Initial commit of "math in cryptography"

This part contains reasons why xor is frequently used in cryptography.

Author: showalski

_posts/2019-03-25-welcome-to-jekyll.markdown

@@ -1,25 +1,25 @@
 ---
 layout: post
-title:  "Welcome to Jekyll!"
+lang: zh
+title:  "密码学中的数据知识"
 date:   2019-03-25 21:30:00 +0800
-categories: [jekyll]
+categories: [cryptography]
 ---
-You’ll find this post in your `_posts` directory. Go ahead and edit it and re-build the site to see your changes. You can rebuild the site in many different ways, but the most common way is to run `jekyll serve`, which launches a web server and auto-regenerates your site when a file is updated.
 
-To add new posts, simply add a file in the `_posts` directory that follows the convention `YYYY-MM-DD-name-of-post.ext` and includes the necessary front matter. Take a look at the source for this post to get an idea about how it works.
+### 异或运算
 
-Jekyll also offers powerful support for code snippets:
+异或运算在密码学中被大量使用，它具有如下性质：
 
-{% highlight ruby %}
-def print_hi(name)
-  puts "Hi, #{name}"
-end
-print_hi('Tom')
-#=> prints 'Hi, Tom' to STDOUT.
-{% endhighlight %}
+* 均匀分布：
 
-Check out the [Jekyll docs][jekyll-docs] for more info on how to get the most out of Jekyll. File all bugs/feature requests at [Jekyll’s GitHub repo][jekyll-gh]. If you have questions, you can ask them on [Jekyll Talk][jekyll-talk].
+    * 假设随机变量X和Y在集合$$\{0, 1\}^n$$上随机分布，X独立于Y，那么变量$$Z = X \oplus Y$$在空间$$\{0, 1\}^n$$上也是均匀分布的。集合$$U = \{0, 1\}^2 = \{00, 01, 10, 11\}$$，即集合U由四个长度为两个bit的二进制的数组成的集合。
 
-[jekyll-docs]: http://jekyllrb.com/docs/home
-[jekyll-gh]:   https://github.com/jekyll/jekyll
-[jekyll-talk]: https://talk.jekyllrb.com/
+    * 设m和k为nbit数，k在集合$$U = \{0, 1\}^n$$上均匀分布，那么$$c = m \oplus k$$也是均匀分布的。证明参考[连接][xor_dis_proof]。这里的c为密文，m为明文，k为密钥。
+
+* [同等概率][xor_prob]:设a和b是{0, 1}空间中的随机变量，即a和b为0或1。根据真值表计算：a & b得到0的概率为75%，得到1个概率为25%；$$a \mid b$$得到0的概率为25%，得到1的概率为75%；而$$a \oplus b$$得到0或者1的概率都是50%。
+
+* 可逆性：设c为密文，m为明文，k为密钥，有等式$$c = m \oplus k, m = c \oplus k$$，即在使用同一个密钥时，异或运算既可以得到密文也可得到明文。这种加解密的方式在one time pad中用到。[参考][xor_reverse]
+
+[xor_prob]: https://stackoverflow.com/questions/5889238/why-is-xor-the-default-way-to-combine-hashes
+[xor_dis_proof]: https://math.stackexchange.com/questions/441329/how-to-prove-uniform-distribution-of-m-oplus-k-if-k-is-uniformly-distributed
+[xor_reverse]: https://stackoverflow.com/questions/1379952/why-is-xor-used-in-cryptography

category/bash.md

@@ -1,6 +1,6 @@
 ---
 layout: posts_by_category
-categories: bash
-title: Bash
-permalink: /category/bash
----
+categories: cryptography
+title: Cryptography
+permalink: /category/cryptography
+---