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Overview on class groups Reducing the defining polynomial

Reducing number field defining polynomials: An application to class group computations

Alexandre Gélin

Laboratoire d’Informatique de Paris 6, UPMC, Sorbonne Universités

09/11/2016

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial

Number fields

K number field ⇒ finite extension of Q ⇒ ∃T ∈ Z[X] monic s.t. K = Q[X]/(T). T is a defining polynomial of K.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial

Number fields

K number field ⇒ finite extension of Q ⇒ ∃T ∈ Z[X] monic s.t. K = Q[X]/(T). T is a defining polynomial of K. Two interesting structures: Group of ideals Group of units

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial

Number fields

K number field ⇒ finite extension of Q ⇒ ∃T ∈ Z[X] monic s.t. K = Q[X]/(T). T is a defining polynomial of K. Two interesting structures: Group of ideals Quotient by principal ideals ⇒ class group Cl(OK) Group of units Finitely generated ⇒ fundamental units

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial

Number fields

K number field ⇒ finite extension of Q ⇒ ∃T ∈ Z[X] monic s.t. K = Q[X]/(T). T is a defining polynomial of K. Two interesting structures: Group of ideals Quotient by principal ideals ⇒ class group Cl(OK) Group of units Finitely generated ⇒ fundamental units Aim: Compute the structure of the class group.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial State of the art General strategy for computation

Outline

1

Overview on class groups State of the art General strategy for computation

2

Reducing the defining polynomial Theoretical results Our algorithm

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial State of the art General strategy for computation

Subexponential L-notation :

LN (0, c) ≈ (log N)c LN (1, c) ≈ Nc LN (α, c) = exp

• (c + o(1))(log N)α(log log N)1−α

.

1969 Shanks: quadratic number fields in O(|∆K|

1 5 ).

1989 Hafner and McCurley: imaginary quadratic number fields in L|∆K|( 1

2,

√ 2). 1990 Buchmann: all number fields with fixed degree in L|∆K|( 1

2, 1.7).

2014 Biasse and Fieker: all number fields in L|∆K|( 2

3 + ε) in general

and L|∆K|( 1

2) if n ≤ log(|∆K|)3/4−ε.

2014 Biasse and Fieker: number fields defined by a good polynomial in L|∆K|(a), 1

3 ≤ a < 1 2.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial State of the art General strategy for computation

Index calculus

1 Factor base

Fix a factor base composed of small elements.

2 Relation collection

Collect some relations between those small elements, corresponding to linear equations.

3 Linear algebra

Deduce the sought result performing linear algebra on the system built.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial State of the art General strategy for computation

The factor base

B = {prime ideals in OK of norm below B} B determined so that B generates the whole class group. Minkowski’s bound: every class contains an ideal of norm smaller than MK =

• |∆K|

4 π r2 n! nn . Bach’s bound: assuming GRH, classes of ideals of norm less than 12(log |∆K|)2 generate the class group.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial State of the art General strategy for computation

The factor base

B = {prime ideals in OK of norm below B} B determined so that B generates the whole class group. Minkowski’s bound: every class contains an ideal of norm smaller than MK =

• |∆K|

4 π r2 n! nn . Bach’s bound: assuming GRH, classes of ideals of norm less than 12(log |∆K|)2 generate the class group. Practically B = L|∆K|(β, cb).

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial State of the art General strategy for computation

Relation collection

B = (p1, · · · , pN) Surjective morphism: ZN

φ

− → I

π

− → Cl(OK) (e1, · · · , eN) − →

• i pei

i

− →

• i[pi]ei

Cl(OK) ≃ ZN/

• (e1, · · · , eN) ∈ ZN |

i pei i = (α)OK

• Alexandre Gélin

Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial State of the art General strategy for computation

Relation collection

B = (p1, · · · , pN) Surjective morphism: ZN

φ

− → I

π

− → Cl(OK) (e1, · · · , eN) − →

• i pei

i

− →

• i[pi]ei

Cl(OK) ≃ ZN/

• (e1, · · · , eN) ∈ ZN |

i pei i = (α)OK

• Idea:

1 Pick at random A =

i pvi i .

2 Find a reduced ideal A′ in the same class. 3 If A′ splits on B (⇔ A′ =

i pv′

i

i ) then

A(A′)−1 =

i pvi−v′

i

i

is principal.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial State of the art General strategy for computation

Linear algebra

Relations stored in a matrix of size about N × N. Structure of the class group given by the Smith Normal Form

• f the matrix.

First compute Hermite Normal Form with a premultiplier because we need kernel vectors. Storjohann and Labahn algorithm, runtime in Nω+1 (2 ≤ ω ≤ 3 exponent of matrix multiplication)

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial State of the art General strategy for computation

Verification

We find a tentative class group H, but the class group Cl(OK) may be only a quotient of H. ⇒ Need an approximation of the class number hK = |Cl(OK)|.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial State of the art General strategy for computation

Verification

We find a tentative class group H, but the class group Cl(OK) may be only a quotient of H. ⇒ Need an approximation of the class number hK = |Cl(OK)|. Class number formula + Euler Product: hKRegK = EP · wK ·

• |∆K|

2r1 · (2π)r2 .

