Schemes for Deterministic Polynomial Factoring

(1)

Schemes for Deterministic Polynomial Factoring

G´ abor Ivanyos ¹ Marek Karpinski ² Nitin Saxena ³

1

Computer and Automation Research Institute, Budapest

2

Dept. of Computer Science, University of Bonn

3

Hausdorff Center for Mathematics, Bonn

ISSAC 2009 Seoul

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Polynomial Factoring The Problem

Outline

Polynomial Factoring The Problem

GRH Connection

Our Algorithm Tensor Powers Schemes Matchings

2 / 20

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Polynomial Factoring over Finite Fields

• Given a polynomial f (x ) ∈ F q [x ] we want a nontrivial factor.

• It is not only a fundamental problem but also has practical applications: coding theory, integer factoring algorithms, computer algebra, ...

• Berlekamp (1967) showed that the problem reduces in deterministic polynomial time to the problem of: factoring a degree n polynomial with n distinct roots in a prime field F p .

3 / 20

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Polynomial Factoring over Finite Fields

• Given a polynomial f (x ) ∈ F q [x ] we want a nontrivial factor.

• It is not only a fundamental problem but also has practical applications: coding theory, integer factoring algorithms, computer algebra, ...

• Berlekamp (1967) showed that the problem reduces in deterministic polynomial time to the problem of: factoring a degree n polynomial with n distinct roots in a prime field F p .

3 / 20

(5)

Polynomial Factoring over Finite Fields

• Given a polynomial f (x ) ∈ F q [x ] we want a nontrivial factor.

• It is not only a fundamental problem but also has practical applications: coding theory, integer factoring algorithms, computer algebra, ...

• Berlekamp (1967) showed that the problem reduces in deterministic polynomial time to the problem of: factoring a degree n polynomial with n distinct roots in a prime field F p .

3 / 20

(6)

Polynomial Factoring Methods

• Let f (x ) be the input polynomial of degree n with distinct n roots in F p .

• The best deterministic algorithm known takes time O ^∼ (n ² √

p) (Shoup 1990).

• The really useful algorithms - Berlekamp (1970), Cantor &

Zassenhaus (1981), von zur Gathen & Shoup (1992), Kaltofen

& Shoup (1995) - are all randomized and take poly (n log p) time.

• It is an open question to derandomize them.

4 / 20

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Polynomial Factoring Methods

• Let f (x ) be the input polynomial of degree n with distinct n roots in F p .

• The best deterministic algorithm known takes time O ^∼ (n ² √

p) (Shoup 1990).

• The really useful algorithms - Berlekamp (1970), Cantor &

Zassenhaus (1981), von zur Gathen & Shoup (1992), Kaltofen

& Shoup (1995) - are all randomized and take poly (n log p) time.

• It is an open question to derandomize them.

4 / 20

(8)

Polynomial Factoring Methods

• Let f (x ) be the input polynomial of degree n with distinct n roots in F p .

• The best deterministic algorithm known takes time O ^∼ (n ² √

p) (Shoup 1990).

• The really useful algorithms - Berlekamp (1970), Cantor &

Zassenhaus (1981), von zur Gathen & Shoup (1992), Kaltofen

& Shoup (1995) - are all randomized and take poly (n log p) time.

• It is an open question to derandomize them.

4 / 20

(9)

Polynomial Factoring Methods

• Let f (x ) be the input polynomial of degree n with distinct n roots in F p .

• The best deterministic algorithm known takes time O ^∼ (n ² √

p) (Shoup 1990).

• The really useful algorithms - Berlekamp (1970), Cantor &

Zassenhaus (1981), von zur Gathen & Shoup (1992), Kaltofen

& Shoup (1995) - are all randomized and take poly (n log p) time.

• It is an open question to derandomize them.

4 / 20

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Polynomial Factoring GRH Connection

Outline

Polynomial Factoring The Problem

GRH Connection

Our Algorithm Tensor Powers Schemes Matchings

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Riemann Hypothesis & Polynomial Factoring

• Generalized Riemann Hypothesis (GRH) has been useful in understanding the deterministic complexity of polynomial factoring, albeit only in special cases.

• Under GRH, any polynomial f (x ) ∈ Z[x] can be factored modulo p deterministically in:

• poly (log p, |Gal _Q (f )|) time (R´ onyai 1989);

• poly (log p, n) time if Gal _Q (f ) is solvable (Evdokimov 1989);

• poly (log p, n) time if Gal _Q (f ) is abelian (Huang 1985).

• Computing √

ⁿ

a(mod p) in poly (log p, n) time (Adleman et. al.

1977).

• These results were based on analytic/algebraic number theory.

6 / 20

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Riemann Hypothesis & Polynomial Factoring

• Generalized Riemann Hypothesis (GRH) has been useful in understanding the deterministic complexity of polynomial factoring, albeit only in special cases.

• Under GRH, any polynomial f (x ) ∈ Z[x] can be factored modulo p deterministically in:

• poly (log p, |Gal _Q (f )|) time (R´ onyai 1989);

• poly (log p, n) time if Gal _Q (f ) is solvable (Evdokimov 1989);

• poly (log p, n) time if Gal _Q (f ) is abelian (Huang 1985).

• Computing √

ⁿ

a(mod p) in poly (log p, n) time (Adleman et. al.

1977).

• These results were based on analytic/algebraic number theory.

6 / 20

(13)

Riemann Hypothesis & Polynomial Factoring

• Generalized Riemann Hypothesis (GRH) has been useful in understanding the deterministic complexity of polynomial factoring, albeit only in special cases.

• Under GRH, any polynomial f (x ) ∈ Z[x] can be factored modulo p deterministically in:

• poly (log p, |Gal _Q (f )|) time (R´ onyai 1989);

• poly (log p, n) time if Gal _Q (f ) is solvable (Evdokimov 1989);

• poly (log p, n) time if Gal _Q (f ) is abelian (Huang 1985).

• Computing √

ⁿ

a(mod p) in poly (log p, n) time (Adleman et. al.

1977).

• These results were based on analytic/algebraic number theory.

6 / 20

(14)

Riemann Hypothesis & Polynomial Factoring

• Generalized Riemann Hypothesis (GRH) has been useful in understanding the deterministic complexity of polynomial factoring, albeit only in special cases.

• Under GRH, any polynomial f (x ) ∈ Z[x] can be factored modulo p deterministically in:

• poly (log p, |Gal _Q (f )|) time (R´ onyai 1989);

• poly (log p, n) time if Gal _Q (f ) is solvable (Evdokimov 1989);

• poly (log p, n) time if Gal _Q (f ) is abelian (Huang 1985).

• Computing √

ⁿ

a(mod p) in poly (log p, n) time (Adleman et. al.

1977).

