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| author | Andrew Guschin <guschin.drew@gmail.com> | 2022-12-11 12:43:48 +0400 |
|---|---|---|
| committer | Andrew Guschin <guschin.drew@gmail.com> | 2022-12-11 12:43:48 +0400 |
| commit | 032e1fa27ff9faa9a5d82624dbfc5fe8a48c339c (patch) | |
| tree | 40083b63867a6517254e9b27a3a81c8f9aac5d37 | |
| parent | a28703315e4628b41e1d7d1d0628189b44b656b8 (diff) | |
Добавлена 5 лабораторная
| -rw-r--r-- | lab5/Cargo.lock | 1917 | ||||
| -rw-r--r-- | lab5/Cargo.toml | 8 | ||||
| -rw-r--r-- | lab5/src/check_window.rs | 90 | ||||
| -rw-r--r-- | lab5/src/forge_window.rs | 91 | ||||
| -rw-r--r-- | lab5/src/generate_window.rs | 128 | ||||
| -rw-r--r-- | lab5/src/main.rs | 44 | ||||
| -rw-r--r-- | lab5/src/sign_window.rs | 101 | ||||
| -rw-r--r-- | lab5/src/utils.rs | 32 | ||||
| -rw-r--r-- | lab5/src/window_state.rs | 6 | ||||
| -rw-r--r-- | reports/lab5/lab5.pdf | bin | 0 -> 162192 bytes | |||
| -rw-r--r-- | reports/lab5/lab5.tex | 135 | ||||
| -rwxr-xr-x | reports/lab5/maker.sh | 35 |
12 files changed, 2587 insertions, 0 deletions
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"6e99be55648b3ae2a52342f9a870c0e138709a3493261ce9b469afe6e4df6d8a" +dependencies = [ + "gethostname", + "nix 0.22.3", + "winapi", + "winapi-wsapoll", +] + +[[package]] +name = "xcursor" +version = "0.3.4" +source = "registry+https://github.com/rust-lang/crates.io-index" +checksum = "463705a63313cd4301184381c5e8042f0a7e9b4bb63653f216311d4ae74690b7" +dependencies = [ + "nom", +] + +[[package]] +name = "xml-rs" +version = "0.8.4" +source = "registry+https://github.com/rust-lang/crates.io-index" +checksum = "d2d7d3948613f75c98fd9328cfdcc45acc4d360655289d0a7d4ec931392200a3" diff --git a/lab5/Cargo.toml b/lab5/Cargo.toml new file mode 100644 index 0000000..944b5d7 --- /dev/null +++ b/lab5/Cargo.toml @@ -0,0 +1,8 @@ +[package] +name = "lab5" +version = "0.1.0" +edition = "2021" + +[dependencies] +eframe = "0.19.0" +fastrand = "1.8.0" diff --git a/lab5/src/check_window.rs b/lab5/src/check_window.rs new file mode 100644 index 0000000..216f1dc --- /dev/null +++ b/lab5/src/check_window.rs @@ -0,0 +1,90 @@ +use crate::utils::{adler64, powmod}; +use crate::window_state::WindowState; +use eframe::egui; + +pub struct Window { + message: String, + signature: String, + key: String, + correct: Option<bool>, + error: Option<String>, +} + +impl Default for Window { + fn default() -> Self { + Self { + message: String::new(), + signature: String::new(), + key: String::new(), + correct: None, + error: None, + } + } +} + +impl Window { + fn check(&mut self) { + let split = self.key.split(",").collect::<Vec<&str>>(); + if split.len() != 2 { + self.correct = None; + self.error = Some("Неверный формат ключа".to_string()); + return; + } + let y = match split[0].parse::<u64>() { + Ok(num) => num, + Err(_) => { + self.correct = None; + self.error = Some("Не удалось прочитать открытый ключ".to_string()); + return; + } + }; + let p = match split[1].parse::<u64>() { + Ok(num) => num, + Err(_) => { + self.correct = None; + self.error = Some("Не удалось прочитать модуль".to_string()); + return; + } + }; + + let signature = match