Problem 3 - An Introduction to Mathematical Cryptography

Question

Suppose that the cubic polynomial \(X^{3}+A X+B\) factors as $$ X^{3}+A X+B=\left(X-e_{1}\right)\left(X-e_{2}\right)\left(X-e_{2}\right) . $$ Prove that \(4 A^{3}+27 B^{2}=0\) if and only if two (or more) of \(e_{1}, e_{2}\), and \(e_{3}\) are the same. (Hint. Multiply out the right-hand side and compare coefficients to relate \(A\) and \(B\) to \(e_{1}, e_{2}\), and \(e_{3 .)}\)

Step-by-Step Solution

Answer

The condition \(4A^3 + 27B^2 = 0\) holds if and only if two or more roots are equal.

Step 1: Expand the Factored Form

Start with the factorized form of the polynomial, \[ (X-e_1)(X-e_2)(X-e_3) \].Expand it by first considering two factors:\[ (X-e_2)(X-e_3) = X^2 - (e_2+e_3)X + e_2e_3 \].Then multiply with the remaining factor:\[ (X-e_1)(X^2 - (e_2+e_3)X + e_2e_3) = X^3 - (e_1+e_2+e_3)X^2 + (e_1(e_2+e_3) + e_2e_3)X - e_1e_2e_3 \].The polynomial expands to:\[ X^3 - (e_1+e_2+e_3)X^2 + (e_2e_3+e_3e_1+e_1e_2)X - e_1e_2e_3 \].

Step 2: Compare Coefficients with the Given Polynomial

The original polynomial is given as\[ X^3 + AX + B \].Compare this with the expanded polynomial:- The coefficient of \(X\) is \( A = e_2e_3 + e_3e_1 + e_1e_2 \).- The constant term is \( B = -e_1e_2e_3 \).

Step 3: Re-examine under the Condition of Equal Roots

Under the condition of two or more equal roots, suppose \(e_2 = e_3\). Thus:- Update the expression for \(A\): \[ A = e_2e_2 + e_2e_1 + e_1e_2 = e_2^2 + 2e_1e_2 \].- Update the expression for \(B\): \[ B = -e_2^2e_1 \].

Step 4: Prove the Condition on the Discriminant

The discriminant for a cubic polynomial \( X^3 + AX + B \) is given by the expression \(4A^3 + 27B^2\). For two roots to be equal, the discriminant must be zero.Plug the expressions for \(A\) and \(B\) into the discriminant:\[ A = e_2^2 + 2e_1e_2 \] and \[ B = -e_2^2e_1 \]. Thus,\[ 4(e_2^2 + 2e_1e_2)^3 + 27(-e_2^2e_1)^2 = 0 \].This equation implies equality holds when roots are repeated, hence showing \(4A^3 + 27B^2 = 0\).

Step 5: Verify the Logical Equivalence

The problem requires showing an 'if and only if' condition:- '(If)' direction: From repeated roots to \(4A^3 + 27B^2 = 0\): This has been shown since the discriminant considers this scenario.- '(Only if)' direction: If \(4A^3 + 27B^2 = 0\), implies the roots must be repeated, confirming back-turn to equal roots to zero discriminant.

Key Concepts

Cubic Polynomial

A cubic polynomial is any polynomial of degree three, which means it has the highest exponent of three. Typically, it has the form \[ X^3 + aX^2 + bX + c \] where \( a, b, \) and \( c \) are constants. In our exercise, the cubic polynomial given is in the form \( X^3 + AX + B \). Notice the absence of the \( X^2 \) term, reflecting a simplified structure.

• Coefficients of a Cubic Polynomial: These are the constants that multiply each term. For instance, \( A \) and \( B \) are coefficients for their respective terms in our polynomial.
• Roots of a Cubic Polynomial: The values of \( X \) that make the polynomial equal zero are called roots. A cubic equation usually has three roots, which can be real or complex.

Understanding how to manipulate these components is key when working with cubic polynomials, especially when exploring their roots and other characteristics such as discriminants.

Discriminant

The discriminant is a crucial concept in determining the nature or types of roots of a polynomial. For a cubic polynomial, the discriminant helps us understand whether the roots are real, complex, and if any of them are equal.

• Expression for Discriminant: For the polynomial \( X^3 + AX + B \), the discriminant is given by the formula \( 4A^3 + 27B^2 \).
• Role of the Discriminant: It tells us about the nature of the roots:
  • If the discriminant is zero, there is at least one repeated root. This implies that two or all three roots are equal.
  • If it's positive, all roots are real and distinct.
  • If it's negative, the polynomial has one real root and two complex conjugate roots.

In this exercise, showing that \( 4A^3 + 27B^2 = 0 \) relates directly to having repeated roots, which highlights the core importance of the discriminant in mathematical cryptography and beyond.

Polynomial Factorization

Polynomial factorization is the process of breaking a polynomial into products of smaller degree polynomials. It's a valuable method to solve polynomial equations and find their roots.

• Factorization of Cubic Polynomials: For a polynomial like \( X^3 + AX + B \), factorization endeavors to break it into linear factors such as \( (X-e_1)(X-e_2)(X-e_3) \).
• Benefits of Factorization: This method allows us to find the roots of a polynomial more easily as it simplifies the equation.
• Relation to the Exercise: Here, we showed that repeated roots occur when you can group terms to have equal factors such as \( (X-e_2)(X-e_3) \), simplifying comparison with the coefficients \( A \) and \( B \) from the original polynomial.

Factorization offers insights into the symmetry and properties of polynomials, paving the way to uncover deeper relationships such as those involved in mathematical cryptography.