• These results were based on analytic/algebraic number theory.

6 / 20

(15)

Riemann Hypothesis & Polynomial Factoring

• Generalized Riemann Hypothesis (GRH) has been useful in understanding the deterministic complexity of polynomial factoring, albeit only in special cases.

• Under GRH, any polynomial f (x ) ∈ Z[x] can be factored modulo p deterministically in:

• poly (log p, |Gal _Q (f )|) time (R´ onyai 1989);

• poly (log p, n) time if Gal _Q (f ) is solvable (Evdokimov 1989);

• poly (log p, n) time if Gal _Q (f ) is abelian (Huang 1985).

• Computing √

ⁿ

a(mod p) in poly (log p, n) time (Adleman et. al.

1977).

• These results were based on analytic/algebraic number theory.

6 / 20

(16)

Riemann Hypothesis & Polynomial Factoring

• Generalized Riemann Hypothesis (GRH) has been useful in understanding the deterministic complexity of polynomial factoring, albeit only in special cases.

• Under GRH, any polynomial f (x ) ∈ Z[x] can be factored modulo p deterministically in:

• poly (log p, |Gal _Q (f )|) time (R´ onyai 1989);

• poly (log p, n) time if Gal _Q (f ) is solvable (Evdokimov 1989);

• poly (log p, n) time if Gal _Q (f ) is abelian (Huang 1985).

• Computing √

ⁿ

a(mod p) in poly (log p, n) time (Adleman et. al.

1977).

• These results were based on analytic/algebraic number theory.

6 / 20

(17)

Riemann Hypothesis & Polynomial Factoring

• Generalized Riemann Hypothesis (GRH) has been useful in understanding the deterministic complexity of polynomial factoring, albeit only in special cases.

• Under GRH, any polynomial f (x ) ∈ Z[x] can be factored modulo p deterministically in:

• poly (log p, |Gal _Q (f )|) time (R´ onyai 1989);

• poly (log p, n) time if Gal _Q (f ) is solvable (Evdokimov 1989);

• poly (log p, n) time if Gal _Q (f ) is abelian (Huang 1985).

• Computing √

ⁿ

a(mod p) in poly (log p, n) time (Adleman et. al.

1977).

• These results were based on analytic/algebraic number theory.

6 / 20

(18)

Riemann Hypothesis & Polynomial Factoring

• Generalized Riemann Hypothesis (GRH) has been useful in understanding the deterministic complexity of polynomial factoring, albeit only in special cases.

• Under GRH, any polynomial f (x ) ∈ Z[x] can be factored modulo p deterministically in:

• poly (log p, |Gal _Q (f )|) time (R´ onyai 1989);

• poly (log p, n) time if Gal _Q (f ) is solvable (Evdokimov 1989);

• poly (log p, n) time if Gal _Q (f ) is abelian (Huang 1985).

• Computing √

ⁿ

a(mod p) in poly (log p, n) time (Adleman et. al.

1977).

• These results were based on analytic/algebraic number theory.

6 / 20

(19)

Riemann Hypothesis & Polynomial Factoring

• There are also results based on GRH and combinatorial tricks, a degree n polynomial f (x ) can be nontrivially factored in deterministic:

• poly (log p, n ^r ) time if r |n (R´ onyai 1987);

• poly (log p, n ^{log n} ) time (Evdokimov 1994).

• It is this line of research that we consider in this work.

• We greatly generalize the combinatorial object associated with these polynomial factoring algorithms

• and using a recent algebraic-combinatorics development (Hanaki & Uno 2006) prove:

poly (log p, n ^r ) time if n is prime and (n − 1) is r -smooth.

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Riemann Hypothesis & Polynomial Factoring

• There are also results based on GRH and combinatorial tricks, a degree n polynomial f (x ) can be nontrivially factored in deterministic:

• poly (log p, n ^r ) time if r |n (R´ onyai 1987);

• poly (log p, n ^{log n} ) time (Evdokimov 1994).

• It is this line of research that we consider in this work.

• We greatly generalize the combinatorial object associated with these polynomial factoring algorithms

• and using a recent algebraic-combinatorics development (Hanaki & Uno 2006) prove:

poly (log p, n ^r ) time if n is prime and (n − 1) is r -smooth.

7 / 20

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Riemann Hypothesis & Polynomial Factoring

• There are also results based on GRH and combinatorial tricks, a degree n polynomial f (x ) can be nontrivially factored in deterministic:

• poly (log p, n ^r ) time if r |n (R´ onyai 1987);

• poly (log p, n ^{log n} ) time (Evdokimov 1994).

• It is this line of research that we consider in this work.

• We greatly generalize the combinatorial object associated with these polynomial factoring algorithms

• and using a recent algebraic-combinatorics development (Hanaki & Uno 2006) prove:

poly (log p, n ^r ) time if n is prime and (n − 1) is r -smooth.

7 / 20

(22)

Riemann Hypothesis & Polynomial Factoring

• There are also results based on GRH and combinatorial tricks, a degree n polynomial f (x ) can be nontrivially factored in deterministic:

• poly (log p, n ^r ) time if r |n (R´ onyai 1987);

• poly (log p, n ^{log n} ) time (Evdokimov 1994).

• It is this line of research that we consider in this work.

• We greatly generalize the combinatorial object associated with these polynomial factoring algorithms

• and using a recent algebraic-combinatorics development (Hanaki & Uno 2006) prove:

poly (log p, n ^r ) time if n is prime and (n − 1) is r -smooth.

7 / 20

(23)

Riemann Hypothesis & Polynomial Factoring

• There are also results based on GRH and combinatorial tricks, a degree n polynomial f (x ) can be nontrivially factored in deterministic:

• poly (log p, n ^r ) time if r |n (R´ onyai 1987);

• poly (log p, n ^{log n} ) time (Evdokimov 1994).

• It is this line of research that we consider in this work.

• We greatly generalize the combinatorial object associated with these polynomial factoring algorithms

• and using a recent algebraic-combinatorics development (Hanaki & Uno 2006) prove:

poly (log p, n ^r ) time if n is prime and (n − 1) is r -smooth.

7 / 20

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Riemann Hypothesis & Polynomial Factoring

• There are also results based on GRH and combinatorial tricks, a degree n polynomial f (x ) can be nontrivially factored in deterministic:

• poly (log p, n ^r ) time if r |n (R´ onyai 1987);

• poly (log p, n ^{log n} ) time (Evdokimov 1994).

• It is this line of research that we consider in this work.

• We greatly generalize the combinatorial object associated with these polynomial factoring algorithms

• and using a recent algebraic-combinatorics development (Hanaki & Uno 2006) prove:

poly (log p, n ^r ) time if n is prime and (n − 1) is r -smooth.

