The Mathematics of Ciphers: Number Theory and RSA by S. C. Coutinho

By S. C. Coutinho

This e-book is an advent to the algorithmic points of quantity concept and its functions to cryptography, with unique emphasis at the RSA cryptosys-tem. It covers a few of the widely used subject matters of undemanding quantity idea, all with an algorithmic twist. The textual content additionally comprises many fascinating historic notes.

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During a brute-force attack, every output that is only 7 bits is stored in a “candidates” file, which will need to be checked by an actual human to determine what the correct result is. False-positive results are extremely rare for language texts. 6 × 1016, keys to check. When measuring cryptographic security, you can’t be afraid of large numbers. While 255 sounds out of reach, in fact, it’s quite doable. Brute-force algorithms are easily parallelizable, and with the Internet, tens of thousands of computers are easily pooled for such attacks, thereby enabling trillions of keys to be tested per second.

Two others called bitCount and bitLength need a bit more explaining. For positive integers the results are what you’d expect: bitLength returns the minimal number of bits required to represent the integer, and BitCount returns the number of one-bits in the representation. For negative numbers, bitLength gives a minimal length excluding the sign bit, and bitCount returns the number of zeros in the representation. Creating and Converting BigInteger objects can be converted by using a string representation of a number, a native type, or with a byte array.

The initialization penalty can somewhat be avoided by directly specifying the internal array instead of having it generated (of course, then a key of 4168 bytes instead of at most 56 bytes must be saved and stored). In overall structure, it’s not that different from DES. The subkey array is eighteen 32-bit subkeys (compare to DES with sixteen 48-bit subkeys). There are four S-Boxes, each with 256 entries (compare to DES, which has eight S-Boxes with 64 entries). The most important difference is that Blowfish dynamically creates S-Box values during initialization, while DES uses fixed values.

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