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This. Seriously, their algorithms and mathematics are public and under constant scrutiny from the entire crytographic community. The vulnerabilities in RSA are known, sha already has a third version ready if a systemic weakness in 128->512 bit sha1/2 is revealed, and AES may require 512 bit keys for guaranteed security in the future, but seems solid.

They can't backdoor a math function because all 3 have been implemented by dozens of libraries and programs independently.

AES is only defined for 128, 192, or 256 bit keys. You'd need to switch to a different block cipher like Blowfish (up to 448 bit keys), RC2 (up to 1024 bit keys), or RC5 (up to 2048 bit keys) to have a larger keyspace.

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