Parallel Gröbner Basis Algorithms over Finite Fields

Christian Eder and Jean-Charles Faugère

University of Kaiserslautern

October 14, 2016

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Table of Contents

1. Gröbner Bases and Buchberger’s Algorithm

2. Faugère’s F4 Algorithm

3. Specialized Linear Algebra for Gröbner Basis Computation

4. GBLA – A Gröbner Basis Linear Algebra Library

5. GB – A Gröbner Basis Library

6. Some benchmarks

7. Outlook

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Buchberger’s criterion

S-polynomialsLet f ̸= 0,g ̸= 0 ∈ R and let λ = lcm (lt (f) , lt (g)) be the least common multiple oflt (f) and lt (g). The S-polynomial between f and g is given by

spol (f ,g) ..=λ

lt (f) f −λ

lt (g)g.

Buchberger’s criterion [2]Let I = ⟨f1, . . . , fm⟩ be an ideal in R. A finite subset G ⊂ R is a Gröbner basis for Iif G ⊂ I and for all f ,g ∈ G : spol (f ,g) G−→ 0.

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Buchberger’s criterion

S-polynomialsLet f ̸= 0,g ̸= 0 ∈ R and let λ = lcm (lt (f) , lt (g)) be the least common multiple oflt (f) and lt (g). The S-polynomial between f and g is given by

spol (f ,g) ..=λ

lt (f) f −λ

lt (g)g.

Buchberger’s criterion [2]Let I = ⟨f1, . . . , fm⟩ be an ideal in R. A finite subset G ⊂ R is a Gröbner basis for Iif G ⊂ I and for all f ,g ∈ G : spol (f ,g) G−→ 0.

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Buchberger’s algorithm

Input: Ideal I = ⟨f1, . . . , fm⟩Output: Gröbner basis G for I1. G← ∅2. G← G ∪ {fi} for all i ∈ {1, . . . ,m}3. Set P← {spol

(fi, fj

)| fi, fj ∈ G, i > j}

4. Choose p ∈ P, P← P \ {p}(a) If p G−→ 0 ▶▶ no new information

Go on with the next element in P.(b) If p G−→ q ̸= 0 ▶▶ new information

Build new S-pair with q and add them to P.Add q to G.Go on with the next element in P.

5. When P = ∅ we are done and G is a Gröbner basis for I.

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Buchberger’s algorithm

Input: Ideal I = ⟨f1, . . . , fm⟩Output: Gröbner basis G for I1. G← ∅2. G← G ∪ {fi} for all i ∈ {1, . . . ,m}3. Set P← {spol

(fi, fj

)| fi, fj ∈ G, i > j}

4. Choose p ∈ P, P← P \ {p}

(a) If p G−→ 0 ▶▶ no new informationGo on with the next element in P.

(b) If p G−→ q ̸= 0 ▶▶ new informationBuild new S-pair with q and add them to P.Add q to G.Go on with the next element in P.

5. When P = ∅ we are done and G is a Gröbner basis for I.

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Buchberger’s algorithm

Input: Ideal I = ⟨f1, . . . , fm⟩Output: Gröbner basis G for I1. G← ∅2. G← G ∪ {fi} for all i ∈ {1, . . . ,m}3. Set P← {spol

(fi, fj

)| fi, fj ∈ G, i > j}

4. Choose p ∈ P, P← P \ {p}(a) If p G−→ 0 ▶▶ no new information

Go on with the next element in P.(b) If p G−→ q ̸= 0 ▶▶ new information

Build new S-pair with q and add them to P.Add q to G.Go on with the next element in P.

5. When P = ∅ we are done and G is a Gröbner basis for I.

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Faugère’s F4 algorithm

Faugère’s F4 algorithm

Input: Ideal I = ⟨f1, . . . , fm⟩Output: Gröbner basis G for I1. G← ∅2. G← G ∪ {fi} for all i ∈ {1, . . . ,m}3. Set P← {(af ,bg) | f ,g ∈ G}4. d← 05. while P ̸= ∅:

(a) d← d+ 1(b) Pd ← Select (P), P← P \ Pd(c) Ld ← {af ,bg | (af ,bg) ∈ Pd}(d) Ld ← Symbolic Preprocessing(Ld,G)(e) Fd ← Reduction(Ld,G)(f) for h ∈ Fd:

▶ If lt (h) /∈ L(G) (all other h are “useless”):▷ P← P ∪ {new pairs with h}▷ G← G ∪ {h}

6. Return G

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Faugère’s F4 algorithm

Input: Ideal I = ⟨f1, . . . , fm⟩Output: Gröbner basis G for I1. G← ∅2. G← G ∪ {fi} for all i ∈ {1, . . . ,m}3. Set P← {(af ,bg) | f ,g ∈ G}4. d← 05. while P ̸= ∅:

(a) d← d+ 1(b) Pd ← Select (P), P← P \ Pd(c) Ld ← {af ,bg | (af ,bg) ∈ Pd}(d) Ld ← Symbolic Preprocessing(Ld,G)(e) Fd ← Reduction(Ld,G)(f) for h ∈ Fd:

▶ If lt (h) /∈ L(G) (all other h are “useless”):▷ P← P ∪ {new pairs with h}▷ G← G ∪ {h}

6. Return G

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Differences to Buchberger

1. Select a subset Pd of P, not only one element.2. Do a symbolic preprocessing:

Search and store reducers, but do not reduce.3. Do a full reduction of Pd at once:

Reduce a subset of R by a subset of R

If Select (P) selects only one pair F4 is just Buchberger’s algorithm.Usually one chooses the normal selection strategy,

i.e. all pairs of lowest degree.

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Differences to Buchberger

1. Select a subset Pd of P, not only one element.2. Do a symbolic preprocessing:

Search and store reducers, but do not reduce.3. Do a full reduction of Pd at once:

Reduce a subset of R by a subset of R

If Select (P) selects only one pair F4 is just Buchberger’s algorithm.Usually one chooses the normal selection strategy,

i.e. all pairs of lowest degree.

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Symbolic preprocessing

Input: L,G finite subsets of ROutput: a finite subset of R1. F← L2. D← L(F) (S-pairs already reduce lead terms)3. while T(F) ̸= D:

(a) Choose m ∈ T(F) \ D, D← D ∪ {m}.(b) If m ∈ L(G) ⇒ ∃ g ∈ G and λ ∈ R such that λ lt (g) = m

▷ F← F ∪ {λg}

4. Return F

We optimize this soon!

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Symbolic preprocessing

Input: L,G finite subsets of ROutput: a finite subset of R1. F← L2. D← L(F) (S-pairs already reduce lead terms)3. while T(F) ̸= D:

(a) Choose m ∈ T(F) \ D, D← D ∪ {m}.(b) If m ∈ L(G) ⇒ ∃ g ∈ G and λ ∈ R such that λ lt (g) = m

▷ F← F ∪ {λg}

4. Return F

We optimize this soon!

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Reduction

Input: L finite subsets of ROutput: a finite subset of R1. M← Macaulay matrix of L2. M← Gaussian Elimination of M (Linear algebra)3. F← polynomials from rows of M4. Return F

Macaulay matrix: columns =̂ monomials (sorted by monomial order <)rows =̂ coefficients of polynomials in L

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Reduction

Input: L finite subsets of ROutput: a finite subset of R1. M← Macaulay matrix of L2. M← Gaussian Elimination of M (Linear algebra)3. F← polynomials from rows of M4. Return F

Macaulay matrix: columns =̂ monomials (sorted by monomial order <)rows =̂ coefficients of polynomials in L

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Example: Cyclic-4

R = Q[a,b, c,d], < denotes DRL and we use the normal selection strategy for Select (P).I = ⟨f1, . . . , f4⟩, where

f1 = abcd− 1,f2 = abc+ abd+ acd+ bcd,f3 = ab+ bc+ ad+ cd,f4 = a+ b+ c+ d.