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial State of the art General strategy for computation

Verification

We find a tentative class group H, but the class group Cl(OK) may be only a quotient of H. ⇒ Need an approximation of the class number hK = |Cl(OK)|. Class number formula + Euler Product: hKRegK = EP · wK ·

• |∆K|

2r1 · (2π)r2 . From the relations, we can also deduce a candidate for an approximation of RegK and perform the verification step.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

Outline

1

Overview on class groups State of the art General strategy for computation

2

Reducing the defining polynomial Theoretical results Our algorithm

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

What is a good polynomial ?

We want a polynomial that defines a fixed number field: The degree is fixed, We want the coefficients as small as possible.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

What is a good polynomial ?

We want a polynomial that defines a fixed number field: The degree is fixed, We want the coefficients as small as possible. Definition Let T = akXk ∈ Z[X]. The height of T is defined as the maximal norm of its coefficients, namely H(T) = max

k

|ak|.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

What is a good polynomial ?

We want a polynomial that defines a fixed number field: The degree is fixed, We want the coefficients as small as possible. Definition Let T = akXk ∈ Z[X]. The height of T is defined as the maximal norm of its coefficients, namely H(T) = max

k

|ak|. Proposition For every defining polynomial T of a degree-n number field K, the discriminants satisfy |∆K| ≤ |∆(T)| ≤ n2nH(T)2n−2.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

Classes from Biasse and Fieker work

Definition Let n0, d0 > 0 and 0 < α < 1

2.

Cn0,d0,α =

• K = Q[X]/(T)
• deg(T) = n0(log |∆K|)α(1 + o(1))

log H(T) = d0(log |∆K|)1−α(1 + o(1))

• Theorem

There exists an L|∆K|(a) algorithm for class group computation for a = max

• α, 1 − α

2

• .

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

Minimal height

If K ∈ Cn0,d0,α, there exists T such that H(T) = |∆K|

κ n ,

with κ = n0d0(1 + o(1)).

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

Minimal height

If K ∈ Cn0,d0,α, there exists T such that H(T) = |∆K|

κ n ,

with κ = n0d0(1 + o(1)). Proposition For every number field K, there exists a defining polynomial T s.t. H(T) ≤ 3n |∆K| n

• n

2n−2

.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

Minimal height

If K ∈ Cn0,d0,α, there exists T such that H(T) = |∆K|

κ n ,

with κ = n0d0(1 + o(1)). Proposition For every number field K, there exists a defining polynomial T s.t. H(T) ≤ 3n |∆K| n

• n

2n−2

. Definition Let n0, d0 > 0, 0 < α < 1 and 1 − α ≤ γ ≤ 1.

Dn0,d0,α,γ =

• K = Q[X]

(T)

• deg(T) ≤ n0
• log |∆K|

log log |∆K|

α log H(T) ≤ d0(log |∆K|)γ(log log |∆K|)1−γ

• Alexandre Gélin

Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

Minimal height

If K ∈ Cn0,d0,α, there exists T such that H(T) = |∆K|

κ n ,

with κ = n0d0(1 + o(1)). Proposition For every number field K, there exists a defining polynomial T s.t. H(T) ≤ 3n |∆K| n

• n

2n−2

. Definition Let n0, d0 > 0, 0 < α < 1 and 1 − α ≤ γ ≤ 1.

Dn0,d0,α,γ =

• K = Q[X]

(T)

• deg(T) ≤ n0
• log |∆K|

log log |∆K|

α log H(T) ≤ d0(log |∆K|)γ(log log |∆K|)1−γ

• Every number field belongs to such a class Dn0,d0,α,γ.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

Prior reduction algorithm

Cohen and Diaz y Diaz minimize the size of T = (X − τj), defined as S(T) =

• |τj|2.

Equivalent to find a short vector in the lattice OK, because OK is generated by the vectors [σ1(τj), · · · , σn(τj)] Examples: Input Output x3 − 5955x2 + 18142x − 607593 x3 − x2 − 2100x + 38117 x3 − 269463x2 + 752031x − 518157 x3 − x2 − 1307x − 13359 x3 − 482665x2 + 773338x − 308749 x3 − x2 − 3210x + 61325 x3 − 456191x2 + 958783x − 499681 x3 − x2 − 936x − 7616

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

Prior reduction algorithm

Cohen and Diaz y Diaz minimize the size of T = (X − τj), defined as S(T) =

• |τj|2.

