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Factoring Polynomials of the form f (x

n

) ∈ F

q

[x ]

Fabio E. Brochero Mart´ınez

joint work with Lucas da Silva Reis

CIMPA Research School Algebraic Methods in Coding Theory

Universidade Federal de Minas Gerais Instituto de Ciˆencias Exatas Departamento de Matem´atica

July 11, 2017

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 1 / 18

(2)

Motivations

A [n,k]q-code C is calledcyclic if it is invariant by the shift permutation, i.e.,

if (a1,a2, . . . ,an)∈ C then the shift (an,a1, . . . ,an−1) is also inC. Since Fnq is isomorphic toRn= hxFnq−1i[x] , subspaces ofRn invariant by a shift are ideals andRn is a principal ideal domain, it follows that each ideal is generated by a polynomial g(x)∈ Rn, whereg is a divisor of xn−1.

Codes generated by a polynomial of the form xh(x)n−1, whereh is an irreducible factor of xn−1, are called minimal cyclic codes.

(3)

Motivations

A [n,k]q-code C is calledcyclic if it is invariant by the shift permutation, i.e.,

if (a1,a2, . . . ,an)∈ C then the shift (an,a1, . . . ,an−1) is also inC.

Since Fnq is isomorphic toRn= hxFnq−1i[x] , subspaces ofRn invariant by a shift are ideals andRn is a principal ideal domain, it follows that each ideal is generated by a polynomial g(x)∈ Rn, whereg is a divisor of xn−1.

Codes generated by a polynomial of the form xh(x)n−1, whereh is an irreducible factor of xn−1, are called minimal cyclic codes.

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 2 / 18

(4)

Motivations

A [n,k]q-code C is calledcyclic if it is invariant by the shift permutation, i.e.,

if (a1,a2, . . . ,an)∈ C then the shift (an,a1, . . . ,an−1) is also inC.

SinceFnq is isomorphic toRn= hxFnq−1i[x] , subspaces ofRn invariant by a shift are ideals andRn is a principal ideal domain, it follows that each ideal is generated by a polynomial g(x)∈ Rn, whereg is a divisor of xn−1.

Codes generated by a polynomial of the form xh(x)n−1, whereh is an irreducible factor of xn−1, are called minimal cyclic codes.

(5)

Motivations

A [n,k]q-code C is calledcyclic if it is invariant by the shift permutation, i.e.,

if (a1,a2, . . . ,an)∈ C then the shift (an,a1, . . . ,an−1) is also inC.

SinceFnq is isomorphic toRn= hxFnq−1i[x] , subspaces ofRn invariant by a shift are ideals andRn is a principal ideal domain, it follows that each ideal is generated by a polynomial g(x)∈ Rn, whereg is a divisor of xn−1.

Codes generated by a polynomial of the form xh(x)n−1, whereh is an irreducible factor of xn−1, are called minimal cyclic codes.

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 2 / 18

(6)

The polynomial xn−1∈Fq[x] splits into monic irreducible factors as xn−1 =f1f2· · ·fr by the Chinese Remainder Theorem

Rn= Fq[x]

hxn−1i '

r

M

j=1

Fq[x]

hfji

so every primitive idempotent generates a maximal ideal of Rnand also one component of this direct sum.

Lemma

Let Fq be a finite field with q elements and n be a positive integer such that gcd(q,n) = 1. Then every primitive idempotent of the group algebra Rn is of the form

ef =−((f)0)

n ·xn−1 f , where f(x)∈Fq[x] is an irreducible factor of xn−1.

(7)

The polynomial xn−1∈Fq[x] splits into monic irreducible factors as xn−1 =f1f2· · ·fr by the Chinese Remainder Theorem

Rn= Fq[x]

hxn−1i '

r

M

j=1

Fq[x]

hfji

so every primitive idempotent generates a maximal ideal of Rnand also one component of this direct sum.

Lemma

Let Fq be a finite field with q elements and n be a positive integer such that gcd(q,n) = 1. Then every primitive idempotent of the group algebra Rn is of the form

ef =−((f)0)

n ·xn−1 f , where f(x)∈Fq[x] is an irreducible factor of xn−1.

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 3 / 18

(8)

The polynomial xn−1∈Fq[x] splits into monic irreducible factors as xn−1 =f1f2· · ·fr by the Chinese Remainder Theorem

Rn= Fq[x]

hxn−1i '

r

M

j=1

Fq[x]

hfji

so every primitive idempotent generates a maximal ideal of Rnand also one component of this direct sum.

Lemma

Let Fq be a finite field with q elements and n be a positive integer such that gcd(q,n) = 1. Then every primitive idempotent of the group algebra Rn is of the form

ef =−((f)0)

n ·xn−1 f ,

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Example

It is well known that

xn−1 =Y

d|n

Φd(x),

in any field, where Φd(x) denotes thed-th cyclotomic polynomial.

In addition Φd(x) can be factor in ordϕ(d)

dq irreducible factor of degree orddq. Then Φd(x) is an irredutible polynomial if and only if orddq=ϕ(d) if and only if

1 d = 2 andq is odd

2 d = 4 andq≡3 (mod 4)

3 d =pk,p is a odd prime and hgi=U(Zpk)

4 d = 2pk,p is a odd prime andhgi=U(Z2pk)

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 4 / 18

(10)

Example

It is well known that

xn−1 =Y

d|n

Φd(x),

in any field, where Φd(x) denotes thed-th cyclotomic polynomial.

