fix: under review -> submitted

.gitignore

@@ -17,10 +17,11 @@ package-lock.json
 
 # IDE configurations
 .idea
-.vscode
+.vscode/*
 !.vscode/settings.json
 !.vscode/extensions.json
+!.vscode/tasks.json
 
 # Misc
-_sass/dist
+_sass/vendors
 assets/js/dist

Gemfile

@@ -2,7 +2,7 @@
 
 source "https://rubygems.org"
 
-gem "jekyll-theme-chirpy", "~> 7.2", ">= 7.2.4"
+gem "jekyll-theme-chirpy", "~> 7.3", ">= 7.3.1"
 
 gem "html-proofer", "~> 5.0", group: :test
 

_includes/js-selector.html

@@ -62,7 +62,7 @@
 
 {% capture script %}/assets/js/dist/{{ js }}.min.js{% endcapture %}
 
-<script src="{{ script | relative_url }}"></script>
+<script defer src="{{ script | relative_url }}"></script>
 
 {% if page.math %}
   <!-- MathJax -->
@@ -143,7 +143,7 @@
       }
     };
   </script>
-  <script src="https://cdnjs.cloudflare.com/polyfill/v3/polyfill.min.js?features=es6"></script>
+  <script async src="https://cdnjs.cloudflare.com/polyfill/v3/polyfill.min.js?features=es6"></script>
   <script id="MathJax-script" async src="{{ site.data.origin[type].mathjax.js | relative_url }}"></script>
 {% endif %}
 

_includes/post-summary.html

@@ -0,0 +1,39 @@
+{%- comment -%}
+  Get the post's description or body content.
+
+  Arguments:
+    full_text: If true, return the full content. Default is false.
+    max_length: The maximum length of the returned content. Default is 200.
+{%- endcomment -%}
+
+{%- if post.description and include.full_text != true -%}
+  {{- post.description -}}
+{%- else -%}
+  {%- comment -%} Remove the line numbers from the code snippet. {%- endcomment -%}
+
+  {%- assign content = post.content -%}
+
+  {%- if content contains '<td class="rouge-gutter gl"><pre class="lineno">' -%}
+    {%- assign content = content
+      | replace: '<td class="rouge-gutter gl"><pre class="lineno">',
+        '<!-- <td class="rouge-gutter gl"><pre class="lineno">'
+    -%}
+    {%- assign content = content | replace: '</td><td class="rouge-code">', '</td> --><td class="rouge-code">' -%}
+  {%- endif -%}
+
+  {%- assign content = content
+    | markdownify
+    | strip_html
+    | newline_to_br
+    | replace: '<br />', ' '
+    | strip_newlines
+    | strip
+  -%}
+
+  {%- unless include.full_text -%}
+    {%- assign max_length = include.max_length | default: 200 -%}
+    {%- assign content = content | truncate: max_length -%}
+  {%- endunless -%}
+
+  {{- content -}}
+{%- endif -%}

