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

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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 2ⁿ in modulus².

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 decomposition ofpas a sum of two squares. Evaluate the complexity of this algorithm³. 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.

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(wherer² ≡ −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 α²+β² ≡ −1 modp.

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

11. Let Λ⊂R⁴ 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< a²+b²+c²+d² <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 decomposition 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.

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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 f⁰ ∈ N⁴.

2. Suppose that f = f₁· · ·f_r is the decomposition of f as a product of irreducible monic polynomials of Fp[X]. Show that N admits

f₁⁰f

f₁ , . . . ,f_r⁰f f_r as a F^p-basis.

3. Prove that g = Pd−1

i=0 g_iXⁱ is an element of N if and only if G = (g₀, . . . , 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·(f⁰)⁻¹ modf.

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

F 8. Let q =pⁿ 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_iX⁻ⁱ

=

∞

X

i=ω

−i q−1

s_iX^−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.

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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.

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