Upload to Canvas a PDF of your work for the following problems.

Assignment: due Monday, 11:59 pm

  1. Consider an equation of a real elliptic curve with an \(x^2\) term: \(y^2 = x^3 + ax^2 + bx + c\). Substitute \(x = u - a/3\) and simplify to show that the resulting equation has no \(u^2\) term. (This exercise proves that the \(x^2\) term can be eliminated via a change of variables.)

  2. Use Pollard’s \(p-1\) method to factor the number n given below. Show work. (Of course, gmp has a function called factorize, but beware of using this function so close to Halloween.)

library(gmp)
n <- as.bigz("32815976131045173956403096853277193054971618642813813139023877173")
  1. Let \(E\) be the group defined by the elliptic curve \(y^2 = x^3 -3x + 3\) modulo 7.
    1. List all the points in \(E\). Explain how you know you have found all the points.
    2. Show that the number of points in \(E\) satisfies Hasse’s Theorem.
  2. Let \(E\) be the group defined by the elliptic curve \(y^2 = x^3 - x\) modulo 71. Determine how many points \(E\) contains. (You don’t need to find them all; just determine how many there are.) Show your calculations. Hint: Since \(71 \bmod 4 = 3\), you can use the theorem on square roots to find all of the values of \(x^3-x\) that have square roots.