Store and bank safe are two real-world applications for polynomial equations. Assuming a bank manager wants to keep money in a safe and they will not be available at the time of delivery. The manager may have to request that his tellers open the safe. However, the manager may not trust his employees enough to lend them a key since he is afraid that they may steal some money. Therefore, a polynomial is used in creating a combination of passcodes, with each person having a portion (Piccoli & Rizzo, 2020). When the three people with the codes monitor each other, neither will they do anything fishy. As a result, the safe can only be accessed when the three people put in their passcodes, preventing any theft.
Assume I would like some kk shares to be able to retrieve the passcode, which is an integer NN. The key is a k1k1 degree polynomial with NN as the constant term. So, for example, if we want three bankers to be able to access the safe and the passcode is 10431043, we may construct the secret polynomial 3X253X+10433X253X+1043 (Piccoli & Rizzo, 2020. They will become a number on this polynomial—for example, if there are six bankers, you may assign each one of the mentioned sequences: (3,2663), (2,2117), (1,1577), (1,515), (2,7), (3,523). (−3,2663),(−2,2117),(−1,1577),(1,515),(2,−7),(3,−523).
The difficulty resides in every teller’s ability to deduce the initial quadratic polynomial from their one-point. No two tellers can agree on the original quadratic polynomial. However, if any three of these come together, they may deduce that a distinct quadratic polynomial is running through all three locations, so the passcode is 10431043.
Reference
De Piccoli, A., Visconti, A., & Rizzo, O. G. (2020). Polynomial multiplication over binary finite fields: new upper bounds. Journal of Cryptographic Engineering, 10(3), 197-210.