self.signature.parse::<u64>() { + Ok(num) => num, + Err(_) => { + self.correct = None; + self.error = Some("Не удалось прочитать подпись".to_string()); + return; + } + }; + + let h = adler64(self.message.as_bytes()) % p; + self.correct = Some(powmod(signature, h, p) == y); + } +} + +impl WindowState for Window { + fn get_name(&self) -> &str { + "Проверить подпись" + } + + fn update(&mut self, ui: &mut egui::Ui) { + ui.label("Сообщение:"); + let resp1 = ui.add(egui::TextEdit::multiline(&mut self.message)); + ui.label("Подпись:"); + let resp2 = ui.add(egui::TextEdit::singleline(&mut self.signature)); + ui.label("Открытый ключ:"); + let resp3 = ui.add(egui::TextEdit::singleline(&mut self.key)); + if resp1.changed() || resp2.changed() || resp3.changed() { + self.correct = None; + } + + if ui.button("Проверить подпись").clicked() { + self.check(); + } + + if let Some(true) = &self.correct { + ui.label("Подпись верна"); + } else if let Some(false) = &self.correct { + ui.label("Подпись не соответствует сообщению"); + } + } +} diff --git a/lab5/src/forge_window.rs b/lab5/src/forge_window.rs new file mode 100644 index 0000000..72347b8 --- /dev/null +++ b/lab5/src/forge_window.rs @@ -0,0 +1,91 @@ +use crate::utils::{adler64, inverse_mod, powmod}; +use crate::window_state::WindowState; +use eframe::egui; + +pub struct Window { + message: String, + key: String, + signed: Option<u64>, + error: Option<String>, +} + +impl Default for Window { + fn default() -> Self { + Self { + message: String::new(), + key: String::new(), + signed: None, + error: None, + } + } +} + +impl Window { + fn sign(&mut self) { + let split = self.key.split(",").collect::<Vec<&str>>(); + if split.len() != 2 { + self.signed = None; + self.error = Some("Неверный формат ключа".to_string()); + return; + } + let y = match split[0].parse::<u64>() { + Ok(num) => num, + Err(_) => { + self.signed = None; + self.error = Some("Не удалось прочитать открытый ключ".to_string()); + return; + } + }; + let p = match split[1].parse::<u64>() { + Ok(num) => num, + Err(_) => { + self.signed = None; + self.error = Some("Не удалось прочитать модуль".to_string()); + return; + } + }; + + let h = adler64(self.message.as_bytes()) % p; + let h1 = inverse_mod(h as i64, p as i64) as u64; + self.signed = Some(powmod(y, h1, p)); + } +} + +impl WindowState for Window { + fn get_name(&self) -> &str { + "Подделать подпись" + } + + fn update(&mut self, ui: &mut egui::Ui) { + ui.label("Сообщение для подписи:"); + let resp1 = ui.add(egui::TextEdit::multiline(&mut self.message)); + ui.label("Открытый ключ:"); + let resp2 = ui.add(egui::TextEdit::singleline(&mut self.key)); + if resp1.changed() || resp2.changed() { + self.signed = None; + self.error = None; + } + + if ui.button("Подписать").clicked() { + self.sign(); + } + + if let Some(signature) = &self.signed { + let mut tmp = signature.to_string(); + ui.horizontal(|ui| { + ui.label("Подпись:"); + ui.add_enabled(false, egui::TextEdit::singleline(&mut tmp)); + }); + if ui.button("Скопировать подпись").clicked() { + ui.output().copied_text = tmp.clone(); + } + if ui.button("Скопировать сообщение").clicked() { + ui.output().copied_text = self.message.clone(); + } + } + + if let Some(error) = &self.error { + ui.label(error); + } + } +} diff --git a/lab5/src/generate_window.rs b/lab5/src/generate_window.rs new file mode 100644 index 0000000..25beb22 --- /dev/null +++ b/lab5/src/generate_window.rs @@ -0,0 +1,128 @@ +use crate::utils::powmod; +use crate::window_state::WindowState; +use eframe::egui; + +#[derive(PartialEq, Eq)] +enum State { + Idle, + Requested, + Generated, +} + +pub struct Window { + state: State, + p: u64, + g: u64, + a: u64, + y: u64, +} + +impl Default for Window { + fn default() -> Self { + Self { + state: State::Idle, + p: 0, + g: 0, + a: 0, + y: 0, + } + } +} + +fn min_order(x: u64, order: u64) -> u64 { + let mut min = 0; + let mut pow = x; + for power in 1..order { + if pow == 1 { + min = power; + break; + } + pow = (pow * x) % order; + } + return min; +} + +impl Window { + fn generate(&mut self) { + let primes = erathosphene(10000); + let i = fastrand::usize(4..primes.len()); + self.p = primes[i as usize]; + + for i in 1..self.p { + if min_order(i, self.p) == self.p - 1 { + self.g = i; + break; + } + } + + self.a = fastrand::u64(1..=self.p - 1); + self.y = powmod(self.g, self.a, self.p); + } +} + +fn erathosphene(n: u64) -> Vec<u64> { + let mut is_prime = vec![true; n as usize]; + let mut p = 2; + while p * p < n { + let mut cur = p * p; + while cur < n { + is_prime[cur as usize] = false; + cur += p; + } + while !is_prime[(p + 1) as usize] { + p += 1; + } + p += 1 + } + let mut primes = Vec::new(); + is_prime.iter().enumerate().for_each(|(i, p)| { + if *p { + primes.push(i as u64); + } + }); + return primes; +} + +impl WindowState for Window { + fn get_name(&self) -> &str { + "Генерация ключей" + } + + fn update(&mut self, ui: &mut egui::Ui) { + if ui.button("Сгенерировать").clicked() { + self.state = State::Requested; + } + if self.state == State::Requested { + self.generate(); + self.state = State::Generated; + } + if self.state == State::Generated { + ui.horizontal(|ui| { + ui.label("Модуль p:"); + let mut tmp = self.p.to_string(); + ui.add_enabled(false, egui::TextEdit::singleline(&mut tmp)); + }); + ui.horizontal(|ui| { + ui.label("Порождающий элемент g группы Z_p:"); + let mut tmp = self.g.to_string(); + ui.add_enabled(false, egui::TextEdit::singleline(&mut tmp)); + }); + ui.horizontal(|ui| { + ui.label("Секретный ключ a:"); + let mut tmp = self.a.to_string(); + ui.add_enabled(false, egui::TextEdit::singleline(&mut tmp)); + }); + ui.horizontal(|ui| { + ui.label("Публичный ключ y:"); + let mut tmp = self.y.to_string(); + ui.add_enabled(false, egui::TextEdit::singleline(&mut tmp)); + }); + if ui.button("Скопировать публичный ключ").clicked() { + ui.output().copied_text = format!("{},{}", self.y, self.p); + } + if ui.button("Скопировать секретный ключ").clicked() { + ui.output().copied_text = format!