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class="col docItemCol_VOVn"><div class="docItemContainer_Djhp"><article><nav class="theme-doc-breadcrumbs breadcrumbsContainer_Z_bl" aria-label="Breadcrumbs"><ul class="breadcrumbs" itemscope="" itemtype="https://schema.org/BreadcrumbList"><li class="breadcrumbs__item"><a aria-label="Home page" class="breadcrumbs__link" href="/"><svg viewBox="0 0 24 24" class="breadcrumbHomeIcon_YNFT"><path d="M10 19v-5h4v5c0 .55.45 1 1 1h3c.55 0 1-.45 1-1v-7h1.7c.46 0 .68-.57.33-.87L12.67 3.6c-.38-.34-.96-.34-1.34 0l-8.36 7.53c-.34.3-.13.87.33.87H5v7c0 .55.45 1 1 1h3c.55 0 1-.45 1-1z" fill="currentColor"></path></svg></a></li><li itemscope="" itemprop="itemListElement" itemtype="https://schema.org/ListItem" class="breadcrumbs__item"><a class="breadcrumbs__link" itemprop="item" href="/algorithms/category/hash-tables/"><span itemprop="name">Hash Tables</span></a><meta itemprop="position" content="1"></li><li itemscope="" itemprop="itemListElement" itemtype="https://schema.org/ListItem" class="breadcrumbs__item"><a class="breadcrumbs__link" itemprop="item" href="/algorithms/hash-tables/breaking/"><span itemprop="name">Breaking Hash Table</span></a><meta itemprop="position" content="2"></li><li itemscope="" itemprop="itemListElement" itemtype="https://schema.org/ListItem" class="breadcrumbs__item breadcrumbs__item--active"><span class="breadcrumbs__link" itemprop="name">Possible Mitigations</span><meta itemprop="position" content="3"></li></ul></nav><div class="tocCollapsible_ETCw theme-doc-toc-mobile tocMobile_ITEo"><button type="button" class="clean-btn tocCollapsibleButton_TO0P">On this page</button></div><div class="theme-doc-markdown markdown"><header><h1>Possible Mitigations</h1></header><p>There are multiple ways the issues created above can be mitigated. Still we can
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only make it better, we cannot guarantee the ideal time complexity…</p>
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<p>For the sake of simplicity (and referencing an article by <em>Neal Wu</em> on the same
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topic; in references below) I will use the C++ to describe the mitigations.</p>
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<h2 class="anchor anchorWithStickyNavbar_LWe7" id="random-seed">Random seed<a href="#random-seed" class="hash-link" aria-label="Direct link to Random seed" title="Direct link to Random seed"></a></h2>
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<p>One of the options how to avoid this kind of an attack is to introduce a random
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seed to the hash. That way it is not that easy to choose the <em>nasty</em> numbers.</p>
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<div class="language-cpp codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-cpp codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token keyword" style="color:#00009f">struct</span><span class="token plain"> </span><span class="token class-name">custom_hash</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> size_t </span><span class="token keyword" style="color:#00009f">operator</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">(</span><span class="token keyword" style="color:#00009f">uint64_t</span><span class="token plain"> x</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">const</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">return</span><span class="token plain"> x </span><span class="token operator" style="color:#393A34">+</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">7529</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">}</span><span class="token punctuation" style="color:#393A34">;</span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg viewBox="0 0 24 24" class="copyButtonSuccessIcon_LjdS"><path fill="currentColor" d="M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z"></path></svg></span></button></div></div></div>
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<p>As you may have noticed, this is not very helpful, since it just shifts the
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issue by some number. Better option is to use a shift from random number
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generator:</p>
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<div class="language-cpp codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-cpp codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token keyword" style="color:#00009f">struct</span><span class="token plain"> </span><span class="token class-name">custom_hash</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> size_t </span><span class="token keyword" style="color:#00009f">operator</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">(</span><span class="token keyword" style="color:#00009f">uint64_t</span><span class="token plain"> x</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">const</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">static</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">const</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">uint64_t</span><span class="token plain"> FIXED_RANDOM </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> chrono</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token