简单RSA

RSA加密需要用到质数

# Prime Number Sieve
# http://inventwithpython.com/hacking (BSD Licensed)

import math


def isPrime(num):
    # Returns True if num is a prime number, otherwise False.

    # Note: Generally, isPrime() is slower than primeSieve().

    # all numbers less than 2 are not prime
    if num < 2:
        return False

    # see if num is divisible by any number up to the square root of num
    for i in range(2, int(math.sqrt(num)) + 1):
        if num % i == 0:
            return False
    return True


def primeSieve(sieveSize):
    # Returns a list of prime numbers calculated using
    # the Sieve of Eratosthenes algorithm.

    sieve = [True] * sieveSize
    sieve[0] = False # zero and one are not prime numbers
    sieve[1] = False

    # create the sieve
    for i in range(2, int(math.sqrt(sieveSize)) + 1):
        pointer = i * 2
        while pointer < sieveSize:
            sieve[pointer] = False
            pointer += i

    # compile the list of primes
    primes = []
    for i in range(sieveSize):
        if sieve[i] == True:
            primes.append(i)

    return primes

两个函数，第一个是常规的质数判断。第二个是埃拉托色尼筛选法

方法：

• （1）先把1删除（现今数学界1既不是质数也不是合数）
• （2）读取队列中当前最小的数2，然后把2的倍数删去
• （3）读取队列中当前最小的数3，然后把3的倍数删去
• （4）读取队列中当前最小的数5，然后把5的倍数删去
• （5）读取队列中当前最小的数7，然后把7的倍数删去
• （6）如上所述直到需求的范围内所有的数均删除或读取

检测较大质数——拉宾米勒质数检验，运用高等数学，且并非万无一失

# Primality Testing with the Rabin-Miller Algorithm
# http://inventwithpython.com/hacking (BSD Licensed)

import random


def rabinMiller(num):
    # Returns True if num is a prime number.

    s = num - 1
    t = 0
    while s % 2 == 0:
        # keep halving s until it is even (and use t
        # to count how many times we halve s)
        s = s // 2
        t += 1

    for trials in range(5): # try to falsify num's primality 5 times
        a = random.randrange(2, num - 1)
        v = pow(a, s, num)
        if v != 1: # this test does not apply if v is 1.
            i = 0
            while v != (num - 1):
                if i == t - 1:
                    return False
                else:
                    i = i + 1
                    v = (v ** 2) % num
    return True


def isPrime(num):
    # Return True if num is a prime number. This function does a quicker
    # prime number check before calling rabinMiller().

    if (num < 2):
        return False # 0, 1, and negative numbers are not prime

    # About 1/3 of the time we can quickly determine if num is not prime
    # by dividing by the first few dozen prime numbers. This is quicker
    # than rabinMiller(), but unlike rabinMiller() is not guaranteed to
    # prove that a number is prime.
    lowPrimes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997]

    if num in lowPrimes:
        return True

    # See if any of the low prime numbers can divide num
    for prime in lowPrimes:
        if (num % prime == 0):
            return False

    # If all else fails, call rabinMiller() to determine if num is a prime.
    return rabinMiller(num)


def generateLargePrime(keysize=1024):
    # Return a random prime number of keysize bits in size.
    while True:
        num = random.randrange(2**(keysize-1), 2**(keysize))
        if isPrime(num):
            return num

简单RSA加密范例：

# RSA Key Generator
# http://inventwithpython.com/hacking (BSD Licensed)

import random, sys, os, rabinMiller, cryptomath


def main():
    # create a public/private keypair with 1024 bit keys
    print('Making key files...')
    makeKeyFiles('al_sweigart', 1024)
    print('Key files made.')

def generateKey(keySize):
    # Creates a public/private key pair with keys that are keySize bits in
    # size. This function may take a while to run.

