Cryptography, Information Theory, and Error-Correction. Aiden A. Bruen

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Cryptography, Information Theory, and Error-Correction - Aiden A. Bruen


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The ideal case is when p minus 1 equals 2 q, where q itself is prime. For example, take p equals 11 so q equals 5 period In the ideal case, p has the greatest possible number of different generators (for its size) so that it is easy to find a generator. However, such primes p, known as Sophie‐Germain primes, are conjectured to be rare. In any event, only a finite number are known to exist.

      Using the Diffie–Hellman idea, it is possible to construct a public‐key cryptosystem called the El Gamal Cryptosystem.

      El Gamal Cryptosystem

      As before, we are given a prime p and a generator g. Each participant bold upper B has, as a private key, a secret integer b (which can be assumed to lie between 2 and p minus 2 right-parenthesis and a public key beta equals Rem left-parenthesis g Superscript b Baseline right-parenthesis. Suppose bold upper A wants to send bold upper B a secret message m, which is in the form of a positive integer less than p. Let the integer a be the private key for bold upper A. bold upper A has, for a public key, alpha equals Rem left-parenthesis g Superscript a Baseline right-parenthesis. bold upper A also computes the key k equals Rem left-parenthesis left-parenthesis g Superscript b Baseline right-parenthesis Superscript a Baseline right-parenthesis, obtained by getting the remainder upon raising beta comma the public key of bold upper B, to the power a, and dividing by p. (As in the DH key‐exchange, bold upper B can also find k by raising alpha to the power of b and dividing by p to get the remainder.)

      Finally, bold upper A transmits the cipher text upper C equals Rem left-parenthesis k m right-parenthesis to bold upper B (as well as alpha right-parenthesis. From alpha, bold upper B can calculate k. Since p is a prime bold upper B can calculate k Superscript negative 1, where Rem left-parenthesis k k Superscript negative 1 Baseline right-parenthesis equals 1. Then bold upper B calculates m equals Rem left-parenthesis k Superscript negative 1 Baseline left-parenthesis k m right-parenthesis right-parenthesis. This is the El Gamal Cryptosystem.

      The RSA digital signature protocol is relatively easy because e and d are both defined on the same set, namely StartSet 0 comma 1 comma 2 comma ellipsis comma m minus 1 EndSet. For a more complicated digital signature example, we present the El Gamal Digital Signature Scheme.

      The El Gamal digital signature scheme

      bold upper A wants to send a signed message upper M to bold upper B. bold upper A begins with a large prime p and a generator g (a primitive root) for that prime. Then bold upper A chooses an integer a such that 2 less-than-or-equal-to a less-than-or-equal-to p minus 2. The public key of Скачать книгу