found a proof that inverting RSA is as hard as factoring. , the largest publicly known factored RSA number had 829 bits (250 decimal digits, RSA-250). Its factorization, by a state-of-the-art distributed implementation, took about 2,700 CPU-years. In practice, RSA keys are typically 1024 to 4096 bits long. In 2003, RSA Security estimated that 1024-bit keys were likely to become crackable by 2010. As of 2020, it is not known whether such keys can be cracked, but minimum recommendations have moved to at least 2048 bits. It is generally presumed that RSA is secure if is sufficiently large, outside of quantum computing. If is 300 bits or shorter, it can be factored in a few hours in a personal computer, using software already freely available. Keys of 512 bits have been shown to be practically breakable in 1999, when RSA-155 was factored by using several hundred computers, and these are now factored in a few weeks using common hardware. Exploits using 512-bit code-signing certificates that may have been factored were reported in 2011. A theoretical hardware device named TWIRL, described by Shamir and Tromer in 2003, called into question the security of 1024-bit keys. In 1994, Peter Shor showed that a quantum computer – if one could ever be practically created for the purpose – would be able to factor in polynomial time, breaking RSA; see Shor's algorithm. Faulty key generation Finding the large primes and is usually done by testing random numbers of the correct size with probabilistic primality tests that quickly eliminate virtually all of the nonprimes. The numbers and should not be "too close", lest the Fermat factorization for be successful. If is less than (, which even for "small" 1024-bit values of is ), solving for and is trivial. Furthermore, if either or has only small prime factors, can be factored quickly by Pollard's p − 1 algorithm, and hence such values of or should be discarded. It is important that the private exponent be large enough. Michael J. Wiener showed that if is between and (which is quite typical) and , then can be computed efficiently from and . There is no known attack against small public exponents such as , provided that the proper padding is used. Coppersmith's attack has many applications in attacking RSA specifically if the public exponent is small and if the encrypted message is short and not padded. 65537 is