We start with G = {f4} and P1 = {(f3,bf4)}, thus L1 = {f3,bf4}.Let us do symbolic preprocessing:

T(L1) = {ab,b2,bc, ad,bd, cd

,d2

}L1 = {f3,bf4

,df4

}

b2 /∈ L(G), bc /∈ L(G), d lt (f4) = ad, all others also /∈ L(G),

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Example: Cyclic-4

R = Q[a,b, c,d], < denotes DRL and we use the normal selection strategy for Select (P).I = ⟨f1, . . . , f4⟩, where

f1 = abcd− 1,f2 = abc+ abd+ acd+ bcd,f3 = ab+ bc+ ad+ cd,f4 = a+ b+ c+ d.

We start with G = {f4} and P1 = {(f3,bf4)}, thus L1 = {f3,bf4}.

Let us do symbolic preprocessing:

T(L1) = {ab,b2,bc, ad,bd, cd

,d2

}L1 = {f3,bf4

,df4

}

b2 /∈ L(G), bc /∈ L(G), d lt (f4) = ad, all others also /∈ L(G),

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Example: Cyclic-4

R = Q[a,b, c,d], < denotes DRL and we use the normal selection strategy for Select (P).I = ⟨f1, . . . , f4⟩, where

f1 = abcd− 1,f2 = abc+ abd+ acd+ bcd,f3 = ab+ bc+ ad+ cd,f4 = a+ b+ c+ d.

We start with G = {f4} and P1 = {(f3,bf4)}, thus L1 = {f3,bf4}.Let us do symbolic preprocessing:

T(L1) = {ab,b2,bc, ad,bd, cd

,d2

}L1 = {f3,bf4

,df4

}

b2 /∈ L(G), bc /∈ L(G), d lt (f4) = ad, all others also /∈ L(G),

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Example: Cyclic-4

R = Q[a,b, c,d], < denotes DRL and we use the normal selection strategy for Select (P).I = ⟨f1, . . . , f4⟩, where

f1 = abcd− 1,f2 = abc+ abd+ acd+ bcd,f3 = ab+ bc+ ad+ cd,f4 = a+ b+ c+ d.

We start with G = {f4} and P1 = {(f3,bf4)}, thus L1 = {f3,bf4}.Let us do symbolic preprocessing:

T(L1) = {ab,b2,bc, ad,bd, cd

,d2

}L1 = {f3,bf4

,df4

}

b2 /∈ L(G),

bc /∈ L(G), d lt (f4) = ad, all others also /∈ L(G),

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Example: Cyclic-4

R = Q[a,b, c,d], < denotes DRL and we use the normal selection strategy for Select (P).I = ⟨f1, . . . , f4⟩, where

f1 = abcd− 1,f2 = abc+ abd+ acd+ bcd,f3 = ab+ bc+ ad+ cd,f4 = a+ b+ c+ d.

We start with G = {f4} and P1 = {(f3,bf4)}, thus L1 = {f3,bf4}.Let us do symbolic preprocessing:

T(L1) = {ab,b2,bc, ad,bd, cd

,d2

}L1 = {f3,bf4

,df4

}

b2 /∈ L(G), bc /∈ L(G),

d lt (f4) = ad, all others also /∈ L(G),

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Example: Cyclic-4

R = Q[a,b, c,d], < denotes DRL and we use the normal selection strategy for Select (P).I = ⟨f1, . . . , f4⟩, where

f1 = abcd− 1,f2 = abc+ abd+ acd+ bcd,f3 = ab+ bc+ ad+ cd,f4 = a+ b+ c+ d.

We start with G = {f4} and P1 = {(f3,bf4)}, thus L1 = {f3,bf4}.Let us do symbolic preprocessing:

T(L1) = {ab,b2,bc, ad,bd, cd,d2}L1 = {f3,bf4,df4}

b2 /∈ L(G), bc /∈ L(G), d lt (f4) = ad,

all others also /∈ L(G),

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Example: Cyclic-4

R = Q[a,b, c,d], < denotes DRL and we use the normal selection strategy for Select (P).I = ⟨f1, . . . , f4⟩, where

f1 = abcd− 1,f2 = abc+ abd+ acd+ bcd,f3 = ab+ bc+ ad+ cd,f4 = a+ b+ c+ d.

We start with G = {f4} and P1 = {(f3,bf4)}, thus L1 = {f3,bf4}.Let us do symbolic preprocessing:

T(L1) = {ab,b2,bc, ad,bd, cd,d2}L1 = {f3,bf4,df4}

b2 /∈ L(G), bc /∈ L(G), d lt (f4) = ad, all others also /∈ L(G),

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Example: Cyclic-4

Now reduction:Convert polynomial data L1 to Macaulay Matrix M1

0 0 0 1 1 1 1

1 0 1 1 0 1 0

1 1 1 0 1 0 0

ab b2 bc ad bd cd d2

df4f3

bf4

Gaussian Elimination of M1:

0 0 0 1 1 1 1

1 0 1 0 −1 0 −1

0 1 0 0 2 0 1

ab b2 bc ad bd cd d2

df4f3

bf4

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Example: Cyclic-4

Now reduction:Convert polynomial data L1 to Macaulay Matrix M1

0 0 0 1 1 1 1

1 0 1 1 0 1 0

1 1 1 0 1 0 0

ab b2 bc ad bd cd d2

df4f3

bf4

Gaussian Elimination of M1:

0 0 0 1 1 1 1

1 0 1 0 −1 0 −1

0 1 0 0 2 0 1

ab b2 bc ad bd cd d2

df4f3

bf4

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Example: Cyclic-4

Convert matrix data back to polynomial structure F1:

0 0 0 1 1 1 1

1 0 1 0 −1 0 −1

0 1 0 0 2 0 1

ab b2 bc ad bd cd d2

df4f3

bf4

F1 =

ad+ bd+ cd+ d2︸ ︷︷ ︸f5

, ab+ bc− bd− d2︸ ︷︷ ︸f6

,b2 + 2bd+ d2︸ ︷︷ ︸f7

lt (f5) , lt (f6) ∈ L(G), so

G← G ∪ {f7}.

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Example: Cyclic-4

Convert matrix data back to polynomial structure F1:

0 0 0 1 1 1 1

1 0 1 0 −1 0 −1

0 1 0 0 2 0 1

ab b2 bc ad bd cd d2

df4f3

bf4

F1 =

ad+ bd+ cd+ d2︸ ︷︷ ︸f5

, ab+ bc− bd− d2︸ ︷︷ ︸f6

,b2 + 2bd+ d2︸ ︷︷ ︸f7

lt (f5) , lt (f6) ∈ L(G), so

G← G ∪ {f7}.

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Example: Cyclic-4

Next round:

G = {f4, f7}, P2 = {(f2,bcf4)} , L2 = {f2,bcf4} .

We can simplify the computations:

lt (bcf4) = abc = lt (cf6) .

f6 possibly better reduced than f4. (f6 is not in G!)

=⇒ L2 = {f2, cf6}

Symbolic preprocessing:

T(L2) = {abc,bc2, abd, acd,bcd, cd2}L2 = {f2, cf6, }

bc2 /∈ L(G), abd = lt (bdf4), but also abd = lt (bf5)!

Let us investigate this in more detail.

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Example: Cyclic-4

Next round:

G = {f4, f7}, P2 = {(f2,bcf4)} , L2 = {f2,bcf4} .

We can simplify the computations:

lt (bcf4) = abc = lt (cf6) .

f6 possibly better reduced than f4. (f6 is not in G!)

=⇒ L2 = {f2, cf6}

Symbolic preprocessing:

T(L2) = {abc,bc2, abd, acd,bcd, cd2}L2 = {f2, cf6, }

bc2 /∈ L(G), abd = lt (bdf4), but also abd = lt (bf5)!

Let us investigate this in more detail.

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Example: Cyclic-4

Next round:

G = {f4, f7}, P2 = {(f2,bcf4)} , L2 = {f2,bcf4} .

We can simplify the computations:

lt (bcf4) = abc = lt (cf6) .

f6 possibly better reduced than f4. (f6 is not in G!)