plain">steady_clock</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token function" style="color:#d73a49">now</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">time_since_epoch</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">count</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">return</span><span class="token plain"> x </span><span class="token operator" style="color:#393A34">+</span><span class="token plain"> FIXED_RANDOM</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">}</span><span class="token punctuation" style="color:#393A34">;</span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg viewBox="0 0 24 24" class="copyButtonSuccessIcon_LjdS"><path fill="currentColor" d="M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z"></path></svg></span></button></div></div></div>
|
||
<p>In this case the hash is using a high-precision clock to shift the number, which
|
||
is much harder to break.</p>
|
||
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="better-random-seed">Better random seed<a href="#better-random-seed" class="hash-link" aria-label="Direct link to Better random seed" title="Direct link to Better random seed"></a></h2>
|
||
<p>Building on the previous solution, we can do some <em>bit magic</em> instead of the
|
||
shifting:</p>
|
||
<div class="language-cpp codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-cpp codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token keyword" style="color:#00009f">struct</span><span class="token plain"> </span><span class="token class-name">custom_hash</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> size_t </span><span class="token keyword" style="color:#00009f">operator</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">(</span><span class="token keyword" style="color:#00009f">uint64_t</span><span class="token plain"> x</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">const</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">static</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">const</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">uint64_t</span><span class="token plain"> FIXED_RANDOM </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> chrono</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token plain">steady_clock</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token function" style="color:#d73a49">now</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">time_since_epoch</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">count</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> x </span><span class="token operator" style="color:#393A34">^=</span><span class="token plain"> FIXED_RANDOM</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">return</span><span class="token plain"> x </span><span class="token operator" style="color:#393A34">^</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">x </span><span class="token operator" style="color:#393A34">>></span><span class="token plain"> </span><span class="token number" style="color:#36acaa">16</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">}</span><span class="token punctuation" style="color:#393A34">;</span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg viewBox="0 0 24 24" class="copyButtonSuccessIcon_LjdS"><path fill="currentColor" d="M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z"></path></svg></span></button></div></div></div>
|
||
<p>This not only shifts the number, it also manipulates the underlying bits of the
|
||
hash. In this case we're also applying the <code>XOR</code> operation.</p>
|
||
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="adjusting-the-hash-function">Adjusting the hash function<a href="#adjusting-the-hash-function" class="hash-link" aria-label="Direct link to Adjusting the hash function" title="Direct link to Adjusting the hash function"></a></h2>
|
||
<p>Another option is to switch up the hash function.</p>
|
||
<p>For example Rust uses <a href="https://en.wikipedia.org/wiki/SipHash" target="_blank" rel="noopener noreferrer"><em>SipHash</em></a> by
|
||
default.</p>
|
||
<p>On the other hand, you can usually specify your own hash function, here we will
|
||
follow the article by <em>Neal</em> that uses so-called <em><code>splitmix64</code></em>.</p>
|
||
<div class="language-cpp codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-cpp codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token keyword" style="color:#00009f">static</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">uint64_t</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">splitmix64</span><span class="token punctuation" style="color:#393A34">(</span><span class="token keyword" style="color:#00009f">uint64_t</span><span class="token plain"> x</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// http://xorshift.di.unimi.it/splitmix64.c</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> x </span><span class="token operator" style="color:#393A34">+=</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">0x9e3779b97f4a7c15</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> x </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">x </span><span class="token operator" style="color:#393A34">^</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">x </span><span class="token operator" style="color:#393A34">>></span><span class="token