    # Step 1: Create two prime numbers, p and q. Calculate n = p * q.
    print('Generating p prime...')
    p = rabinMiller.generateLargePrime(keySize)
    print('Generating q prime...')
    q = rabinMiller.generateLargePrime(keySize)
    n = p * q

    # Step 2: Create a number e that is relatively prime to (p-1)*(q-1).
    print('Generating e that is relatively prime to (p-1)*(q-1)...')
    while True:
        # Keep trying random numbers for e until one is valid.
        e = random.randrange(2 ** (keySize - 1), 2 ** (keySize))
        if cryptomath.gcd(e, (p - 1) * (q - 1)) == 1:
            break

    # Step 3: Calculate d, the mod inverse of e.
    print('Calculating d that is mod inverse of e...')
    d = cryptomath.findModInverse(e, (p - 1) * (q - 1))

    publicKey = (n, e)
    privateKey = (n, d)

    print('Public key:', publicKey)
    print('Private key:', privateKey)

    return (publicKey, privateKey)


def makeKeyFiles(name, keySize):
    # Creates two files 'x_pubkey.txt' and 'x_privkey.txt' (where x is the
    # value in name) with the the n,e and d,e integers written in them,
    # delimited by a comma.

    # Our safety check will prevent us from overwriting our old key files:
    if os.path.exists('%s_pubkey.txt' % (name)) or os.path.exists('%s_privkey.txt' % (name)):
        sys.exit('WARNING: The file %s_pubkey.txt or %s_privkey.txt already exists! Use a different name or delete these files and re-run this program.' % (name, name))

    publicKey, privateKey = generateKey(keySize)

    print()
    print('The public key is a %s and a %s digit number.' % (len(str(publicKey[0])), len(str(publicKey[1]))))
    print('Writing public key to file %s_pubkey.txt...' % (name))
    fo = open('%s_pubkey.txt' % (name), 'w')
    fo.write('%s,%s,%s' % (keySize, publicKey[0], publicKey[1]))
    fo.close()

    print()
    print('The private key is a %s and a %s digit number.' % (len(str(publicKey[0])), len(str(publicKey[1]))))
    print('Writing private key to file %s_privkey.txt...' % (name))
    fo = open('%s_privkey.txt' % (name), 'w')
    fo.write('%s,%s,%s' % (keySize, privateKey[0], privateKey[1]))
    fo.close()


# If makeRsaKeys.py is run (instead of imported as a module) call
# the main() function.
if __name__ == '__main__':
    main()

# RSA Cipher
# http://inventwithpython.com/hacking (BSD Licensed)

import sys

# IMPORTANT: The block size MUST be less than or equal to the key size!
# (Note: The block size is in bytes, the key size is in bits. There
# are 8 bits in 1 byte.)
DEFAULT_BLOCK_SIZE = 128 # 128 bytes
BYTE_SIZE = 256 # One byte has 256 different values.

def main():
    # Runs a test that encrypts a message to a file or decrypts a message
    # from a file.
    filename = 'encrypted_file.txt' # the file to write to/read from
    mode = 'encrypt' # set to 'encrypt' or 'decrypt'

    if mode == 'encrypt':
        message = '''"Journalists belong in the gutter because that is where the ruling classes throw their guilty secrets." -Gerald Priestland "The Founding Fathers gave the free press the protection it must have to bare the secrets of government and inform the people." -Hugo Black'''
        pubKeyFilename = 'al_sweigart_pubkey.txt'
        print('Encrypting and writing to %s...' % (filename))
        encryptedText = encryptAndWriteToFile(filename, pubKeyFilename, message)

        print('Encrypted text:')
        print(encryptedText)

    elif mode == 'decrypt':
        privKeyFilename = 'al_sweigart_privkey.txt'
        print('Reading from %s and decrypting...' % (filename))
        decryptedText = readFromFileAndDecrypt(filename, privKeyFilename)

        print('Decrypted text:')
        print(decryptedText)


def getBlocksFromText(message, blockSize=DEFAULT_BLOCK_SIZE):
    # Converts a string message to a list of block integers. Each integer
    # represents 128 (or whatever blockSize is set to) string characters.