Equivalent to find a short vector in the lattice OK, because OK is generated by the vectors [σ1(τj), · · · , σn(τj)] Examples: Input Output x3 + 6381x2 + 4378x − 1216 x3 − x2 − 3537064x + 2193757452 x3 − 9681x2 − 5434x − 6901 x3 − 31246021x − 67226458585 x3 − 6665x2 − 4318x − 2977 x3 + 336681x − 419200237 x3 − 6018x2 − 1387x + 6161 x3 − 12073495x − 16147208593

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

Our algorithm

Goal: Find the monic polynomial TF of minimal height defining K.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

Our algorithm

Goal: Find the monic polynomial TF of minimal height defining K. Idea: Introduce weighted lattices and look for small vectors in them.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

Our algorithm

Goal: Find the monic polynomial TF of minimal height defining K. Idea: Introduce weighted lattices and look for small vectors in them. Let θF root of TF ← → v(θF ) = [σ1(θF ), · · · , σn(θF )] ∈ OK.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

Our algorithm

Goal: Find the monic polynomial TF of minimal height defining K. Idea: Introduce weighted lattices and look for small vectors in them. Let θF root of TF ← → v(θF ) = [σ1(θF ), · · · , σn(θF )] ∈ OK. Let c > 1 and bF = [bF

1 , · · · , bF n ] defined by bF j = ⌈logc |σj(θF )|⌉.

We introduce a weighted copy of OK in Cn, generated by:

• Ωi =

σ1(ωi) cbF

1

, · · · , σn(ωi) cbF

n

• .

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

Our algorithm

Goal: Find the monic polynomial TF of minimal height defining K. Idea: Introduce weighted lattices and look for small vectors in them. Let θF root of TF ← → v(θF ) = [σ1(θF ), · · · , σn(θF )] ∈ OK. Let c > 1 and bF = [bF

1 , · · · , bF n ] defined by bF j = ⌈logc |σj(θF )|⌉.

We introduce a weighted copy of OK in Cn, generated by:

• Ωi =

σ1(ωi) cbF

1

, · · · , σn(ωi) cbF

n

• .

By construction, |˜ v(θF )i| ≤ 1 and ˜ v(θF )2 ≤ √n.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

Differences between the two algorithms

Shape of the vectors found by the algorithm of Cohen: Shape of the vectors we find: As the constant coefficient of the polynomial is the product of all the roots, we prefer vectors of the second family.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

Final results

If K ∈ Dn0,d0,α,γ, we find the minimal defining polynomial T in time L|∆K|(α).

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

Final results

If K ∈ Dn0,d0,α,γ, we find the minimal defining polynomial T in time L|∆K|(α). If γ = 1 − α, we can apply the algorithm of Biasse and Fieker and find the class group in L|∆K|(a), a = max

• α, 1−α

2

• .

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

Final results

If K ∈ Dn0,d0,α,γ, we find the minimal defining polynomial T in time L|∆K|(α). If γ = 1 − α, we can apply the algorithm of Biasse and Fieker and find the class group in L|∆K|(a), a = max

• α, 1−α

2

• .

Theorem Under GRH and smoothness heuristics, for every K ∈ Dn0,d0,α,γ, α < 1

2, there exists an L∆K(a) algorithm for class group computation

with a = max

• α, γ

2

• .

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

State of the art [BF14]

General case: L|∆K| 1

2

• L|∆K|

2

3 + ε

• a

1 3 1 2 2 3 1 4 1 3 1 2 2 3 3 4

1 α First general subexponential algorithm.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

State of the art [BF14]

Special case: L|∆K| 1

2

• L|∆K|

2

3 + ε

• L|∆K|
• max(α, 1−α

2 )

• a

1 3 1 2 2 3 1 4 1 3 1 2 2 3 3 4

1 α Only if K is defined by T such that H(T) = L|∆K| (1 − α).

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

This work

General case: depending on γ L|∆K|

• max(α, γ

2)

• a

1 3 1 2 2 3 1 4 1 3 1 2 2 3 3 4

1 α Without any condition.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

Practically

K is defined by the polynomial

x5−2x4−8001397580x3−31542753393650x2+3636653302451131875x+4818547529425280067500

. Magma V2.22-2 finds the class group – assuming GRH – in about 285 seconds. With our implementation, we reduce this defining polynomial to T = x5 − 5843635x4 + 931633x2 + 6577x − 8570.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

Practically

K is defined by the polynomial

x5−2x4−8001397580x3−31542753393650x2+3636653302451131875x+4818547529425280067500

. Magma V2.22-2 finds the class group – assuming GRH – in about 285 seconds. With our implementation, we reduce this defining polynomial to T = x5 − 5843635x4 + 931633x2 + 6577x − 8570. Magma V2.19-10 has class group computation not as optimized as in V2.22, but works with the input polynomial: with T: about 140 seconds, with the “reduced” one: about 3240 seconds.

Alexandre Gélin Reducing number field defining polynomials

Overview on class groups Reducing the defining polynomial Theoretical results Our algorithm

Thanks

Merci

Alexandre Gélin Reducing number field defining polynomials