_includes/sidebar.html

@@ -0,0 +1,97 @@
+<!-- The Side Bar -->
+
+<aside aria-label="Sidebar" id="sidebar" class="d-flex flex-column align-items-end">
+  <header class="profile-wrapper">
+    <a href="{{ '/' | relative_url }}" id="avatar" class="rounded-circle">
+      {%- if site.avatar != empty and site.avatar -%}
+        {%- capture avatar_url -%}
+          {% include media-url.html src=site.avatar %}
+        {%- endcapture -%}
+        <img src="{{- avatar_url -}}" width="112" height="112" alt="avatar" onerror="this.style.display='none'">
+      {%- endif -%}
+    </a>
+
+    <a class="site-title d-block" href="{{ '/' | relative_url }}">{{ site.title }}</a>
+    <p class="site-subtitle fst-italic mb-0">{{ site.tagline }}</p>
+  </header>
+  <!-- .profile-wrapper -->
+
+  <nav class="flex-column flex-grow-1 w-100 ps-0">
+    <ul class="nav">
+      {% for tab in site.tabs %}
+        <li class="nav-item{% if tab.url == page.url %}{{ " active" }}{% endif %}">
+          <a href="{{ tab.url | relative_url }}" class="nav-link">
+            <i class="fa-fw {{ tab.icon }}"></i>
+            {% capture tab_name %}{{ tab.url | split: '/' }}{% endcapture %}
+
+            <span>{{ site.data.locales[include.lang].tabs.[tab_name] | default: tab.title | upcase }}</span>
+          </a>
+        </li>
+        <!-- .nav-item -->
+      {% endfor %}
+    </ul>
+  </nav>
+
+  <div class="sidebar-bottom d-flex flex-wrap  align-items-center w-100">
+    {% unless site.theme_mode %}
+      <button type="button" class="btn btn-link nav-link" aria-label="Switch Mode" id="mode-toggle">
+        <i class="fas fa-adjust"></i>
+      </button>
+
+      {% if site.data.contact.size > 0 %}
+        <span class="icon-border"></span>
+      {% endif %}
+    {% endunless %}
+
+    {% for entry in site.data.contact %}
+      {%- assign url = null -%}
+
+      {% case entry.type %}
+        {% when 'github', 'twitter' %}
+          {%- unless site[entry.type].username -%}
+            {%- continue -%}
+          {%- endunless -%}
+          {%- capture url -%}
+            https://{{ entry.type }}.com/{{ site[entry.type].username }}
+          {%- endcapture -%}
+        {% when 'email' %}
+          {%- unless site.social.email -%}
+            {%- continue -%}
+          {%- endunless -%}
+          {%- assign email = site.social.email | split: '@' -%}
+          {%- capture url -%}
+            javascript:location.href = 'mailto:' + ['{{ email[0] }}','{{ email[1] }}'].join('@')
+          {%- endcapture -%}
+        {% when 'rss' %}
+          {% assign url = '/feed.xml' | relative_url %}
+        {% else %}
+          {% assign url = entry.url %}
+      {% endcase %}
+
+      {% if url %}
+        <a
+          href="{{ url }}"
+          aria-label="{{ entry.type }}"
+          {% assign link_types = '' %}
+
+          {% unless entry.noblank %}
+            target="_blank"
+            {% assign link_types = 'noopener noreferrer' %}
+          {% endunless %}
+
+          {% if entry.type == 'mastodon' %}
+            {% assign link_types = link_types | append: ' me' | strip %}
+          {% endif %}
+
+          {% unless link_types == empty %}
+            rel="{{ link_types }}"
+          {% endunless %}
+        >
+          <i class="{{ entry.icon }}"></i>
+        </a>
+      {% endif %}
+    {% endfor %}
+  </div>
+  <!-- .sidebar-bottom -->
+</aside>
+<!-- #sidebar -->

