Extended Euclidean Algorithm and Inverse Modulo Tutorial

Summary

The transcript explains the process of finding the modular inverse of 17 modulo 43. It begins by setting up the equation 17X ≡ 1 (mod 43) and then uses the Euclidean algorithm to determine the greatest common divisor (GCD). The extended Euclidean algorithm is applied to express 1 as a linear combination of 43 and 17. Through a series of substitutions and simplifications, the inverse is found to be -5, which corresponds to 38 modulo 43. The explanation is methodical, illustrating each step of the algorithm to solve for X.

Takeaways

• 🔢 The script discusses the concept of finding the modular multiplicative inverse, specifically for the number 17 modulo 43.
• ➗ To find the inverse, the script demonstrates the use of division, suggesting that 17/X ≡ 1 (mod 43) is the starting point.
• 📉 The Euclidean algorithm is introduced as a method to find the greatest common divisor (GCD), which is necessary to confirm the existence of an inverse.
• 🔄 The extended Euclidean algorithm is used to express the inverse in terms of the original numbers, which involves a series of substitutions and simplifications.
• 📝 The script walks through the Euclidean algorithm step by step, showing how to reduce 43 and 17 to find their GCD.
• 🔍 It's highlighted that an inverse exists only if the GCD of the two numbers is 1, which is confirmed in the case of 43 and 17.
• 🔄 The process of back substitution in the Euclidean algorithm is explained to express the equation in terms of the variables used in the original question.
• 🧮 The script provides a detailed calculation, showing how to manipulate the equations to isolate the inverse.
• 🔢 The final step involves simplifying the equation to find that the inverse of 17 modulo 43 is -5, which is then adjusted to be within the modular range (0 to 42).
• 🎯 The conclusion is that the modular inverse of 17 modulo 43 is 38, as it is the number that, when multiplied by 17 and taken modulo 43, equals 1.

Q & A

What is the concept of inverse modulo numbers?

Inverse modulo numbers refer to finding a number X such that when it is multiplied by another given number (say 17), the result is congruent to 1 modulo a third number (in this case, 43). It is written as 17X ≡ 1 (mod 43).

How do you find the inverse modulo of a number?

To find the inverse modulo of a number, you use the Extended Euclidean Algorithm. It involves expressing the numbers in terms of each other and finding coefficients that satisfy the equation 1 = a*m + b*n, where 'a' is the inverse you're looking for, 'm' and 'n' are the original numbers, and 'b' is a coefficient.

Why is it necessary for the GCD of the two numbers to be 1 to find an inverse modulo?

The GCD (Greatest Common Divisor) must be 1 because it indicates that the two numbers are coprime (relatively prime). Only coprime numbers have an inverse modulo each other. If the GCD is greater than 1, the numbers are not coprime and an inverse modulo does not exist.

What is the Euclidean algorithm and how does it relate to finding modulo inverses?

The Euclidean algorithm is a method for finding the GCD of two numbers. It is used in the Extended Euclidean algorithm to express one number as a linear combination of the other two, which is necessary for finding the inverse modulo.

How does the Extended Euclidean algorithm work?

The Extended Euclidean algorithm works by applying the Euclidean algorithm iteratively and keeping track of the coefficients of the remainders. It rewrites the equation at each step until it finds an equation of the form 1 = a*m + b*n, where 'a' is the inverse modulo.

What is the significance of the equation 1 = 43 - 17*2 in the context of the script?

This equation is a step in the Extended Euclidean algorithm. It expresses 1 as a linear combination of 43 and 17, which is crucial for finding the inverse modulo of 17 modulo 43.

Why is it important to keep track of the coefficients during the Extended Euclidean algorithm?

Keeping track of the coefficients allows you to express the final equation in terms of the original numbers, which is necessary to find the inverse modulo. The coefficients represent the 'a' and 'b' values in the equation 1 = a*m + b*n.

What does it mean when the script says 'sub in' an equation?

In the context of the script, 'sub in' means to substitute one equation into another to simplify and eventually express the equation in terms of the original numbers, which helps in finding the inverse modulo.

How do you ensure the final inverse modulo result is between 0 and the modulo number minus one?

After finding the inverse modulo, you may need to adjust it to be within the range of 0 to (modulo number - 1). This is done by adding or subtracting multiples of the modulo number until the result falls within the desired range.

What is the final result for the inverse of 17 modulo 43 according to the script?

The final result for the inverse of 17 modulo 43 is 38, as it is the number that when multiplied by 17 gives a result congruent to 1 modulo 43.