    messageBytes = message.encode('ascii') # convert the string to bytes

    blockInts = []
    for blockStart in range(0, len(messageBytes), blockSize):
        # Calculate the block integer for this block of text
        blockInt = 0
        for i in range(blockStart, min(blockStart + blockSize, len(messageBytes))):
            blockInt += messageBytes[i] * (BYTE_SIZE ** (i % blockSize))
        blockInts.append(blockInt)
    return blockInts


def getTextFromBlocks(blockInts, messageLength, blockSize=DEFAULT_BLOCK_SIZE):
    # Converts a list of block integers to the original message string.
    # The original message length is needed to properly convert the last
    # block integer.
    message = []
    for blockInt in blockInts:
        blockMessage = []
        for i in range(blockSize - 1, -1, -1):
            if len(message) + i < messageLength:
                # Decode the message string for the 128 (or whatever
                # blockSize is set to) characters from this block integer.
                asciiNumber = blockInt // (BYTE_SIZE ** i)
                blockInt = blockInt % (BYTE_SIZE ** i)
                blockMessage.insert(0, chr(asciiNumber))
        message.extend(blockMessage)
    return ''.join(message)


def encryptMessage(message, key, blockSize=DEFAULT_BLOCK_SIZE):
    # Converts the message string into a list of block integers, and then
    # encrypts each block integer. Pass the PUBLIC key to encrypt.
    encryptedBlocks = []
    n, e = key

    for block in getBlocksFromText(message, blockSize):
        # ciphertext = plaintext ^ e mod n
        encryptedBlocks.append(pow(block, e, n))
    return encryptedBlocks


def decryptMessage(encryptedBlocks, messageLength, key, blockSize=DEFAULT_BLOCK_SIZE):
    # Decrypts a list of encrypted block ints into the original message
    # string. The original message length is required to properly decrypt
    # the last block. Be sure to pass the PRIVATE key to decrypt.
    decryptedBlocks = []
    n, d = key
    for block in encryptedBlocks:
        # plaintext = ciphertext ^ d mod n
        decryptedBlocks.append(pow(block, d, n))
    return getTextFromBlocks(decryptedBlocks, messageLength, blockSize)


def readKeyFile(keyFilename):
    # Given the filename of a file that contains a public or private key,
    # return the key as a (n,e) or (n,d) tuple value.
    fo = open(keyFilename)
    content = fo.read()
    fo.close()
    keySize, n, EorD = content.split(',')
    return (int(keySize), int(n), int(EorD))


def encryptAndWriteToFile(messageFilename, keyFilename, message, blockSize=DEFAULT_BLOCK_SIZE):
    # Using a key from a key file, encrypt the message and save it to a
    # file. Returns the encrypted message string.
    keySize, n, e = readKeyFile(keyFilename)

    # Check that key size is greater than block size.
    if keySize < blockSize * 8: # * 8 to convert bytes to bits
        sys.exit('ERROR: Block size is %s bits and key size is %s bits. The RSA cipher requires the block size to be equal to or greater than the key size. Either decrease the block size or use different keys.' % (blockSize * 8, keySize))


    # Encrypt the message
    encryptedBlocks = encryptMessage(message, (n, e), blockSize)

    # Convert the large int values to one string value.
    for i in range(len(encryptedBlocks)):
        encryptedBlocks[i] = str(encryptedBlocks[i])
    encryptedContent = ','.join(encryptedBlocks)

    # Write out the encrypted string to the output file.
    encryptedContent = '%s_%s_%s' % (len(message), blockSize, encryptedContent)
    fo = open(messageFilename, 'w')
    fo.write(encryptedContent)
    fo.close()
    # Also return the encrypted string.
    return encryptedContent


def readFromFileAndDecrypt(messageFilename, keyFilename):
    # Using a key from a key file, read an encrypted message from a file
    # and then decrypt it. Returns the decrypted message string.
    keySize, n, d = readKeyFile(keyFilename)


    # Read in the message length and the encrypted message from the file.
    fo = open(messageFilename)
    content = fo.read()
    messageLength, blockSize, encryptedMessage = content.split('_')
    messageLength = int(messageLength)
    blockSize = int(blockSize)

    # Check that key size is greater than block size.
    if keySize < blockSize * 8: # * 8 to convert bytes to bits
        sys.exit('ERROR: Block size is %s bits and key size is %s bits. The RSA cipher requires the block size to be equal to or greater than the key size. Did you specify the correct key file and encrypted file?' % (blockSize * 8, keySize))

    # Convert the encrypted message into large int values.
    encryptedBlocks = []
    for block in encryptedMessage.split(','):
        encryptedBlocks.append(int(block))

    # Decrypt the large int values.
    return decryptMessage(encryptedBlocks, messageLength, (n, d), blockSize)


# If rsaCipher.py is run (instead of imported as a module) call
# the main() function.
if __name__ == '__main__':
    main()