MIT OpenCourseWare http ocw mit edu 6 080 6 089 Great Ideas in Theoretical Computer Science Spring 2008 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 6 080 6 089 GITCS April 8 2008 Lecture 15 Lecturer Scott Aaronson 1 Scribe Ti any Wang Administrivia Midterms have been graded and the class average was 67 Grades will be normalized so that the average roughly corresponds to a B The solutions will be posted on the website soon Pset4 will be handed out on Thursday 2 2 1 Recap Probabilistic Computation We previously examined probabilistic computation methods and the di erent probabilistic com plexity classes as seen in Figure 1 BPP RP coRP ZPP P Figure by MIT OpenCourseWare Figure 1 Probabilistic Complexity Classes P Polynomial time Problems that can be solved deterministically in polynomial time ZPP Zero error Probabilistic Polynomial Expected Polynomial time Problems that can be solved e ciently but with 50 chance that the algorithm does not produce an answer and must be run again If the algorithm does produce an answer it is guaranteed to be correct RP Randomized Polynomial time Problems for which if the answer is NO the algorithm always outputs NO Otherwise if the answer is YES the algorithm outputs YES at least 50 of the time Hence there is an asymmetry between YES and NO outputs coRP Complement of RP These are problems for which there s a polynomial time algorithm that always outputs YES if the answer is YES and outputs NO at least 50 of the time if the answer 15 1 is NO BPP Bounded error Probabilistic Polynomial time Problems where if the answer is YES the algorithm accepts with probability 23 and if the answer is NO the algorithm accepts with prob ability 13 2 2 Ampli cation and Cherno Bound The question that arises is whether the boundary values 13 and 23 have any particular signi cance One of the nice things about using a probabilistic algorithm is that as long as there is a noticeable gap between the probability of accepting if the answer is YES and the probability of accepting if the answer is NO that gap can be ampli ed by repeatedly running the algorithm For example if you have an algorithm that outputs a wrong answer with Pr 13 then you can repeat the algorithm hundreds of times and just take the majority answer The probability of obtaining a wrong answer becomes astronomically small there s a much greater chance of an asteroid destroying your computer This notion of ampli cation can be proven using a tool known as the Cherno Bound The Cherno Bound states that given a set of independent events the number of events that will hap pen is heavily concentrated about the expected value of the number of occurring events So given an algorithm that outputs a wrong answer with Pr 13 repeating the algorithm 10 000 times would produce an expected number of 3333 3 wrong answers The number of wrong an swers will not be exactly the expected value but the probability of getting a number far from the expected value say 5 000 is very small 2 3 P vs BPP There exists a fundamental question as to whether every probabilistic algorithm can be replaced by a deterministic one or derandomized In terms of complexity the question is whether P BPP which is almost as deep a question as P NP There is currently a very strong belief that derandomization is possible in general but no one yet knows how to prove it 3 Derandomization Although derandomization has yet to be proven in the general case it has been proven for some spectacular special cases cases where for decades the only known e cient solutions came from randomized algorithms Indeed this has been one of the big success stories in theoretical computer science in the last 10 years 15 2 3 1 AKS Primality Test In 2002 Agrawal Kayal and Saxena of the Indian Institute of Technology Kanpur developed a deterministic polynomial time algorithm for testing whether an integer is prime or composite also known as the AKS primality test For several decades prior there existed good algorithms to test primality but all were proba bilistic The problem was rst known to be in the class RP and then later shown to be in the class ZPP It was also shown that the problem was in the class P but only assuming that the General ized Riemann Hypothesis was true The problem was also known to be solvable deterministically in nO logloglogn time which is slightly more than polynomial Ultimately it was nice to have the nal answer and the discovery was an exciting thing to be alive for in the world of theoretical computer science The basic idea behind AKS is related to Pascal s Triangle As seen in Figure 2 in every primenumbered row the numbers in Pascal s Triangle are all a multiple of the row number On the other hand in every composite numbered row the numbers are not all multiples of the row number Figure 2 Pascal s Triangle and Prime Numbers So to test the primality of an integer N can we just check whether or not all the numbers in the N th row of Pascal s Triangle are multiples of N The problem is that there are exponentially many numbers to check and checking all of them would be no more e cient than trial division Looking at the expression x a N which has coe cients determined by the N th row of Pascal s Triangle AKS noticed that the relationship x a N xN aN mod N holds if and only if N is prime This is because if N is prime then all the middle coe cients will be divisible by N and therefore disappear when we reduce mod N while if N is composite then some middle coe cients will not be divisible by N What this means is that the primality testing problem can be mapped to an instance of the polynomial identity testing problem given two algebraic formulas decide whether or not they represent the same polynomial In order to determine whether x a N xN aN mod N one approach would be to plug in many random values of a and see if we get the same result each time However since the number of terms would still be exponential we need to evaluate the expression not only mod N but also mod a random polynomial x a N xN aN 15 3 mod N xr 1 It turns out that this solution method works on the other hand it still depends on the use of randomness the thing we re trying to eliminate The tour de force of the AKS paper was to show that if N is composite then it is only necessary to try some small number of deterministically chosen values of a and r until a pair is found such that the equation is not satis ed This immediately leads to a method for