_layouts/home.html

@@ -0,0 +1,150 @@
+---
+layout: default
+---
+
+{% include lang.html %}
+
+{% assign all_pinned = site.posts | where: 'pin', 'true' %}
+{% assign all_normal = site.posts | where_exp: 'item', 'item.pin != true and item.hidden != true' %}
+
+{% assign posts = '' | split: '' %}
+
+<!-- Pagination fallbacks -->
+{% assign per_page = paginator.per_page | default: site.paginate | default: 10 %}
+{% assign page_num = paginator.page | default: 1 %}
+
+<!-- Get pinned posts on current page -->
+
+{% assign visible_start = page_num | minus: 1 | times: per_page %}
+{% assign visible_end = visible_start | plus: per_page %}
+
+{% if all_pinned.size > visible_start %}
+  {% if all_pinned.size > visible_end %}
+    {% assign pinned_size = paginator.per_page %}
+  {% else %}
+    {% assign pinned_size = all_pinned.size | minus: visible_start %}
+  {% endif %}
+
+  {% for i in (visible_start..all_pinned.size) limit: pinned_size %}
+    {% assign posts = posts | push: all_pinned[i] %}
+  {% endfor %}
+{% else %}
+  {% assign pinned_size = 0 %}
+{% endif %}
+
+<!-- Get normal posts on current page -->
+
+{% assign paginator_posts = paginator.posts | default: site.posts %}
+{% assign normal_size = paginator_posts | size | minus: pinned_size %}
+
+{% if normal_size > 0 %}
+  {% if pinned_size > 0 %}
+    {% assign normal_start = 0 %}
+  {% else %}
+    {% assign normal_start = visible_start | minus: all_pinned.size %}
+  {% endif %}
+
+  {% assign normal_end = normal_start | plus: normal_size | minus: 1 %}
+
+  {% assign normal_end = 10 %}
+
+  {% for i in (normal_start..normal_end) %}
+    {% assign posts = posts | push: all_normal[i] %}
+  {% endfor %}
+{% endif %}
+
+
+
+<div id="post-list" class="flex-grow-1 px-xl-1">
+  {% for post in posts %}
+    <article class="card-wrapper card">
+      <a href="{{ post.url | relative_url }}" class="post-preview row g-0 flex-md-row-reverse">
+        {% assign card_body_col = '12' %}
+
+        {% if post.image %}
+          {% assign src = post.image.path | default: post.image %}
+
+          {% if post.media_subpath %}
+            {% unless src contains '://' %}
+              {% assign src = post.media_subpath
+                | append: '/'
+                | append: src
+                | replace: '///', '/'
+                | replace: '//', '/'
+              %}
+            {% endunless %}
+          {% endif %}
+
+          {% if post.image.lqip %}
+            {% assign lqip = post.image.lqip %}
+
+            {% if post.media_subpath %}
+              {% unless lqip contains 'data:' %}
+                {% assign lqip = post.media_subpath
+                  | append: '/'
+                  | append: lqip
+                  | replace: '///', '/'
+                  | replace: '//', '/'
+                %}
+              {% endunless %}
+            {% endif %}
+
+            {% assign lqip_attr = 'lqip="' | append: lqip | append: '"' %}
+          {% endif %}
+
+          {% assign alt = post.image.alt | xml_escape | default: 'Preview Image' %}
+
+          <div class="col-md-5">
+            <img src="{{ src }}" alt="{{ alt }}" {{ lqip_attr }}>
+          </div>
+
+          {% assign card_body_col = '7' %}
+        {% endif %}
+
+        <div class="col-md-{{ card_body_col }}">
+          <div class="card-body d-flex flex-column">
+            <h1 class="card-title my-2 mt-md-0">{{ post.title }}</h1>
+
+            <div class="card-text content mt-0 mb-3">
+              <p>{% include post-summary.html %}</p>
+            </div>
+
+            <div class="post-meta flex-grow-1 d-flex align-items-end">
+              <div class="me-auto">
+                <!-- posted date -->
+                <i class="far fa-calendar fa-fw me-1"></i>
+                {% include datetime.html date=post.date lang=lang %}
+
+                <!-- categories -->
+                {% if post.categories.size > 0 %}
+                  <i class="far fa-folder-open fa-fw me-1"></i>
+                  <span class="categories">
+                    {% for category in post.categories %}
+                      {{ category }}
+                      {%- unless forloop.last -%},{%- endunless -%}
+                    {% endfor %}
+                  </span>
+                {% endif %}
+              </div>
+
+              {% if post.pin %}
+                <div class="pin ms-1">
+                  <i class="fas fa-thumbtack fa-fw"></i>
+                  <span>{{ site.data.locales[lang].post.pin_prompt }}</span>
+                </div>
+              {% endif %}
+            </div>
+            <!-- .post-meta -->
+          </div>
+          <!-- .card-body -->
+        </div>
+      </a>
+    </article>
+  {% endfor %}
+</div>
+<!-- #post-list -->
+
+{% assign total_pages = paginator.total_pages | default: 2 %}
+{% if total_pages > 1 and paginator %}
+  {% include post-paginator.html %}
+{% endif %}

