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authorAndrew Guschin <guschin.drew@gmail.com>2022-12-04 15:52:05 +0400
committerAndrew Guschin <guschin.drew@gmail.com>2022-12-04 15:52:05 +0400
commit1155995f9ef0e44b839e43c2d9d609d2e6cfaa4f (patch)
tree2aeb32ed8cc6ed083c2bca9bff464b38201e8152 /cryptography
parent2a36eaec0ad895b5866d9cc6cad8976bbb9ae9d3 (diff)
Дописаны недостающие части лекций и добавлены рисунки
Diffstat (limited to 'cryptography')
-rw-r--r--cryptography/cryptography.pdfbin701428 -> 949458 bytes
-rw-r--r--cryptography/images/lecture11/characteristic.pdf937
-rw-r--r--cryptography/images/lecture13/linear_registry.pdf1105
-rw-r--r--cryptography/images/lecture9/disk_after_rotation.pdf2857
-rw-r--r--cryptography/images/lecture9/gamma.pdf554
-rw-r--r--cryptography/images/lecture9/start_disk.pdf2554
-rw-r--r--cryptography/lectures/lecture10.tex2
-rw-r--r--cryptography/lectures/lecture11.tex14
-rw-r--r--cryptography/lectures/lecture13.tex45
-rw-r--r--cryptography/lectures/lecture9.tex68
10 files changed, 8125 insertions, 11 deletions
diff --git a/cryptography/cryptography.pdf b/cryptography/cryptography.pdf
index 3f46cf1..d155bef 100644
--- a/cryptography/cryptography.pdf
+++ b/cryptography/cryptography.pdf
Binary files differ
diff --git a/cryptography/images/lecture11/characteristic.pdf b/cryptography/images/lecture11/characteristic.pdf
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diff --git a/cryptography/lectures/lecture10.tex b/cryptography/lectures/lecture10.tex
index 351980a..13aaa68 100644
--- a/cryptography/lectures/lecture10.tex
+++ b/cryptography/lectures/lecture10.tex
@@ -1,6 +1,6 @@
% Лекция 10 (07.11.22)
-\begin{example}
+\begin{example}\hfill
\begin{enumerate}
\item
Любая группа является также и квазигруппой, так как $$a \cdot x = b \iff x
diff --git a/cryptography/lectures/lecture11.tex b/cryptography/lectures/lecture11.tex
index f6fc1ec..5d6b149 100644
--- a/cryptography/lectures/lecture11.tex
+++ b/cryptography/lectures/lecture11.tex
@@ -105,7 +105,11 @@ y \in Y$ выполняется равенство $p(x/y) = p_X(x)$.
На рисунке приведена рабочая характеристика шифра простой замены (пунктир
--- имеется несколько возможных решений; по мере увеличения объёма перехвата
количество необходимой работы быстро уменьшается).
-\textbf{TODO: рисунок 1}
+\begin{figure}[H]
+ \centering
+ \includegraphics[width=0.6\textwidth]{lecture11/characteristic.pdf}
+ \caption{Характеристика шифра простой замены}
+\end{figure}
Практическую стойкость шифра обычно оценивают с помощью величины
$W_\text{д}(\infty)$, которую можно назвать \emph{достигнутой оценкой} рабочей
@@ -179,4 +183,10 @@ $p_\text{н} = \max \set{p_\text{им}, p_\text{подм}}$.
Обозначим через $I(Y, K)$ \emph{взаимную информацию} между $Y$ и $K$, то есть
величину, определяемую формулой $I(Y, K) = H(Y) - H(Y/K)$.
-% TODO: Дописать
+\emph{Совершенная имитостойкость} (то есть теоретически наилучшая защита от
+имитации или подмены) определяется как равенство $\log p_\text{н} = -I(Y, K)$.
