= ax5 10.3 Winter Dining (*) (8 Points) Natalie operates a restaurant offering both indoor and outdoor dining. She adapts the fol- lowing pandemic safety measure: For indoor dining, all customers except one are required to show a proof of vaccination against COVID-19. The intention is that the exception is reserved for the most lucrative customers. This is why Natalie introduces the following system: At the beginning of each week, she randomly picks a polynomial a(x) of the form a(2) + Bx3 + (2a +B) Za[2], where q 2 11 is a (known) prime. On day i e {1,...,7} of the week, every visiting restau- rant customer receives the loyalty token t; = a(i) (printed on the receipt). The idea is that the reward token a can be recovered with sufficiently many loyalty tokens. The first customer to present the reward token is allowed to dine indoors without a vaccination certificate. a) Let q = 11. Karen explains to her friend: As a(x) is of degree at most 5, I can compute the reward token from 6 loyalty tokens. So I need to eat 6 times outdoors in the cold to get the reward token. Despite the freezing weather, Karen sticks to her plan and obtains 6 loyalty tokens. Unfortunately, a mishap occurs when she attempts to sanitize the receipts?. Only two loyalty tokens are left, t2 = 3 and t4 = 4. Is it possible to compute the reward token from these two loyalty tokens only? If so, carry out the computation and find a. If not, justify why it is not possible. b) Let q = 13. Prove or disprove the following claim: If only a single loyalty token t Z for some i E {1,..., 7} is known, then any reward token a Zq is possible. 'It turns out that microwave-based sanitization methods are suboptimal for paper... = ax5 10.3 Winter Dining (*) (8 Points) Natalie operates a restaurant offering both indoor and outdoor dining. She adapts the fol- lowing pandemic safety measure: For indoor dining, all customers except one are required to show a proof of vaccination against COVID-19. The intention is that the exception is reserved for the most lucrative customers. This is why Natalie introduces the following system: At the beginning of each week, she randomly picks a polynomial a(x) of the form a(2) + Bx3 + (2a +B) Za[2], where q 2 11 is a (known) prime. On day i e {1,...,7} of the week, every visiting restau- rant customer receives the loyalty token t; = a(i) (printed on the receipt). The idea is that the reward token a can be recovered with sufficiently many loyalty tokens. The first customer to present the reward token is allowed to dine indoors without a vaccination certificate. a) Let q = 11. Karen explains to her friend: As a(x) is of degree at most 5, I can compute the reward token from 6 loyalty tokens. So I need to eat 6 times outdoors in the cold to get the reward token. Despite the freezing weather, Karen sticks to her plan and obtains 6 loyalty tokens. Unfortunately, a mishap occurs when she attempts to sanitize the receipts?. Only two loyalty tokens are left, t2 = 3 and t4 = 4. Is it possible to compute the reward token from these two loyalty tokens only? If so, carry out the computation and find a. If not, justify why it is not possible. b) Let q = 13. Prove or disprove the following claim: If only a single loyalty token t Z for some i E {1,..., 7} is known, then any reward token a Zq is possible. 'It turns out that microwave-based sanitization methods are suboptimal for paper