7 / 20

(25)

Riemann Hypothesis & Polynomial Factoring

• There are also results based on GRH and combinatorial tricks, a degree n polynomial f (x ) can be nontrivially factored in deterministic:

• poly (log p, n ^r ) time if r |n (R´ onyai 1987);

• poly (log p, n ^{log n} ) time (Evdokimov 1994).

• It is this line of research that we consider in this work.

• We greatly generalize the combinatorial object associated with these polynomial factoring algorithms

• and using a recent algebraic-combinatorics development (Hanaki & Uno 2006) prove:

poly (log p, n ^r ) time if n is prime and (n − 1) is r -smooth.

7 / 20

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Our Algorithm Tensor Powers

Outline

Polynomial Factoring The Problem

GRH Connection

Our Algorithm Tensor Powers Schemes Matchings

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Tensor Powers

• Let the input be f (x ) ∈ F p of degree n having distinct roots α ₁ , . . . , α _n ∈ F p .

• Under GRH we can construct a field k ⊇ F p that has s-th roots of unity for all s ∈ [m]. (parameter m will be fixed later.)

• We have a natural associated algebra A := k[X ]/(f (X )). A is isomorphic to k ⁿ , the direct sum of n copies of the algebra k.

• A ^⊗s , for s ∈ [m], is the s-th tensor power of A. A ^⊗s is isomorphic to k ⁿ

^s

.

• Lemma: These tensor powers can be computed (in basis form over k) in deterministic poly (log p, n ^m ) time.

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Tensor Powers

• Let the input be f (x ) ∈ F p of degree n having distinct roots α ₁ , . . . , α _n ∈ F p .

• Under GRH we can construct a field k ⊇ F p that has s-th roots of unity for all s ∈ [m]. (parameter m will be fixed later.)

• We have a natural associated algebra A := k[X ]/(f (X )). A is isomorphic to k ⁿ , the direct sum of n copies of the algebra k.

• A ^⊗s , for s ∈ [m], is the s-th tensor power of A. A ^⊗s is isomorphic to k ⁿ

^s

.

• Lemma: These tensor powers can be computed (in basis form over k) in deterministic poly (log p, n ^m ) time.

9 / 20

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Tensor Powers

• Let the input be f (x ) ∈ F p of degree n having distinct roots α ₁ , . . . , α _n ∈ F p .

• Under GRH we can construct a field k ⊇ F p that has s-th roots of unity for all s ∈ [m]. (parameter m will be fixed later.)

• We have a natural associated algebra A := k[X ]/(f (X )). A is isomorphic to k ⁿ , the direct sum of n copies of the algebra k.

• A ^⊗s , for s ∈ [m], is the s-th tensor power of A. A ^⊗s is isomorphic to k ⁿ

^s

.

• Lemma: These tensor powers can be computed (in basis form over k) in deterministic poly (log p, n ^m ) time.

9 / 20

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Tensor Powers

• Let the input be f (x ) ∈ F p of degree n having distinct roots α ₁ , . . . , α _n ∈ F p .

• Under GRH we can construct a field k ⊇ F p that has s-th roots of unity for all s ∈ [m]. (parameter m will be fixed later.)

• We have a natural associated algebra A := k[X ]/(f (X )). A is isomorphic to k ⁿ , the direct sum of n copies of the algebra k.

• A ^⊗s , for s ∈ [m], is the s-th tensor power of A. A ^⊗s is isomorphic to k ⁿ

^s

.

• Lemma: These tensor powers can be computed (in basis form over k) in deterministic poly (log p, n ^m ) time.

9 / 20

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Tensor Powers

• Let the input be f (x ) ∈ F p of degree n having distinct roots α ₁ , . . . , α _n ∈ F p .

• Under GRH we can construct a field k ⊇ F p that has s-th roots of unity for all s ∈ [m]. (parameter m will be fixed later.)

• We have a natural associated algebra A := k[X ]/(f (X )). A is isomorphic to k ⁿ , the direct sum of n copies of the algebra k.

• A ^⊗s , for s ∈ [m], is the s-th tensor power of A. A ^⊗s is isomorphic to k ⁿ

^s

.

• Lemma: These tensor powers can be computed (in basis form over k) in deterministic poly (log p, n ^m ) time.

9 / 20

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Tensor Powers

• Let the input be f (x ) ∈ F p of degree n having distinct roots α ₁ , . . . , α _n ∈ F p .

• Under GRH we can construct a field k ⊇ F p that has s-th roots of unity for all s ∈ [m]. (parameter m will be fixed later.)

• We have a natural associated algebra A := k[X ]/(f (X )). A is isomorphic to k ⁿ , the direct sum of n copies of the algebra k.

• A ^⊗s , for s ∈ [m], is the s-th tensor power of A. A ^⊗s is isomorphic to k ⁿ

^s

.

• Lemma: These tensor powers can be computed (in basis form over k) in deterministic poly (log p, n ^m ) time.

9 / 20

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Tensor Powers

• Let the input be f (x ) ∈ F p of degree n having distinct roots α ₁ , . . . , α _n ∈ F p .

• Under GRH we can construct a field k ⊇ F p that has s-th roots of unity for all s ∈ [m]. (parameter m will be fixed later.)

• We have a natural associated algebra A := k[X ]/(f (X )). A is isomorphic to k ⁿ , the direct sum of n copies of the algebra k.

• A ^⊗s , for s ∈ [m], is the s-th tensor power of A. A ^⊗s is isomorphic to k ⁿ

^s

.

• Lemma: These tensor powers can be computed (in basis form over k) in deterministic poly (log p, n ^m ) time.

9 / 20

(34)

Tensor Powers

• Let the input be f (x ) ∈ F p of degree n having distinct roots α ₁ , . . . , α _n ∈ F p .

• Under GRH we can construct a field k ⊇ F p that has s-th roots of unity for all s ∈ [m]. (parameter m will be fixed later.)

• We have a natural associated algebra A := k[X ]/(f (X )). A is isomorphic to k ⁿ , the direct sum of n copies of the algebra k.

• A ^⊗s , for s ∈ [m], is the s-th tensor power of A. A ^⊗s is isomorphic to k ⁿ

^s

.

• Lemma: These tensor powers can be computed (in basis form over k) in deterministic poly (log p, n ^m ) time.

9 / 20

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Step 0: Initiation

• Initiate the decomposition of the tensor powers A ^⊗s , for all s ∈ [m], into ideals.

• Aut _k (A ^⊗s ) contains Symm _s . For σ ∈ Symm _s the corresponding algebra automorphism action is:

(b _i

₁

⊗ · · · ⊗ b _i

_s

) ^σ = b _i

_1σ

⊗ · · · ⊗ b _i

_sσ

.

