<p>What you are looking for is the <a href="https://en.wikipedia.org/wiki/Accumulator_(cryptography)" rel="nofollow noreferrer">Accumulators</a>. Currently, they are very popular with <a href="https://www.youtube.com/results?search_query=rsa%20accumulator" rel="nofollow noreferrer">digital coins @youtube</a></p>

<p>From Wikipedia;</p>

<blockquote>
  <p>A cryptographic accumulator is a one-way membership function. It answers a query as to whether a potential candidate is a member of a set without revealing the individual members of the set. </p>
</blockquote>

<p>For example this <a href="https://arxiv.org/pdf/0905.1307v1.pdf" rel="nofollow noreferrer">paper</a>;</p>

<blockquote>
  <p>We show how to use the RSA one-way accumulator to realize an efficient and dynamic authenticated dictionary, where untrusted directories provide cryptographically verifiable answers
  to membership queries on a set maintained by a trusted source</p>
</blockquote>

<p>With a Straightforward Accumulator-Based Scheme;</p>

<ul>
<li>Query: When asking for a proof of membership.</li>
<li>Verification: check the validity of the answer.</li>
<li>Updates: Insertion and deletions</li>
</ul>

<p>are available.</p>
is there a hashing function that satisfies the following
<p>is there a hashing algorithm that satisfies the following?</p>

<p><code>let "hash_funct" be a hashing function that takes two args, and returns a hash value. so all the following will be true</code></p>

<p><code>Hash1 = hash_funct(arg1, arg2) &lt;=&gt; hash_funct(Hash1, arg1) = hash_funct(Hash1, arg2) = Hash1;</code></p>

<p>Can anyone point me to this Algorithm? or if it doesn't exist, can anyone collaborate with me to invent it?</p>

<p>more explanation:</p>

<p>imagine a set <code>S={A,B,C,D}</code>, and the Hashing function above.</p>

<p>if we can make: <code>Hash1 = hash_funct(A,B,C,D)</code>, then we can check if an element <code>X</code> is in the set by checking the hash result of <code>hash_funct(Hash1,X) == Hash1 ? belogns to the set : doesn't belong</code></p>

<p>with this property we make checking the exisitance of an element in a set O(1) instead of O(NlogN)</p>