_layouts/post.html

@@ -10,7 +10,9 @@ script_includes:
   - comment
 ---
 
-{% include lang.html %} {% include toc-status.html %}
+{% include lang.html %}
+
+{% include toc-status.html %}
 
 <article class="px-1" data-toc="{{ enable_toc }}">
   <header>
@@ -22,36 +24,40 @@ script_includes:
     <div class="post-meta text-muted">
       <!-- published date -->
       <span>
-        {{ site.data.locales[lang].post.posted }} {% include datetime.html
+        {{ site.data.locales[lang].post.posted }}
-        date=page.date tooltip=true lang=lang %}
+        {% include datetime.html date=page.date tooltip=true lang=lang %}
       </span>
 
       <!-- lastmod date -->
       {% if page.last_modified_at and page.last_modified_at != page.date %}
         <span>
-        {{ site.data.locales[lang].post.updated }} {% include datetime.html
+          {{ site.data.locales[lang].post.updated }}
-        date=page.last_modified_at tooltip=true lang=lang %}
+          {% include datetime.html date=page.last_modified_at tooltip=true lang=lang %}
         </span>
       {% endif %}
 
       <div class="d-flex justify-content-between">
         <!-- author(s) -->
         <span>
-          {% if page.author %} {% assign authors = page.author %} {% elsif
+          {% if page.author %}
-          page.authors %} {% assign authors = page.authors %} {% endif %} {{
+            {% assign authors = page.author %}
-          site.data.locales[lang].post.written_by }}
+          {% elsif page.authors %}
+            {% assign authors = page.authors %}
+          {% endif %}
+
+          {{ site.data.locales[lang].post.written_by }}
 
           <em>
-            {% if authors %} {% for author in authors %} {% if
+            {% if authors %}
-            site.data.authors[author].url -%}
+              {% for author in authors %}
-            <a href="{{ site.data.authors[author].url }}"
+                {% if site.data.authors[author].url -%}
-              >{{ site.data.authors[author].name }}</a
+                  <a href="{{ site.data.authors[author].url }}">{{ site.data.authors[author].name }}</a>
-            >
+                {%- else -%}
-            {%- else -%} {{ site.data.authors[author].name }} {%- endif %} {%
+                  {{ site.data.authors[author].name }}
-            unless forloop.last %}{{ '</em
+                {%- endif %}
-          >,
+                {% unless forloop.last %}{{ '</em>, <em>' }}{% endunless %}
-          <em
+              {% endfor %}
-            >' }}{% endunless %} {% endfor %} {% else %}
+            {% else %}
               <a href="{{ site.social.links[0] }}">{{ site.social.name }}</a>
             {% endif %}
           </em>
@@ -59,8 +65,7 @@ script_includes:
 
         <div>
           <!-- pageviews -->
-          {% if site.pageviews.provider and
+          {% if site.pageviews.provider and site.analytics[site.pageviews.provider].id %}
-          site.analytics[site.pageviews.provider].id %}
             <span>
               <em id="pageviews">
                 <i class="fas fa-spinner fa-spin small"></i>
@@ -77,37 +82,22 @@ script_includes:
   </header>
 
   {% if enable_toc %}
-  <div
+    <div id="toc-bar" class="d-flex align-items-center justify-content-between invisible">
-    id="toc-bar"
-    class="d-flex align-items-center justify-content-between invisible"
-  >
       <span class="label text-truncate">{{ page.title }}</span>
       <button type="button" class="toc-trigger btn me-1">
         <i class="fa-solid fa-list-ul fa-fw"></i>
       </button>
     </div>
 
-  <button
+    <button id="toc-solo-trigger" type="button" class="toc-trigger btn btn-outline-secondary btn-sm">
-    id="toc-solo-trigger"
+      <span class="label ps-2 pe-1">{{- site.data.locales[lang].panel.toc -}}</span>
-    type="button"
-    class="toc-trigger btn btn-outline-secondary btn-sm"
-  >
-    <span class="label ps-2 pe-1"
-      >{{- site.data.locales[lang].panel.toc -}}</span
-    >
       <i class="fa-solid fa-angle-right fa-fw"></i>
     </button>
 
