Salazar, C., & Macris, C. (2022, August), Constructing Quadratic Equations modulo N to emphasize the differences between real and modular arithmetic operations  Paper presented at 2022 ASEE Annual Conference & Exposition, Minneapolis, MN. 10.18260/1-2--40838

\bibitem{asee_peer_40838}
  Salazar, C., \& Macris, C. (2022, August), \emph{Constructing Quadratic Equations modulo N to emphasize the differences between real and modular arithmetic operations}
  Paper presented at 2022 ASEE Annual Conference \& Exposition, Minneapolis, MN.
  10.18260/1-2--40838

Carlos Salazar and Constantine Macris.  "Constructing Quadratic Equations modulo N to emphasize the differences between real and modular arithmetic operations".  2022 ASEE Annual Conference & Exposition, Minneapolis, MN, 2022, August.  ASEE Conferences, 2022.  https://peer.asee.org/40838 Internet.  07 May, 2024

\bibitem{asee_peer_40838}
  Carlos Salazar and Constantine Macris.
  "Constructing Quadratic Equations modulo N to emphasize the differences between real and modular arithmetic operations".
  \emph{2022 ASEE Annual Conference \& Exposition, Minneapolis, MN, 2022, August}.
  ASEE Conferences, 2022.
  https://peer.asee.org/40838 Internet.  07 May, 2024

@INPROCEEDINGS{asee_peer_40838
author = "Carlos Salazar and Constantine Macris"
title = "Constructing Quadratic Equations modulo N to emphasize the differences between real and modular arithmetic operations"
booktitle = "2022 ASEE Annual Conference \& Exposition"
year = "2022"
month = "August"
address = "Minneapolis, MN"
publisher = "ASEE Conferences"
note = {https://peer.asee.org/40838}
number = {10.18260/1-2--40838}
}

TY  - CPAPER
AB  - Cadets majoring in Cybersecurity at the United States Coast Guard Academy are typically enrolled in the department’s Introductory Cryptography course during the Fall of their Junior year. Many of these students struggled to understand the need to use modular operations to solve quadratic equations modulo N. This was because homework assignments requiring the solution of these equations typically employed equations for which regular real operations were perfectly satisfactory. Quadratic equations modulo N are assigned to help the students recognize the similarities between modular and real arithmetic while also helping them see the differences. These equations are also introduced to prepare the students for the study of elliptic curves. Three specific differences between modular quadratic equations and real arithmetic quadratic equations were identified. Modular arithmetic equations can fail to have any solutions if the modulus for an otherwise acceptable equation does not permit a multiplicative inverse required for solution. Modular quadratic equations can also have solutions where real quadratic equations would fail to have any solutions (complex solutions). And finally, modular quadratic equations with composite moduli can have many more than two solutions. All three of these differences can be exploited to prevent the students from solving a quadratic equation using real arithmetic methods and thus compel them to use modular techniques. It was observed on the homework assignments that a significant fraction of the students failed to use modular techniques to solve the quadratic equations that were assigned and it was suspected that many of those that did were unsure of why they needed to employ them. After constructing quadratic equations that took advantage of the peculiar properties of modular arithmetic the students were unable to use real arithmetic techniques and had to resort to modular methods thus reinforcing the need to use those methods to solve the modular equations.
AU  - Carlos Salazar
AU  - Constantine Macris
CY  - Minneapolis, MN
DA  - 2022/08/23
PB  - ASEE Conferences
TI  - Constructing Quadratic Equations modulo N to emphasize the differences between real and modular arithmetic operations
UR  - https://peer.asee.org/40838
DO  - 10.18260/1-2--40838
ER  -