=⇒ L2 = {f2, cf6}

Symbolic preprocessing:

T(L2) = {abc,bc2, abd, acd,bcd, cd2}L2 = {f2, cf6, }

bc2 /∈ L(G), abd = lt (bdf4), but also abd = lt (bf5)!

Let us investigate this in more detail.

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Example: Cyclic-4

Next round:

G = {f4, f7}, P2 = {(f2,bcf4)} , L2 = {f2,bcf4} .

We can simplify the computations:

lt (bcf4) = abc = lt (cf6) .

f6 possibly better reduced than f4. (f6 is not in G!)

=⇒ L2 = {f2, cf6}

Symbolic preprocessing:

T(L2) = {abc,bc2, abd, acd,bcd, cd2}L2 = {f2, cf6, }

bc2 /∈ L(G),

abd = lt (bdf4), but also abd = lt (bf5)!

Let us investigate this in more detail.

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Example: Cyclic-4

Next round:

G = {f4, f7}, P2 = {(f2,bcf4)} , L2 = {f2,bcf4} .

We can simplify the computations:

lt (bcf4) = abc = lt (cf6) .

f6 possibly better reduced than f4. (f6 is not in G!)

=⇒ L2 = {f2, cf6}

Symbolic preprocessing:

T(L2) = {abc,bc2, abd, acd,bcd, cd2}L2 = {f2, cf6, }

bc2 /∈ L(G), abd = lt (bdf4), but also abd = lt (bf5)!

Let us investigate this in more detail.

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Example: Cyclic-4

Next round:

G = {f4, f7}, P2 = {(f2,bcf4)} , L2 = {f2,bcf4} .

We can simplify the computations:

lt (bcf4) = abc = lt (cf6) .

f6 possibly better reduced than f4. (f6 is not in G!)

=⇒ L2 = {f2, cf6}

Symbolic preprocessing:

T(L2) = {abc,bc2, abd, acd,bcd, cd2}L2 = {f2, cf6, }

bc2 /∈ L(G), abd = lt (bdf4), but also abd = lt (bf5)!

Let us investigate this in more detail.

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Interlude – Simplify

IdeaReplace u · f by (wv) · g where vg ∈ Fi for a previous reduction step.⇒ Reuse rows that are reduced but not “in” G.

Note▶ Tries to reuse all rows from old matrices.⇒ We need to keep them in memory.

▶ We also simplify generators of S-pairs, as we have donein our example: (f2,bcf4) =⇒ (f2, cf6).

▶ One can also choose “better” reducers by other properties, not only “lastreduced one”.

In our example:Choose bf5 as reducer, not bdf4.

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Interlude – Simplify

IdeaReplace u · f by (wv) · g where vg ∈ Fi for a previous reduction step.⇒ Reuse rows that are reduced but not “in” G.

Note▶ Tries to reuse all rows from old matrices.⇒ We need to keep them in memory.

▶ We also simplify generators of S-pairs, as we have donein our example: (f2,bcf4) =⇒ (f2, cf6).

▶ One can also choose “better” reducers by other properties, not only “lastreduced one”.

In our example:Choose bf5 as reducer, not bdf4.

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Interlude – Simplify

IdeaReplace u · f by (wv) · g where vg ∈ Fi for a previous reduction step.⇒ Reuse rows that are reduced but not “in” G.

Note▶ Tries to reuse all rows from old matrices.⇒ We need to keep them in memory.

▶ We also simplify generators of S-pairs, as we have donein our example: (f2,bcf4) =⇒ (f2, cf6).

▶ One can also choose “better” reducers by other properties, not only “lastreduced one”.

In our example:Choose bf5 as reducer, not bdf4.

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Example: Cyclic-4

Symbolic preprocessing - now with simplify:

T(L2) = {abc,bc2, abd, acd,bcd, cd2

,b2d, c2d, . . .

}L2 = {f2, cf6

,bf5, cf5,df7

}

bc2 /∈ L(G),

abd = lt (bf5), and so on.

Now try to exploit the special structure of the Macaulay matrices.

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Example: Cyclic-4

Symbolic preprocessing - now with simplify:

T(L2) = {abc,bc2, abd, acd,bcd, cd2

,b2d, c2d, . . .

}L2 = {f2, cf6

,bf5, cf5,df7

}

bc2 /∈ L(G), abd = lt (bf5),

and so on.

Now try to exploit the special structure of the Macaulay matrices.

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Example: Cyclic-4

Symbolic preprocessing - now with simplify:

T(L2) = {abc,bc2, abd, acd,bcd, cd2,b2d, c2d

, . . .

}L2 = {f2, cf6,bf5

, cf5,df7

}

bc2 /∈ L(G), abd = lt (bf5),

and so on.

Now try to exploit the special structure of the Macaulay matrices.

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Example: Cyclic-4

Symbolic preprocessing - now with simplify:

T(L2) = {abc,bc2, abd, acd,bcd, cd2,b2d, c2d, . . . }L2 = {f2, cf6,bf5, cf5,df7}

bc2 /∈ L(G), abd = lt (bf5), and so on.

Now try to exploit the special structure of the Macaulay matrices.

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Example: Cyclic-4

Symbolic preprocessing - now with simplify:

T(L2) = {abc,bc2, abd, acd,bcd, cd2,b2d, c2d, . . . }L2 = {f2, cf6,bf5, cf5,df7}

bc2 /∈ L(G), abd = lt (bf5), and so on.

Now try to exploit the special structure of the Macaulay matrices.

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Specialized Linear Algebra for Gröbner Basis Computation

Idea by Faugère & Lachartre

Specialize Linear Algebra for reduction steps in GB computations.

1 3 0 0 7 1 01 0 4 1 0 0 50 1 6 0 8 0 10 1 0 0 0 7 00 0 0 0 1 3 1

S-pair

S-pair

reducer

Try to exploit underlying GB structure.

Main ideaDo a static reordering before the Gaussian Elimination to achieve

a better initial shape. Invert the reordering afterwards.

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Idea by Faugère & Lachartre

Specialize Linear Algebra for reduction steps in GB computations.

1 3 0 0 7 1 01 0 4 1 0 0 50 1 6 0 8 0 10 1 0 0 0 7 00 0 0 0 1 3 1

S-pair

S-pair

reducer

Try to exploit underlying GB structure.

Main ideaDo a static reordering before the Gaussian Elimination to achieve

a better initial shape. Invert the reordering afterwards.

17 / 50

Idea by Faugère & Lachartre

Specialize Linear Algebra for reduction steps in GB computations.

1 3 0 0 7 1 01 0 4 1 0 0 50 1 6 0 8 0 10 1 0 0 0 7 00 0 0 0 1 3 1

S-pair

S-pair

reducer

Try to exploit underlying GB structure.

Main ideaDo a static reordering before the Gaussian Elimination to achieve

a better initial shape. Invert the reordering afterwards.

17 / 50

Idea by Faugère & Lachartre

Specialize Linear Algebra for reduction steps in GB computations.

1 3 0 0 7 1 01 0 4 1 0 0 50 1 6 0 8 0 10 1 0 0 0 7 00 0 0 0 1 3 1

S-pair

S-pair

reducer

Try to exploit underlying GB structure.

Main ideaDo a static reordering before the Gaussian Elimination to achieve

a better initial shape. Invert the reordering afterwards.

17 / 50

Idea by Faugère & Lachartre

Specialize Linear Algebra for reduction steps in GB computations.

1 3 0 0 7 1 01 0 4 1 0 0 50 1 6 0 8 0 10 1 0 0 0 7 00 0 0 0 1 3 1

S-pair

S-pair

reducer

Try to exploit underlying GB structure.

Main ideaDo a static reordering before the Gaussian Elimination to achieve

a better initial shape. Invert the reordering afterwards.

17 / 50

Idea by Faugère & Lachartre

Specialize Linear Algebra for reduction steps in GB computations.

1 3 0 0 7 1 01 0 4 1 0 0 50 1 6 0 8 0 10 1 0 0 0 7 00 0 0 0 1 3 1

S-pair

S-pair

reducer

Try to exploit underlying GB structure.