     <dialog id="toc-popup" class="p-0">
-    <div
+      <div class="header d-flex flex-row align-items-center justify-content-between">
-      class="header d-flex flex-row align-items-center justify-content-between"
-    >
         <div class="label text-truncate py-2 ms-4">{{- page.title -}}</div>
-      <button
+        <button id="toc-popup-close" type="button" class="btn mx-1 my-1 opacity-75">
-        id="toc-popup-close"
-        type="button"
-        class="btn mx-1 my-1 opacity-75"
-      >
           <i class="fas fa-close"></i>
         </button>
       </div>
@@ -115,7 +105,9 @@ script_includes:
     </dialog>
   {% endif %}
 
-  <div class="content">{{ content }}</div>
+  <div class="content">
+    {{ content }}
+  </div>
 
   <div class="post-tail-wrapper text-muted">
     <!-- categories -->
@@ -123,11 +115,9 @@ script_includes:
       <div class="post-meta mb-3">
         <i class="far fa-folder-open fa-fw me-1"></i>
         {% for category in page.categories %}
-      <a
+          <a href="{{ site.baseurl }}/categories/{{ category | slugify | url_encode }}/">{{ category }}</a>
-        href="{{ site.baseurl }}/categories/{{ category | slugify | url_encode }}/"
+          {%- unless forloop.last -%},{%- endunless -%}
-        >{{ category }}</a
+        {% endfor %}
-      >
-      {%- unless forloop.last -%},{%- endunless -%} {% endfor %}
       </div>
     {% endif %}
 
@@ -147,16 +137,21 @@ script_includes:
     {% endif %}
 
     <div
-      class="post-tail-bottom d-flex justify-content-between align-items-center mt-5 pb-2"
+      class="
+        post-tail-bottom
+        d-flex justify-content-between align-items-center mt-5 pb-2
+      "
     >
       <div class="license-wrapper">
-        {% if site.data.locales[lang].copyright.license.template %} {% capture
+        {% if site.data.locales[lang].copyright.license.template %}
-        _replacement %}
+          {% capture _replacement %}
         <a href="{{ site.data.locales[lang].copyright.license.link }}">
           {{ site.data.locales[lang].copyright.license.name }}
         </a>
-        {% endcapture %} {{ site.data.locales[lang].copyright.license.template |
+        {% endcapture %}
-        replace: ':LICENSE_NAME', _replacement }} {% endif %}
+
+          {{ site.data.locales[lang].copyright.license.template | replace: ':LICENSE_NAME', _replacement }}
+        {% endif %}
       </div>
 
