In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further limited to prime numbers, the process is called prime factorization. Integer factorization has been a subject of study by many mathematicians for a long time. Finding a prime factor of a composite number is considered a difficult problem and is often used in cryptography. John M. Pollard has invented two essential methods for factorizing large positive integers: \(\rho \) (rho) method and \( p \) − 1 method. Pollard’s \(\rho \) mehod is effective for large numbers that have at least one small prime factor. The idea is to first construct a sequence of integers \( (x_i) \) modulo \( p \), where \( p \) is the prime factor we are trying to find. Then we need to find the period of this sequence and finally determine the prime factor \( p \).This factorization algorithm is inefficient if the period of the sequence is large. Mathematicians like Floyd and Brent further improved Pollard’s ρ method by using their cycle finding algorithms. Floyd’s variant is based on the parallel calculation of the sequences \( (x_i) \) and \((x_{2i})\) and the calculation of \(\gcd{(x_{2k}-x_k, n)} \). For the Brent variant, we compute the sequence \( (x_i) \) and gcd \({(x_i-x_{2^k-1}, n)} \) for \(k\in\mathbb{N}\) i \(3 \cdot 2^{k-1} \leq i \leq 2^{k+1}-1\). In both variants we are done if we get that gcd > 1, but it may happen that the algorithms do not give us a prime factor as the output. Methods can be further accelerated by periodically calculating the greatest common divisor. Analogous to the factorization method, Pollard invented the \(\rho \) method for calculating discrete logarithms. The discrete logarithm problem for a cyclic group G concerns finding the number k such that \(h=g^k\) is valid for some element \(h\in G\), where g is a generator of the group. This problem is also considered difficult and is applied in cryptography. We have explained Pollard’s \(\rho \) method for the discrete logarithm problem for the group \(\mathbb{Z}_p^*\), where \( p \) is an odd prime number. Pollard’s \( p \) − 1 method is based on Little Fermat’s theorem. It is successful for large numbers for which it is true that the factorization of the number \( p \) −1 consists only of small prime factors, where \( p \) is the prime factor of the number we want to factorize. The idea is to find a multiple m of \( p \) − 1 and use the Euclidean algorithm to calculate the factor \( p \). So if we get that gcd \(\gcd{(a^m-1, n)}\) is non-trivial, we have found a factor of n. This method may also not give us a prime factor as a output. If for the output we get that the greatest common divisor is equal to 1, we can apply the second phase of the algorithm, the so-called large prime variation.