Merge overleaf-2026-01-29-0628 into main

Author: gogo9th

00-main.tex

@@ -297,7 +297,7 @@ \section*{Preface}
 
 %$ $
 
-This book is available both as an \href{https://arxiv.org/abs/2503.05136}{\textbf{arXiv PDF}} and on an \href{https://fhetextbook.github.io} {\textbf{auto-generated website}} (powered by \href{https://www.kodymirus.cz/overleaf-html-sample/main.html}{make4ht}). We provide a Python demo FHE library (TFHE, BFV, BGV, CKKS) for educational purposes, which is available at \href{https://github.com/fhetextbook/fhe-textbook}{\texttt{https://github.com/fhetextbook/fhe-textbook}}.
+This book is available both as an \href{https://arxiv.org/abs/2503.05136}{\textbf{arXiv PDF}} and on an \href{https://fhetextbook.github.io} {\textbf{auto-generated dynamic website}} (powered by \href{https://www.kodymirus.cz/overleaf-html-sample/main.html}{make4ht}). We provide a Python Demo FHE library (TFHE, BFV, BGV, CKKS) for educational purposes, which is available at \href{https://github.com/fhetextbook/fhe-textbook}{\texttt{https://github.com/fhetextbook/fhe-textbook}}.
 Please report any errors regarding the book draft on the \href{https://github.com/fhetextbook/fhe-textbook/issues}{\textbf{Issues Board}}.
 
 

README.md

@@ -1,10 +1,12 @@
+# The Beginner's Textbook for Fully Homomorphic Encryption
+
 #### ArXiv's PDF Link: [https://arxiv.org/abs/2503.05136](https://arxiv.org/abs/2503.05136)
 
 #### Dynamic Website Version: [https://fhetextbook.github.io](https://fhetextbook.github.io)
 
 -  Please post any bugs or errors regarding the draft to the Issues board or create a pull request.
 
-## Python Demo FHE Library Quickstart
+## Python FHE Demo Library Quickstart
 
 We implemented this library for educational purposes. Because it is written in pure Python and does not use an RNS-optimized backend, computations are relatively slow. For this reason, the default parameters are intentionally small—for example, the ring dimension / polynomial modulus degree ($N$) and ciphertext modulus ($Q$) are set to small values (e.g., $N=4$, $Q = 2^{30}$).
 

c03-glwe-mult-plain.tex

@@ -102,13 +102,11 @@ \subsection{Gadget Decomposition for Noise Suppression}
 
 $= \textsf{GLWE}_{S, \sigma}\left(\left(\Lambda_1\cdot\dfrac{q}{\beta} + \Lambda_2\cdot\dfrac{q}{\beta^2} + \cdots + \Lambda_l\cdot\dfrac{q}{\beta^l}\right)\cdot \Delta M  + E_{\textit{all}} \right)$ \textcolor{red}{ $\rhd$ where $E_{\textit{all}} = \sum\limits_{i=1}^l \Lambda_iE_i$}
 
-$= \textsf{GLWE}_{S, \sigma}\left(\Lambda \cdot \Delta M + E_{\textit{all}}\right)$
-
-$= \Lambda \cdot \textsf{GLWE}_{S, \sigma}( \Delta M + E_{\textit{all}})$
+$= \textsf{GLWE}_{S, \sigma}\left(\Lambda \cdot \Delta M + E_{\textit{all}}\right)$ \textcolor{red}{ $\rhd$ whose decryption is $\Lambda\cdot M$}
 
 $ $
 
-While the computation results are the same, as we decompose $\Lambda$ into smaller plaintext polynomials $\Lambda_1, \Lambda_2, \cdots, \Lambda_l$, the noise generated by each of $l$ plaintext-to-ciphertext multiplications becomes smaller.  Given the noise of each GLWE ciphertext in the GLev ciphertext is $E_i$, the final noise of the ciphertext-to-plaintext multiplication is $E_{\textit{all}} = \sum\limits_{i=1}^{l}\Lambda_i\cdot E_i$, which is much smaller than $\Lambda \cdot E$, because 
+While the decrypted results are the same, as we decompose $\Lambda$ into smaller plaintext polynomials $\Lambda_1, \Lambda_2, \cdots, \Lambda_l$, the noise generated by each of $l$ plaintext-to-ciphertext multiplications becomes smaller.  Given the noise of each GLWE ciphertext in the GLev ciphertext is $E_i$, the final noise of the ciphertext-to-plaintext multiplication is $E_{\textit{all}} = \sum\limits_{i=1}^{l}\Lambda_i\cdot E_i$, which is much smaller than $\Lambda \cdot E$, because 
 the coefficients of each decomposed polynomial $\Lambda_i$ are significantly smaller than those of $\Lambda$ (i.e., 
 $\|\Lambda_i\|_\infty \le \beta/2$, whereas $\|\Lambda\|_\infty$ can be as large as $q/2$). This is visually depicted in~\autoref{fig:decomp2}.
 
@@ -119,6 +117,7 @@ \subsection{Gadget Decomposition for Noise Suppression}
   \label{fig:decomp2}
 \end{figure}
 
-However, we cannot use this decomposition technique to the resulting ciphertext again, because the output of this algorithm is a GLWE ciphertext and converting it into a GLev ciphertext without decrypting it costs much noise and computation time (as we need to multiply the GLWE ciphertext by $\dfrac{q}{\beta}, \dfrac{q}{\beta^2}, \cdots, \dfrac{q}{\beta^l})$.  
 
-As a more efficient technique to re-initialize the noise $E$, we will describe TFHE's noise bootstrapping technique in \autoref{subsec:tfhe-noise-bootstrapping}.
+
+
+Meanwhile, for the technique to repeatedly re-initialize the noise $E$ of regular ciphertexts, we will describe TFHE's noise bootstrapping technique in \autoref{subsec:tfhe-noise-bootstrapping}.

d07-python-demo-library.tex

@@ -1 +1 @@
-We provide our Python demo FHE library (TFHE, BFV, CKKS, BGV) for educational purposes. The source code and the manual are available at \href{https://github.com/fhetextbook/fhe-textbook}{\texttt{https://github.com/fhetextbook/fhe-textbook}}.
+We provide our Python Demo FHE library (TFHE, BFV, CKKS, BGV) for educational purposes. The source code and the manual are available at \href{https://github.com/fhetextbook/fhe-textbook}{\texttt{https://github.com/fhetextbook/fhe-textbook}}.