$1\mathrm{.}3.$ We consider the long$-$term security of the Advanced Encryption Standard $($AES$)$ with a key length of $128-$bit with respect to exhaustive key$-$search attacks. AES is perhaps the most widely used symmetric cipher at this time.$1.$ Assume that an attacker has a special purpose application specific integrated cir$-$cuit $($ASIC$)$ which checks $5108$ keys per second, and she has a budget of $$1$ million. One ASIC costs $$50,$ and we assume $100\%$ overhead for integrating