• These nontrivial automorphisms of A ^⊗s (when s > 1) help decompose these algebras under GRH (R´ onyai 1992).

• Thus, for all 2 ≤ s ≤ m, we can compute mutually orthogonal t _s ≥ 2 ideals I _s,i of A ^⊗s s.t. A ^⊗s = I _s,1 + · · · + I _s,t

_s

.

• Next try out the following refinements:

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Step 0: Initiation

• Initiate the decomposition of the tensor powers A ^⊗s , for all s ∈ [m], into ideals.

• Aut _k (A ^⊗s ) contains Symm _s . For σ ∈ Symm _s the corresponding algebra automorphism action is:

(b _i

₁

⊗ · · · ⊗ b _i

_s

) ^σ = b _i

_1σ

⊗ · · · ⊗ b _i

_sσ

.

• These nontrivial automorphisms of A ^⊗s (when s > 1) help decompose these algebras under GRH (R´ onyai 1992).

• Thus, for all 2 ≤ s ≤ m, we can compute mutually orthogonal t _s ≥ 2 ideals I _s,i of A ^⊗s s.t. A ^⊗s = I _s,1 + · · · + I _s,t

_s

.

• Next try out the following refinements:

10 / 20

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Step 0: Initiation

• Initiate the decomposition of the tensor powers A ^⊗s , for all s ∈ [m], into ideals.

• Aut _k (A ^⊗s ) contains Symm _s . For σ ∈ Symm _s the corresponding algebra automorphism action is:

(b _i

₁

⊗ · · · ⊗ b _i

_s

) ^σ = b _i

_1σ

⊗ · · · ⊗ b _i

_sσ

.

• These nontrivial automorphisms of A ^⊗s (when s > 1) help decompose these algebras under GRH (R´ onyai 1992).

• Thus, for all 2 ≤ s ≤ m, we can compute mutually orthogonal t _s ≥ 2 ideals I _s,i of A ^⊗s s.t. A ^⊗s = I _s,1 + · · · + I _s,t

_s

.

• Next try out the following refinements:

10 / 20

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Step 0: Initiation

• Initiate the decomposition of the tensor powers A ^⊗s , for all s ∈ [m], into ideals.

• Aut _k (A ^⊗s ) contains Symm _s . For σ ∈ Symm _s the corresponding algebra automorphism action is:

(b _i

₁

⊗ · · · ⊗ b _i

_s

) ^σ = b _i

_1σ

⊗ · · · ⊗ b _i

_sσ

.

• These nontrivial automorphisms of A ^⊗s (when s > 1) help decompose these algebras under GRH (R´ onyai 1992).

• Thus, for all 2 ≤ s ≤ m, we can compute mutually orthogonal t _s ≥ 2 ideals I _s,i of A ^⊗s s.t. A ^⊗s = I _s,1 + · · · + I _s,t

_s

.

• Next try out the following refinements:

10 / 20

(39)

Step 0: Initiation

• Initiate the decomposition of the tensor powers A ^⊗s , for all s ∈ [m], into ideals.

• Aut _k (A ^⊗s ) contains Symm _s . For σ ∈ Symm _s the corresponding algebra automorphism action is:

(b _i

₁

⊗ · · · ⊗ b _i

_s

) ^σ = b _i

_1σ

⊗ · · · ⊗ b _i

_sσ

.

• These nontrivial automorphisms of A ^⊗s (when s > 1) help decompose these algebras under GRH (R´ onyai 1992).

• Thus, for all 2 ≤ s ≤ m, we can compute mutually orthogonal t _s ≥ 2 ideals I _s,i of A ^⊗s s.t. A ^⊗s = I _s,1 + · · · + I _s,t

_s

.

• Next try out the following refinements:

10 / 20

(40)

Step 0: Initiation

• Initiate the decomposition of the tensor powers A ^⊗s , for all s ∈ [m], into ideals.

• Aut _k (A ^⊗s ) contains Symm _s . For σ ∈ Symm _s the corresponding algebra automorphism action is:

(b _i

₁

⊗ · · · ⊗ b _i

_s

) ^σ = b _i

_1σ

⊗ · · · ⊗ b _i

_sσ

.

• These nontrivial automorphisms of A ^⊗s (when s > 1) help decompose these algebras under GRH (R´ onyai 1992).

• Thus, for all 2 ≤ s ≤ m, we can compute mutually orthogonal t _s ≥ 2 ideals I _s,i of A ^⊗s s.t. A ^⊗s = I _s,1 + · · · + I _s,t

_s

.

• Next try out the following refinements:

10 / 20

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Step 1: Homogeneity

• Suppose the current tensor power decomposition into orthogonal nonzero ideals is: A ^⊗s = I _s,1 + · · · + I _s,t

_s

, for all s ∈ [m].

• If t ₁ > 1 then I _1,1 is a nontrivial ideal of A. We can efficiently find the unity e(x ) of I 1,1 .

• Output the gcd (e(x ), f (x )), it is assured to be a nontrivial factor.

• Else t ₁ = 1 and we try the following refinements on A ^⊗s (s > 1):

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Step 1: Homogeneity

• Suppose the current tensor power decomposition into orthogonal nonzero ideals is: A ^⊗s = I _s,1 + · · · + I _s,t

_s

, for all s ∈ [m].

• If t ₁ > 1 then I _1,1 is a nontrivial ideal of A. We can efficiently find the unity e(x ) of I 1,1 .

• Output the gcd (e(x ), f (x )), it is assured to be a nontrivial factor.

• Else t ₁ = 1 and we try the following refinements on A ^⊗s (s > 1):

11 / 20

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Step 1: Homogeneity

• Suppose the current tensor power decomposition into orthogonal nonzero ideals is: A ^⊗s = I _s,1 + · · · + I _s,t

_s

, for all s ∈ [m].

• If t ₁ > 1 then I _1,1 is a nontrivial ideal of A. We can efficiently find the unity e(x ) of I 1,1 .

• Output the gcd (e(x ), f (x )), it is assured to be a nontrivial factor.

• Else t ₁ = 1 and we try the following refinements on A ^⊗s (s > 1):

11 / 20

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Step 1: Homogeneity

• Suppose the current tensor power decomposition into orthogonal nonzero ideals is: A ^⊗s = I _s,1 + · · · + I _s,t

_s

, for all s ∈ [m].

• If t ₁ > 1 then I _1,1 is a nontrivial ideal of A. We can efficiently find the unity e(x ) of I 1,1 .

• Output the gcd (e(x ), f (x )), it is assured to be a nontrivial factor.

• Else t ₁ = 1 and we try the following refinements on A ^⊗s (s > 1):

11 / 20

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Steps 2 & 3: Compatibility & Regularity

• Suppose the current tensor power decomposition into orthogonal nonzero ideals is: A ^⊗s = I _s,1 + · · · + I _s,t

_s

, for all s ∈ [m].