       {% include post-sharing.html lang=lang %}

_posts/mathematics/2025-08-22-group-structure-of-z-2n-z-star.md

@@ -0,0 +1,146 @@
+---
+share: true
+toc: true
+math: true
+categories:
+  - Mathematics
+path: _posts/mathematics
+tags:
+  - math
+  - cryptography
+title: Group Structure of $(\mathbb{Z}/2^n \mathbb{Z})^*$
+date: 2025-08-22
+github_title: 2025-08-22-group-structure-of-z-2n-z-star
+---
+
+To compute the rotation automorphism homomorphically, we use the fact that $(\Z/2^n\Z)^* \simeq \span{-1, 5}$. I couldn't find a clear proof of this result online, so I just accepted the fact although it wasn't very satisfying.
+
+After more than a year, I got a chance to revisit the rotation automorphism and I figured that I should clear things up once and for all. So I decided to compile a proof, drawn from many sources.
+
+## Theorem
+
+> **Theorem 1.** $(\Z/2^n \Z)^*$ is the direct product of a cyclic group of order $2$ and cyclic group of order $2^{n-2}$, for all $n \geq 2$.
+
+The above theorem is from Corollary 20 (2) of Section 9.5 in Abstract Algebra, 3rd Edition, Dummit and Foote.
+
+> **Theorem 2.** $(\Z/2^n\Z)^* \simeq \span{-1, 5}$ for $n \geq 3$.
+
+## Observations
+
+### Order of $5$ Modulo $2^n$
+
+> **Proposition.** $5^{2^{n-3}} \equiv 1 + 2^{n-1} \pmod {2^n}$ for $n \geq 3$.
+
+*Proof*. This is an easy proof with induction. Omitted.
+
+> **Lemma.** $5$ has order $2^{n-2}$ in $(\Z/2^n \Z)^*$, for $n \geq 2$.
+
+*Proof*. We will use strong induction. For $n = 2, 3$, the lemma can be checked by direct computation. Now assume that the order of $5$ is $2^{k-2}$ in $(\Z/2^k\Z)^*$, for all $3 \leq k \leq n$.
+
+Let $r$ be the order of $5$ modulo $2^{n+1}$. Then $2^{n-2} \mid r$. This is from the fact that
+
+$$
+5^r \equiv 1 \pmod {2^{n+1}} \implies 5^r \equiv 1 \pmod {2^n}.
+$$
+
+Therefore $r$ must be a multiple of $2^{n-2}$. (order of $5$ modulo $2^n$) But from the above proposition, $5^{2^{n-2}} \equiv 1 + 2^n \pmod {2^{n+1}}$, so $r \neq 2^{n-2}$. The next candidate of $r$ is $2^{n-1}$ since it should be a multiple of $2^{n-2}$. Observe that
+
+$$
+5^{2^{n-1}} = \paren{5^{2^{n-2}}}^2 = (1 + 2^{n})^2 \equiv 1 \pmod {2^{n+1}},
+$$
+
+completing the proof.
+
+### Group is Not Cyclic
+
+> **Proposition.** Let $G = \span{x}$ be a cyclic group of finite order $n < \infty$. For each divisor $a$ of $n$, there exists a unique subgroup of $G$ with order $a$.
+
+*Proof*. Since $a \mid n$, set $d = n /a$. Then $\span{x^d}$ is a subgroup of order $a$, showing existence.
+
+For uniqueness, suppose $H \neq \span{x^d}$ is another subgroup of $G$ with order $a$. Since subgroups of cyclic groups are also cyclic, $H = \span{x^k}$ where $k$ is the smallest positive integer with $x^k \in H$. Then from
+
+$$
+\frac{n}{d} = a = \abs{H} = \frac{n}{\gcd(n, k)},
+$$
+
+$d = \gcd(n, k)$. So $k$ is a multiple of $d$, resulting in $x^k \in \span{x^d}$. Therefore, $H \leq \span{x^d}$, but the two groups have the same order, so $H = \span{x^d}$.
+
+> **Lemma.** $(\Z/2^n \Z)^*$ is not cyclic for any $n \geq 3$.
+
+*Proof*. $(\Z/2^n\Z)^*$ has two distinct subgroups of order $2$. For $n \geq 3$,
+
+$$
+(2^n - 1)^2 \equiv (-1)^2 \equiv 1 \pmod {2^n}
+$$
+
+and
+
+$$
+(2^{n-1}-1)^2 = 2^{2n-2} - 2^n + 1 \equiv 1 \pmod {2^n}.
