how to make barcode in c#.net The algorithm of Figure 17 therefore has the following probabilistic behavior in Software

Painting QR Code in Software The algorithm of Figure 17 therefore has the following probabilistic behavior

The algorithm of Figure 17 therefore has the following probabilistic behavior
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Figure 18 An algorithm for testing primality, with low error probability function primality2(N ) Input: Positive integer N Output: yes/no Pick positive integers a1 , a2 , , ak < N at random if aN 1 1 (mod N ) for all i = 1, 2, , k: i return yes else: return no
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This probability of error drops exponentially fast, and can be driven arbitrarily low by choosing k large enough Testing k = 100 values of a makes the probability of failure at most 2 100 , which is miniscule: far less, for instance, than the probability that a random cosmic ray will sabotage the computer during the computation!
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The smallest Carmichael number is 561 It is not a prime: 561 = 3 11 17; yet it fools the Fermat test, because a560 1 (mod 561) for all values of a relatively prime to 561 For a long time it was thought that there might be only nitely many numbers of this type; now we know they are in nite, but exceedingly rare There is a way around Carmichael numbers, using a slightly more re ned primality test due to Rabin and Miller Write N 1 in the form 2 t u As before we ll choose a random base a and check the value of aN 1 mod N Perform this computation by rst determining au mod N and then repeatedly squaring, to get the sequence: au mod N, a2u mod N, , a2
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= aN 1 mod N
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If aN 1 1 mod N , then N is composite by Fermat s little theorem, and we re done But if aN 1 1 mod N , we conduct a little follow-up test: somewhere in the preceding sequence, we ran into a 1 for the rst time If this happened after the rst position (that is, if a u mod N = 1), and if the preceding value in the list is not 1 mod N , then we declare N composite In the latter case, we have found a nontrivial square root of 1 modulo N : a number that is not 1 mod N but that when squared is equal to 1 mod N Such a number can only exist if N is composite (Exercise 140) It turns out that if we combine this square-root check with our earlier Fermat test, then at least three-fourths of the possible values of a between 1 and N 1 will reveal a composite N , even if it is a Carmichael number
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We are now close to having all the tools we need for cryptographic applications The nal piece of the puzzle is a fast algorithm for choosing random primes that are a few hundred bits 34
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long What makes this task quite easy is that primes are abundant a random n-bit number has roughly a one-in-n chance of being prime (actually about 1/(ln 2 n ) 144/n) For instance, about 1 in 20 social security numbers is prime! Lagrange s prime number theorem Let (x) be the number of primes x Then (x) x/(ln x), or more precisely, (x) = 1 lim x (x/ ln x) Such abundance makes it simple to generate a random n-bit prime: Pick a random n-bit number N Run a primality test on N If it passes the test, output N ; else repeat the process How fast is this algorithm If the randomly chosen N is truly prime, which happens with probability at least 1/n, then it will certainly pass the test So on each iteration, this procedure has at least a 1/n chance of halting Therefore on average it will halt within O(n) rounds (Exercise 134) Next, exactly which primality test should be used In this application, since the numbers we are testing for primality are chosen at random rather than by an adversary, it is suf cient to perform the Fermat test with base a = 2 (or to be really safe, a = 2, 3, 5), because for random numbers the Fermat test has a much smaller failure probability than the worst-case 1/2 bound that we proved earlier Numbers that pass this test have been jokingly referred to as industrial grade primes The resulting algorithm is quite fast, generating primes that are hundreds of bits long in a fraction of a second on a PC The important question that remains is: what is the probability that the output of the algorithm is really prime To answer this we must rst understand how discerning the Fermat test is As a concrete example, suppose we perform the test with base a = 2 for all numbers N 25 109 In this range, there are about 109 primes, and about 20,000 composites that pass the test (see the following gure) Thus the chance of erroneously outputting a composite is approximately 20,000/109 = 2 10 5 This chance of error decreases rapidly as the length of the numbers involved is increased (to the few hundred digits we expect in our applications)
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