• Consider algebra embeddings ι ^s _j : A ^⊗(s−1) → A ^⊗s mapping b i

1

⊗ · · · ⊗ b _i

_s−1

7→ b _i

₁

⊗ · · · ⊗ b _i

_{j −1}

⊗ 1 ⊗ b _i

_j

⊗ · · · b _i

_s−1

.

• Compatibility: If ι ^s _j (I s−1,i )I s,i

⁰

6= 0, I _s,i

⁰

, then we can efficiently compute a subideal of I _s,i

⁰

and refine the decomposition.

• Regularity: If ι ^s _j (I s−1,i )I s,i

⁰

is not a free module over ι ^s _j (I s−1,i ), then we can efficiently compute a subideal of I _s−1,i and again refine the decomposition.

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(46)

Steps 2 & 3: Compatibility & Regularity

• Suppose the current tensor power decomposition into orthogonal nonzero ideals is: A ^⊗s = I _s,1 + · · · + I _s,t

_s

, for all s ∈ [m].

• Consider algebra embeddings ι ^s _j : A ^⊗(s−1) → A ^⊗s mapping b i

1

⊗ · · · ⊗ b _i

_s−1

7→ b _i

₁

⊗ · · · ⊗ b _i

_{j −1}

⊗ 1 ⊗ b _i

_j

⊗ · · · b _i

_s−1

.

• Compatibility: If ι ^s _j (I s−1,i )I s,i

⁰

6= 0, I _s,i

⁰

, then we can efficiently compute a subideal of I _s,i

⁰

and refine the decomposition.

• Regularity: If ι ^s _j (I s−1,i )I s,i

⁰

is not a free module over ι ^s _j (I s−1,i ), then we can efficiently compute a subideal of I _s−1,i and again refine the decomposition.

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(47)

Steps 2 & 3: Compatibility & Regularity

• Suppose the current tensor power decomposition into orthogonal nonzero ideals is: A ^⊗s = I _s,1 + · · · + I _s,t

_s

, for all s ∈ [m].

• Consider algebra embeddings ι ^s _j : A ^⊗(s−1) → A ^⊗s mapping b i

1

⊗ · · · ⊗ b _i

_s−1

7→ b _i

₁

⊗ · · · ⊗ b _i

_{j −1}

⊗ 1 ⊗ b _i

_j

⊗ · · · b _i

_s−1

.

• Compatibility: If ι ^s _j (I s−1,i )I s,i

⁰

6= 0, I _s,i

⁰

, then we can efficiently compute a subideal of I _s,i

⁰

and refine the decomposition.

• Regularity: If ι ^s _j (I s−1,i )I s,i

⁰

is not a free module over ι ^s _j (I s−1,i ), then we can efficiently compute a subideal of I _s−1,i and again refine the decomposition.

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(48)

Steps 2 & 3: Compatibility & Regularity

• Suppose the current tensor power decomposition into orthogonal nonzero ideals is: A ^⊗s = I _s,1 + · · · + I _s,t

_s

, for all s ∈ [m].

• Consider algebra embeddings ι ^s _j : A ^⊗(s−1) → A ^⊗s mapping b i

1

⊗ · · · ⊗ b _i

_s−1

7→ b _i

₁

⊗ · · · ⊗ b _i

_{j −1}

⊗ 1 ⊗ b _i

_j

⊗ · · · b _i

_s−1

.

• Compatibility: If ι ^s _j (I s−1,i )I s,i

⁰

6= 0, I _s,i

⁰

, then we can efficiently compute a subideal of I _s,i

⁰

and refine the decomposition.

• Regularity: If ι ^s _j (I s−1,i )I s,i

⁰

is not a free module over ι ^s _j (I s−1,i ), then we can efficiently compute a subideal of I _s−1,i and again refine the decomposition.

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(49)

Steps 4 & 5: Invariance & Antisymmetricity

• Suppose the current tensor power decomposition into orthogonal nonzero ideals is: A ^⊗s = I _s,1 + · · · + I _s,t

_s

, for all s ∈ [m].

• Invariance: If I _s,i ^σ ∩ I _s,i

⁰

6= 0, I _s,i

⁰

, then we can efficiently compute a subideal of I s,i

⁰

and hence refine the decomposition.

• Antisymmetricity: If for some σ 6= id : I _s,i ^σ = I _s,i , then σ is an algebra automorphism of I s,i . Thus we can efficiently compute its subideal (under GRH, R´ onyai 1992 ) and again refine the decomposition.

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(50)

Steps 4 & 5: Invariance & Antisymmetricity

• Suppose the current tensor power decomposition into orthogonal nonzero ideals is: A ^⊗s = I _s,1 + · · · + I _s,t

_s

, for all s ∈ [m].

• Invariance: If I _s,i ^σ ∩ I _s,i

⁰

6= 0, I _s,i

⁰

, then we can efficiently compute a subideal of I s,i

⁰

and hence refine the decomposition.

• Antisymmetricity: If for some σ 6= id : I _s,i ^σ = I _s,i , then σ is an algebra automorphism of I s,i . Thus we can efficiently compute its subideal (under GRH, R´ onyai 1992 ) and again refine the decomposition.

13 / 20

(51)

Steps 4 & 5: Invariance & Antisymmetricity

• Suppose the current tensor power decomposition into orthogonal nonzero ideals is: A ^⊗s = I _s,1 + · · · + I _s,t

_s

, for all s ∈ [m].

• Invariance: If I _s,i ^σ ∩ I _s,i

⁰

6= 0, I _s,i

⁰

, then we can efficiently compute a subideal of I s,i

⁰

and hence refine the decomposition.

• Antisymmetricity: If for some σ 6= id : I _s,i ^σ = I _s,i , then σ is an algebra automorphism of I s,i . Thus we can efficiently compute its subideal (under GRH, R´ onyai 1992 ) and again refine the decomposition.

13 / 20

(52)

Our Algorithm Schemes

Outline

Polynomial Factoring The Problem

GRH Connection

Our Algorithm Tensor Powers Schemes Matchings

14 / 20

(53)

The underlying Combinatorics

• Our full algorithm then is to start with Step 0, and then keep repeating Steps 1-5 until a factor of f (x ) is found or these refinements have no effect.

• Let the stable tensor power decomposition into orthogonal nonzero ideals be: A ^⊗s = I s,1 + · · · + I s,t

s

, for all s ∈ [m].

• Let V := {α 1 , . . . , α n } be the roots of f (x ) and V ^s be the set of all s -tuples.