+$$
+
+Both $2^n-1$ and $2^{n-1} - 1$ have order $2$ modulo $2^n$ and they are distinct since $n \geq 3$. By the above proposition, $(\Z/2^n\Z)^*$ cannot be cyclic.
+
+## Proof of Theorem 1
+
+*Proof*. $(\Z/2^n\Z)^*$ is a finitely generated abelian group, so the fundamental theorem of finitely generated abelian groups applies here. We know that group has order $2^{n-1}$, and from the above results,
+
+- $(\Z/2^n \Z)^*$ has an element of order $2^{n-2}$,
+- $(\Z/2^n \Z)^*$ is not cyclic for $n \geq 3$.
+
+Thus, for $n \geq 3$, the only possible case is $(\Z/2^n\Z)^* \simeq \Z _ 2 \times \Z_{2^{n-2}}$. As for $n = 2$, $(\Z/4\Z)^* \simeq \Z_2 \times \Z_1$ is pretty obvious.
+
+*Note*. I'm still looking for an elementary proof that doesn't use the fundamental theorem. This sort of feels like nuking a mosquito.
+
+## More Observations
+
+> **Lemma.** Suppose that $H$ and $K$ are normal subgroups of $G$ and $H \cap K = \braces{1}$. Then $HK \simeq H \times K$.
+
+*Proof*. Construct an isomorphism $\varphi : H \times K \ra HK$ such that $(h, k) \mapsto hk$.
+
+Since $H, K \unlhd G$, observe that $hkh\inv k \inv \in K \cap H = \braces{1}$ and $hk = kh$. Therefore,
+
+$$
+\varphi(h, k) \cdot \varphi(h',k') = hkh'k' = hh' kk' = \varphi\paren{(h, k)\cdot (h', k')}
+$$
+
+and $\varphi$ is a homomorphism.
+
+Next, if $\varphi(h, k) = hk = 1$, we have $h = k\inv \in H\cap K = \braces{1}$. Then $h = k = 1$, showing that $\ker \varphi$ is trivial and $\varphi$ is injective.
+
+Surjectivity of $\varphi$ is trivial. $\varphi$ is an isomorphism and $HK \simeq H \times K$.
+
+> **Proposition.** As subgroups of $(\Z/2^n\Z)^*$, $\span{-1} \cap \span{5} = \braces{1}$ for $n \geq 3$.
+
+*Proof*. It suffices to show that $-1 \notin \span{{5}}$. Suppose that $-1 \in \span{5}$. Since $\span{5}$ is cyclic, it has a unique element of order $2$. Since $5$ has order $2^{n-2}$, it must be the case that $-1 \equiv 5^{2^{n-3}} \pmod {2^n}$.
+
+Then we have
+
+$$
+-1 \equiv 5^{2^{n-3}} \equiv 1 + 2^{n-1} \pmod {2^n},
+$$
+
+which gives $2^{n-1} + 2 \equiv 0 \pmod {2^n}$. But for $n \geq 3$, this is impossible since $0 < 2^{n-1} + 2 < 2^n$. Contradiction.
+
+*Note*. If $-1 \in \span{5}$, then maybe $5$ would have generated the whole group. But the group isn't cyclic, so we have a contradiction?
+
+## Proof of Theorem 2
+
+*Proof*. Since we are dealing with commutative groups, all subgroups are normal. We have $\span{-1}, \span{5} \unlhd (\Z/2^n\Z)^*$ and $\span{-1} \cap \span{5} = \braces{1}$. Therefore,
+
+$$
+(\Z/2^n\Z)^* \simeq \Z_2 \times \Z_{2^{n-2}} = \span{-1} \times \span{5} \simeq \span{-1}\span{5}.
+$$
+
+This means that we can uniquely write all elements of $(\Z/2^n\Z)^*$ as $(-1)^a 5^b$ for ${} 0 \leq a < 2 {}$, $0 \leq b < 2^{n-2}$. From commutativity, this exactly equals the subgroup generated by $-1$ and $5$, which is $\span{-1, 5}$. This concludes the proof.
+
+## Notes
+
+The theorem wasn't so trivial after all, but I'm still happy to have resolved a long overdue task.
+
+## References
+
+- My notes taken from abstract algebra class
+- <https://math.stackexchange.com/q/459815>
+- <https://math.stackexchange.com/q/3881641>
+- <https://math.stackexchange.com/a/4910312/329909>

