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If I want to encrypt some payload with AES-GCM, with the Web Cryptography API, how resistant would this be against side channel attacks?

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The safe academic thing is to assume that no, something using the Web Cryptoraphy API can not be assumed resistant to side-channel attacks.

The Web Cryptography API is, as the acronym states, an Application Programming Interface. What it does is platform-dependent. For example, in a browser environment, the code that implements that API is browser-dependent (including version), and what it interfaces to can be OS-dependent (including version). User-settings may matter, compilation options of the native code often do. Plus, the feasibility of side channel attacks turns out to be CPU-dependent, microcode-of-the-CPU-dependent, setting-of-the-BIOS-or-virtualization-environment-if-any-dependent...

However, the worst is not certain! Actual feasibility of side channel attacks is extremely attacker-capabilities-dependent, and quite often it is pointless to assume that the attacker has a foot in the client machine (for in this case the attacker has options beyond side-channel attacks). And once there is the constraint to be platform-independent, perhaps the Web Cryptoraphy API is the best thing to use: rolling one's own side-channel-resistant implementation in JS is bound to be much less efficient than a native implementation, hard to the point of being next to infeasible, potentially ruined by the execution environment, and rightly recommended against by every expert.

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