("{},{},{}", self.g, self.a, self.p); + } + } + } +} diff --git a/lab5/src/main.rs b/lab5/src/main.rs new file mode 100644 index 0000000..d056ebb --- /dev/null +++ b/lab5/src/main.rs @@ -0,0 +1,44 @@ +mod utils; +mod window_state; + +mod check_window; +mod forge_window; +mod generate_window; +mod sign_window; + +use crate::window_state::WindowState; +use eframe::egui; + +fn main() { + let options = eframe::NativeOptions::default(); + eframe::run_native( + "Криптографические методы защиты информации", + options, + Box::new(|_cc| Box::new(Application::default())), + ); +} + +struct Application { + labs: Vec<Box<dyn WindowState>>, +} + +impl Default for Application { + fn default() -> Self { + Self { + labs: vec![ + Box::new(generate_window::Window::default()), + Box::new(sign_window::Window::default()), + Box::new(check_window::Window::default()), + Box::new(forge_window::Window::default()), + ], + } + } +} + +impl eframe::App for Application { + fn update(&mut self, ctx: &egui::Context, _frame: &mut eframe::Frame) { + for window in &mut self.labs { + egui::Window::new(window.get_name()).show(ctx, |ui| window.update(ui)); + } + } +} diff --git a/lab5/src/sign_window.rs b/lab5/src/sign_window.rs new file mode 100644 index 0000000..1a574e4 --- /dev/null +++ b/lab5/src/sign_window.rs @@ -0,0 +1,101 @@ +use crate::utils::{adler64, inverse_mod, powmod}; +use crate::window_state::WindowState; +use eframe::egui; + +pub struct Window { + message: String, + key: String, + signed: Option<u64>, + error: Option<String>, +} + +impl Default for Window { + fn default() -> Self { + Self { + message: String::new(), + key: String::new(), + signed: None, + error: None, + } + } +} + +impl Window { + fn sign(&mut self) { + let split = self.key.split(",").collect::<Vec<&str>>(); + if split.len() != 3 { + self.signed = None; + self.error = Some("Неверный формат ключа".to_string()); + return; + } + let g = match split[0].parse::<u64>() { + Ok(num) => num, + Err(_) => { + self.signed = None; + self.error = Some("Не удалось прочитать закрытый ключ".to_string()); + return; + } + }; + let a = match split[1].parse::<u64>() { + Ok(num) => num, + Err(_) => { + self.signed = None; + self.error = Some("Не удалось прочитать закрытый ключ".to_string()); + return; + } + }; + let p = match split[2].parse::<u64>() { + Ok(num) => num, + Err(_) => { + self.signed = None; + self.error = Some("Не удалось прочитать модуль".to_string()); + return; + } + }; + + let h = adler64(self.message.as_bytes()) % p; + let h1 = inverse_mod(h as i64, p as i64) as u64; + + let z = (h1 * a) % (p - 1); + self.signed = Some(powmod(g, z, p)); + } +} + +impl WindowState for Window { + fn get_name(&self) -> &str { + "Подпись сообщения" + } + + fn update(&mut self, ui: &mut egui::Ui) { + ui.label("Сообщение для подписи:"); + let resp1 = ui.add(egui::TextEdit::multiline(&mut self.message)); + ui.label("Закрытый ключ:"); + let resp2 = ui.add(egui::TextEdit::singleline(&mut self.key)); + if resp1.changed() || resp2.changed() { + self.signed = None; + self.error = None; + } + + if ui.button("Подписать").clicked() { + self.sign(); + } + + if let Some(signature) = &self.signed { + let mut tmp = signature.to_string(); + ui.horizontal(|ui| { + ui.label("Подпись:"); + ui.add_enabled(false, egui::TextEdit::singleline(&mut tmp)); + }); + if ui.button("Скопировать подпись").clicked() { + ui.output().copied_text = tmp.clone(); + } + if ui.button("Скопировать сообщение").clicked() { + ui.output().copied_text = self.message.clone(); + } + } + + if let Some(error) = &self.error { + ui.label(error); + } + } +} diff --git a/lab5/src/utils.rs b/lab5/src/utils.rs new