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August 19th, 2018, 02:25 AM   #1
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Thoughts about factorising an RSA modulus

As I've pondered my likely inability to factorise a 2048 bit RSA modulus that would give me control of some equipment that I own, I've wondered whether anything can be made of the public exponent. No doubt I'm traversing a well worn path to failure, but anyway.

I've reduced(?) the problem to finding pairs of n1, n2 that satisfy

N = E(n1^2 - n2^2) - 2(n1 * k1 - n2 * k2)

where N, E (prime), k1, and k2 are known. In this particular instance N is composite (it isn't always), but unlike the original public modulus, was not created with the intention that it be the product of large primes, so there is at least the possibility that it has a factor that is of a tractable size. So far an elliptic curve factorisation program hasn't found it, but it's only been a few hours.

But I'm wondering whether knowing the factor is going to help anyway. Is there a way to leverage knowledge of a factor of N to get n1 and n2 more quickly?
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August 19th, 2018, 03:23 AM   #2
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What is the equipment you referred to?
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August 19th, 2018, 04:56 AM   #3
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What is the equipment you referred to?
I'd rather not say at this stage. However, if you're concerned about legal issues, I'm in Australia. The Australian equivalent of the DCMA clearly allows me to break any protection scheme for the purpose of achieving interoperability, or where the protection scheme limits my use of the device in which the protected software is embedded.
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August 19th, 2018, 04:58 AM   #4
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So far an elliptic curve factorisation program hasn't found it.
Actually, N is always divisible by 2. I'm trying to factorise N/2.
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August 21st, 2018, 08:07 PM   #5
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But I'm wondering whether knowing the factor is going to help anyway. Is there a way to leverage knowledge of a factor of N to get n1 and n2 more quickly?
I'll take that as a "no". Anyway, the factorisation program made no headway in factorising N over several days, and I gave up.
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