_tabs/about.md

@@ -1,16 +1,15 @@
 ---
 # the default layout is 'page'
 icon: fas fa-info-circle
-order: 4
+order: 0
 ---
 
 # Sungchan Yi
 
-- Last updated: 2025-07-18
+- Last updated: 2025-11-20
 - Email: [calofmijuck at snu dot ac dot kr](mailto:calofmijuck@snu.ac.kr)
-- [CV (pdf)](https://zxcvber.com/files/cv-eng.pdf)
 
-**Research Interests**: computer architecture, hardware-software co-design, security, formal verification
+**Research Interests**: computer architecture, hardware-software co-design, cryptography, formal verification
 
 ## Education
 
@@ -33,7 +32,7 @@ order: 4
 
 [**Cryptography & Privacy Lab**](https://crypto.snu.ac.kr), *Undergraduate Research Assistant* (Jan. 2024 ~ Feb. 2024)
 
-- Implemented BFV homomorphic encryption scheme and its bootstrapping over a large message space
+- Implemented generalized BFV scheme with bootstrapping based on CKKS bootstrapping techniques
 - Analyzed automorphism group of the plaintext space and determined their effects on ciphertext
 
 [**Software Foundations Lab**](https://sf.snu.ac.kr), *Undergraduate Research Assistant* (Jul. 2023 ~ Oct. 2023)
@@ -60,12 +59,13 @@ order: 4
 
 ## Publications
 
+- **Sungchan Yi**, Keun Soo Lim, Hoyeon Jo, Sungjun Jung, Minwoo Kwak, Dongoh Kim, Luigi Cussigh, Seong Hoon Seo, Jinkyu Jeong, Jae W. Lee, "Software-Defined Manycores via Hardware-Managed Preemptive Coroutine Scheduling", _International Symposium on Computer Architecture (ISCA)_, 2026. (Submitted)
 - **[ACCV'24]** Soosung Kim, Yeonhong Park, Hyunseung Lee, **Sungchan Yi**, and Jae W. Lee, "ReLUifying Smooth Functions: Low-Cost Knowledge Distillation to Obtain High-Performance ReLU Networks", _Asian Conference on Computer Vision (ACCV)_, Hanoi, Vietnam, December 2024. ([PDF](https://arc.snu.ac.kr/pubs/ACCV24_ReLU.pdf))
-- **Sungchan Yi**, "Secure IAM on AWS with Multi-Account Strategy", _Undergraduate Thesis_, December 2023. **Received best undergraduate paper award.** ([PDF](https://arxiv.org/pdf/2501.02203))
+- **Sungchan Yi**, "Secure IAM on AWS with Multi-Account Strategy", _Undergraduate Thesis_, December 2023. **Outstanding Undergraduate Thesis Award.** ([PDF](https://arxiv.org/pdf/2501.02203))
 
 ## Honors & Awards
 
-**Best Undergraduate Paper Award**
+**Outstanding Undergraduate Thesis Award**
 
 - Title: Secure IAM on AWS using Multi-Account Strategy
 

_tabs/archives.md

@@ -1,5 +1,5 @@
 ---
 layout: archives
 icon: fas fa-archive
-order: 3
+order: 4
 ---

_tabs/blog.md

@@ -0,0 +1,13 @@
+---
+title: Blog
+layout: home
+icon: fas fa-pencil
+order: 1
+# pagination:
+#   enabled: true
+#   per_page: 10
+#   collection: posts
+#   permalink: '/page/:num/'
+---
+
+This page serves as the site "Home" (posts listing). It is now a tab in the sidebar.

_tabs/categories.md

@@ -1,5 +1,5 @@
 ---
 layout: categories
 icon: fas fa-stream
-order: 1
+order: 2
 ---

_tabs/tags.md

@@ -1,5 +1,5 @@
 ---
 layout: tags
 icon: fas fa-tags
-order: 2
+order: 3
 ---

assets/css/jekyll-theme-chirpy.scss

@@ -1,7 +1,7 @@
 ---
 ---
 
-@import 'main
+@use 'main
 {%- if jekyll.environment == 'production' -%}
   .bundle
 {%- endif -%}

index.html

@@ -1,4 +1,13 @@
 ---
-layout: home
+layout: page
-# Index page
+title: About
 ---
+
+{%- assign about_tab = site.tabs | where: "url", "/about/" | first -%}
+
+{%- if about_tab -%}
+	{{ about_tab.content }}
+{%- else -%}
+	<!-- Fallback: no about tab found -->
+	<h1>About</h1>
+{%- endif -%}