We describe a new simple but more powerful form of linear cryptanalysis. It appears to break AES (and undoubtably other cryptosystems too, e.g. SKIPJACK). The break is "nonconstructive," i.e. we make it plausible (e.g. prove it in certain approximate probabilistic models) that a small algorithm for quickly determining AES-256 keys from plaintext-ciphertext pairs exists – but without constructing the algorithm! The attack's runtime is comparable to performing 64^w encryptions where w is the (unknown) minimum Hamming weight in certain binary linear error-correcting codes (BLECCs) associated with AES-256. If w<43 then our attack is faster than exhaustive key search. Probably w<10. (Also there should be ciphertext-only attacks if the plaintext is natural English.)

Even if this break breaks due to the underlying models inadequately approximating the real world, we explain how AES still could contain "trapdoors" which would make cryptanalysis unexpectedly easy for anybody who knew the trapdoor. If AES's designers had inserted such a trapdoor, it could be very easy for them to convince us of that. But if none exist, then it is probably infeasibly difficult for them to convince us of that.We then discuss how to use the theory of BLECCs tobuild cryptosystems provably1. not containing trapdoors of this sort,2. secure against our strengthened form of linear cryptanalysis,3. secure against ``differential'' cryptanalysis,4. secure against D.J.Bernstein's timing attack.Using this technique we prove a fundamental theorem:it is possible to thus-encrypt $n$ bits with security$2^{cn}$, via an circuit $Q_n$ containing $le c n$two-input logic gatesand operating in $le c log n$ gate-delays, where the three $c$s denote(possibly different) positive constants and $Q_n$ is constructiblein polynomial$(n)$ time.At the end we give tables of useful binary codes.