Address review comments

* remove 'unique' qualification for elements of a set (by definition,
all elements in a set are unique...)
* rename `S_p` -> `n`

docs/alba.js

@@ -7,7 +7,7 @@ document.addEventListener('DOMContentLoaded', () => {
   const n_f = document.getElementById('n_f');
   const item = document.getElementById('item');
   const proof_size = document.getElementById('proof_size');
-  const s_p = document.getElementById('s_p');
+  const n = document.getElementById('n');
   const proof_proba = document.getElementById('proof_proba');
 
   function U(n_p, n_f) {
@@ -87,7 +87,7 @@ document.addEventListener('DOMContentLoaded', () => {
         x: {
           type: 'linear',
           title: {
-            text: 'S_p',
+            text: 'n',
             display: true
           },
         }
@@ -106,9 +106,10 @@ document.addEventListener('DOMContentLoaded', () => {
   function updateProofProba() {
     const n_p_v = Number(n_p.value);
     const n_f_v = Number(n_f.value);
-    const s_p_v = Number(s_p.value);
+    const n_v = Number(n.value);
     const u = U(n_p_v, n_f_v);
-    const proba = probabilityOfProof(u, n_p_v, n_f_v, s_p_v);
+    const proba = probabilityOfProof(u, n_p_v, n_f_v, n_v);
+    // expected prover time is n_p + O (u^2), we neglect the n_p part here
     proof_proba.value = proba.toExponential(4);
   }
 
@@ -124,7 +125,7 @@ document.addEventListener('DOMContentLoaded', () => {
   n_p.addEventListener('change', updateChart);
   n_f.addEventListener('change', updateChart);
   item.addEventListener('change', updateChart);
-  s_p.addEventListener('change', updateProofProba);
+  n.addEventListener('change', updateProofProba);
 
   updateProofSize();
   updateProofProba();

docs/src/intro.md

@@ -2,7 +2,7 @@
 
 ## Definition
 
-_Approximate Lower Bound Arguments_ are a form of cryptographic certificates that allow a _prover_ to convince a _verifier_ they know some set of elements by providing only a small subset of those elements. More formally, given some set of _unique_ elements, and some bounds \\(n_p\\) and \\(n_f\\), where \\(n_p > n_f\\), about the size of this set, ALBA provides an algorithm that allows to build a _proof_ they know _more than_ \\(n_f\\) elements, where the proof size is a constant value derived from the \\(\frac{n_p}{n_f}\\) ratio.
+_Approximate Lower Bound Arguments_ are a form of cryptographic certificates that allow a _prover_ to convince a _verifier_ they know some set of elements by providing only a small subset of those elements. More formally, given some set of elements, and some bounds \\(n_p\\) and \\(n_f\\), where \\(n_p > n_f\\), about the size of this set, ALBA provides an algorithm that allows to build a _proof_ they know _at least_ \\(n_f\\) elements, where the proof size is a constant value derived from the \\(\frac{n_p}{n_f}\\) ratio.
 
 The algorithm provides the following guarantees, given the prover has \\(S_p\\) elements and a security parameter \\(\lambda\\):
 

docs/src/simulation.md

@@ -1,6 +1,6 @@
 # ALBA Parameters Simulation
 
-ALBA provides strong security guarantees about the capability to generate a proof given some number of items \\(S_p\\):
+ALBA provides strong security guarantees about the capability to generate a proof given some number of items \\(n\\):
 
 * a honest prover with at least \\(n_p\\) items has a probability of
   failing to generate a proof lower than \\(2^{-128}\\),
@@ -10,7 +10,7 @@ ALBA provides strong security guarantees about the capability to generate a proo
 
 This page provides a simple and user-friendly way of estimating how
 various ALBA parameters impact the size of the proof and probability
-of having a proof, given a number of items \\(S_p\\) varying between
+of having a proof, given a number of items \\(n\\) varying between
 the two parameters' bounds. It is expected to be a quick way to
 estimate what would be "good" values for those parameters in a
 specific context, depending on security requirements and expected
@@ -21,17 +21,17 @@ One can define:
 * \\(n_p\\): The size of the _honest set_ of items.
 * \\(n_f\\): The size of the _faulty set_ of items. \\(n_f\\) must be lower than \\(n_p\\).
 * Item size: The size of a single item, this is used to compute the full size of a proof.
-* \\(S_p\\): The number of items actually available to an adversary, or put differently, how much adversarial stake should we expect.
+* \\(n\\): The number of items actually available to an adversary, or put differently, how much adversarial stake should we expect.
 
 From these parameters, we display a chart showing the probability of
-an adversary being able to compute a valid proof, given some
-number of items \\(S_p\\) available between \\(n_p\\) and
+an adversary being able to compute a single valid proof, given some
+number of items \\(n\\) available between \\(n_p\\) and
 \\(n_f\\). The formula used, initially proposed in this [GitHub
 discussion](https://github.com/cardano-scaling/alba/discussions/17)
 is:
 
 \\[
- r \cdot {\left( \frac{S_p}{n_p} \right)}^u\cdot qd
+ r \cdot {\left( \frac{n}{n_p} \right)}^u\cdot qd
 \\]
 
 where _u_, _r_, _q_, and _d_ are defined in the aforementioned
@@ -43,8 +43,7 @@ We also display two specific values:
   item size, which is given by the formula from section 3.2.2,
   Corollary 3, from the [ALBA
   paper](https://iohk.io/en/research/library/papers/approximate-lower-bound-arguments/),
-* the probability of having a proof for some specific number of \\(S_p\\) computed.
-
+* the probability of having a proof for some specific number of \\(n\\) (computed.
 
 <div class="col col-md-8">
   <div class="row">
@@ -59,7 +58,7 @@ We also display two specific values:
     </div>
     <div class="col col-md-3">
         <label for="n_f" class="form-label">\(n_f\)<span class="help" data-bs-toggle="tooltip" title="Expected number of faulty set of items.">?</span></label>
-        <input type="number" step="10" class="form-control" id="n_f" value="200">
+        <input type="number" step="10" class="form-control" id="n_f" value="400">
     </div>
     <div class="col col-md-3">
         <label for="item" class="form-label">Item size<span class="help" data-bs-toggle="tooltip" title="Size of a single item.">?</span></label>
@@ -72,8 +71,8 @@ We also display two specific values:
   </div>
   <div class="row g-3" style="width: 95%; margin: auto;">
     <div class="col col-md-3">
-        <label for="s_p" class="form-label">\(S_p\)<span class="help" data-bs-toggle="tooltip" title="Number of items actually available.">?</span></label>
-        <input type="number" step="10" class="form-control" id="s_p" value="300">
+        <label for="n" class="form-label">\(n\)<span class="help" data-bs-toggle="tooltip" title="Number of items actually available.">?</span></label>
+        <input type="number" step="10" class="form-control" id="n" value="700">
     </div>
     <div class="col col-md-3">
         <label for="proof_proba" class="form-label">Proof probability<span class="help" data-bs-toggle="tooltip" title="Probability of having a proof for some number of items.">?</span></label>