plain"> </span><span class="token number" style="color:#36acaa">30</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">*</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">0xbf58476d1ce4e5b9</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> x </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">x </span><span class="token operator" style="color:#393A34">^</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">x </span><span class="token operator" style="color:#393A34">>></span><span class="token plain"> </span><span class="token number" style="color:#36acaa">27</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">*</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">0x94d049bb133111eb</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">return</span><span class="token plain"> x </span><span class="token operator" style="color:#393A34">^</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">x </span><span class="token operator" style="color:#393A34">>></span><span class="token plain"> </span><span class="token number" style="color:#36acaa">31</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">}</span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg viewBox="0 0 24 24" class="copyButtonSuccessIcon_LjdS"><path fill="currentColor" d="M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z"></path></svg></span></button></div></div></div>
|
||
<p>As you can see, this definitely doesn't do identity on the integers <!-- -->😄</p>
|
||
<p>Another example would be
|
||
<a href="https://github.com/openjdk/jdk/blob/dc256fbc6490f8163adb286dbb7380c10e5e1e06/src/java.base/share/classes/java/util/HashMap.java#L320-L339" target="_blank" rel="noopener noreferrer"><code>HashMap::hash()</code></a>
|
||
function in Java:</p>
|
||
<div class="language-java codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-java codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token comment" style="color:#999988;font-style:italic">/**</span><br></span><span class="token-line" style="color:#393A34"><span class="token comment" style="color:#999988;font-style:italic"> * Computes key.hashCode() and spreads (XORs) higher bits of hash</span><br></span><span class="token-line" style="color:#393A34"><span class="token comment" style="color:#999988;font-style:italic"> * to lower. Because the table uses power-of-two masking, sets of</span><br></span><span class="token-line" style="color:#393A34"><span class="token comment" style="color:#999988;font-style:italic"> * hashes that vary only in bits above the current mask will</span><br></span><span class="token-line" style="color:#393A34"><span class="token comment" style="color:#999988;font-style:italic"> * always collide. (Among known examples are sets of Float keys</span><br></span><span class="token-line" style="color:#393A34"><span class="token comment" style="color:#999988;font-style:italic"> * holding consecutive whole numbers in small tables.) So we</span><br></span><span class="token-line" style="color:#393A34"><span class="token comment" style="color:#999988;font-style:italic"> * apply a transform that spreads the impact of higher bits</span><br></span><span class="token-line" style="color:#393A34"><span class="token comment" style="color:#999988;font-style:italic"> * downward. There is a tradeoff between speed, utility, and</span><br></span><span class="token-line" style="color:#393A34"><span class="token comment" style="color:#999988;font-style:italic"> * quality of bit-spreading. Because many common sets of hashes</span><br></span><span class="token-line" style="color:#393A34"><span class="token comment" style="color:#999988;font-style:italic"> * are already reasonably distributed (so don't benefit from</span><br></span><span class="token-line" style="color:#393A34"><span class="token comment" style="color:#999988;font-style:italic"> * spreading), and because we use trees to handle large sets of</span><br></span><span class="token-line" style="color:#393A34"><span class="token comment" style="color:#999988;font-style:italic"> * collisions in bins, we just XOR some shifted bits in the</span><br></span><span class="token-line" style="color:#393A34"><span class="token comment" style="color:#999988;font-style:italic"> * cheapest possible way to reduce systematic lossage, as well as</span><br></span><span class="token-line" style="color:#393A34"><span class="token comment" style="color:#999988;font-style:italic"> * to incorporate impact of the highest bits that would otherwise</span><br></span><span class="token-line" style="color:#393A34"><span class="token comment" style="color:#999988;font-style:italic"> * never be used in index calculations because of table bounds.</span><br></span><span class="token-line" style="color:#393A34"><span class="token comment" style="color:#999988;font-style:italic"> */</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token keyword" style="color:#00009f">static</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">final</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">hash</span><span class="token punctuation" style="color:#393A34">(</span><span class="token class-name">Object</span><span class="token plain"> key</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">int</span><span class="token plain"> h</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">return</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">key </span><span class="token operator" style="color:#393A34">==</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">null</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">?