flags  << 0) |
            (right_flags << 2) |
            (has_value   << 4) |
            (prune_left  << 5) |
            (prune_right << 6) |
            (prune_value << 7)

The <code>left_flags</code> and <code>right_flags</code> are special
2-bit enumeration fields. A value of 0 indicates that the node does
not branch in this direction, and the corresponding <code>left</code>
or <code>right</code> branch is missing (replaced with the empty
string in the node serialization). A value of 1 indicates a single bit
key prefix for this branch, implicitly 0 for <code>left</code> and 1
for <code>right</code>. A 2 indicates up to 7 bits of additional skip
prefix (beyond the implicit first bit, making 8 bits total) are stored
in a compact single-byte format. A 3 indicates a skip prefix with
greater than 7 additional bits, stored length-prefix encoded.

The single bit <code>has_value</code> indicates whether the node
stores a data bytestring, the value associated with its key prefix.
Since keys may be any value or length, including one key being a
prefix of another, it is possible for interior nodes in addition to
leaf nodes to have values associated with them, and therefore an
explicit value-existence bit is required.

The remaining three bits are used for proof extraction, and are masked
away prior to hash operations. <code>prune_left</code> indicates that
the entire left branch has been pruned. <code>prune_right</code> has
similar meaning for the right branch. If <code>has_value</code> is
set, <code>prune_value</code> may be set to exclude the node's value
from encoded proof. This is necessary field for interior nodes, since
it is possible that their values may be pruned while their children
are not.

The <code>value</code> field is only present if the bit
<code>flags.has_value</code> is set, in which case it is a
<code>VARCHAR</code> bytestring:

    switch flags.has_value:
      case 0:
        value := ε
      case 1:
        value := VARCHAR(node.value)

The <code>extra</code> field is always present, and takes on a
bytestring value defined by the particular application. Use of the
<code>extra</code> field is application dependent, and will not be
covered in this specification. It can be set to the empty bytestring
(serialized as a single zero byte) if the application has no use for
the <code>extra</code> field.

    value := VARCHAR(calculate_extra(node))

The <code>left</code> and <code>right</code> non-terminals are only
present if the corresponding <code>flags.left_flags</code> or
<code>flags.right_flags</code> are non-zero. The format depends on the
value of this flags setting:

    switch branch_flags:
      case 0:
        branch := ε
      case 1:
        branch := branch_node_or_hash
      case 2:
        prefix  = prefix >> 1
        branch := int_to_byte(1 << len(prefix) | bits_to_int(prefix)) ||
                  branch_node_or_hash
      case 3:
        prefix  = prefix >> 1
        branch := VARINT(len(prefix) - 9) ||
                  bits_to_string(prefix) ||
                  branch_node_or_hash

<code>branch_flags</code> is a stand-in meant to describe either
<code>left_flags</code> or <code>right_flags</code>, and likewise
everywhere else in the above pseudocode <code>branch</code> can be
replaced with either <code>left</code> or <code>right</code>.

<code>prefix</code> is the key bits between the current node and the
next branching, terminal, and/or leaf node, including the implicit
leading bit for the branch (0 for the left branch, 1 for the right
branch). In the above code, <code>len(prefix)</code> returns the
number of bits in the bitstring, and <code>prefix >> 1</code> drops
the first bit reducing the size of the bitstring by one and
renumbering the indices accordingly.

The function <code>int_to_byte</code> takes an integer in the range
[0, 255] and returns the octet representing that value. This is a NOP
in many languages, but present in this pseudocode so as to be explicit
about what is going on.

The function <code>bits_to_int</code> interprets a sequence of bits as
a little-endian integer value. This is analogous to the following
pseudocode:

    def bits_to_int(bits):
        result = 0
        for idx in 1..len(bits):
            if bits[idx] == 1:
                result |= 1<<idx

The function <code>bits_to_string</code> serializes a sequence of bits
into a binary string. It uses little-endian bit and byte order, as
demonstrated by the following pseudocode:

    def bits_to_string(bits):
        bytes = [0] * ceil(len(bits) / 8)
        for idx in 1..len(bits):
            if bits[idx] == 1:
                bytes[idx / 8] |= 1 << idx % 8
        return map(int_to_byte, bytes)

<code>branch_node_or_hash</code> is either the serialized child node
or its SHA-256 hash and associated meta-data. Context determines which
value to use: during digest calculations, disk/database serialization,
and when the branch is pruned the hash value is used and serialized in
the same way as other SHA-256 values in the bitcoin protocol (note
however that it is single-SHA-256, not the double-SHA-256 more
commonly used in bitcoin). The number of terminal (value-containing)
nodes and the serialized size in bytes of the fully unpruned branch
are suffixed to the branch hash. When serializing a proof or
snapshotting tree state and the branch is not pruned, the serialized
child node is included directly and the count and size are omitted as
they can be derived from the serialization.

