ance: http://en.wikipedia.org/wiki/Enigma_machine, www.codesandciphers.org.uk/enigma/index.htm, www.mlb.co.jp/linux/science/genigma/enigma-referat/enigma-referat.html www.enigmahistory.org/simulators.html 25.2 Encryption and decryption with binary numbers possibilities. Yet the code was eventually cracked in 1932, again illustrating the unlimited resource of human ingenuity! A second historical turn in cryptography history was marked by the advent of digital computers and, hence, the possibility not only to substitute letter symbols into each other, but to intermingle their bits, and intermingle all the bits of all plaintext symbols, notwith- standing the extended keyboard alphabet of 28 = 256 ASCII word symbols. Moreover, binary operations can be introduced (such as XOR and polynomial multiplication) to make cryptanalysis even more intractable. These new classes of cryptosystem yield ciphertexts under the form of near-perfect pseudo-random bit sequences, thus, betraying no specific patterns or block frequencies on which previous cryptanalysis fed. A third historical turn was marked by the invention of public-key cryptography or PKC (see further), which is basically cryptography without key exchange. As we shall see, the strength or confidence in PKC is based on the difficulty of factorizing large numbers into primes. But, as we already know, the difficulty could be alleviated, should practical quantum computing be developed, with Shor's factorization algorithm, which was described in Chapter 20. The above illustrates that the notion of cryptosystem invulnerability is all relative, especially considering the power of supercomputers and the ever-increasing sophistica- tion of cryptanalysis. Encryption and decryption with binary numbers This section reviews the basics of what could be called "modern" cryptography. As we shall see, the principles used in modern cryptography are not just the same older ones implemented with powerful computers and bigger keys, like "super-Enigma" machines. Rather, and as mentioned ear