Project Specification Generation of truly random numbers with known factorization

Project Specification Generation of truly random numbers with known factorizationAnupama Paila, Pavan Burri, Rohan MisquithMotivationThe number field sieve of factoring 1024-bit RSA keys has many steps involved, one of them being the ‘relation collection’ step. This step consists of two parts, the first part is called Sieving and is used to generate smooth random numbers (numbers whose factors are less than a given boundary), these smooth numbers are factored in the second part by special factoring methods like ECM, p-1 and rho. Now Sieving is a complex process and takes a lot of time to generate smooth big numbers. Our project involves generating these smooth numbers so that they can be processed by part two directly.ObjectiveThe main objective of our project is to generate random smooth numbers using software implementation. The first part involves the implementation of algorithms proposed by Adam Kalai and Eric Bach, the principle involved in both the algorithms is to generate a random integer with known factorization, the second part of the of the project is to include logic in both the algorithms so as to output a random number whose factors are less than a certain boundary, this part will result in the rejection of a lot of random numbers. Lastly we will perform various experiments to evaluate the performance and optimize the execution time. The originality of the project lies in efficient implementation of proposed algorithms by appropriate choice of cryptographic libraries, random number generators, algorithms for primality testing and optimization of code to decrease the execution time.Language: CIDE: Code::Blocks / Compiler: GNU GCCPlatform: Windows XP/UNIXAdditional libraries: GMPProject ContentsAlgorithmsOur goal is to generate a uniformly random number “r” between 1 and N, along with its prime factorization. Since there are no known efficient factoring algorithms, one cannot simply generate a random number and then factor it. Eric Bach's award-winning dissertation introduces and solves this problem. Adam Kalai presents a significantly simpler algorithm and analysis for the same problem; however, it is less efficientThe inputs to both the algorithms are “x” and sigma () in terms of no of bits Where N=2x, B=2(smooth boundary) and output is the random number “r”Implementation of the algorithmsAdam Kalai's algorithmIn this Algorithm a uniform random number “r” is generated between 1 and N, this is done by first generating numbers n1, n2, n3…..in such a way that n1 is uniformly random between 1 and N, n2 is uniformly random between 1 and n1, n3 is uniformly random between 1 and n2…and so on. Prime numbers are picked from the set {n1, n2, n3….} and if their product(r, the product of all primes from the set) is less than N, then it is output as the random number r with probability r/N, otherwise the whole process is repeated. To minimize the bias that is cause in picking n1, n2, n3 etc. r is output with the probability r/N. The pseudo code for this algorithm can be got from Adam Kalai’s paper[5].The function used to generate n1, n2, n3 etc is mpz_urandomm()The Miller-Rabin primality test would be used to check if n1, n2, n3 etc are prime. The GMP library has a built in Miller-Rabin function, mpz_probab_prime_p(), which we will be using in our program.Eric Bach’s Algorithm:The Algorithm outputs a uniformly random number r in the range (N/2, N].Given the value of N as input. The output r will be a product of know primes which are chosen during the execution of the algorithm. The paper by Eric Bach [7] suggests the choice of algorithms for primality testing in order to reduce the complexity of execution.The implementation of the Algorithm is divided as three processes:Process A: Acceptance Rejection method.Process F: Factor Generation.Process R: Random Factorization Generation.The algorithm describes the implementation of acceptance rejection method eliminates the bias and makes the output random. During the course of the project comprehensive analysis will be done on the randomness of the outputs generated by the algorithm. Analysis of the Mathematics provided by the paper is chosen as the main criteria. Other criterion may be chosen during the course of the project.Modifications of the Kalai's algorithm and the Bach's algorithmKalai and Bach’s algorithm output a random number ‘r’. Modifications are to be made to the algorithm in order to make the random number smooth. Smooth random numbers have factors less than a certain boundary; in this case it would be 2.In our initial implementation we would check the factors of the random number ‘r’ just before outputting ‘r’, (i.e.) after calculating r/N probability(in case of Adam Kalai), if wenumbers ‘r’. We may later decide to implement this check when the factors are generated, in this case we would make sure that n1, n2, n3 are less than 2, if they are greater then they would be recomputed, In this case, we would get a smooth random number the first time itself. If the program terminates, then the smooth random number check would return false, else it would return true and display the random number and its prime factors.Input/OutputThe user will have to input ‘x’ and ‘sigma - ’The output would be a random number ‘r’ which lies in between 1 and 2x, the factors of r would be less than 2. Thus the random number ‘r’ could be called smooth. The time taken to calculate the random number would be displayed and the number of times the program had to run to get a smooth ‘r’ would be displayed. All inputs/outputs will be in decimal notation.Example:InputPlease enter the number of bits for N: xPlease enter the number of bits for sigma: sigmaPlease enter the number of random numbers you want:Output*Random Number: rThe factors of the random number are.........Prime numbers of the set {n1, n2, n3…..} would be displayed hereTime taken to find Random Number is: t second(s)Number of iterations to get a smooth number: L*The random numbers, their factors and L will be stored in three different text files, randomNumbers.txt, randomFactors.txt, smoothCount.txt respectivelyDistribution of task among team membersTask Task1 Task2 Task3 Task4 Task5 Task6Study of the two algorithms Implementation ofAdamKalai’sAlgorithmImplementationof Eric Bach’sAlgorithm andpresentation ofset of slidesdescribing therandomness ofboth theAlgorithms.Modificationof theAlgorithms toreject ‘r’ if itsfactors aregreater thanthe boundary2Comparison ofthe softwareimplementationsof the twoalgorithmsTestingrandomnessof outputusingstatisticaltest forrandomnessComplete Date October 5th October25thNovember 14th November 18th November 21st November28thResponsible Person(s) Pavan,Rohan,AnupamaRohan Anupama andPavan.Rohan andAnupamaPavan, Rohan,AnupamaPavan andAnupamaDeliverables1. Basic implementation of Adam Kalai’s Algorithm2. Basic implementation of Eric Bach’s Algorithm3. Modified implementation of Kalai’s Algorithm4. Set of slides discussing randomness of the output generated byBach and Kalai’s algorithms5. Modified implementation of Bach’s Algorithm6. Results of comparison of the Kalai’s and Bach’s Algorithm7. Results of further test suggested by the instructor8. Project report + slides for presentation9. Project presentationTentative table of contents of our final report:1. Abstract2. Introduction3. Importance of random factored numbers4. Explanation of the algorithms5. Explanation and Justification of the randomness6. Generating the numbers given upper limit7. Primarily testing for the numbers generated8. The pseudocode9. Our implementation of the pseudocode10. Testing and results11. Conclusion11. Future work12. ReferencesList of literature:[1] - Good source for random number generator algorithms.[2] / - Good source for primality testing algorithms.[3] /bps/linked_list.html - Tutorial on Linked List in C[4] /question697.htm- How random numbers are generated by computers[5] Adam Kalai, Generating Random Factored Numbers, Easily[6] E. Bach and J. Shallit, Algorithmic Number Theory, MIT Press, Cambridge, 1996.[7] E. Bach, How to Generate Factored Random Numbers, SIAM J. Computing, 17 (1988), pp. 179-193.[8] /gmp/ - The GMP homepage.[9] /gmp/manual/ - The GMP manual.[10] / - Homepage of the IDE used for this project.[11] /SmoothNumber.html - Smooth Numbers。