For almost 35 years, Schönhage-Strassen's algorithm has been the fastest algorithm known for multiplying integers, with a time complexity $O(n \cdot \log n \cdot \log \relax \log n)$ for multiplying $n$-bit inputs. In 2007, Fürer proved that there exists $K>1$ and an algorithm performing this operation in $O(n \cdot \log n \cdot K^{\log^* n})$. Recent work by Harvey, van der Hoeven, and Lecerf showed that this complexity estimate can be improved in order to get $K=8$, and conjecturally $K=4$. We obtain here the same result $K=4$ using simple modular arithmetic as a building block, and a careful complexity analysis. We obtain a similar result $K=4$ using an alternative somewhat simpler algorithm, which relies on arithmetic modulo generalized Fermat primes.