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Algebraic Aspects of Cryptography (Algorithms and - download pdf or read online

By Neal Koblitz

ISBN-10: 3540634460

ISBN-13: 9783540634461

From the stories: "This is a textbook in cryptography with emphasis on algebraic equipment. it's supported via many workouts (with solutions) making it acceptable for a path in arithmetic or laptop technology. [...] total, this is often a very good expository textual content, and may be very beneficial to either the scholar and researcher." Mathematical reports

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F) The n-th Fibonacci number. 1 == 1 , fz 1 , and fn+1 fn + fn-1 for n == 2, 3, . . ) · == · · == 3. Find a simple function g(n) such that the length of the n-th Fibonacci number is asymptotically equal to g(n). 4. Show that at least n/2 of the numbers 1 , 2, 3, . , n have length equal to or greater than log 2 n - 1 . Then show that n! has length at least equal to 1" (log2 n- 2), and that for large n this is greater than C' n In n for a suitable constant C' . . 5. Use Stirling's formula to find a simple function g(n) such that the length i s asymptotically equal to g(n).

The Integer Factorization problem is also in co-NP, although to see this requires more thought. If the answer to the above question is "no", the extraterrestrial gives you the complete prime factorization of N, from which you can immediately see that there is no prime factor ::; k. At the same time she must also give you a certificate with which you can verify in polynomial time that each of the prime factors really is prime. 20 of [Rosen 1 993]). Example 4. 9. The Traveling Salesrep decision problem is almost certainly not in P (we'll have more to say about that later).

4 From Polynomial Time to Exponential Time We now make a definition that is fundamental in the study of algorithms. 1. An algorithm to perform a computation is said to be a polynomial time algorithm if there exists an integer d such that the number of bit operations required to perform the algorithm on integers of total length at most k is O(kd). Thus, the usual arithmetic operations +, -, x , -;- are examples of polynomial time algorithms; so is conversion from one base to another. On the other hand, computation of n!

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Algebraic Aspects of Cryptography (Algorithms and Computation in Mathematics) by Neal Koblitz


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