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We give a new proof of F\"urer's bound for the cost of multiplying n-bitintegers in the bit complexity model. Unlike F\"urer, our method does notrequire constructing special coefficient rings with "fast" roots of unity.Moreover, we prove the more explicit bound O(n log n K^(log^* n))$ with K = 8.We show that an optimised variant of F\"urer's algorithm achieves only K = 16,suggesting that the new algorithm is faster than F\"urer's by a factor of2^(log^* n). Assuming standard conjectures about the distribution of Mersenneprimes, we give yet another algorithm that achieves K = 4.

Even faster integer multiplication