New Methods of Generating Primes Secure against Both P − 1 and P + 1 Methods ∗

Since cryptosystem based on the problem of factoring the composite number N can be attacked with P − 1 and P +1 methods, it is required that P −1 and P +1 should be difficult to be factored into many small primes, where we assume that the P is a factor of N . In this paper, first, we consider the distribution of secure primes against both P −1 and P +1 methods. Second, we propose two efficient algorithms for generating secure primes against both P −1 and P +1 methods by extending the trial division method. key words: prime generation, P − 1 method, P + 1 method, trial division method