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Is there a kind of analysis which would enable me to get AES key which was used for encrypting arbitrary data while I have both original and encrypted version of data ?

The problem is that the AES key is burned into coprocessor but I have an interface to tell it to encrypt supplied data (therefore I can generate as much sample plaintext and encrypted data for possible analysis as it would be required).

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No. This is known as a known-plaintext attack (or a chosen-plaintext attack if you are not only aware of but can select the plaintexts), and is a type of attack that AES is highly resistant to: there are no known attacks of either type that are faster than brute force.

If you've got access to the encryption coprocessor (and a good electronics lab), you may be able to perform side-channel attacks to recover the key, by analyzing the activity of the coprocessor as it performs encryption.

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    No publicly known attacks. – Ben Aug 18 '14 at 8:45
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The last resort attack in this case is the way some of the early BluRay keys were obtained, that is burning away the outer layers of the chip with HF and reading off the crypto key from the chip itself.

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