+
+В общем случае не известно, при каких условиях существуют шифры, обеспечивающие
+совершенную имитостойкость, хотя имеются соответствующие примеры, которые
+свидетельствуют о том, что криптостойкость и имитостойкость являются
+независимыми свойствами шифра.
diff --git a/cryptography/lectures/lecture13.tex b/cryptography/lectures/lecture13.tex
index bd2b326..cc9ee50 100644
--- a/cryptography/lectures/lecture13.tex
+++ b/cryptography/lectures/lecture13.tex
@@ -78,7 +78,11 @@ P$, заданная на множестве целых неотрицатель
\end{equation*}
ЛРП реализуется схемой линейного регистра сдвига, изображённой на рисунке.
-\textbf{TODO: Рисунок 1}
+\begin{figure}[H]
+ \centering
+ \includegraphics[width=0.8\textwidth]{lecture13/linear_registry.pdf}
+ \caption{Линейный регистр сдвига}
+\end{figure}
В очередном такте работы регистра значения, содержащиеся в ячейках накопителя,
умножаются на соответствующие коэффициента $f_j$ и суммируются, после чего
@@ -133,6 +137,43 @@ $P$. Тогда функцией \emph{<<след>> из поля $Q$ в пол
\end{equation*}
\begin{theorem}
- Пусть $F(x) = x^m - \sum_{j = 0}^{m - 1}
+ Пусть $F(x) = x^m - \sum_{j = 0}^{m - 1} f_j \cdot x^j$ --- неприводимый
+ многочлен над полем $P$ степени $m$, $\theta$ --- корень $F(x)$ в поле $Q$.
+ Тогда для ЛРП $\set{u(i)}$ с характеристическим многочленом $F(X)$
+ существует единственная константа $\alpha \in P$ такая, что
+ \begin{equation*}
+ u(i) = \text{tr}_q^{q^m} (\alpha \cdot \theta^i),\, i \geq 0
+ \end{equation*}
\end{theorem}
+Пусть $u$ --- ЛРП минимального периода над полем $P$ и $\nu(\alpha_1, \dots,
+\alpha_k)$ --- число решений системы уравнений
+\begin{equation*}
+ \begin{cases}
+ u(i + j) = \alpha_j,\, j = \overline{1, k} \\
+ 0 \leq i < q^m - 1
+ \end{cases}
+\end{equation*}
+то есть число появлений мультиграммы $\alpha_1, \dots, \alpha_k$ на периоде
+последовательности $u$.
+
+% NOTE: Утверждение 5
+\begin{statement}
+ Пусть $F(x)$ --- примитивный многочлен степени $m$ над полем $P$ и $\theta$
+ --- корень $F(x)$ в поле $Q$. Тогда любая ненулевая мультиграмма $(\alpha_1,
+ \dots, \alpha_k)$ встречается на периоде ЛРП $u$ ровно $\nu(\alpha_1, \dots,
+ \alpha_k) = q^{m - 1}$ раз, а число вхождений нулевой мультиграммы на единицу
+ меньше.
+\end{statement}
+
+Таким образом, ЛРП над полем позволяет обеспечить первые два из трёх требований
+к псевдослучайным последовательностям, используемым при построении управляющих
+блоков поточных шифрсистем.
+
+За счёт выбора закона рекурсии можно также гарантировать достаточную величину
+периода получаемой псевдослучайной последовательности и хорошие статистические
+качества. Вместе с тем аналитическое строение ЛРП оказывается достаточно
+простым.
+
+Для определения начального вектора по некоторому отрезку последовательности
+достаточно решить систему линейных уравнений.
diff --git a/cryptography/lectures/lecture9.tex b/cryptography/lectures/lecture9.tex
index 8d285c4..76c9261 100644
--- a/cryptography/lectures/lecture9.tex
+++ b/cryptography/lectures/lecture9.tex
@@ -17,11 +17,17 @@ $$X = \begin{pmatrix}
Для того, чтобы выписать подстановку, реализуемую диском после поворота на угол
$\frac{2 \pi}{n}$ посмотрим на соответствующие рисунки.