Main ideaDo a static reordering before the Gaussian Elimination to achieve

a better initial shape. Invert the reordering afterwards.

17 / 50

Idea by Faugère & Lachartre

Specialize Linear Algebra for reduction steps in GB computations.

1 3 0 0 7 1 01 0 4 1 0 0 50 1 6 0 8 0 10 1 0 0 0 7 00 0 0 0 1 3 1

S-pair

S-pair

reducer

Try to exploit underlying GB structure.

Main ideaDo a static reordering before the Gaussian Elimination to achieve

a better initial shape. Invert the reordering afterwards.

17 / 50

Idea by Faugère & Lachartre

1st step: Sort pivot and non-pivot columns

1 3 0 0 7 1 01 0 4 1 0 0 50 1 6 0 8 0 10 1 0 0 0 7 00 0 0 0 1 3 1

Pivot column Non-Pivot column

1 3 7 0 0 1 01 0 0 4 1 0 50 1 8 6 0 0 90 1 0 0 0 7 00 0 1 0 0 3 1

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Idea by Faugère & Lachartre

1st step: Sort pivot and non-pivot columns

1 3 0 0 7 1 01 0 4 1 0 0 50 1 6 0 8 0 10 1 0 0 0 7 00 0 0 0 1 3 1

Pivot column

Non-Pivot column

1 3 7 0 0 1 01 0 0 4 1 0 50 1 8 6 0 0 90 1 0 0 0 7 00 0 1 0 0 3 1

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Idea by Faugère & Lachartre

1st step: Sort pivot and non-pivot columns

1 3 0 0 7 1 01 0 4 1 0 0 50 1 6 0 8 0 10 1 0 0 0 7 00 0 0 0 1 3 1

Pivot column

Non-Pivot column

1 3 7 0 0 1 01 0 0 4 1 0 50 1 8 6 0 0 90 1 0 0 0 7 00 0 1 0 0 3 1

18 / 50

Idea by Faugère & Lachartre

1st step: Sort pivot and non-pivot columns

1 3 0 0 7 1 01 0 4 1 0 0 50 1 6 0 8 0 10 1 0 0 0 7 00 0 0 0 1 3 1

Pivot column Non-Pivot column

1 3 7 0 0 1 01 0 0 4 1 0 50 1 8 6 0 0 90 1 0 0 0 7 00 0 1 0 0 3 1

18 / 50

Idea by Faugère & Lachartre

1st step: Sort pivot and non-pivot columns

1 3 0 0 7 1 01 0 4 1 0 0 50 1 6 0 8 0 10 1 0 0 0 7 00 0 0 0 1 3 1

Pivot column Non-Pivot column

1 3 7 0 0 1 01 0 0 4 1 0 50 1 8 6 0 0 90 1 0 0 0 7 00 0 1 0 0 3 1

18 / 50

Idea by Faugère & Lachartre

1st step: Sort pivot and non-pivot columns

1 3 0 0 7 1 01 0 4 1 0 0 50 1 6 0 8 0 10 1 0 0 0 7 00 0 0 0 1 3 1

Pivot column Non-Pivot column

1 3 7 0 0 1 01 0 0 4 1 0 50 1 8 6 0 0 90 1 0 0 0 7 00 0 1 0 0 3 1

18 / 50

Idea by Faugère & Lachartre

2nd step: Sort pivot and non-pivot rows

1 3 7 0 0 1 01 0 0 4 1 0 50 1 8 6 0 0 90 1 0 0 0 7 00 0 1 0 0 3 1

Pivot row Non-Pivot row

1 0 0 4 1 0 50 1 0 0 0 7 00 0 1 0 0 3 11 3 7 0 0 1 00 1 8 6 0 0 9

19 / 50

Idea by Faugère & Lachartre

2nd step: Sort pivot and non-pivot rows

1 3 7 0 0 1 01 0 0 4 1 0 50 1 8 6 0 0 90 1 0 0 0 7 00 0 1 0 0 3 1

Pivot row

Non-Pivot row

1 0 0 4 1 0 50 1 0 0 0 7 00 0 1 0 0 3 11 3 7 0 0 1 00 1 8 6 0 0 9

19 / 50

Idea by Faugère & Lachartre

2nd step: Sort pivot and non-pivot rows

1 3 7 0 0 1 01 0 0 4 1 0 50 1 8 6 0 0 90 1 0 0 0 7 00 0 1 0 0 3 1

Pivot row Non-Pivot row

1 0 0 4 1 0 50 1 0 0 0 7 00 0 1 0 0 3 11 3 7 0 0 1 00 1 8 6 0 0 9

19 / 50

Idea by Faugère & Lachartre

2nd step: Sort pivot and non-pivot rows

1 3 7 0 0 1 01 0 0 4 1 0 50 1 8 6 0 0 90 1 0 0 0 7 00 0 1 0 0 3 1

Pivot row Non-Pivot row

1 0 0 4 1 0 50 1 0 0 0 7 00 0 1 0 0 3 11 3 7 0 0 1 00 1 8 6 0 0 9

19 / 50

Idea by Faugère & Lachartre

2nd step: Sort pivot and non-pivot rows

1 3 7 0 0 1 01 0 0 4 1 0 50 1 8 6 0 0 90 1 0 0 0 7 00 0 1 0 0 3 1

Pivot row Non-Pivot row

1 0 0 4 1 0 50 1 0 0 0 7 00 0 1 0 0 3 11 3 7 0 0 1 00 1 8 6 0 0 9

19 / 50

Idea by Faugère & Lachartre

3rd step: Reduce lower left part to zero

1 0 0 4 1 0 50 1 0 0 0 7 00 0 1 0 0 3 11 3 7 0 0 1 00 1 8 6 0 0 9

1 0 0 4 1 0 50 1 0 0 0 7 00 0 1 0 0 3 10 0 0 7 10 3 100 0 0 6 0 2 1

20 / 50

Idea by Faugère & Lachartre

3rd step: Reduce lower left part to zero

1 0 0 4 1 0 50 1 0 0 0 7 00 0 1 0 0 3 11 3 7 0 0 1 00 1 8 6 0 0 9

1 0 0 4 1 0 50 1 0 0 0 7 00 0 1 0 0 3 10 0 0 7 10 3 100 0 0 6 0 2 1

20 / 50

Idea by Faugère & Lachartre

4th step: Reduce lower right part

1 0 0 4 1 0 50 1 0 0 0 7 00 0 1 0 0 3 10 0 0 7 10 3 100 0 0 6 0 2 1

1 0 0 4 1 0 50 1 0 0 0 7 00 0 1 0 0 3 10 0 0 7 0 6 30 0 0 0 4 1 5

5th step: Remap columns and get new polynomials for GB out of lower right part.

21 / 50

Idea by Faugère & Lachartre

4th step: Reduce lower right part

1 0 0 4 1 0 50 1 0 0 0 7 00 0 1 0 0 3 10 0 0 7 10 3 100 0 0 6 0 2 1

1 0 0 4 1 0 50 1 0 0 0 7 00 0 1 0 0 3 10 0 0 7 0 6 30 0 0 0 4 1 5

5th step: Remap columns and get new polynomials for GB out of lower right part.

21 / 50

Idea by Faugère & Lachartre

4th step: Reduce lower right part

1 0 0 4 1 0 50 1 0 0 0 7 00 0 1 0 0 3 10 0 0 7 10 3 100 0 0 6 0 2 1

1 0 0 4 1 0 50 1 0 0 0 7 00 0 1 0 0 3 10 0 0 7 0 6 30 0 0 0 4 1 5

5th step: Remap columns and get new polynomials for GB out of lower right part.

21 / 50

“Real world” matrices?