• Lemma: The ideal I s,i implicitly defines a subset of V ^s : Supp(I _s,i ) := V ^s \ {¯ v ∈ V ^s | ∀a ∈ I _s,i , a(¯ v ) = 0}

• Thus, a decomposition of A ^⊗s induces a partition P _s of V ^s . The equivalence classes in P _s will be called colors.

15 / 20

(54)

The underlying Combinatorics

• Our full algorithm then is to start with Step 0, and then keep repeating Steps 1-5 until a factor of f (x ) is found or these refinements have no effect.

• Let the stable tensor power decomposition into orthogonal nonzero ideals be: A ^⊗s = I s,1 + · · · + I s,t

s

, for all s ∈ [m].

• Let V := {α 1 , . . . , α n } be the roots of f (x ) and V ^s be the set of all s -tuples.

• Lemma: The ideal I s,i implicitly defines a subset of V ^s : Supp(I _s,i ) := V ^s \ {¯ v ∈ V ^s | ∀a ∈ I _s,i , a(¯ v ) = 0}

• Thus, a decomposition of A ^⊗s induces a partition P _s of V ^s . The equivalence classes in P _s will be called colors.

15 / 20

(55)

The underlying Combinatorics

• Our full algorithm then is to start with Step 0, and then keep repeating Steps 1-5 until a factor of f (x ) is found or these refinements have no effect.

• Let the stable tensor power decomposition into orthogonal nonzero ideals be: A ^⊗s = I s,1 + · · · + I s,t

s

, for all s ∈ [m].

• Let V := {α 1 , . . . , α n } be the roots of f (x ) and V ^s be the set of all s -tuples.

• Lemma: The ideal I s,i implicitly defines a subset of V ^s : Supp(I _s,i ) := V ^s \ {¯ v ∈ V ^s | ∀a ∈ I _s,i , a(¯ v ) = 0}

• Thus, a decomposition of A ^⊗s induces a partition P _s of V ^s . The equivalence classes in P _s will be called colors.

15 / 20

(56)

The underlying Combinatorics

• Our full algorithm then is to start with Step 0, and then keep repeating Steps 1-5 until a factor of f (x ) is found or these refinements have no effect.

• Let the stable tensor power decomposition into orthogonal nonzero ideals be: A ^⊗s = I s,1 + · · · + I s,t

s

, for all s ∈ [m].

• Let V := {α 1 , . . . , α n } be the roots of f (x ) and V ^s be the set of all s -tuples.

• Lemma: The ideal I s,i implicitly defines a subset of V ^s : Supp(I _s,i ) := V ^s \ {¯ v ∈ V ^s | ∀a ∈ I _s,i , a(¯ v ) = 0}

• Thus, a decomposition of A ^⊗s induces a partition P _s of V ^s . The equivalence classes in P _s will be called colors.

15 / 20

(57)

The underlying Combinatorics

• Our full algorithm then is to start with Step 0, and then keep repeating Steps 1-5 until a factor of f (x ) is found or these refinements have no effect.

• Let the stable tensor power decomposition into orthogonal nonzero ideals be: A ^⊗s = I s,1 + · · · + I s,t

s

, for all s ∈ [m].

• Let V := {α 1 , . . . , α n } be the roots of f (x ) and V ^s be the set of all s -tuples.

• Lemma: The ideal I s,i implicitly defines a subset of V ^s : Supp(I _s,i ) := V ^s \ {¯ v ∈ V ^s | ∀a ∈ I _s,i , a(¯ v ) = 0}

• Thus, a decomposition of A ^⊗s induces a partition P _s of V ^s . The equivalence classes in P _s will be called colors.

15 / 20

(58)

The underlying Combinatorics

• Our full algorithm then is to start with Step 0, and then keep repeating Steps 1-5 until a factor of f (x ) is found or these refinements have no effect.

• Let the stable tensor power decomposition into orthogonal nonzero ideals be: A ^⊗s = I s,1 + · · · + I s,t

s

, for all s ∈ [m].

• Let V := {α 1 , . . . , α n } be the roots of f (x ) and V ^s be the set of all s -tuples.

• Lemma: The ideal I s,i implicitly defines a subset of V ^s : Supp(I _s,i ) := V ^s \ {¯ v ∈ V ^s | ∀a ∈ I _s,i , a(¯ v ) = 0}

• Thus, a decomposition of A ^⊗s induces a partition P _s of V ^s . The equivalence classes in P _s will be called colors.

15 / 20

(59)

Compatibility, Regularity & Invariance

• Consider projection π _i ^s : V ^s → V ^s−1 as the map that drops the i -th coordinate.

• Compatibility step ensures that if two tuples in V ^s have the same color then, for every projection π ^s _i , the projected tuples have the same color as well.

• Regularity step ensures that the number of preimages in V ^s of an (s − 1)-tuple depends only on the color of the tuple.

• Invariance step ensures that the partitions P _s are invariant under the action of the corresponding symmetric group.

• A {P ₁ , . . . , P m } satisfying these three conditions we call an m-scheme on n points.

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(60)

Compatibility, Regularity & Invariance

• Consider projection π _i ^s : V ^s → V ^s−1 as the map that drops the i -th coordinate.

• Compatibility step ensures that if two tuples in V ^s have the same color then, for every projection π ^s _i , the projected tuples have the same color as well.

• Regularity step ensures that the number of preimages in V ^s of an (s − 1)-tuple depends only on the color of the tuple.

• Invariance step ensures that the partitions P _s are invariant under the action of the corresponding symmetric group.

• A {P ₁ , . . . , P m } satisfying these three conditions we call an m-scheme on n points.

16 / 20

(61)

Compatibility, Regularity & Invariance

• Consider projection π _i ^s : V ^s → V ^s−1 as the map that drops the i -th coordinate.

• Compatibility step ensures that if two tuples in V ^s have the same color then, for every projection π ^s _i , the projected tuples have the same color as well.

• Regularity step ensures that the number of preimages in V ^s of an (s − 1)-tuple depends only on the color of the tuple.

• Invariance step ensures that the partitions P _s are invariant under the action of the corresponding symmetric group.

• A {P ₁ , . . . , P m } satisfying these three conditions we call an m-scheme on n points.

16 / 20

(62)

Compatibility, Regularity & Invariance

• Consider projection π _i ^s : V ^s → V ^s−1 as the map that drops the i -th coordinate.

• Compatibility step ensures that if two tuples in V ^s have the same color then, for every projection π ^s _i , the projected tuples have the same color as well.

• Regularity step ensures that the number of preimages in V ^s of an (s − 1)-tuple depends only on the color of the tuple.

• Invariance step ensures that the partitions P _s are invariant under the action of the corresponding symmetric group.

• A {P ₁ , . . . , P m } satisfying these three conditions we call an m-scheme on n points.