-% TODO: Рисунок --- начальное положение диска.
-\textbf{TODO: рис1}
-
-% TODO: Рисунок --- положение диска после поворота.
-\textbf{TODO: рис2}
+\begin{figure}[H]
+ \centering
+ \includegraphics[width=0.9\textwidth]{lecture9/start_disk.pdf}
+ \caption{Начальное положение диска}
+\end{figure}
+
+\begin{figure}[H]
+ \centering
+ \includegraphics[width=0.9\textwidth]{lecture9/disk_after_rotation.pdf}
+ \caption{Положение диска после поворота}
+\end{figure}
Так как диск сдвигается как твёрдое тело, символ открытого текста поступающий
на него с входной розетки, проходит затем по имеющимся в диске соединениям,
@@ -98,7 +104,57 @@ X_2 \cdot T^{\gamma_2 - \gamma_3} \cdot \dots T^{\gamma_{N - 1} - \gamma_N}
Криптоанализ дисковых шифраторов является весьма сложной задачей.
\subsection{Шифры гаммирования}
-\textbf{TODO: Дописать}
+
+Во второй половине XIX века появился весьма устойчивый способ усложнения
+числовых кодов --- гаммирование.
\paragraph{Шифр Виженера.}
+
+Исторически первый шифр гаммирования совпадал, по сути, с шифром Виженера,
+однако без использования самой таблицы Виженера (квадрат, каждая строка и каждый
+столбец которой --- некоторая перестановка знаков данного алфавита).
+
+Французский криптограф Б. Виженер опубликовал свой метод в <<Трактате о шифрах>>
+в 1585 году. С тех пор на протяжении трёх столетий шифр Виженера считался
+нераскрываемым, пока с ним не справился Ф. Казински (в 1863 году).
+
+Открытый текст разбивается на блоки длины $n > 1$. Задаётся ключ ---
+последовательность из $n$ натуральных чисел $(a_1, a_2, \dots, a_n)$. В каждом
+блоке первая буква циклически сдвигается вправо на $a_1$ позиций в алфавите,
+вторая --- на $a_2$, $\dots$, последняя --- на $a_n$.
+
\paragraph{Табличное гаммирование.}
+
+Латинский квадрат --- таблица $n \times n$, заполненная $n$ различными символами
+таким образом, чтобы в каждой строке и в каждом столбце встречались все $n$
+символов (каждый по одному разу).
+
+\begin{table}[H]
+ \centering
+ \begin{tabular}{|c|c|c|}
+ \hline
+ 1 & 2 & 3 \\ \hline
+ 2 & 3 & 1 \\ \hline
+ 3 & 1 & 2 \\ \hline
+ \end{tabular}
+\end{table}
+
+Шифр табличного гаммирования в алфавите $A = \set{a_1, \dots, a_n}$
+определяется произвольным латинским квадратом $L$ на $A$ и способом получения
+последовательности букв из $A$, называемой \emph{гаммой шифра}.
+
+\begin{figure}[H]
+ \centering
+ \includegraphics[width=0.45\textwidth]{lecture9/gamma.pdf}
+ \caption{Гамма шифра}
+\end{figure}
+
+Буква $a_i$ открытого текста под действием гаммы $a_j$ переходит в букву $a_k$
+шифрованного текста, содержащуюся в $j$-й строке и $i$-м столбце квадрата $L$
+(подразумевается, что строки и столбцы в $L$ занумерованы в соответствии с
+порядковым следованием букв в алфавите $A$).
+
+\emph{Квазигруппой} называют пару $(Q, \cdot)$ из непустого множества $Q$ с
+бинарной операцией $\cdot : Q \times Q \to Q$, удовлетворяющей следующему
+условию: для любых элементов $a, b \in Q$ найдутся единственные элементы
+$x, y \in Q$, такие что $a \cdot x = b,\, y \cdot a = b$.
generated by cgit v1.2.3 (git 2.25.1) at 2025-12-17 09:15:48 +0000