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<h1 id="what-are-interchangable-passphrases-in-vault-and-what-does-that-mean-in-practice">What are &ldquo;interchangable passphrases&rdquo; in <code>vault</code>, and what does that mean in practice?<a class="headerlink" href="#what-are-interchangable-passphrases-in-vault-and-what-does-that-mean-in-practice" title="Permanent link">&para;</a></h1>
<h2 id="what-are-interchangable-passphrases">What are &ldquo;interchangable passphrases&rdquo;?<a class="headerlink" href="#what-are-interchangable-passphrases" title="Permanent link">&para;</a></h2>
<p>The &ldquo;vault&rdquo; derivation scheme internally uses PBKDF2-HMAC-SHA1<sup id="fnref:1"><a class="footnote-ref" href="#fn:1">1</a></sup> to turn
the master passphrase<sup id="fnref:2"><a class="footnote-ref" href="#fn:2">2</a></sup> into a pseudo-random bit sequence, which then
drives the actual passphrase derivation.
In this context, the master passphrase is passed directly as a key to
HMAC-SHA1, and because HMAC-SHA1 requires keys of exactly 64 bytes size, the
key is thus subject to the HMAC key mapping procedure.
Because the mapping of infinitely many arbitrarily sized keys to 64-byte
sized keys cannot be one-to-one, there exist pairs of keys that behave
identically when passed into (PBKDF2-)HMAC-SHA1, i.e., the keys (master
passphrases) are &ldquo;interchangable&rdquo; from the vault scheme&rsquo;s perspective.</p>
<p>Fundamentally, this is an issue of <em>encoding</em>: the master passphrase is
interpreted as an encoding of the HMAC-SHA1 key, and this encoding is not
unique, so the effective space of HMAC-SHA1 keys is reduced through the
presence of &ldquo;non-canonical&rdquo; encodings of keys.</p>
<h2 id="what-is-the-hmac-key-mapping-procedure">What is the HMAC key mapping procedure?<a class="headerlink" href="#what-is-the-hmac-key-mapping-procedure" title="Permanent link">&para;</a></h2>
<details class="abstract" open="open">
<summary>HMAC key mapping procedure</summary>
<p>Let <var>MP</var> denote the master passphrase, and let <var>K</var>
denote the HMAC key candidate.  Let <var>B</var> denote the block size
of HMAC-SHA1 in bytes, i.e., <code>64</code>.  At the beginning,
<var>K</var> = <var>MP</var>.</p>
<ol>
<li>If <var>K</var> (= <var>MP</var>) is larger than <var>B</var>, set
    <var>K</var> to <code>SHA1(K)</code>.  This updates <var>K</var> for all
    further steps below.</li>
<li>If <var>K</var> is smaller than <var>B</var>, append as many NUL
    bytes as necessary to extend <var>K</var> to size <var>B</var>.</li>
<li>Use <var>K</var> as the HMAC key.</li>
</ol>
</details>
<h2 id="what-effect-does-the-hmac-key-mapping-procedure-have-on-key-security">What effect does the HMAC key mapping procedure have on key security?<a class="headerlink" href="#what-effect-does-the-hmac-key-mapping-procedure-have-on-key-security" title="Permanent link">&para;</a></h2>
<p>The key space shrinks to 99.6% of its original size.
But since it started out as astronomically large (2<sup>512</sup>), it <em>still</em> is
astronomically large.</p>
<details class="example">
<summary>Mathematical details: key space</summary>
<table>
<thead>
<tr>
<th style="text-align: left;">variant</th>
<th style="text-align: left;">key space size</th>
<th style="text-align: left;">fraction of total size</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align: left;">64-byte keys only</td>
<td style="text-align: left;">256<sup>64</sup> = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096</td>
<td style="text-align: left;">99.609375%</td>
</tr>
<tr>
<td style="text-align: left;">full key size up to 64 bytes</td>
<td style="text-align: left;">(256<sup>65</sup> – 1) / (256 – 1) = 13460387568883548460748825096238025916214579019888834136067575410167731732152266768867764001296969715641757473316629133406121545097483615318772604492382465</td>
<td style="text-align: left;">100%</td>
</tr>
</tbody>
</table>
<p>The key space sizes can be calculated using the following formulas.  Let
<var>q</var> = <code>256</code> denote the alphabet size of binary strings, and let
<var>n</var> denote the string length.  The total count of all strings
of size <var>n</var> is <var>q</var><sup><var>n</var></sup>, and the total count
of all strings up to (and including) size <var>n</var> is
(<var>q</var><sup><var>n</var> + 1</sup> – 1) / (<var>q</var> – 1) per
the formula for geometric series.</p>
<p>Verification:</p>
<div class="highlight"><pre><span></span><code><span class="gp">$ </span><span class="c1"># using GNU bc 1.07.1</span>
<span class="gp">$ </span>