</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">0</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">:</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">h </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> key</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">hashCode</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">^</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">h </span><span class="token operator" style="color:#393A34">>>></span><span class="token plain"> </span><span class="token number" style="color:#36acaa">16</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">}</span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg viewBox="0 0 24 24" class="copyButtonSuccessIcon_LjdS"><path fill="currentColor" d="M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z"></path></svg></span></button></div></div></div>
|
||
<p>You can notice that they try to include the upper bits of the hash by using
|
||
<code>XOR</code>, this would render our attack in the previous part helpless.</p>
|
||
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="combining-both">Combining both<a href="#combining-both" class="hash-link" aria-label="Direct link to Combining both" title="Direct link to Combining both"></a></h2>
|
||
<p>Can we make it better? Of course! Use multiple mitigations at the same time. In
|
||
our case, we will both inject the random value <strong>and</strong> use the <em><code>splitmix64</code></em>:</p>
|
||
<div class="language-cpp codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-cpp codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token keyword" style="color:#00009f">struct</span><span class="token plain"> </span><span class="token class-name">custom_hash</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">static</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">uint64_t</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">splitmix64</span><span class="token punctuation" style="color:#393A34">(</span><span class="token keyword" style="color:#00009f">uint64_t</span><span class="token plain"> x</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token comment" style="color:#999988;font-style:italic">// http://xorshift.di.unimi.it/splitmix64.c</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> x </span><span class="token operator" style="color:#393A34">+=</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">0x9e3779b97f4a7c15</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> x </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">x </span><span class="token operator" style="color:#393A34">^</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">x </span><span class="token operator" style="color:#393A34">>></span><span class="token plain"> </span><span class="token number" style="color:#36acaa">30</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">*</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">0xbf58476d1ce4e5b9</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> x </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">x </span><span class="token operator" style="color:#393A34">^</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">x </span><span class="token operator" style="color:#393A34">>></span><span class="token plain"> </span><span class="token number" style="color:#36acaa">27</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token operator" style="color:#393A34">*</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">0x94d049bb133111eb</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">return</span><span class="token plain"> x </span><span class="token operator" style="color:#393A34">^</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">x </span><span class="token operator" style="color:#393A34">>></span><span class="token plain"> </span><span class="token number" style="color:#36acaa">31</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain" style="display:inline-block"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> size_t </span><span class="token keyword" style="color:#00009f">operator</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">(</span><span class="token keyword" style="color:#00009f">uint64_t</span><span class="token plain"> x</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">const</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">{</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">static</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">const</span><span class="token plain"> </span><span class="token keyword" style="color:#00009f">uint64_t</span><span class="token plain"> FIXED_RANDOM </span><span class="token operator" style="color:#393A34">=</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> chrono</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token plain">steady_clock</span><span class="token double-colon punctuation" style="color:#393A34">::</span><span class="token function" style="color:#d73a49">now</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">time_since_epoch</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">.