What would the existence of an exascale supercomputer mean for today's popular encryption/hashing algorithms?

Nothing, nothing at all.

Suppose, for example that your exascale computer could do exa-AES-ops... 10^18 AES encryptions per second. It would take that computer 1.7E20 seconds to brute force half of the AES-128 key space. That's 5.4E12 years, to achieve a 50% chance of recovering a single key.

And if that weren't the case, you could always step up to 192 or 256-bit keys. In "Applied Cryptography", in the chapter on key length, Bruce Schneier analyzed thermodynamic limitations on brute force key search. He calculated the amount of energy required for a perfectly efficient computer to merely increment a counter through all of its values. That's not to actually do anything useful like perform an AES operation and a comparison to test a particular key, but merely to count through all possible keys. Such a computer, running at the ambient temperature of the universe, would consume 4.4E-6 ergs to set or clear a single bit. Consuming the entire output of our star for a year, and cycling through the states in an order chosen to minimize bit flips rather than just counting sequentially, would provide enough energy for this computer to count through 2^187. The entire output of the sun for 32 years gets us up to 2^192. To run a perfectly-efficient computer through 2^256 states, you'd need to capture all of the energy from approximately **137 billion supernovae**[*]. To brute force a 256-bit key you'd need to not only change your counter to each value, you'd then need to perform an AES operation.

Raw computing power is not and never will be the way to break modern crypto systems[**]. To break them you need to either exploit unknown weaknesses in the algorithms (which means you have to be smarter than the world's academic cryptographers), or exploit defects in the implementation (e.g. side channel attacks) or find other ways to get the keys -- attack the key management. The last option is always the best, though implementation defects are also quite productive. Neither of them benefit significantly from having massive computational resources available.

[*] Schneier didn't take into account reversible computing in his calculation. A cleverly-constructed perfectly-efficient computer could make use of reversible circuits everywhere they can work, and a carefully-constructed algorithm could make use of as much reversibility as possible. With that, it might be feasible to lower the energy requirements significantly, maybe even several orders of magnitude (though that would be tough). We're still talking energy requirements involving the total energy output of many supernovae.

[**] Another possibility is to change the question entirely by creating computers that don't operate sequentially, but instead test all possible answers at once. Quantum computers. Their practical application to the complex messiness of block ciphers is questionable, though the mathematical simplicity of public key encryption is easy to implement on QCs. Assuming we ever manage to build them on the necessary scale. If we do, we can expect an intense new focus on protocols built around symmetric cryptography, I expect.