integer factorization algorithmsの例文

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1. These schemes are therefore termed computationally secure; theoretical advances, e . g ., improvements in integer factorization algorithms, and faster computing technology require these solutions to be continually adapted.
2. Congruences of squares are extremely useful in integer factorization algorithms and are extensively used in, for example, the quadratic sieve, general number field sieve, continued fraction factorization, and Dixon's factorization.
3. Conversely, because finding square roots modulo a composite number turns out to be probabilistic polynomial-time equivalent to factoring that number, any integer factorization algorithm can be used efficiently to identify a congruence of squares.
4. The block Lanczos algorithm is amongst the most efficient methods known for finding nullspaces, which is the final stage in integer factorization algorithms such as the quadratic sieve and number field sieve, and its development has been entirely driven by this application.
5. In general, if a protocol is proven secure, attacks to that protocol must either be outside what was proven, or break one of the assumptions in the proof; for instance if the proof relies on the hardness of integer factorization, to break this assumption one must discover a fast integer factorization algorithm.