In this Saxon Algebra 2 investigation, Grade 10 students explore cryptography by building substitution ciphers and using the VLOOKUP function in a spreadsheet to encrypt and decrypt messages. Students then advance to matrix multiplication as a second layer of encryption, organizing cipher values into 3×3 matrices and applying an encoding matrix to produce a more secure encrypted message. The lesson also covers how to reverse the process using inverse matrices and a graphing calculator to fully decrypt an encoded string.
Section 1
📘 Understanding Cryptography
A number replaces each letter of the alphabet, so that all messages consist of a string of numbers.
Next, you’ll build your own substitution cipher using a spreadsheet, turning abstract rules into a practical encryption tool.
Section 2
Substitution Cipher
A substitution cipher is a classic encryption method where each letter is replaced by a number based on a secret key. This transforms a readable message into a string of numbers. To decode it, someone needs the exact same key that was used for encryption, which maps the numbers back to their original letters. This is a foundational concept in cryptography.
Encrypting "HI": Using the cipher where H=15 and I=17, the message becomes the number string "15 17".
The word "SECRET" is encrypted as "12 9 5 10 9 14" based on the provided cipher key.
Think of it like a secret decoder ring! Each letter gets a secret number buddy. To send a message, you just swap out the letters for their number partners, turning words into a string of code. To read it, you simply swap them back using the key.
Section 3
Matrix Multiplication for Security
To add a powerful layer of security, the number string from a substitution cipher is organized into matrices. These "cipher matrices" are then multiplied by a secret "encoding matrix." This mathematical operation thoroughly scrambles the original numbers, making the code exponentially harder to crack without knowing the exact encoding matrix used for the encryption process.
Cipher matrix multiplied by encoding matrix creates a new encrypted matrix .
The string "14 4 6 12 9 5" is first arranged into a cipher matrix before multiplication.
Swapping numbers is easy to crack. For real security, we put numbers into a grid (a matrix) and scramble them by multiplying with a secret encoding grid. This jumbles the numbers in a complex way, making it much harder for anyone to guess the original message.
Section 4
Decrypting with an Inverse Matrix
Decrypting a message encoded with matrix multiplication requires using the inverse of the original encoding matrix. Each encrypted matrix is multiplied by this inverse matrix. This calculation perfectly reverses the scrambling process, restoring the numbers to their simple substitution cipher form. These numbers can then be translated back into letters using the original cipher key.
To decrypt, multiply the encrypted matrix by the inverse of the encoding matrix : .
If is encrypted, multiplying by returns the original cipher matrix .
How do you unscramble the matrix-encrypted message? You use the ultimate undo button: the inverse matrix! Multiplying the scrambled message by the inverse of the secret encoding grid perfectly reverses the encryption, giving you back the original numbers, which you can then translate to letters.
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Section 1
📘 Understanding Cryptography
A number replaces each letter of the alphabet, so that all messages consist of a string of numbers.
Next, you’ll build your own substitution cipher using a spreadsheet, turning abstract rules into a practical encryption tool.
Section 2
Substitution Cipher
A substitution cipher is a classic encryption method where each letter is replaced by a number based on a secret key. This transforms a readable message into a string of numbers. To decode it, someone needs the exact same key that was used for encryption, which maps the numbers back to their original letters. This is a foundational concept in cryptography.
Encrypting "HI": Using the cipher where H=15 and I=17, the message becomes the number string "15 17".
The word "SECRET" is encrypted as "12 9 5 10 9 14" based on the provided cipher key.
Think of it like a secret decoder ring! Each letter gets a secret number buddy. To send a message, you just swap out the letters for their number partners, turning words into a string of code. To read it, you simply swap them back using the key.
Section 3
Matrix Multiplication for Security
To add a powerful layer of security, the number string from a substitution cipher is organized into matrices. These "cipher matrices" are then multiplied by a secret "encoding matrix." This mathematical operation thoroughly scrambles the original numbers, making the code exponentially harder to crack without knowing the exact encoding matrix used for the encryption process.
Cipher matrix multiplied by encoding matrix creates a new encrypted matrix .
The string "14 4 6 12 9 5" is first arranged into a cipher matrix before multiplication.
Swapping numbers is easy to crack. For real security, we put numbers into a grid (a matrix) and scramble them by multiplying with a secret encoding grid. This jumbles the numbers in a complex way, making it much harder for anyone to guess the original message.
Section 4
Decrypting with an Inverse Matrix
Decrypting a message encoded with matrix multiplication requires using the inverse of the original encoding matrix. Each encrypted matrix is multiplied by this inverse matrix. This calculation perfectly reverses the scrambling process, restoring the numbers to their simple substitution cipher form. These numbers can then be translated back into letters using the original cipher key.
To decrypt, multiply the encrypted matrix by the inverse of the encoding matrix : .
If is encrypted, multiplying by returns the original cipher matrix .
How do you unscramble the matrix-encrypted message? You use the ultimate undo button: the inverse matrix! Multiplying the scrambled message by the inverse of the secret encoding grid perfectly reverses the encryption, giving you back the original numbers, which you can then translate to letters.
Book overview
Jump across lessons in the current chapter without opening the full course modal.
Continue this chapter