    if branch_pruned or SER_HASH:
        branch_node_or_hash := SHA-256(branch) ||
                               count(branch) ||
                               size(branch)
    else:
        branch_node_or_hash := serialize(branch)

As an example, here is the serialization of a prefix tree mapping the
names men and women of science to the year of their greatest publication:

    >>> dict = AuthTree()
    >>> dict['Curie'] = VARINT(1898)
    >>> dict('Einstein') = VARINT(1905)
    >>> dict['Fleming'] = VARINT(1928)
    >>> dict['中本'] = VARINT(2009)
    >>> dict.serialize()
    # An bytestring, broken out into parts:

    # . Root node:
    0x0e # left_flags: 2, right_flags: 3, has_value: 1
    0x00 # extra: ε

    # .l Inner node: 0b01000
    0x11 # 0b01000
    0x07 # left_flags: 3, right_flags: 1
    0x00 # extra: ε

    # .l.l Inner node: 0b01000011 0b01110101 0b01110010 0b01101001
    #                  'C'        'u'        'r'        'i'
    #                  0b01100101
    #                  'e'
    0x1abb3a599a02 # 0b01101110101011100100110100101100101
    0x10           # has_value: 1
    0x00           # extra: ε
    0x03fd6a07     # value: VARINT(1911)

    # .l.r Inner node: 0b010001
    0x0f # left_flags: 3, right_flags: 3
    0x00 # extra: ε

    # .l.r.l Inner node: 0b01000101 0b01101001 0b01101110 0b01110011
    #                    'E'        'i'        'n'        's'
    #                    0b01110100 0b01100101 0b01101001 0b01101110
    #                    't'        'e'        'i'        'n'
    0x312ded9c5d4c2ded00 # 0b1011010010110111
                         # 0b0011100110111010
                         # 0b0011001010110100
                         # 0b101101110
    0x10                 # has_value: 1
    0x00                 # extra: ε
    0x03fd7107           # value: VARINT(1905)

    # .l.r.r Inner node: 0b01000110 0b01101100 0b01100101 0b01101101
    #                    'F'        'l'        'e'        'm'
    #                    0b01101001 0b01101110 0b01100111
    #                    'i'        'n'        'g'
    0x296c4c6d2dedcc01 # 0b0011011000110010
                       # 0b1011011010110100
                       # 0b10110111001100111
    0x10               # has_value: 1
    0x00               # extra: ε
    0x03fd8807         # value: VARINT(1928)

    # .r Inner node: 0b11100100 0b10111000 0b10101101
    #                '中'
    #                0b11100110 0b10011100 0b10101100
    #                '本'
    0x27938edab39c1a # 0b1100100101110001
                     # 0b0101101111001101
                     # 0b001110010101100
    0x10             # has_value: 1
    0x00             # extra: ε
    0x03fdd907       # value: VARINT(2009)

==Hashing==

There are two variations of the authenticated prefix tree presented in
this draft BIP. They differ only in the way in which hash values of a
node and its left/right branches are constructed. The variations,
discussed below, tradeoff computational resources for the ability to
compose operational proofs. Whether the performance hit is
significant, and whether or not the added features are worth the
tradeoff depends very much on the application.

===Variation 1: Level-compressed hashing===

In this variation the referenced child node's hash is used in
construction of an interior node's hash digest. The interior node is
serialized just as described (using the child node's digest instead of
inline serialization), the resulting bytestring is passed through one
round of SHA-256, and the digest that comes out of that is the hash
value of the node. This is very efficient to calculate, requiring the
absolute minimum number of SHA-256 hash operations, and achieving
level-compression of computational resources in addition to reduction
of space usage.

For example:

    >>> dict = AuthTree()
    >>> dict['a'] = 0xff
    >>> dict.serialize()
    0x0200c3100001ff
    >>> dict.root
    AuthTreeNode(
        left_prefix = 0b01100001,
        left_hash   =
0xbafa0e2bba3396c5e9804b6cbe61be82bc442c1121aed81f8d5de36e9b20dc2f,
        left_count  = 1,
        left_size   = 4)
    >>> dict.hash
    0xb4837376022a7c9ddaa7d685ad183bcbd5d16c362b81fa293a7b9e911766cf3c

Assuming uniform distribution of key values, level-compressed hashing
has time-complexity logarithmic with respect to the number of keys in
the prefix tree. The disadvantage is that it is not possible in
general to "rebase" an operational proof on top of a sibling,
particularly if that sibling deletes branches that result in
reorganization and level compression of internal nodes used by the
rebased proof.

