By Richard E. Blahut

Today's pervasive computing and communications networks have created an excessive want for safe and trustworthy cryptographic structures. Bringing jointly a desirable mix of themes in engineering, arithmetic, machine technology, and informatics, this e-book offers the undying mathematical conception underpinning cryptosystems either previous and new. significant branches of classical and glossy cryptography are mentioned intimately, from easy block and circulation cyphers via to structures in accordance with elliptic and hyperelliptic curves, followed by means of concise summaries of the required mathematical heritage. useful facets similar to implementation, authentication and protocol-sharing also are lined, as are the prospective pitfalls surrounding a number of cryptographic equipment. Written in particular with engineers in brain, and delivering an effective grounding within the correct algorithms, protocols and methods, this insightful advent to the principles of recent cryptography is perfect for graduate scholars and researchers in engineering and computing device technology, and practitioners all in favour of the layout of protection structures for communications networks.

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**Additional resources for Cryptography and Secure Communication**

**Example text**

How many zeros does the polynomial x 2 − 1 have in Z 15 ? Repeat this analysis for Z 17 . In what way are the results different? Is there an explanation? a Prove that n i2 = i=1 n(n + 1)(2n + 1) . 6 b Is there any value of n for which this sum is a square? c Express the condition that ni=1 i 2 is a square in terms of the (rational) zeros of a bivariate polynomial p(x, y) of degree three. d Ponder but don’t answer the following question: Are there an infinite number of n for which ni=1 i 2 is a square?

7 Cryptanalysis known ciphertext attack is always a threat to a public-key cryptosystem, which will be discussed in later chapters. 1 for a cryptosystem that uses an elementary modulo-two (exclusive-or) addition of a binary key to a binary dataword. If the same key k is used twice to form the two ciphertexts y1 = x 1 + k and y2 = x 2 + k, then the cryptanalyst can add the two ciphertexts to form y1 + y2 = x 1 + x 2 . Because x 2 is presumably a message with a known structure, it will be easier to recover x 1 corrupted by x 2 than to recover x 1 hidden by k.

If the keys are m-bit binary numbers, then #K = 2m , provided all bits are effectively used. One way of trying to improve the security is to encrypt twice, using two keys. Thus, encrypt by using y = ek (ek (x)) and decrypt by using x = dk (dk (y)). This is called an iterated cipher or, if the two ciphers are of different kinds, a cascade cipher. The product keyspace K2 then has the apparent size (#K)2 or (#K1 )(#K2 ). If the keys are each an m-bit binary number, then (#K)2 = 22m . However, ensuring that these double keys are all different might not be enough.