file mode 100644 index 0000000..34f87b3 --- /dev/null +++ b/lab5/src/utils.rs @@ -0,0 +1,32 @@ +pub fn powmod(x: u64, p: u64, m: u64) -> u64 { + let mut res = x; + for _ in 1..p { + res = (res * x) % m; + } + return res; +} + +pub fn adler64(buf: &[u8]) -> u64 { + let mod64 = 4294967291u64; + let mut s1 = 1u64; + let mut s2 = 0u64; + + for n in buf { + s1 = (s1 + *n as u64) % mod64; + s2 = (s2 + s1) % mod64; + } + return (s2 << 32) | s1; +} + +pub fn inverse_mod(x: i64, m: i64) -> i64 { + let (_, x, _) = extended_gcd(x, m); + return (x % m + m) % m; +} + +pub fn extended_gcd(a: i64, b: i64) -> (i64, i64, i64) { + if a == 0 { + return (b, 0, 1); + } + let (d, x, y) = extended_gcd(b % a, a); + return (d, y - (b / a) * x, x); +} diff --git a/lab5/src/window_state.rs b/lab5/src/window_state.rs new file mode 100644 index 0000000..ba8e57e --- /dev/null +++ b/lab5/src/window_state.rs @@ -0,0 +1,6 @@ +use eframe::egui; + +pub trait WindowState { + fn get_name(&self) -> &str; + fn update(&mut self, ui: &mut egui::Ui); +} diff --git a/reports/lab5/lab5.pdf b/reports/lab5/lab5.pdf Binary files differnew file mode 100644 index 0000000..1798093 --- /dev/null +++ b/reports/lab5/lab5.pdf diff --git a/reports/lab5/lab5.tex b/reports/lab5/lab5.tex new file mode 100644 index 0000000..5505297 --- /dev/null +++ b/reports/lab5/lab5.tex @@ -0,0 +1,135 @@ +\documentclass[a4paper,oneside]{article} + +\usepackage[utf8]{inputenc} +\usepackage[T2A]{fontenc} +\usepackage[english,russian]{babel} + +\usepackage{amsmath} +\usepackage{mathtools} +\usepackage{amsfonts} +\usepackage{enumitem} +\usepackage{amsthm} +\usepackage{minted} +\setminted{fontsize=\small, breaklines=true, style=emacs, linenos} +\usepackage{graphicx} +\graphicspath{ {./images/} } +\usepackage{float} +\usepackage{algorithm} +\floatname{algorithm}{Алгоритм} + +\newcommand{\Z}{\mathbb{Z}} +\DeclarePairedDelimiter\ceil{\lceil}{\rceil} +\DeclarePairedDelimiter\floor{\lfloor}{\rfloor} + +\newtheorem{theorem}{Теорема} +\newtheorem*{theorem*}{Теорема} + +% --- Определение --- % +\theoremstyle{definition} +\newtheorem{definition}{Определение} +\newtheorem*{definition*}{Определение} +% ------------------- % + +\theoremstyle{definition} +\newtheorem*{example}{Пример} + + +\title{{Криптографические методы защиты информации}\\{Лабораторная работа №5}} +\author{Гущин Андрей, 431 группа, 1 подгруппа, 2 вариант} +\date{\the\year{} г.} + +\begin{document} +\maketitle + +\section{Схема} + +Рассмотрим следующее упрощение схемы Эль-Гамаля. Пусть $p$ --- простое число, +$g$ --- порождающий элемент мультипликативной группы $Z^\cdot_p$. Эти данные +открыты. + +Пользователь выбирает секретное число $1 \leq a \leq p - 1$ и открывает значение +$y = g^a \pmod{p}$. + +К сообщению $m \in \{ 0, 1 \}^*$ применяется известная хэш-функция, дающая +значение $h = h(m) \in Z_p$. + +Допустим, что $\gcd(h, p - 1) = 1$. Этого можно добиться, внеся необходимые +коррективы в случае невыполнения. + +При генерации подписи пользователь вычисляет значение +\begin{equation} + z = h^{-1} a \pmod{(p - 1)} + \label{eq:zpow} +\end{equation} + +В качестве подписи под документом $m$ предлагается использовать $g^z$. + +Проверка правильности подписи элементарна: +\begin{equation} + (g^z)^h = g^{h^{-1} h a} = g^a \pmod{p} + \label{eq:check} +\end{equation} + +Подпись принимается, если полученное значение совпадает с $y$. + +Необходимо объяснить почему упомянутая схема неприемлема. + + +\section{Неверность схемы} + +Заметим, что в выражении \ref{eq:check} подразумевается, что значения $z$ +и $h$ находятся в одном кольце и поэтому можно написать $z \cdot h = +h^{-1} \cdot a \cdot h = h^{-1} \cdot h \cdot a = a$. + +Но, также можно заметить, что при вычислении $z$ в выражении \ref{eq:zpow} +умножаемые значения берутся по модулю $(p - 1)$. То есть, значение $z$ +является элементом мультипликативной группы $Z^\cdot_{(p - 1)}$. При этом +значение $h$ является элементом группы $Z^\cdot_p$ по условию. + +Таким образом, в общем случае мы можем сказать, что +\begin{equation*} + z \cdot h + = (h^{-1} \cdot a \pmod{(p - 1)}) \cdot h + \neq h^{-1} \cdot a \cdot h +\end{equation*} + +Например, пусть $p = 31, a = 19, h = 11$. При этом $h^{-1} = 17$, где обратное +берётся в группе $Z^\cdot_p$. + +\begin{align*} + z &= 17 \cdot 19 \pmod{30} = 23 \\ + z \cdot h &= 23 \cdot 11 \pmod{31} = 5 \\ + h^{-1} \cdot a \cdot h &= 17 \cdot 19 \cdot 11 \pmod{31} = 19 +\end{align*} + +Получаем +\begin{equation*} + z \cdot h = 5 \neq 19 = h^{-1} \cdot a \cdot h +\end{equation*} + +Таким образом, получаем, что сгенерированную подпись невозможно проверить и +данная схема является неприменимой и неприемлемой. + +\section{Взлом схемы} + +Будем считать схему подписи \emph{неприемлемой}, если имеется возможность +создать подпись для произвольного сообщения не имея при этом секретного ключа. + +Рассмотрим выражение \ref{eq:check}. Можно заметить, что $(g^z)^h = y$. Также +можем получить +\begin{equation*} + \left( \left( g^z \right)^h \right)^{h^{-1}} = y^{h^{-1}} +\end{equation*} + +То есть +\begin{equation*} + g^z = y^{h^{-1}} +\end{equation*} + +Но значение $y$ является частью открытого ключа, а $h^{-1}$ вычисляется из +публично известной хэш-функции для произвольного сообщения. + +Таким образом, можно создать подпись для произвольного сообщения не имея при +этом секретного ключа, а основываясь только на публичных данных. + +\end{document} diff --git a/reports/lab5/maker.sh b/reports/lab5/maker.sh new file mode 100755 index 0000000..e847acf --- /dev/null +++ b/reports/lab5/maker.sh @@ -0,0 +1,35 @@ +#!/bin/sh + +watch() { + [ -z "$1" ] && echo "Необходимо указать название основного документа" && help + set -o xtrace + latexmk -pdf -f -shell-escape -interaction=nonstopmode -pvc $1 +} + +doc() { + [ -z "$1" ] && echo "Необходимо указать название основного документа" && help + set -o xtrace + latexmk -pdf -f -shell-escape -interaction=nonstopmode $1 +} + +clean() { + set -o xtrace + rm -rf _minted-* + find . -name "*.aux" -exec rm {} \; + rm -f *.dvi *.fdb_latexmk *.fls *.log *.out *.toc +} + +help() { + echo "Использование:" + echo "./maker.sh watch <main-doc>.tex -> Запуск процесса, пересобирающего документ при изменениях" + echo "./maker.sh doc <main-doc>.tex -> Пересобрать документ" + echo "./maker.sh clean -> Удаление сгенерированных файлов" + exit 1 +} + +case "$1" in + watch) watch $2 ;; + doc) doc $2 ;; + clean) clean ;; + *) help ;; +esac |