Serious Cryptography (2018)
by Jean-Philippe Aumasson
Factoring large numbers isn't always hard. For example, take the 1024-bit number N, which is equal to the following:⏎ 179769313486231590772930519078902473361797697894230657273430081157739343819933842986982557174198257278917258638193709265819186026626180659730665062710995556578639447715608415186895652841691982921107202317165369124890481512388558039053427125099290315449262324709315263256083132540461407052872832790915388014592⏎ For 1024-bit numbers used in RSA encryption or signature schemes where N = pq, we expect the best factoring algorithms to need around 270 operations, as we discussed earlier. But you can factor this sample number in seconds using SageMath, a piece of Python-based mathematical software. Using SageMath's factor() function on my 2015 MacBook, it took less than five seconds to find the following factorization:⏎ 2^800 x 641 x 6700417 x 167773885276849215533569 x 37414057161322375957408148834323969🏁
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Factoring large numbers isn't always hard. For example, take the 1024-bit number N, which is equal to the following:⏎ 179769313486231590772930519078902473361797697894230657273430081157739343819933842986982557174198257278917258638193709265819186026626180659730665062710995556578639447715608415186895652841691982921107202317165369124890481512388558039053427125099290315449262324709315263256083132540461407052872832790915388014592⏎ For 1024-bit numbers used in RSA encryption or signature schemes where N = pq, we expect the best factoring algorithms to need around 270 operations, as we discussed earlier. But you can factor this sample number in seconds using SageMath, a piece of Python-based mathematical software. Using SageMath's factor() function on my 2015 MacBook, it took less than five seconds to find the following factorization:⏎ 2^800 x 641 x 6700417 x 167773885276849215533569 x 37414057161322375957408148834323969🏁