16 / 20

(63)

Compatibility, Regularity & Invariance

• Consider projection π _i ^s : V ^s → V ^s−1 as the map that drops the i -th coordinate.

• Compatibility step ensures that if two tuples in V ^s have the same color then, for every projection π ^s _i , the projected tuples have the same color as well.

• Regularity step ensures that the number of preimages in V ^s of an (s − 1)-tuple depends only on the color of the tuple.

• Invariance step ensures that the partitions P _s are invariant under the action of the corresponding symmetric group.

• A {P ₁ , . . . , P m } satisfying these three conditions we call an m-scheme on n points.

16 / 20

(64)

Homogeneity & Antisymmetricity Revisited

• Examples of m-schemes are abundant in algebraic- combinatorics, but the ones troubling us are rarer:

• Homogeneity step ensures that |P ₁ | = 1, i.e. P ₁ = {V }.

• Antisymmetricity step ensures that for every regular color P ∈ P _s and σ 6= id: P ^σ 6= P. A color is regular if each tuple in it has distinct coordinates.

• Lemma: Our algorithm either factors f (x ) or arranges the roots in a homogeneous, antisymmetric m-scheme (in deterministic poly (log p, n ^m ) time under GRH).

17 / 20

(65)

Homogeneity & Antisymmetricity Revisited

• Examples of m-schemes are abundant in algebraic- combinatorics, but the ones troubling us are rarer:

• Homogeneity step ensures that |P ₁ | = 1, i.e. P ₁ = {V }.

• Antisymmetricity step ensures that for every regular color P ∈ P _s and σ 6= id: P ^σ 6= P. A color is regular if each tuple in it has distinct coordinates.

• Lemma: Our algorithm either factors f (x ) or arranges the roots in a homogeneous, antisymmetric m-scheme (in deterministic poly (log p, n ^m ) time under GRH).

17 / 20

(66)

Homogeneity & Antisymmetricity Revisited

• Examples of m-schemes are abundant in algebraic- combinatorics, but the ones troubling us are rarer:

• Homogeneity step ensures that |P ₁ | = 1, i.e. P ₁ = {V }.

• Antisymmetricity step ensures that for every regular color P ∈ P _s and σ 6= id: P ^σ 6= P. A color is regular if each tuple in it has distinct coordinates.

• Lemma: Our algorithm either factors f (x ) or arranges the roots in a homogeneous, antisymmetric m-scheme (in deterministic poly (log p, n ^m ) time under GRH).

17 / 20

(67)

Homogeneity & Antisymmetricity Revisited

• Examples of m-schemes are abundant in algebraic- combinatorics, but the ones troubling us are rarer:

• Homogeneity step ensures that |P ₁ | = 1, i.e. P ₁ = {V }.

• Antisymmetricity step ensures that for every regular color P ∈ P _s and σ 6= id: P ^σ 6= P. A color is regular if each tuple in it has distinct coordinates.

• Lemma: Our algorithm either factors f (x ) or arranges the roots in a homogeneous, antisymmetric m-scheme (in deterministic poly (log p, n ^m ) time under GRH).

17 / 20

(68)

Homogeneity & Antisymmetricity Revisited

• Examples of m-schemes are abundant in algebraic- combinatorics, but the ones troubling us are rarer:

• Homogeneity step ensures that |P ₁ | = 1, i.e. P ₁ = {V }.

• Antisymmetricity step ensures that for every regular color P ∈ P _s and σ 6= id: P ^σ 6= P. A color is regular if each tuple in it has distinct coordinates.

• Lemma: Our algorithm either factors f (x ) or arranges the roots in a homogeneous, antisymmetric m-scheme (in deterministic poly (log p, n ^m ) time under GRH).

17 / 20

(69)

Our Algorithm Matchings

Outline

Polynomial Factoring The Problem

GRH Connection

Our Algorithm Tensor Powers Schemes Matchings

18 / 20

(70)

A New Step 6: Matchings

• It may happen that our algorithm gets stuck in a

homogeneous, antisymmetric m-scheme Π. What do we do then?

• We then look for a certain color in Π called a matching: A regular color P ∈ P s ∈ Π such that for some 1 ≤ i < j ≤ s : π ^s _i (P) = π _j ^s (P) and |π ^s _i (P)| = |P|.

• Such a matching color P has π ^s _i (π _j ^s ) ⁻¹ as automorphism, which can be used to refine the tensor power decomposition.

• Schemes Conjecture: Every homogeneous, antisymmetric 5-scheme has a matching.

• We have proved this conjecture for the only 5-schemes we know: group schemes.

19 / 20

(71)

A New Step 6: Matchings

• It may happen that our algorithm gets stuck in a

homogeneous, antisymmetric m-scheme Π. What do we do then?

• We then look for a certain color in Π called a matching: A regular color P ∈ P s ∈ Π such that for some 1 ≤ i < j ≤ s : π ^s _i (P) = π _j ^s (P) and |π ^s _i (P)| = |P|.

• Such a matching color P has π ^s _i (π _j ^s ) ⁻¹ as automorphism, which can be used to refine the tensor power decomposition.

• Schemes Conjecture: Every homogeneous, antisymmetric 5-scheme has a matching.

• We have proved this conjecture for the only 5-schemes we know: group schemes.

19 / 20

(72)

A New Step 6: Matchings

• It may happen that our algorithm gets stuck in a

homogeneous, antisymmetric m-scheme Π. What do we do then?

• We then look for a certain color in Π called a matching: A regular color P ∈ P s ∈ Π such that for some 1 ≤ i < j ≤ s : π ^s _i (P) = π _j ^s (P) and |π ^s _i (P)| = |P|.

• Such a matching color P has π ^s _i (π _j ^s ) ⁻¹ as automorphism, which can be used to refine the tensor power decomposition.

• Schemes Conjecture: Every homogeneous, antisymmetric 5-scheme has a matching.

• We have proved this conjecture for the only 5-schemes we know: group schemes.

19 / 20

(73)

A New Step 6: Matchings

• It may happen that our algorithm gets stuck in a

homogeneous, antisymmetric m-scheme Π. What do we do then?

• We then look for a certain color in Π called a matching: A regular color P ∈ P s ∈ Π such that for some 1 ≤ i < j ≤ s : π ^s _i (P) = π _j ^s (P) and |π ^s _i (P)| = |P|.

• Such a matching color P has π ^s _i (π _j ^s ) ⁻¹ as automorphism, which can be used to refine the tensor power decomposition.

• Schemes Conjecture: Every homogeneous, antisymmetric 5-scheme has a matching.

• We have proved this conjecture for the only 5-schemes we know: group schemes.

19 / 20

(74)

A New Step 6: Matchings

• It may happen that our algorithm gets stuck in a

homogeneous, antisymmetric m-scheme Π. What do we do then?

