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Side Channel Attacks on AES- Leakage Test

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I have read the paper about testing a device for side channel leakage or if it is even possible to perform such an attack on the device. The idea is to perform a TVLA, which performs a welchs t-test to figure out the leakage. My question is: would it be also possible to perform TVLA even if the key is unknown(but constant) or/and the key cannot be changed? In the case of this test we have to adjust the key/plaintext values to certain values but it is not always possible on the hardware you are testing (in most cases you dont know the key). I just wonder if the values have to be exact the same as the testvectors mentioned in the paper, or can they just follow the same scheme (random vs fixed), but be different values? Another question would be if it is possible to perform the TVLA without control over the plaintext (when its generated randomly)?

https://www.rambus.com/wp-content/uploads/2015/08/TVLA-DTR-with-AES.pdf

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