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I am given $N$, which is Bob's encryption key. I was able to successfully factor $N$ into $N=p*q$.

Alice encrypts a messages for bob and sends
$a = [1,1,1,0,0,0,1,0,1,1,0,...]$
$b = 1545164156...$

where $b<N$. Given this information, how can I decrypt what the message Alice sent to Bob? I am very confused with the process for Blum-Blum-Shub Cryptosystem so any explanation as to how it works would be much appreciated.

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Richard Feynman once quipped "Nobody understands quantum physics". Well, nobody understands the BBS Cryptosystem either!(kidding). Actually this algorithm is very slow and there are other problems with using it ( see Ritter's Cryptoglossary for a good exposition on this). My advice is to not waste your time with it. Use something else like the Alternating Step Generator as the key generator for a stream cipher. There is no public cryptanalysis (break) for this generator as long as each shift register is 128 bits long. –  William Hird Dec 11 '13 at 9:41
BBS is a pseudo-random generator, not a cryptosystem... –  fkraiem Jun 5 at 8:58
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