===Variation 2: Proof-updatable hashing===

In this variation, level-compressed branches are expanded into a
series of chained single-branch internal nodes, each including the
hash of its direct child. For a brach with a prefix N bits in length,
this requires N chained hashes. Thanks to node-compression (excluding
empty branches from the serialization), it is possible for each hash
operation + padding to fit within a single SHA-256 block.

Note that the serialization semantics are unchanged! The variation
only changes the procedure for calculating the hash values of interior
nodes. The serialization format remains the same (modulo differing
hash values standing in for pruned branches).

Using the above example, calling <code>dict.hash</code> causes the
following internal nodes to be constructed:

    >>> node1 = AuthTreeNode(
        right_prefix = 0b1,
        right_hash   =
0xbafa0e2bba3396c5e9804b6cbe61be82bc442c1121aed81f8d5de36e9b20dc2f,
        right_count  = 1,
        right_size   = 4)
    >>> node2 = AuthTreeNode( left_prefix=0b0,  left_hash=node1.hash,
 left_count=1,  left_size=4)
    >>> node3 = AuthTreeNode( left_prefix=0b0,  left_hash=node2.hash,
 left_count=1,  left_size=4)
    >>> node4 = AuthTreeNode( left_prefix=0b0,  left_hash=node3.hash,
 left_count=1,  left_size=4)
    >>> node5 = AuthTreeNode( left_prefix=0b0,  left_hash=node4.hash,
 left_count=1,  left_size=4)
    >>> node6 = AuthTreeNode(right_prefix=0b1, right_hash=node5.hash,
right_count=1, right_size=4)
    >>> node7 = AuthTreeNode(right_prefix=0b1, right_hash=node6.hash,
right_count=1, right_size=4)
    >>> node8 = AuthTreeNode( left_prefix=0b0,  left_hash=node7.hash,
 left_count=1,  left_size=4,
                              value=0xff)
    >>> dict.hash == node8.hash
    True
    >>> dict.hash
    0xc3a9328eff06662ed9ff8e82aa9cc094d05f70f0953828ea8c643c4679213895

The advantage of proof-updatable hashing is that any operational proof
may be "rebased" onto the tree resulting from a sibling proof, using
only the information locally available in the proofs, even in the
presence of deletion operations that result in level-compression of
the serialized form. The disadvantage is performance: validating an
updatable proof requires a number of hash operations lower-bounded by
the length of the key in bits.

==Inclusion proofs==

An inclusion proof is a prefix tree pruned to contain a subset of its
keys. The serialization of an inclusion proof takes the following form:

    inclusion_proof := variant || root_hash || root_node || checksum

Where <code>variant</code> is a single-byte value indicating the
presence of level-compression (0 for proof-updatable hashing, 1 for
level-compressed hashing). <code>root_hash</code> is the Merkle
compression hash of the tree, the 32-byte SHA-256 hash of the root
node. <code>tree</code> is the possibly pruned, serialized
representation of the tree. And finally, <code>checksum</code> is the
first 4 bytes of the SHA-256 checksum of <code>variant</code>,
<code>root_hash</code>, and <code>root_node</code>.

For ease of transport, the standard envelope for display of an
inclusion proof is internet-standard base64 encoding in the following
format:

- -----BEGIN INCLUSION PROOF-----
ATzPZheRnns6KfqBKzZs0dXLOxithdan2p18KgJ2c4O0DgARBwAauzpZmgIQAAP9agcPADEt7Zxd
TC3tABAAA/1xBylsTG0t7cwBEAAD/YgHJ5OO2rOcGhAAA/3ZByEg+2g=
- -----END INCLUSION PROOF-----

Decoded, it looks like this:

    0x01 # Level-compressed hashing
    # Merkle root:
    0x3ccf6617919e7b3a29fa812b366cd1d5cb3b18ad85d6a7da9d7c2a02767383b4
    # Serialized tree (unpruned):
    0x0e001107001abb3a599a02100003fd6a070f00312ded9c5d4c2ded00100003fd
    0x7107296c4c6d2dedcc01100003fd880727938edab39c1a100003fdd907
    # Checksum:
    0x2120fb68