<span class="gp">$ </span><span class="nv">BC_LINE_LENGTH</span><span class="o">=</span><span class="m">0</span><span class="w"> </span>bc<span class="w"> </span>&lt;&lt;<span class="s1">&#39;HERE&#39;</span>
<span class="go">&gt; # The total count of size 64 byte strings.</span>
<span class="go">&gt; 256^64</span>
<span class="go">&gt; # The total count of byte strings of size 64 or less.</span>
<span class="go">&gt; (256^65 - 1) / (256 - 1)</span>
<span class="go">&gt; # The fraction of the former within the latter.</span>
<span class="go">&gt; scale = 8</span>
<span class="go">&gt; (256^64) / ((256^65 - 1) / (256 - 1))</span>
<span class="go">&gt; HERE</span>
<span class="go">13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096</span>
<span class="go">13460387568883548460748825096238025916214579019888834136067575410167731732152266768867764001296969715641757473316629133406121545097483615318772604492382465</span>
<span class="go">.99609375</span>
<span class="gp">$ </span>
<span class="gp">$ </span><span class="nv">BC_LINE_LENGTH</span><span class="o">=</span><span class="m">0</span><span class="w"> </span>bc<span class="w"> </span>&lt;&lt;<span class="s1">&#39;HERE&#39;</span>
<span class="go">&gt; # The fraction of unusable keys.</span>
<span class="go">&gt; scale = 160</span>
<span class="go">&gt; (256^64 - 1) / (256^65 - 1)</span>
<span class="go">&gt; 1 / 256</span>
<span class="go">&gt; HERE</span>
<span class="go">.0039062499999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999997097</span>
<span class="go">.0039062500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000</span>
</code></pre></div>
</details>
<p>In particular, assuming a sufficiently secure master passphrase, this
mapping procedure is still cryptographically secure against attackers
without possession of the master passphrase if the hashing function (here:
SHA1) is secure against preimage attacks:</p>
<ul>
<li>
<p>The attacker can attempt to guess a NUL-extended version of the
    passphrase if it is shorter than or equal in length to 64 bytes.  This
    has the same computational cost as guessing the master passphrase
    directly, which is cryptographically secure by assumption.</p>
</li>
<li>
<p>The attacker can attempt to guess a hashed and NUL-extended version of
    the passphrase if it is larger than 64 bytes.  This amounts to carrying
    out a preimage attack against the SHA1 digest of the master passphrase,
    which is also cryptographically secure by assumption.</p>
</li>
</ul>
<h2 id="what-effect-does-the-hmac-key-mapping-procedure-have-on-derivepassphrase">What effect does the HMAC key mapping procedure have on <code>derivepassphrase</code>?<a class="headerlink" href="#what-effect-does-the-hmac-key-mapping-procedure-have-on-derivepassphrase" title="Permanent link">&para;</a></h2>
<p><code>derivepassphrase vault</code> does not check for interchangable passphrases, and
will happily accept any (non-empty) passphrase it is given.
The <a class="autorefs autorefs-internal" title="            Vault" href="../../reference/derivepassphrase.vault/#derivepassphrase.vault.Vault"><code>derivepassphrase.vault.Vault</code></a> class does not check for
interchangable passphrases either, and will happily accept any passphrase it
is given, even empty ones.</p>
<p>Most interchangable variations of a master passphrase contain binary
characters such as NUL, or even arbitrary byte sequences, which may be hard
to type in or impossible to express in certain storage formats.  As such, it
is unlikely&mdash;but otherwise supported&mdash;that the user would want to enter
or store a different, interchangable version of their master passphrase in
the first place.</p>
<div class="footnote">
<hr />
<ol>
<li id="fn:1">
<p>PBKDF2 is a key derivation function, published in <a href="https://datatracker.ietf.org/doc/html/rfc2898">RFC 2898</a>.
  It uses a pseudo-random function such as HMAC-SHA1 (hashed message
  authentication code, specified in <a href="https://datatracker.ietf.org/doc/html/rfc2104">RFC 2104</a> and using SHA1 as the
  underlying hash function) when processing its input.  PBKDF2 passes the
  key on to its pseudo-random function, and otherwise only depends on the
  output of the pseudo-random function, not on the key.&#160;<a class="footnote-backref" href="#fnref:1" title="Jump back to footnote 1 in the text">&#8617;</a></p>
</li>
<li id="fn:2">
<p>If you use a master SSH key, it is first converted to an &ldquo;equivalent
  master passphrase&rdquo;.&#160;<a class="footnote-backref" href="#fnref:2" title="Jump back to footnote 2 in the text">&#8617;</a></p>
</li>
</ol>
</div>

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