By Nigel Smart
In this introductory textbook the writer explains the main issues in cryptography. he's taking a contemporary process, the place defining what's intended by way of "secure" is as vital as growing whatever that achieves that aim, and protection definitions are imperative to the dialogue throughout.
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Extra info for Cryptography Made Simple
There is an algorithm due to Adleman and Huang which, unlike the ECPP method, is guaranteed to terminate with a proof of primality on input of a prime number. It is based on a generalization of elliptic curves called hyperelliptic curves and has never (to my knowledge) been implemented. The fact that it has never been implemented is not only due to the far more complicated mathematics involved, but is also due to the fact that while the hyperelliptic variant is mathematically guaranteed to produce a proof, the ECPP method will always do so in practice for less work eﬀort.
The function π(X) counts the number of primes less than X, where we have the approximation π(X) ≈ X . log X This means primes are quite common. For example, the number of primes less than 21024 is about 21014 . The Prime Number Theorem also allows us to estimate the probability of a random number being prime: if p is a number chosen at random then the probability it is prime is about 1 . log p So a random number p of 1024 bits in length will be a prime with probability ≈ 1 1 ≈ . log p 709 So on average we need to select 354 odd numbers of size 21024 before we ﬁnd one which is prime.
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). 2 2 1 0 0 1 We now try to ﬁnd a (non-zero) binary vector z such that z·A=0 (mod 2). In our example we can take z = (0, 1, 1) since ⎛ 0 1 1 ⎞ 1 1 1 ⎜ ⎟ ⎝ 0 0 1 ⎠= 0 0 1 0 0 0 (mod 2). This solution vector z = (0, 1, 1) tells us that multiplying the last two equations together will produce a square modulo N . 44 2. PRIMALITY TESTING AND FACTORING Finding the vector z is done using a variant of Gaussian Elimination.
Cryptography Made Simple by Nigel Smart