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Is there a point where increasing the complexity of an encryption algorithm will make it easier to break using a timing attack?

Or is there no connection here at all?

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Timing attacks rely on operations which do not always take the same time to execute, depending on the processed data. For instance, on a typical software platform (say, a PC) implementing SHA-256, all operations are 32-bit additions or rotations or bitwise combinations which take a constant time to execute, regardless of the actual operand values. SHA-256 is thus largely immune to timing attacks, because processing time, down to the nanosecond, depends only on the length of the input message, not on the values of the bits constituting the message. An attacker observing from the outside a SHA-256 computation would learn nothing about the hashed message, except possibly an estimate of the message length. On the other hand, a simple RSA implementation will leak information because the modular exponentiation incurs data-dependent and key-dependent costs (in a programmer's point of view, the code if full of "if" which operate on values computed from the input data and key).

"Complexity" of the algorithm is mostly irrelevant for that question. You can make a very complex algorithm out of operations which have no timing-related issue. Complexity is, in itself, a bad thing, since it makes analysis more complex (it is much harder to know whether the result is secure or not) and implies a higher implementation cost (in development time, and, often, in code size and execution speed).

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