• We then look for a certain color in Π called a matching: A regular color P ∈ P s ∈ Π such that for some 1 ≤ i < j ≤ s : π ^s _i (P) = π _j ^s (P) and |π ^s _i (P)| = |P|.

• Such a matching color P has π ^s _i (π _j ^s ) ⁻¹ as automorphism, which can be used to refine the tensor power decomposition.

• Schemes Conjecture: Every homogeneous, antisymmetric 5-scheme has a matching.

• We have proved this conjecture for the only 5-schemes we know: group schemes.

19 / 20

(75)

A New Step 6: Matchings

• It may happen that our algorithm gets stuck in a

homogeneous, antisymmetric m-scheme Π. What do we do then?

• We then look for a certain color in Π called a matching: A regular color P ∈ P s ∈ Π such that for some 1 ≤ i < j ≤ s : π ^s _i (P) = π _j ^s (P) and |π ^s _i (P)| = |P|.

• Such a matching color P has π ^s _i (π _j ^s ) ⁻¹ as automorphism, which can be used to refine the tensor power decomposition.

• Schemes Conjecture: Every homogeneous, antisymmetric 5-scheme has a matching.

• We have proved this conjecture for the only 5-schemes we know: group schemes.

19 / 20

(76)

Towards the Conjecture

• It is clear that the conjecture implies a deterministic polynomial time factoring under GRH.

• We have the following partial results:

1. Every homogeneous, antisymmetric (m + 1)-scheme on n points has a matching if n is prime and (n − 1) is m-smooth.

2. Every homogeneous, antisymmetric m-scheme on n points has a matching if m = d ² ₃ log ₂ ne.

• Result (1) uses a recent classification result of Hanaki & Uno (2006) about 3-schemes, and it implies the main result of the paper.

• Result (2) follows by a matrix calculation.

• Further development of representation theory for m-schemes?

• Other examples of homogoneous, antisymmetric 4-schemes?

20 / 20

(77)

Towards the Conjecture

• It is clear that the conjecture implies a deterministic polynomial time factoring under GRH.

• We have the following partial results:

1. Every homogeneous, antisymmetric (m + 1)-scheme on n points has a matching if n is prime and (n − 1) is m-smooth.

2. Every homogeneous, antisymmetric m-scheme on n points has a matching if m = d ² ₃ log ₂ ne.

• Result (1) uses a recent classification result of Hanaki & Uno (2006) about 3-schemes, and it implies the main result of the paper.

• Result (2) follows by a matrix calculation.

• Further development of representation theory for m-schemes?

• Other examples of homogoneous, antisymmetric 4-schemes?

20 / 20

(78)

Towards the Conjecture

• It is clear that the conjecture implies a deterministic polynomial time factoring under GRH.

• We have the following partial results:

1. Every homogeneous, antisymmetric (m + 1)-scheme on n points has a matching if n is prime and (n − 1) is m-smooth.

2. Every homogeneous, antisymmetric m-scheme on n points has a matching if m = d ² ₃ log ₂ ne.

• Result (1) uses a recent classification result of Hanaki & Uno (2006) about 3-schemes, and it implies the main result of the paper.

• Result (2) follows by a matrix calculation.

• Further development of representation theory for m-schemes?

• Other examples of homogoneous, antisymmetric 4-schemes?

20 / 20

(79)

Towards the Conjecture

• It is clear that the conjecture implies a deterministic polynomial time factoring under GRH.

• We have the following partial results:

1. Every homogeneous, antisymmetric (m + 1)-scheme on n points has a matching if n is prime and (n − 1) is m-smooth.

2. Every homogeneous, antisymmetric m-scheme on n points has a matching if m = d ² ₃ log ₂ ne.

• Result (1) uses a recent classification result of Hanaki & Uno (2006) about 3-schemes, and it implies the main result of the paper.

• Result (2) follows by a matrix calculation.

• Further development of representation theory for m-schemes?

• Other examples of homogoneous, antisymmetric 4-schemes?

20 / 20

(80)

Towards the Conjecture

• It is clear that the conjecture implies a deterministic polynomial time factoring under GRH.

• We have the following partial results:

1. Every homogeneous, antisymmetric (m + 1)-scheme on n points has a matching if n is prime and (n − 1) is m-smooth.

2. Every homogeneous, antisymmetric m-scheme on n points has a matching if m = d ² ₃ log ₂ ne.

• Result (1) uses a recent classification result of Hanaki & Uno (2006) about 3-schemes, and it implies the main result of the paper.

• Result (2) follows by a matrix calculation.

• Further development of representation theory for m-schemes?

• Other examples of homogoneous, antisymmetric 4-schemes?

20 / 20

(81)

Towards the Conjecture

• It is clear that the conjecture implies a deterministic polynomial time factoring under GRH.

• We have the following partial results:

1. Every homogeneous, antisymmetric (m + 1)-scheme on n points has a matching if n is prime and (n − 1) is m-smooth.

2. Every homogeneous, antisymmetric m-scheme on n points has a matching if m = d ² ₃ log ₂ ne.

• Result (1) uses a recent classification result of Hanaki & Uno (2006) about 3-schemes, and it implies the main result of the paper.

• Result (2) follows by a matrix calculation.

• Further development of representation theory for m-schemes?

• Other examples of homogoneous, antisymmetric 4-schemes?

20 / 20

(82)

Towards the Conjecture

• It is clear that the conjecture implies a deterministic polynomial time factoring under GRH.

• We have the following partial results:

1. Every homogeneous, antisymmetric (m + 1)-scheme on n points has a matching if n is prime and (n − 1) is m-smooth.

2. Every homogeneous, antisymmetric m-scheme on n points has a matching if m = d ² ₃ log ₂ ne.

• Result (1) uses a recent classification result of Hanaki & Uno (2006) about 3-schemes, and it implies the main result of the paper.

• Result (2) follows by a matrix calculation.

• Further development of representation theory for m-schemes?

• Other examples of homogoneous, antisymmetric 4-schemes?

20 / 20

(83)

Towards the Conjecture

• It is clear that the conjecture implies a deterministic polynomial time factoring under GRH.

• We have the following partial results:

1. Every homogeneous, antisymmetric (m + 1)-scheme on n points has a matching if n is prime and (n − 1) is m-smooth.

2. Every homogeneous, antisymmetric m-scheme on n points has a matching if m = d ² ₃ log ₂ ne.

• Result (1) uses a recent classification result of Hanaki & Uno (2006) about 3-schemes, and it implies the main result of the paper.

• Result (2) follows by a matrix calculation.

• Further development of representation theory for m-schemes?

• Other examples of homogoneous, antisymmetric 4-schemes?

20 / 20

Schemes for Deterministic Polynomial Factoring