What our matrices look like

23 / 50

What our matrices look like

Characteristics of this matrix

▶ F4 computation of homogeneous Katsura-12, degree 6 matrix

▶ Size 55MB

▶ 24,006,869 nonzero elements (density: 5%)

▶ Dimensions:full matrix: 21,182 × 22,207

upper-left: 17,915 × 17,915 known pivotslower-left: 3,267 × 17,915

upper-right: 17,915 × 4,292lower-right: 3,267 × 4,292 new information

24 / 50

What our matrices look like

Characteristics of this matrix

▶ F4 computation of homogeneous Katsura-12, degree 6 matrix

▶ Size 55MB

▶ 24,006,869 nonzero elements (density: 5%)

▶ Dimensions:full matrix: 21,182 × 22,207

upper-left: 17,915 × 17,915 known pivotslower-left: 3,267 × 17,915

upper-right: 17,915 × 4,292lower-right: 3,267 × 4,292 new information

24 / 50

What our matrices look like

Characteristics of this matrix

▶ F4 computation of homogeneous Katsura-12, degree 6 matrix

▶ Size 55MB

▶ 24,006,869 nonzero elements (density: 5%)

▶ Dimensions:full matrix: 21,182 × 22,207

upper-left: 17,915 × 17,915 known pivotslower-left: 3,267 × 17,915

upper-right: 17,915 × 4,292lower-right: 3,267 × 4,292 new information

24 / 50

What our matrices look like

Characteristics of this matrix

▶ F4 computation of homogeneous Katsura-12, degree 6 matrix

▶ Size 55MB

▶ 24,006,869 nonzero elements (density: 5%)

▶ Dimensions:full matrix: 21,182 × 22,207

upper-left: 17,915 × 17,915 known pivotslower-left: 3,267 × 17,915

upper-right: 17,915 × 4,292lower-right: 3,267 × 4,292 new information

24 / 50

What our matrices look like

25 / 50

What our matrices look like

26 / 50

Hybrid Matrix Multiplication A-1B

27 / 50

Hybrid Matrix Multiplication A-1B

28 / 50

Reduce C to zero

29 / 50

Gaussian Elimination on D

30 / 50

New information

31 / 50

GBLA – A Gröbner Basis Linear Algebra Library

Library Overview

▶ Open source library written in plain C.

▶ Specialized linear algebra for GB computations.

▶ Parallel implementation (OpenMP), scaling “nicely” up to 32 cores.

▶ Works over finite fields for 16-bit primes (at the moment).

▶ Several strategies for splicing and reduction.

▶ Includes converter from and to our dedicated matrix format.

▶ Access to huge matrix database: > 500 matrices, > 280GB of data.

http://hpac.imag.fr/gbla

https://www.github.com/ederc/gbla

33 / 50

Library Overview

▶ Open source library written in plain C.

▶ Specialized linear algebra for GB computations.

▶ Parallel implementation (OpenMP), scaling “nicely” up to 32 cores.

▶ Works over finite fields for 16-bit primes (at the moment).

▶ Several strategies for splicing and reduction.

▶ Includes converter from and to our dedicated matrix format.

▶ Access to huge matrix database: > 500 matrices, > 280GB of data.

http://hpac.imag.fr/gbla

https://www.github.com/ederc/gbla

33 / 50

Library Overview

▶ Open source library written in plain C.

▶ Specialized linear algebra for GB computations.

▶ Parallel implementation (OpenMP), scaling “nicely” up to 32 cores.

▶ Works over finite fields for 16-bit primes (at the moment).

▶ Several strategies for splicing and reduction.

▶ Includes converter from and to our dedicated matrix format.

▶ Access to huge matrix database: > 500 matrices, > 280GB of data.

http://hpac.imag.fr/gbla

https://www.github.com/ederc/gbla

33 / 50

Library Overview

▶ Open source library written in plain C.

▶ Specialized linear algebra for GB computations.

▶ Parallel implementation (OpenMP), scaling “nicely” up to 32 cores.

▶ Works over finite fields for 16-bit primes (at the moment).

▶ Several strategies for splicing and reduction.

▶ Includes converter from and to our dedicated matrix format.

▶ Access to huge matrix database: > 500 matrices, > 280GB of data.

http://hpac.imag.fr/gbla

https://www.github.com/ederc/gbla

33 / 50

Library Overview

▶ Open source library written in plain C.

▶ Specialized linear algebra for GB computations.

▶ Parallel implementation (OpenMP), scaling “nicely” up to 32 cores.

▶ Works over finite fields for 16-bit primes (at the moment).

▶ Several strategies for splicing and reduction.

▶ Includes converter from and to our dedicated matrix format.

▶ Access to huge matrix database: > 500 matrices, > 280GB of data.

http://hpac.imag.fr/gbla

https://www.github.com/ederc/gbla

33 / 50

Library Overview

▶ Open source library written in plain C.

▶ Specialized linear algebra for GB computations.

▶ Parallel implementation (OpenMP), scaling “nicely” up to 32 cores.

▶ Works over finite fields for 16-bit primes (at the moment).

▶ Several strategies for splicing and reduction.

▶ Includes converter from and to our dedicated matrix format.

▶ Access to huge matrix database: > 500 matrices, > 280GB of data.

http://hpac.imag.fr/gbla

https://www.github.com/ederc/gbla

33 / 50

Library Overview

▶ Open source library written in plain C.

▶ Specialized linear algebra for GB computations.

▶ Parallel implementation (OpenMP), scaling “nicely” up to 32 cores.

▶ Works over finite fields for 16-bit primes (at the moment).

▶ Several strategies for splicing and reduction.

▶ Includes converter from and to our dedicated matrix format.

▶ Access to huge matrix database: > 500 matrices, > 280GB of data.

http://hpac.imag.fr/gbla

https://www.github.com/ederc/gbla

33 / 50

Library Overview

▶ Open source library written in plain C.

▶ Specialized linear algebra for GB computations.

▶ Parallel implementation (OpenMP), scaling “nicely” up to 32 cores.

▶ Works over finite fields for 16-bit primes (at the moment).

▶ Several strategies for splicing and reduction.

▶ Includes converter from and to our dedicated matrix format.

▶ Access to huge matrix database: > 500 matrices, > 280GB of data.

http://hpac.imag.fr/gbla

https://www.github.com/ederc/gbla

33 / 50

Exploiting block structures in GB matrices

34 / 50

Exploiting block structures in GB matrices

Matrices from GB computations have nonzero entries often grouped in blocks.

Horizontal Pattern If mi,j ̸= 0 then often mi,j+1 ̸= 0.

Vertical Pattern If mi,j ̸= 0 then often mi+1,j ̸= 0.

▶ Can be used to optimize AXPY and TRSM operations in FL reduction.

▶ Horizontal pattern taken care of canonically.

▶ Need to take care of vertical pattern.

35 / 50

Exploiting block structures in GB matrices

Matrices from GB computations have nonzero entries often grouped in blocks.

Horizontal Pattern If mi,j ̸= 0 then often mi,j+1 ̸= 0.Vertical Pattern If mi,j ̸= 0 then often mi+1,j ̸= 0.

▶ Can be used to optimize AXPY and TRSM operations in FL reduction.

▶ Horizontal pattern taken care of canonically.

▶ Need to take care of vertical pattern.

35 / 50

Exploiting block structures in GB matrices

Matrices from GB computations have nonzero entries often grouped in blocks.

Horizontal Pattern If mi,j ̸= 0 then often mi,j+1 ̸= 0.Vertical Pattern If mi,j ̸= 0 then often mi+1,j ̸= 0.

▶ Can be used to optimize AXPY and TRSM operations in FL reduction.

▶ Horizontal pattern taken care of canonically.

▶ Need to take care of vertical pattern.

35 / 50

Exploiting block structures in GB matrices

Matrices from GB computations have nonzero entries often grouped in blocks.

Horizontal Pattern If mi,j ̸= 0 then often mi,j+1 ̸= 0.Vertical Pattern If mi,j ̸= 0 then often mi+1,j ̸= 0.

▶ Can be used to optimize AXPY and TRSM operations in FL reduction.

▶ Horizontal pattern taken care of canonically.

▶ Need to take care of vertical pattern.

