From cefaaca3a21110022d74b682eb2a12763dfa83ab Mon Sep 17 00:00:00 2001 From: Zentrik Date: Fri, 26 Apr 2024 09:09:40 +0100 Subject: [PATCH] CC: Various fixes --- .../04_algebraic_coding_theory.tex | 25 ++++++++++-------- CodingAndCryptography/05_cryptography.tex | 9 ++++--- CodingAndCryptography/cc.pdf | Bin 348405 -> 349740 bytes 3 files changed, 19 insertions(+), 15 deletions(-) diff --git a/CodingAndCryptography/04_algebraic_coding_theory.tex b/CodingAndCryptography/04_algebraic_coding_theory.tex index bb3179f..3c80dda 100644 --- a/CodingAndCryptography/04_algebraic_coding_theory.tex +++ b/CodingAndCryptography/04_algebraic_coding_theory.tex @@ -212,7 +212,7 @@ \subsection{Syndrome decoding} \subsection{Hamming codes} \begin{definition}[Hamming Code] - Let $d \geq 1$, and let $n = 2^d - 1$. + Let $d \geq 2$, and let $n = 2^d - 1$. Let $H$ be the $d \times n$ matrix with columns given by the nonzero elements of $\mathbb F_2^d$. The \vocab{Hamming $(n, n-d)$-linear code} is the (linear) code with parity check matrix $H$. \end{definition} @@ -226,7 +226,8 @@ \subsection{Hamming codes} \end{lemma} \begin{proof} - Any two columns of $H$ are linearly independent, but there are three linearly dependent columns as $n = 2^d-1$. + Any two columns of $H$ are linearly independent as otherwise they would be the same. + $H$ has $3$ columns $100\dots0, 010\dots0, 110\dots0$ which are linearly dependent. Hence, $d(C) = 3$. Hence, $C$ is $\floor*{\frac{3-1}{2}} = 1$-error correcting. A perfect code is one s.t. $\abs{C} = \frac{2^n}{V(n,e)}$. @@ -362,7 +363,7 @@ \subsection{New codes from old (again)} \begin{enumerate} \item One could define $RM(d, 0)$ and $RM(d, d)$ and also define recursively $RM(d, r)$ as a bar product. \item $RM(5, 1)$ was used by NASA for the Mariner 9 mission to Mars. - \item Decoding procedure using `successive majority verdicts' is outline in (Goldie and Pinch, pages 165-167). + \item Decoding procedure using `successive majority verdicts' is outlined in (Goldie and Pinch, pages 165-167). \end{enumerate} \end{remark} @@ -430,11 +431,11 @@ \subsection{Cyclic Codes} \begin{proof} If $g(X) = a_0 + a_1X + \dots + a_{n-1}X^{n-1} \mod (X^n - 1)$, then $Xg(X) = a_{n-1} + a_0X + \dots + a_{n-2}X^{n-1} \mod (X^n - 1)$. - So $\mathcal C$ is cyclic iff (i) and (ii) hold and if (iii)': $g(X) \in C \implies Xg(X) \in C$. - Note (iii)' is the case $f(X) = X$ of (iii). + So $\mathcal C$ is cyclic iff (1) and (2) hold and if (3)': $g(X) \in C \implies Xg(X) \in C$ holds. + Note (3)' is the special case $f(X) = X$ of (3). In general, $f(X) = \sum a_i X^i$ so \begin{align*} - f(X) g(X) &= \sum_i a_i \underbracket{X^i g(X)}_{\in \mathcal{C} \text{ by (iii)}} \in \mathcal{C} \text{ by (ii)} + f(X) g(X) &= \sum_i a_i \underbracket{X^i g(X)}_{\in \mathcal{C} \text{ by (iii)}} \in \mathcal{C} \text{ by (2)} \end{align*} \end{proof} @@ -456,7 +457,7 @@ \subsection{Cyclic Codes} \end{theorem} \begin{proof} - Let $g(X) \in \mathbb F_2[X]$ be a poly poly of least degree representing a $\neq 0$ codeword of $C$. + Let $g(X) \in \mathbb F_2[X]$ be a poly of least degree representing a $\neq 0$ codeword of $C$. Note that $\deg g < n$. Since $C$ is cyclic, $(g) \subseteq C$. \\ Now let $p(X) \in \mathbb F_2[X]$ represent a codeword. @@ -481,12 +482,13 @@ \subsection{Cyclic Codes} \end{lemma} \begin{proof} - \underline{Linear Independence}: Suppose $f(X) g(X) = 0 \mod (X^n - 1)$ for some $f(X) \in \mathbb{F}_2[X]$ with $\deg f < n - k$. + \underline{Linear Independence}: Consider $a_1 g + a_2 Xg + \dots + a_{n - k} X^{n - k - 1} g$ as $f(X) g(X)$. + Suppose $f(X) g(X) = 0 \mod (X^n - 1)$ for some $f(X) \in \mathbb{F}_2[X]$ with $\deg f < n - k$. Then $\deg fg < n$, so $f(X)g(X) = 0$, hence $f(X) = 0$, i.e. every dependence relation is trivial. \underline{Spanning}: Let $p(x) \in \mathbb{F}_2[X]$ represent a codeword. WLOG $\deg p < n$. - Since $g(X)$ is the gen poly, $g(x) \mid p(X)$ i.e. $p(X) = f(X) g(X)$ for some $f(X) \in \mathbb{F}_2[X]$. + Since $g(X)$ is the gen poly, $g(X) \mid p(X)$ i.e. $p(X) = f(X) g(X)$ for some $f(X) \in \mathbb{F}_2[X]$. Also $\deg f = \deg p - \deg q < n - k$, so $p(X)$ lies in the span of $g(X), \dots, X^{n - k - 1} g(X)$. \end{proof} @@ -629,7 +631,8 @@ \subsection{BCH codes} \end{align*} This is a $(\delta - 1) \times n$ matrix. Any collection of $(\delta - 1)$ columns is independent as it forms a Vandermonde matrix. - But any codeword of $C$ is a dependence relation between the columns of $H$. + As any codeword of $C$ is s.t. $Hc = 0$, $c$ satisfies $\delta - 1$ independent linear relations. + % But any codeword of $C$ is a dependence relation between the columns of $H$. Hence every nonzero codeword has weight at least $\delta$, giving $d(C) \geq \delta$. \end{proof} @@ -697,7 +700,7 @@ \subsubsection{Decoding BCH Codes} Recall $e(\alpha^j) = r(\alpha^j)$ for $j = 1, 2, \dots, 2t$. \begin{itemize} - \item Set $\sigma(X) = \sigma_0 + \sigma_1 X + \dots + \sigma_t X^t$ and $\sigma(X)(r(\alpha) X + r(\alpha^2) X^2 \dots + r(\alpha^{2t})X^{2t} + e(\alpha^{2t + 1}) X^{2t + 1} + \dots) = \sum_{i=0}^{t} \omega_i x^i$. + \item Set $\sigma(X) = \sigma_0 + \sigma_1 X + \dots + \sigma_t X^t$ and $\sum_{i=0}^{t} \omega_i X^i = \sigma(X)(r(\alpha) X + r(\alpha^2) X^2 + \dots + r(\alpha^{2t})X^{2t})$ mod $X^{2t + 1}$. \item Coeffs of $X^i$ for $t < i \leq 2i$ are $\sum_{j=0}^t \sigma_j r(\alpha^{i-j}) = 0$ which don't involve any of $e(\alpha^j)X^j$ for all $1 \leq j \leq 2t$. \item So we obtain a system of linear equations \begin{align*} diff --git a/CodingAndCryptography/05_cryptography.tex b/CodingAndCryptography/05_cryptography.tex index 9686f95..3a7b7b3 100644 --- a/CodingAndCryptography/05_cryptography.tex +++ b/CodingAndCryptography/05_cryptography.tex @@ -286,7 +286,7 @@ \subsection{Rabin cryptosystem} Usually, we restrict our messages so that $(m, N) = 1$ and $m > \sqrt{N}$. Receiving ciphertext $c$, we can solve for $x_1, x_2$ s.t. $x_1^2 \equiv c$ mod $p$ and $x_2^2 \equiv c$ mod $q$ using the previous lemma. -Then, applying the Chinese remainder theorem, we can find $x$ s.t. $x \equiv x_1$ mod $p$ and $x \equiv x_2$ mod $q$, hence $x^2 \equiv c$ mod $N$. +Then, we use the Chinese remainder theorem, we can find $x$ s.t. $x \equiv x_1$ mod $p$ and $x \equiv x_2$ mod $q$, hence $x^2 \equiv c$ mod $N$\footnote{By squaring $x^2 \equiv x_1^2$ mod $p$ and $x^2 \equiv x_2^2$ mod $q$ so $x = pn + c = qm + c$. As $p, q$ coprime $x = pqa + c$.}. Indeed, running the Euclidean algorithm on $p, q$ gives integers $r, s$ s.t. $rp + sq = 1$, then we can take $x = sqx_1 + rpx_2$. \begin{lemma} ~\vspace*{-1.5\baselineskip} @@ -299,10 +299,11 @@ \subsection{Rabin cryptosystem} \end{lemma} \begin{proof} - \emph{Part (i).} - $x^2 \equiv y^2$ mod $p$ iff $p \mid (x^2 - y^2) = (x-y)(x+y)$, so either $p \mid x-y$ or $p \mid x+y$, so $x = \pm y$. + \emph{Part (1).} + If there is a solution $x$, $-x$ also works. + If $x, y$ are solutions, then $x^2 \equiv y^2$ mod $p$ so $p \mid (x^2 - y^2) = (x-y)(x+y)$, so either $p \mid x-y$ or $p \mid x+y$, so $x = \pm y$. - \emph{Part (ii).} + \emph{Part (2).} If $x_0$ is a solution, then by the Chinese remainder theorem, there exist solutions $x$ with $x \equiv \pm x_0$ mod $p$ and $x \equiv \pm x_0$ mod $q$. This gives four solutions as required. 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