lecture_2.org

IN-PROGRESS lecture NOTES

• bound for number of quadratic resides in a finite field
• polynomials in a finite field
• lagrange interpolation
• d+1 points uniquely define a d-degreee polynomial which in turn is uniquely defined by its d+1 coefficients
• the difference between the order (of a group / field) and a characteristic (of a field)
• can a field have ene element?

p = 11
F = GF(p)

# maps elements in the field to it's square roots (if any)
# for a given element it's square roots are the inverse of each other (obviously)
square_roots = {}
for y in F:
  square_roots[y] = [];
  for x in F:
    if x * x == y : square_roots[y].append(x);

print(f"GF{p} elements and it's square roots: {square_roots}")

GF11 elements and it's square roots: {0: [0], 1: [1, 10], 2: [], 3: [5, 6], 4: [2, 9], 5: [4, 7], 6: [], 7: [], 8: [], 9: [3, 8], 10: []}