35 / 50

Exploiting block structures in GB matrices

Matrices from GB computations have nonzero entries often grouped in blocks.

Horizontal Pattern If mi,j ̸= 0 then often mi,j+1 ̸= 0.Vertical Pattern If mi,j ̸= 0 then often mi+1,j ̸= 0.

▶ Can be used to optimize AXPY and TRSM operations in FL reduction.

▶ Horizontal pattern taken care of canonically.

▶ Need to take care of vertical pattern.

35 / 50

Exploiting block structures in GB matrices

. . . ......

......

......

1 · · · ∗ ∗ ∗ ∗1 · · · ai,j ai,j+1 ∗. . . ...

......

1 · · · ∗1 ∗

1

A......

......

......

...∗ ∗ · · · ∗ · · · ∗ ∗∗ ∗ · · · bi,k ∗ · · · ∗......

......

......

...∗ ∗ · · · bj,k ∗ · · · ∗∗ ∗ · · · bj+1,k ∗ · · · ∗∗ ∗ · · · ∗ · · · ∗ ∗

B

Exploiting horizontal and vertical patterns in the TRSM step.

36 / 50

Multiline data structure – an example

Consider the following two rows:

r1 = [ 2 3 0 1 4 0 5 ],r2 = [ 1 7 0 0 3 1 2 ].

A sparse vector representation of the two rows would be given by

r1.val = [ 2 3 1 4 5 ],r1.pos = [ 0 1 3 4 6 ],

r2.val = [ 1 7 3 1 2 ],r2.pos = [ 0 1 4 5 6 ].

A multiline vector representation of r1 and r2 is given by

ml.val = [ 2 1 3 7 1 0 4 3 0 1 5 2 ],ml.pos = [ 0 1 3 4 5 6 ].

37 / 50

Multiline data structure – an example

Consider the following two rows:

r1 = [ 2 3 0 1 4 0 5 ],r2 = [ 1 7 0 0 3 1 2 ].

A sparse vector representation of the two rows would be given by

r1.val = [ 2 3 1 4 5 ],r1.pos = [ 0 1 3 4 6 ],

r2.val = [ 1 7 3 1 2 ],r2.pos = [ 0 1 4 5 6 ].

A multiline vector representation of r1 and r2 is given by

ml.val = [ 2 1 3 7 1 0 4 3 0 1 5 2 ],ml.pos = [ 0 1 3 4 5 6 ].

37 / 50

Multiline data structure – an example

Consider the following two rows:

r1 = [ 2 3 0 1 4 0 5 ],r2 = [ 1 7 0 0 3 1 2 ].

A sparse vector representation of the two rows would be given by

r1.val = [ 2 3 1 4 5 ],r1.pos = [ 0 1 3 4 6 ],

r2.val = [ 1 7 3 1 2 ],r2.pos = [ 0 1 4 5 6 ].

A multiline vector representation of r1 and r2 is given by

ml.val = [ 2 1 3 7 1 0 4 3 0 1 5 2 ],ml.pos = [ 0 1 3 4 5 6 ].

37 / 50

Multiline data structure – an example

Consider the following two rows:

r1 = [ 2 3 0 1 4 0 5 ],r2 = [ 1 7 0 0 3 1 2 ].

A sparse vector representation of the two rows would be given by

r1.val = [ 2 3 1 4 5 ],r1.pos = [ 0 1 3 4 6 ],

r2.val = [ 1 7 3 1 2 ],r2.pos = [ 0 1 4 5 6 ].

A multiline vector representation of r1 and r2 is given by

ml.val = [ 2 1 3 7 1 0 4 3 0 1 5 2 ],ml.pos = [ 0 1 3 4 5 6 ].

37 / 50

New order of operations

▶ Number of initially known pivots (i.e. # rows of A and B) is large compared to # rows ofC and D.

▶ Most time of FL reduction is spent in TRSM step A−1B.

▶ Only interested in D resp. rank of M?

Change order of operations

1. Reduce C directly with A (store corresponding data in C).

2. Carry out corresponding operations from B to D using updated C.

3. Reduce D.

This leads to reduced matrices that keep A and B untouched,i.e. the computation has a smaller memory footprint.

38 / 50

New order of operations

▶ Number of initially known pivots (i.e. # rows of A and B) is large compared to # rows ofC and D.

▶ Most time of FL reduction is spent in TRSM step A−1B.

▶ Only interested in D resp. rank of M?

Change order of operations

1. Reduce C directly with A (store corresponding data in C).

2. Carry out corresponding operations from B to D using updated C.

3. Reduce D.

This leads to reduced matrices that keep A and B untouched,i.e. the computation has a smaller memory footprint.

38 / 50

New order of operations

▶ Number of initially known pivots (i.e. # rows of A and B) is large compared to # rows ofC and D.

▶ Most time of FL reduction is spent in TRSM step A−1B.

▶ Only interested in D resp. rank of M?

Change order of operations

1. Reduce C directly with A (store corresponding data in C).

2. Carry out corresponding operations from B to D using updated C.

3. Reduce D.

This leads to reduced matrices that keep A and B untouched,i.e. the computation has a smaller memory footprint.

38 / 50

New order of operations

▶ Number of initially known pivots (i.e. # rows of A and B) is large compared to # rows ofC and D.

▶ Most time of FL reduction is spent in TRSM step A−1B.

▶ Only interested in D resp. rank of M?

Change order of operations

1. Reduce C directly with A (store corresponding data in C).

2. Carry out corresponding operations from B to D using updated C.

3. Reduce D.

This leads to reduced matrices that keep A and B untouched,i.e. the computation has a smaller memory footprint.

38 / 50

New order of operations

▶ Number of initially known pivots (i.e. # rows of A and B) is large compared to # rows ofC and D.

▶ Most time of FL reduction is spent in TRSM step A−1B.

▶ Only interested in D resp. rank of M?

Change order of operations

1. Reduce C directly with A (store corresponding data in C).

2. Carry out corresponding operations from B to D using updated C.

3. Reduce D.

This leads to reduced matrices that keep A and B untouched,i.e. the computation has a smaller memory footprint.

38 / 50

New order of operations

▶ Number of initially known pivots (i.e. # rows of A and B) is large compared to # rows ofC and D.

▶ Most time of FL reduction is spent in TRSM step A−1B.

▶ Only interested in D resp. rank of M?

Change order of operations

1. Reduce C directly with A (store corresponding data in C).

2. Carry out corresponding operations from B to D using updated C.

3. Reduce D.

This leads to reduced matrices that keep A and B untouched,i.e. the computation has a smaller memory footprint.

38 / 50

New order of operations

▶ Number of initially known pivots (i.e. # rows of A and B) is large compared to # rows ofC and D.

▶ Most time of FL reduction is spent in TRSM step A−1B.

▶ Only interested in D resp. rank of M?

Change order of operations

1. Reduce C directly with A (store corresponding data in C).

2. Carry out corresponding operations from B to D using updated C.

3. Reduce D.

This leads to reduced matrices that keep A and B untouched,i.e. the computation has a smaller memory footprint.

38 / 50

New order of operations

▶ Number of initially known pivots (i.e. # rows of A and B) is large compared to # rows ofC and D.

▶ Most time of FL reduction is spent in TRSM step A−1B.

▶ Only interested in D resp. rank of M?

Change order of operations

1. Reduce C directly with A (store corresponding data in C).

2. Carry out corresponding operations from B to D using updated C.

3. Reduce D.

This leads to reduced matrices that keep A and B untouched,i.e. the computation has a smaller memory footprint.

38 / 50

GBLA Matrix formats

▶ Matrices are pretty sparse, but structured.

▶ GBLA supports two matrix formats, both use binary format.

▶ GBLA includes a converter between the two supported formats and can also dump toMagma matrix format.

Table: Old matrix format (legacy version)

Size Length Data Descriptionuint32_t 1 b version numberuint32_t 1 m # rowsuint32_t 1 n # columnsuint32_t 1 p prime / field characteristicuint64_t 1 nnz # nonzero entriesuint16_t nnz data entry in matrixuint32_t nnz cols column index of entryuint32_t m rows length of rows

39 / 50

GBLA Matrix formats

▶ Matrices are pretty sparse, but structured.