</span><span class="token function" style="color:#d73a49">count</span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token keyword" style="color:#00009f">return</span><span class="token plain"> </span><span class="token function" style="color:#d73a49">splitmix64</span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">x </span><span class="token operator" style="color:#393A34">+</span><span class="token plain"> FIXED_RANDOM</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">;</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">}</span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">}</span><span class="token punctuation" style="color:#393A34">;</span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg viewBox="0 0 24 24" class="copyButtonSuccessIcon_LjdS"><path fill="currentColor" d="M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z"></path></svg></span></button></div></div></div>
|
||
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="fallback-for-extreme-cases">Fallback for extreme cases<a href="#fallback-for-extreme-cases" class="hash-link" aria-label="Direct link to Fallback for extreme cases" title="Direct link to Fallback for extreme cases"></a></h2>
|
||
<p>As we have mentioned above, Python resolves the conflicts by probing (it looks
|
||
for empty space somewhere else in the table, but it's deterministic about it, so
|
||
it's not “<em>oops, this is full, let's go one-by-one and find some spot</em>”). In the
|
||
case of C++ and Java, they resolve the conflicts by linked lists, as is the
|
||
usual text-book depiction of the hash table.</p>
|
||
<p>However Java does something more intelligent. Once you go over the threshold of
|
||
conflicts in one spot, it converts the linked list to an RB-tree that is sorted
|
||
by the hash and key respectively.</p>
|
||
<div class="theme-admonition theme-admonition-tip admonition_xJq3 alert alert--success"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><svg viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>tip</div><div class="admonitionContent_BuS1"><p>You may wonder what sense does it make to define an ordering on the tree by the
|
||
hash, if we're dealing with conflicts. Well, there are less buckets than the
|
||
range of the hash, so if we take lower bits, we can have a conflict even though
|
||
the hashes are not the same.</p></div></div>
|
||
<p>You might have noticed that if we get a <strong>really bad</strong> hashing function, this is
|
||
not very helpful. It is not, <strong>but</strong> it can help in other cases.</p>
|
||
<div class="theme-admonition theme-admonition-danger admonition_xJq3 alert alert--danger"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><svg viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>danger</div><div class="admonitionContent_BuS1"><p>As the ordering on the keys of the hash table is not required and may not be
|
||
implemented, the tree may be ordered by just the hash.</p></div></div>
|
||
<hr>
|
||
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="references">References<a href="#references" class="hash-link" aria-label="Direct link to References" title="Direct link to References"></a></h2>
|
||
<ol>
|
||
<li>Neal Wu.
|
||
<a href="https://codeforces.com/blog/entry/62393" target="_blank" rel="noopener noreferrer">Blowing up <code>unordered_map</code>, and how to stop getting hacked on it</a>.</li>
|
||
</ol></div><footer class="theme-doc-footer docusaurus-mt-lg"><div class="theme-doc-footer-tags-row row margin-bottom--sm"><div class="col"><b>Tags:</b><ul class="tags_jXut padding--none margin-left--sm"><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/algorithms/tags/cpp/">cpp</a></li><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/algorithms/tags/python/">python</a></li><li class="tag_QGVx"><a class="tag_zVej tagRegular_sFm0" href="/algorithms/tags/hash-tables/">hash-tables</a></li></ul></div></div><div class="theme-doc-footer-edit-meta-row row"><div class="col"><a href="https://github.com/mfocko/blog/tree/main/algorithms/12-hash-tables/2023-11-28-breaking/02-mitigations.md" target="_blank" rel="noopener noreferrer" class="theme-edit-this-page"><svg fill="currentColor" height="20" width="20" viewBox="0 0 40 40" class="iconEdit_Z9Sw" aria-hidden="true"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z 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class="table-of-contents__link toc-highlight">Adjusting the hash function</a></li><li><a href="#combining-both" class="table-of-contents__link toc-highlight">Combining both</a></li><li><a href="#fallback-for-extreme-cases" class="table-of-contents__link toc-highlight">Fallback for extreme cases</a></li><li><a href="#references" class="table-of-contents__link toc-highlight">References</a></li></ul></div></div></div></div></main></div></div></div><footer class="footer footer--dark"><div class="container container-fluid"><div class="row footer__links"><div class="col footer__col"><div class="footer__title">Git</div><ul class="footer__items clean-list"><li class="footer__item"><a href="https://github.com/mfocko" target="_blank" rel="noopener noreferrer" class="footer__link-item">GitHub<svg width="13.5" height="13.5" aria-hidden="true" viewBox="0 0 24 24" class="iconExternalLink_nPIU"><path fill="currentColor" d="M21 13v10h-21v-19h12v2h-10v15h17v-8h2zm3-12h-10.988l4.035 4-6.977 7.07 2.828 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