By Nigel Smart
In this introductory textbook the writer explains the most important themes in cryptography. he is taking a latest process, the place defining what's intended through "secure" is as vital as developing anything that achieves that aim, and safety definitions are vital to the dialogue throughout.
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Additional resources for Cryptography Made Simple
Discrete Mathematics. Oxford University Press, 1989. H. Rosen. Discrete Mathematics and Its Applications. McGraw-Hill, 1999. CHAPTER 2 Primality Testing and Factoring Chapter Goals • • • • • To explain the basics of primality testing. To describe the most used primality-testing algorithm, namely Miller–Rabin. To examine the relationship between various mathematical problems based on factoring. To explain various factoring algorithms. To sketch how the most successful factoring algorithm works, namely the Number Field Sieve.
The slaves then communicate any relations they ﬁnd to the central master computer which performs the linear algebra step. In such a way the Internet can be turned into a large parallel computer dedicated to factoring numbers. As we have already remarked, the ﬁnal (linear algebra) step often needs to be performed on specialized equipment with large amounts of disk space and RAM, so this ﬁnal computation cannot be distributed over the Internet. 2. Higher-Degree Sieving: The linear sieve is simply not good enough to factor large numbers.
This is where linear algebra can come to our aid. We explain how to automate the process using linear algebra by referring to our previous simple example. Recall that our relations were equivalent to p−1 · q · r−1 = 1 (mod N ), q ·r =1 (mod N ), p ·q ·r =1 (mod N ). 2 2 3 2 To ﬁnd which equations to multiply together to obtain a square, we take a matrix A with #F columns and number of rows equal to the number of relations. Each relation is coded into the matrix as a row, modulo two, which in our example becomes ⎛ ⎞ ⎛ ⎞ −1 1 1 1 1 1 ⎜ ⎟ ⎜ ⎟ A = ⎝ 0 2 3 ⎠ = ⎝ 0 0 1 ⎠ (mod 2).
Cryptography Made Simple by Nigel Smart