本文详细解答了关于密码学和安全通信的习题，涉及排列组合、加密密钥计算、数学证明、指数运算、素数分布等多个方面，探讨了加密算法的周期、安全性等问题。
1.1
- Prove that there are N! distinct ways to arrange a sequence of N distinct symbols.
ans: an=n∗an−1a_n=n*a_{n-1}an=n∗an−1 - How many nontrivial recreational cipher keys on an alphabet of size 26 are there?
ans: N!-1 - How many recreational cipher keys are there satisfying the requirement that no letter is represented by itself?
思路：
ans:
N!−(CN1(N−1)!−CN2(N−2)!+CN3(N−3)!−...+...)=N!2!−N!3!+N!4!−N!5!+...−...N!-(C_{N}^{1}(N-1)!-C_{N}^{2}(N-2)!+C_{N}^{3}(N-3)!-...+...) \\ = \frac{N!}{2!}-\frac{N!}{3!}+\frac{N!}{4!}-\frac{N!}{5!}+...-...N!−(CN1(N−1)!−CN2(N−2)!+CN3(N−3)!−...+...)=2!N!−3!N!+4!N!−5!N!+...−...
当N足够大时，根据泰勒展开得到：
上式≈N!e上式\approx \frac{N!}{e}上式≈eN!
1.2
- Estimate the value of log2(256!)log_2(256!)log
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