==Operational proofs==

An operational proof is a list of insert/update and delete operations
suffixed to an inclusion proof which contains the pathways necessary
to perform the specified operations. The inclusion proof must contain
the key values to be updated or deleted, and the nearest adjacent key
values for each insertion. The serialization of an operational proof
takes the following form:

    operational_proof := variant || root_hash || tree ||
                         VARLIST(delete*) || VARLIST(update*) ||
                         new_hash || checksum

    delete := VARCHAR(key)
    update := VARCHAR(key) || VARCHAR(value)

The first three fields, <code>variant</code>, <code>root_hash</code>,
and <code>tree</code> are the inclusion proof, and take the same
values described in the previous section. <code>deletes</code> is a
list of key values to be deleted; each key value in this list must
exist in the inclusion proof. <code>updates</code> is a list of key,
value mappings which are to be inserted into the tree, possibly
replacing any mapping for the key which already exists; either the key
itself if it exists (update), or the two lexicographically closest
keys on either side if it does not (insert) must be present in the
insertion proof. <code>new_hash</code> is the resulting Merkle root
after the insertion, updates, and deletes are performed, and
<code>checksum</code> is the initial 4 bytes of the SHA-256 hash of
the preceding fields.

Just like inclusion proofs, an operational proof is encoded in base64
for display and transport. Here's the same

- -----BEGIN OPERATIONAL PROOF-----
ATzPZheRnns6KfqBKzZs0dXLOxithdan2p18KgJ2c4O0LgARaIsVaQi/GdhOPOgA8p4Pu4PiEfEg
lcmy3j7bOc7hXw0DLSeTjtqznBoQAAP92QcBMOS4reacrACzuZJbyP7fqIOf5VEk4iarG4+uPoZC
oun8BztQMQBy0LHVeSY=
- -----END OPERATIONAL PROOF-----

Decoded and broken into its constituent fields:

    0x01 # Level-compressed hashing
    # Original Merkle root:
    0x3ccf6617919e7b3a29fa812b366cd1d5cb3b18ad85d6a7da9d7c2a02767383b4
    # Serialized tree (included keys: '中本'):
    0x2e0011688b156908bf19d84e3ce800f29e0fbb83e211f12095c9b2de3edb39ce
    0xe15f0d032d27938edab39c1a100003fdd907
    # Deletion list ['中本']:
    0x01
    0x30e4b8ade69cac
    # Insertion list []:
    0x00
    # New Merkle root:
    0xb3b9925bc8fedfa8839fe55124e226ab1b8fae3e8642a2e9fc073b50310072d0
    # Checksum:
    0xb1d57926

~End of File~
-----BEGIN PGP SIGNATURE-----
Version: GnuPG v1.4.14 (GNU/Linux)
Comment: GPGTools - http://gpgtools.org
Comment: Using GnuPG with Thunderbird - http://www.enigmail.net/

iQIcBAEBAgAGBQJSs6HIAAoJEAdzVfsmodw4gooQAJm7XNsZjgdeTSpKIvUIU38f
tQx2FD08hQdLl48me5mDUbHJgGlYINsKAgoZ8Mqwi/kHEEYhuIlLIX1p6Ovigidb
21BiVoOLdG1egGOwxp17DuwYaDPTppFTlN9TBjZzW6WKc7+4aNvyc1KtrbHIhtj/
04ekFyAn4U5UH0ht7CI79j0u3Kp85p5D4PyYZB2m82mzti6OxpSM4tXlMkDW7ihg
QJwiZSjzejqTd7WF0zr0SLeGVRSN2A0dzUCoVsI98eIa3hkw2N4ae6dRkibyStOT
V8VEDvHArEDlvu8jiryajhsom5mvtOOclNDkVXWAf/Te4gj05iYdTIvNvDEJtqsP
XDbmw6GgV1kBLlLo0mp//t/+wr+nIvy+sVAP+eqtM/0vjaVXBkXxkUMqqNkrtVpB
f3whq7nFahssUMSoWE93jgob1ayAax2XUALVMAXYsJl7b2MqBGlhiTZ8FQZ+TW4A
tIpKeUprPmDvA18rO3SCbmLMQryZqYiH0sRyvUc5kvn3qCRHrISZNkEuK591eS+x
BO1eOluPzVqeXPPSK1jvGeY0FNJtwzbov4nI1mzOvzQHLCvkHn5PhUFCK5tL5tAe
b0Z5qwDV+SvVs7W1R7ejYBzEj77U1zuzZ9AtikOuvy+bNGrkIlpI49EyXHijm7C3
Q6JacTuI0PelYji2gaBJ
=BbDs
-----END PGP SIGNATURE-----