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實驗原理及演算法

1. generate key
   本次模擬利用私鑰（n,sID）生成公鑰，具體過程如下：
   A）選擇一對512bit的素數p，q，計算n = p * q；
   B）計算fn = (p-1) * (q-1)；
   C）根據e * d ≡ 1 (mod fn)計算出公鑰（n,e）。
   CODE

# key generation
def gen_key(p, q):
    n = p * q
    fn =
 
 (p - 1) * (q - 1)
    d = 16337053
    # generate e
    # use e*d = k*fn + 1
    x = fn
    while (x + 1) % d:
        x += fn 
    e = (x+1) // d
    # public key, self key
    return (n, e), (n, d)

2. encryption and decryption
   加密：c ≡ m^e mod n
   解密:：m ≡ c^d mod n
   將plaintext字串轉化為ascii十進位制數字。
   CODE

def encryption(m, pk):
    n = pk[0]
    e = pk[1]
    c = square_mod(m, e, n)
    return c

def decryption(c, sk):
    n = sk[0]
    d = sk[1]
    m = square_mod(c, d, n)
return m

# get bit string of plaintext
    pt = 'test RSA'
    m = int(''.join(format(ord(ch)) for ch in pt))

3. 模重複平方演算法
   利用模重複平方演算法求得大數取模的結果。
   CODE

# b^m (mod m)
def square_mod(base, n, m):
    b = base
    bin_exp = bin(n)[2:][::-1]
    a = 1
    for bit in bin_exp:
        if bit == '1': 
            a = a * b % m
        b = b * b % m
    return a

試驗成功。