Bordeaux 1 University MHT 933 – Master 2

Mathematics Year 2008-2009

Final Exam. 2008 December 19th, 8h-12h

Handwritten lecture notes are allowed as well as the course typescript. You may compose in either English or French.

Exercise 1 (Primes as sums of squares) Part 1

1. Let x =a+bi and y = c+di 6= 0 be two Gaussian integers: x, y ∈Z[i]

where i is a square root of −1. Prove that there exists an element q ∈ Z[i]

such that

|x−qy|^{2} ≤ 1
2|y|^{2}
and show how to compute such a q.

2. Deduce from this an algorithm to compute gcd(u, v)^{1} where u, v are non
zero elements of Z[i].

3. Show that this algorithm has word complexity in O(ne ^{2}) for operands
bounded by 2^{n} in modulus^{2}.

4. Let p ≡ 1 mod 4 be a prime. Let m be the smallest positive quadratic
non-residue mod p and let us put x = m^{(p−1)/4} modp. Show that the com-
putation of gcd(p, x+i) in Z[i] gives a decomposition of p as a sum of two
squares.

5. Prove that this decomposition is essentially unique.

6. Write a deterministic algorithm with inputpand outpout the decomposi-
tion ofpas a sum of two squares. Evaluate the complexity of this algorithm^{3}.
7. We have already seen during the lectures that Minkowski’s Theorem ap-
plied to the free Z-module generated by the columns of

p r 0 1

1Our gcd is not unique: we can multiply it by±1 or±i. This gives four possibilities.

Here, we consider any one of those four possibilities to be “the” gcd.

2This naive algorithm can be improved and it is possible to obtain a word complexity in O(n) (A. Weilert 2000) using a divide and conquer approach.e

3You can assume GRH and use Bach’s bound: m≤2(logp)^{2}.

1

(wherer^{2} ≡ −1 mod p) leads to the existence of such a decomposition. Show
how the LLL algorithm gives a solution, write another algorithm for the same
problem and compare the new complexity to the previous one.

Part 2

From now on, p is a prime such that p≡3 mod 4.

8. Letxbe a quadratic residue modp. Find an easy way to obtain a square root of x modp.

9. Prove that there exist α, β ∈Z such that
α^{2}+β^{2} ≡ −1 modp.

10. Show how to find such a pair thanks to the smallest positive quadratic non-residue m.

11. Let Λ⊂R^{4} be the free Z-module generated by the columns of

p 0 α β 0 p β −α

0 0 1 0

0 0 0 1

.

Prove that there exists (a, b, c, d)∈Λ such that
0< a^{2}+b^{2}+c^{2}+d^{2} <2p,

and deduce from this that p can be written as a sum of four squares. Is this decomposition unique?

12. Explain how this result implies that every non negative integer can be written as a sum of four squares (Lagrange 1770).

13. Show how we can obtain, thanks to LLL-algorithm, a decomposition of p as a sum of four squares.

14. Write a deterministic algorithm with inputp and with output a decom- position of p as a sum of four squares.

15. Assuming GRH and Bach’s bound, compute the word complexity of this algorithm.

Exercise 2 (Niederreiter’s algorithm)

In what follows, pis a prime, f ∈Fp[X] is monic, squarefree and has degree
d ≥ 1. Consider the field of rational functions Fp(X) and the differential
equation (E) over F^{p}(X):

(E) : y^{(p−1)}+y^{p} = 0.

2

Define N as the set of g ∈Fp[X] such thatg/f is a solution of (E).

1. Show that N is a linear subspace of F^{p}[X] (considered as an F^{p}-vector
space), that every g ∈ N has degree< d and that f^{0} ∈ N^{4}.

2. Suppose that f = f_{1}· · ·f_{r} is the decomposition of f as a product of
irreducible monic polynomials of Fp[X]. Show that N admits

f_{1}^{0}f

f_{1} , . . . ,f_{r}^{0}f
f_{r}
as a F^{p}-basis.

3. Prove that g = Pd−1

i=0 g_{i}X^{i} is an element of N if and only if G =
(g_{0}, . . . , gd−1) satisfies an equation

M_{p}(f) + Id_{d}

·G^{t}= 0,
where M_{p}(f) is a matrix of Md×d(Fp) such that

rank M_{p}(f) + Id_{d}

=d−r,

and corresponds to an endomorphism of {P ∈F^{p}[X]; degP < d} to be pre-
cised.

4. Assume thatf is not irreducible over Fp. How can an element ofN yield a nontrivial factor of f? What is the probability for a random element ofN to do this?

5. Write a deterministic algorithm usingN, linear algebra (like in Berlekamp’s algorithm) and giving a non-trivial factor of f.

6. Explicit the matrix M_{p}(f) in the case p= 2 and prove that the previous
algorithm leads to the complete factorization of f over F2 in O(d^{ω}) opera-
tions in F2. What is the advantage of this algorithm on Berlekamp’s?

7. Now, let B ⊂ Fp[X]/hfi be the Berlekamp algebra of f. Consider the map

Φ : N −→ B

g 7−→ g·(f^{0})^{−1} modf.

Prove that Φ is well defined and that it is a vector space isomorphism.

F 8. Let q =p^{n} a power of p. Let us define over the set Fq((X^{−1})) of formal
Laurent series over Fq the Hasse-Teichmuller derivative by:

H^{(q−1)}X^{∞}

i=ω

s_{i}X^{−i}

=

∞

X

i=ω

−i q−1

s_{i}X^{−i−q+1}.

4In this question and in the following one, you can make use of the decomposition f =Q

1≤i≤d(x−λi) where theλi are the distinct roots off in a splitting fieldEoff over Fp.

3

Show how to generalize the previous method to obtain a factorization of a monic squarefree polynomial f ∈Fq[X] thanks to the differential equation

H^{(q−1)}(y) =y^{q}
where y∈F^{q}((X^{−1})).

Exercise 3 (Lenstra’s numbers)

We say that an integer N >0 is a Lenstra number if and only if
a^{N}^{+1} ≡amodN for every a∈Z.

For instance 2 and 6 are two Lenstra numbers.

1. Prove that N is a Lenstra number if and only if it is squarefree and p−1|N for every prime divisor pof N.

2. Show that the set of Lenstra numbers is finite and give the complete list of its elements.

4