▶ GBLA supports two matrix formats, both use binary format.

▶ GBLA includes a converter between the two supported formats and can also dump toMagma matrix format.

Table: Old matrix format (legacy version)

Size Length Data Descriptionuint32_t 1 b version numberuint32_t 1 m # rowsuint32_t 1 n # columnsuint32_t 1 p prime / field characteristicuint64_t 1 nnz # nonzero entriesuint16_t nnz data entry in matrixuint32_t nnz cols column index of entryuint32_t m rows length of rows

39 / 50

GBLA Matrix formats

▶ Matrices are pretty sparse, but structured.

▶ GBLA supports two matrix formats, both use binary format.

▶ GBLA includes a converter between the two supported formats and can also dump toMagma matrix format.

Table: Old matrix format (legacy version)

Size Length Data Descriptionuint32_t 1 b version numberuint32_t 1 m # rowsuint32_t 1 n # columnsuint32_t 1 p prime / field characteristicuint64_t 1 nnz # nonzero entriesuint16_t nnz data entry in matrixuint32_t nnz cols column index of entryuint32_t m rows length of rows

39 / 50

GBLA Matrix formats

▶ Matrices are pretty sparse, but structured.

▶ GBLA supports two matrix formats, both use binary format.

▶ GBLA includes a converter between the two supported formats and can also dump toMagma matrix format.

Table: Old matrix format (legacy version)

Size Length Data Descriptionuint32_t 1 b version numberuint32_t 1 m # rowsuint32_t 1 n # columnsuint32_t 1 p prime / field characteristicuint64_t 1 nnz # nonzero entriesuint16_t nnz data entry in matrixuint32_t nnz cols column index of entryuint32_t m rows length of rows

39 / 50

GBLA Matrix formats

Table: New matrix format (compressing data and cols)

Size Length Data Descriptionuint32_t 1 b version number + information for data type of pdatauint32_t 1 m # rowsuint32_t 1 n # columnsuint32_t 1 p prime / field characteristicuint64_t 1 nnz # nonzero entriesuint16_t nnz data several rows are of type xifjuint32_t nnz cols can be compressed for consecutive elementsuint32_t m rows length of rowsuint32_t m pmap maps rows to pdata

uint64_t 1 k size of compressed colid

uint64_t k colid compression of columns:Single column entry masked via (1 << 31);s consecutive entries starting at column c are stored as “c s”

uint32_t 1 pnb # polynomialsuint64_t 1 pnnz # nonzero coefficients in polynomialsuint32_t pnb prow length of polynomial / row representationxinty_t pnnz pdata coefficients of polynomials

Approximately 1/3rd of memory usage compared to the old format.

40 / 50

GBLA Matrix formats

Table: New matrix format (compressing data and cols)

Size Length Data Descriptionuint32_t 1 b version number + information for data type of pdatauint32_t 1 m # rowsuint32_t 1 n # columnsuint32_t 1 p prime / field characteristicuint64_t 1 nnz # nonzero entriesuint16_t nnz data several rows are of type xifjuint32_t nnz cols can be compressed for consecutive elementsuint32_t m rows length of rowsuint32_t m pmap maps rows to pdata

uint64_t 1 k size of compressed colid

uint64_t k colid compression of columns:Single column entry masked via (1 << 31);s consecutive entries starting at column c are stored as “c s”

uint32_t 1 pnb # polynomialsuint64_t 1 pnnz # nonzero coefficients in polynomialsuint32_t pnb prow length of polynomial / row representationxinty_t pnnz pdata coefficients of polynomials

Approximately 1/3rd of memory usage compared to the old format.

40 / 50

GBLA Matrix formats – Comparison

Table: Storage and time efficiency of the new formatMatrix Size old Size new gzipped old gzipped new Time old Time new

F4-kat14-mat9 2.3GB 0.74GB 1.2GB 0.29GB 230s 66sF5-kat17-mat10 43GB 12GB 24GB 5.3GB 4419s 883s

New format vs. Old format

▶ 1/3rd of memory usage.

▶ 1/4th of memory usage when compressed with gzip.

▶ Compression 4− 5 times faster.

41 / 50

GBLA Matrix formats – Comparison

Table: Storage and time efficiency of the new formatMatrix Size old Size new gzipped old gzipped new Time old Time new

F4-kat14-mat9 2.3GB 0.74GB 1.2GB 0.29GB 230s 66sF5-kat17-mat10 43GB 12GB 24GB 5.3GB 4419s 883s

New format vs. Old format

▶ 1/3rd of memory usage.

▶ 1/4th of memory usage when compressed with gzip.

▶ Compression 4− 5 times faster.

41 / 50

GBLA Matrix formats – Comparison

Table: Storage and time efficiency of the new formatMatrix Size old Size new gzipped old gzipped new Time old Time new

F4-kat14-mat9 2.3GB 0.74GB 1.2GB 0.29GB 230s 66sF5-kat17-mat10 43GB 12GB 24GB 5.3GB 4419s 883s

New format vs. Old format

▶ 1/3rd of memory usage.

▶ 1/4th of memory usage when compressed with gzip.

▶ Compression 4− 5 times faster.

41 / 50

GBLA Matrix formats – Comparison

Table: Storage and time efficiency of the new formatMatrix Size old Size new gzipped old gzipped new Time old Time new

F4-kat14-mat9 2.3GB 0.74GB 1.2GB 0.29GB 230s 66sF5-kat17-mat10 43GB 12GB 24GB 5.3GB 4419s 883s

New format vs. Old format

▶ 1/3rd of memory usage.

▶ 1/4th of memory usage when compressed with gzip.

▶ Compression 4− 5 times faster.

41 / 50

GBLA Matrix formats – Comparison

Table: Storage and time efficiency of the new formatMatrix Size old Size new gzipped old gzipped new Time old Time new

F4-kat14-mat9 2.3GB 0.74GB 1.2GB 0.29GB 230s 66sF5-kat17-mat10 43GB 12GB 24GB 5.3GB 4419s 883s

New format vs. Old format

▶ 1/3rd of memory usage.

▶ 1/4th of memory usage when compressed with gzip.

▶ Compression 4− 5 times faster.

41 / 50

GB – A Gröbner Basis Library

Library Overview

▶ Open source library written in plain C.

▶ Uses GBLA for linear algebra part.

▶ Parallel matrix construction based on OpenMP.

▶ Works over finite fields (restrictions due to GBLA).

▶ DRL and LEX monomial ordering.

▶ Several strategies for simplification.

▶ Available as alpha version in Singular after 4-0-3 release.

https://www.github.com/ederc/gb

43 / 50

Library Overview

▶ Open source library written in plain C.

▶ Uses GBLA for linear algebra part.

▶ Parallel matrix construction based on OpenMP.

▶ Works over finite fields (restrictions due to GBLA).

▶ DRL and LEX monomial ordering.

▶ Several strategies for simplification.

▶ Available as alpha version in Singular after 4-0-3 release.

https://www.github.com/ederc/gb

43 / 50

Library Overview

▶ Open source library written in plain C.

▶ Uses GBLA for linear algebra part.

▶ Parallel matrix construction based on OpenMP.

▶ Works over finite fields (restrictions due to GBLA).

▶ DRL and LEX monomial ordering.

▶ Several strategies for simplification.

▶ Available as alpha version in Singular after 4-0-3 release.

https://www.github.com/ederc/gb

43 / 50

Library Overview

▶ Open source library written in plain C.

▶ Uses GBLA for linear algebra part.

▶ Parallel matrix construction based on OpenMP.

▶ Works over finite fields (restrictions due to GBLA).

▶ DRL and LEX monomial ordering.

▶ Several strategies for simplification.

▶ Available as alpha version in Singular after 4-0-3 release.

https://www.github.com/ederc/gb

43 / 50

Library Overview

▶ Open source library written in plain C.

▶ Uses GBLA for linear algebra part.

▶ Parallel matrix construction based on OpenMP.

▶ Works over finite fields (restrictions due to GBLA).

▶ DRL and LEX monomial ordering.

▶ Several strategies for simplification.

▶ Available as alpha version in Singular after 4-0-3 release.

https://www.github.com/ederc/gb

43 / 50

Library Overview

▶ Open source library written in plain C.

▶ Uses GBLA for linear algebra part.

▶ Parallel matrix construction based on OpenMP.

▶ Works over finite fields (restrictions due to GBLA).

▶ DRL and LEX monomial ordering.

▶ Several strategies for simplification.

▶ Available as alpha version in Singular after 4-0-3 release.

https://www.github.com/ederc/gb

43 / 50

Library Overview

▶ Open source library written in plain C.

▶ Uses GBLA for linear algebra part.

▶ Parallel matrix construction based on OpenMP.

▶ Works over finite fields (restrictions due to GBLA).

▶ DRL and LEX monomial ordering.

▶ Several strategies for simplification.

▶ Available as alpha version in Singular after 4-0-3 release.

https://www.github.com/ederc/gb

43 / 50

Library Overview

▶ Open source library written in plain C.

▶ Uses GBLA for linear algebra part.

▶ Parallel matrix construction based on OpenMP.

▶ Works over finite fields (restrictions due to GBLA).

▶ DRL and LEX monomial ordering.

▶ Several strategies for simplification.

▶ Available as alpha version in Singular after 4-0-3 release.

https://www.github.com/ederc/gb

43 / 50

Katsura–n w.r.t. DRL (single-threaded)

n

Time (log 2) in seconds

12 13 14 150

2

4

6

8

10

12

14

16

Singular 4-0-3FGb v1.68GB v0.1 w/ GBLA v0.2

44 / 50

Cyclic–n w.r.t. DRL (single-threaded)

n

Time (log 2) in seconds

7 8 9 100

2

4

6

8

10

12

14

16

Singular 4-0-3FGb v1.68GB v0.1 w/ GBLA v0.2

Magma v2.21 / Maple 2016

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Cyclic–n w.r.t. DRL (single-threaded)

n

Time (log 2) in seconds

7 8 9 100

2

4

6

8

10

12

14

16

Singular 4-0-3FGb v1.68GB v0.1 w/ GBLA v0.2Magma v2.21 / Maple 2016

45 / 50

Cyclic–10 w.r.t. DRL (n-threads)

n

Time (log 2) in seconds

1 2 4 8 120

2

4

6

8

10

12

14

16

GB v0.1 w/ GBLA v0.2Maple 2016

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Next steps

▶ v0.3 of GBLA.

▶ Optimizing GBLA for floating point and 32-bit unsigned int arithmetic.

▶ Multi-modular computation in GB.

▶ Parallel hashing in GB.

▶ Better handling of memory bounds in GB/GBLA.

▶ FGLM in GB for zero-dimensional system solving.

▶ First steps exploiting heterogeneous CPU/GPU platforms for GBLA.

▶ Deeper investigation on parallelization on networks.

47 / 50

Next steps

▶ v0.3 of GBLA.

▶ Optimizing GBLA for floating point and 32-bit unsigned int arithmetic.

▶ Multi-modular computation in GB.

▶ Parallel hashing in GB.

▶ Better handling of memory bounds in GB/GBLA.

▶ FGLM in GB for zero-dimensional system solving.

▶ First steps exploiting heterogeneous CPU/GPU platforms for GBLA.

▶ Deeper investigation on parallelization on networks.

47 / 50

Next steps

▶ v0.3 of GBLA.

▶ Optimizing GBLA for floating point and 32-bit unsigned int arithmetic.

▶ Multi-modular computation in GB.

▶ Parallel hashing in GB.

▶ Better handling of memory bounds in GB/GBLA.

▶ FGLM in GB for zero-dimensional system solving.

▶ First steps exploiting heterogeneous CPU/GPU platforms for GBLA.

▶ Deeper investigation on parallelization on networks.

47 / 50

Next steps

▶ v0.3 of GBLA.

▶ Optimizing GBLA for floating point and 32-bit unsigned int arithmetic.

▶ Multi-modular computation in GB.

▶ Parallel hashing in GB.

▶ Better handling of memory bounds in GB/GBLA.

▶ FGLM in GB for zero-dimensional system solving.

▶ First steps exploiting heterogeneous CPU/GPU platforms for GBLA.

▶ Deeper investigation on parallelization on networks.

47 / 50

Next steps

▶ v0.3 of GBLA.

▶ Optimizing GBLA for floating point and 32-bit unsigned int arithmetic.

▶ Multi-modular computation in GB.

▶ Parallel hashing in GB.

▶ Better handling of memory bounds in GB/GBLA.

▶ FGLM in GB for zero-dimensional system solving.

▶ First steps exploiting heterogeneous CPU/GPU platforms for GBLA.

▶ Deeper investigation on parallelization on networks.

47 / 50

Next steps

▶ v0.3 of GBLA.

▶ Optimizing GBLA for floating point and 32-bit unsigned int arithmetic.

▶ Multi-modular computation in GB.

▶ Parallel hashing in GB.

▶ Better handling of memory bounds in GB/GBLA.

▶ FGLM in GB for zero-dimensional system solving.

▶ First steps exploiting heterogeneous CPU/GPU platforms for GBLA.

▶ Deeper investigation on parallelization on networks.

47 / 50

Next steps

▶ v0.3 of GBLA.

▶ Optimizing GBLA for floating point and 32-bit unsigned int arithmetic.

▶ Multi-modular computation in GB.

▶ Parallel hashing in GB.

▶ Better handling of memory bounds in GB/GBLA.

▶ FGLM in GB for zero-dimensional system solving.

▶ First steps exploiting heterogeneous CPU/GPU platforms for GBLA.

▶ Deeper investigation on parallelization on networks.

47 / 50

Next steps

▶ v0.3 of GBLA.

▶ Optimizing GBLA for floating point and 32-bit unsigned int arithmetic.

▶ Multi-modular computation in GB.

▶ Parallel hashing in GB.

▶ Better handling of memory bounds in GB/GBLA.

▶ FGLM in GB for zero-dimensional system solving.

▶ First steps exploiting heterogeneous CPU/GPU platforms for GBLA.

▶ Deeper investigation on parallelization on networks.

47 / 50

References

[1] Boyer, B. and Eder, C. and Faugère, J.-C. and Lachartre, S. and Martani, F. GBLA -Gröbner Basis Linear Algebra Package, 2016. Proceedings of the 2016 InternationalSymposium on Symbolic and Algebraic Computation

[2] Buchberger, B. Ein Algorithmus zum Auffinden der Basiselemente desRestklassenringes nach einem nulldimensionalen Polynomideal, 1965. PhD thesis,Universtiy of Innsbruck, Austria

[3] Buchberger, B. A criterion for detecting unnecessary reductions in the construction ofGröbner bases, 1979. EUROSAM ’79, An International Symposium on Symbolic andAlgebraic Manipulation

[4] Buchberger, B. Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory,1985. Multidimensional Systems Theory, D. Reidel Publication Company

[5] Eder, C. and Faugère, J.-C. A survey on signature-based Gröbner basis algorithms,2016. Journal of Symbolic Computation

[6] Faugère, J.-C. A new efficient algorithm for computing Gröbner bases (F4), 1999.Journal of Pure and Applied Algebra

[7] Faugère, J.-C. A new efficient algorithm for computing Gröbner bases without reductionto zero (F5), 2002. Proceedings of the 2002 International Symposium on Symbolic andAlgebraic Computation

[8] Faugère, J.-C. and Lachartre, S. Parallel Gaussian Elimination for Gröbner basescomputations in finite fields, 2010. Proceedings of the 4th International Workshop onParallel and Symbolic Computation

[9] Gebauer, R. and Möller, H. M. On an installation of Buchberger’s algorithm, 1988.Journal of Symbolic Computation

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Thank you!

Questions? Comments?