In addition Φd(x) can be factor in ordϕ(d)

dq irreducible factor of degree orddq.

Then Φd(x) is an irredutible polynomial if and only if orddq=ϕ(d) if and only if

1 d = 2 andq is odd

2 d = 4 andq≡3 (mod 4)

3 d =pk,p is a odd prime and hgi=U(Zpk)

4 d = 2pk,p is a odd prime andhgi=U(Z2pk)

(11)

Example

It is well known that

xn−1 =Y

d|n

Φd(x),

in any field, where Φd(x) denotes thed-th cyclotomic polynomial.

In addition Φd(x) can be factor in ordϕ(d)

dq irreducible factor of degree orddq.

Then Φd(x) is an irredutible polynomial

if and only if orddq=ϕ(d) if and only if

1 d = 2 andq is odd

2 d = 4 andq≡3 (mod 4)

3 d =pk,p is a odd prime and hgi=U(Zpk)

4 d = 2pk,p is a odd prime andhgi=U(Z2pk)

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 4 / 18

(12)

Example

It is well known that

xn−1 =Y

d|n

Φd(x),

in any field, where Φd(x) denotes thed-th cyclotomic polynomial.

In addition Φd(x) can be factor in ordϕ(d)

dq irreducible factor of degree orddq.

Then Φd(x) is an irredutible polynomial if and only if orddq=ϕ(d)

if and only if

1 d = 2 andq is odd

2 d = 4 andq≡3 (mod 4)

3 d =pk,p is a odd prime and hgi=U(Zpk)

4 d = 2pk,p is a odd prime andhgi=U(Z2pk)

(13)

Example

It is well known that

xn−1 =Y

d|n

Φd(x),

in any field, where Φd(x) denotes thed-th cyclotomic polynomial.

In addition Φd(x) can be factor in ordϕ(d)

dq irreducible factor of degree orddq.

Then Φd(x) is an irredutible polynomial if and only if orddq=ϕ(d) if and only if

1 d = 2 andq is odd

2 d = 4 andq≡3 (mod 4)

3 d =pk,p is a odd prime and hgi=U(Zpk)

4 d = 2pk,p is a odd prime andhgi=U(Z2pk)

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 4 / 18

(14)

Question

Determine explicitly every irreducible factor of xn−1∈Fq[x]

In general, Question

Given f(x)∈Fq[x]irreducible polynomial of degree m and order e and n a positive integer, determine explicitly every irreducible factor of f(xn) Question

When f(xn) is an irreducible polynomial and when f(xn) splits into n irreducible factors?

(15)

Question

Determine explicitly every irreducible factor of xn−1∈Fq[x]

In general, Question

Given f(x)∈Fq[x]irreducible polynomial of degree m and order e and n a positive integer, determine explicitly every irreducible factor of f(xn)

Question

When f(xn) is an irreducible polynomial and when f(xn) splits into n irreducible factors?

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 5 / 18

(16)

Question

Determine explicitly every irreducible factor of xn−1∈Fq[x]

In general, Question

Given f(x)∈Fq[x]irreducible polynomial of degree m and order e and n a positive integer, determine explicitly every irreducible factor of f(xn) Question

When f(xn) is an irreducible polynomial and when f(xn) splits into n irreducible factors?

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Theorem (Lidl-Niederreiter Theorem 3.35)

Let n be a positive integer and f(x)∈Fq[x]be an irreducible polynomial of degree m and order e. Then the polynomial f(xn) is irreducible overFq

if and only if the following conditions are satisfied:

1 Every prime divisor of n divides e,

2 gcd(n,(qm−1)/e) = 1

3 if4|n then 4|qm−1.

In addition, in the case where the polynomial f(xn) is irreducible, it has degree mn and order en.

Remark

Observe that the conditions (1) and (2) of Theorem before can be rewritten as

νp(e)≥1 and νp(qm−1) =νp(e) for every prime divisor p of n.

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 6 / 18

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Theorem (Lidl-Niederreiter Theorem 3.35)

Let n be a positive integer and f(x)∈Fq[x]be an irreducible polynomial of degree m and order e. Then the polynomial f(xn) is irreducible overFq

if and only if the following conditions are satisfied:

1 Every prime divisor of n divides e,

2 gcd(n,(qm−1)/e) = 1

3 if4|n then 4|qm−1.

In addition, in the case where the polynomial f(xn) is irreducible, it has degree mn and order en.

Remark

Observe that the conditions (1) and (2) of Theorem before can be rewritten as

νp(e)≥1 and νp(qm−1) =νp(e) for every prime divisor p of n.

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Theorem (Lidl-Niederreiter Theorem 3.35)

Let n be a positive integer and f(x)∈Fq[x]be an irreducible polynomial of degree m and order e. Then the polynomial f(xn) is irreducible overFq

if and only if the following conditions are satisfied:

1 Every prime divisor of n divides e,

2 gcd(n,(qm−1)/e) = 1

3 if4|n then 4|qm−1.

In addition, in the case where the polynomial f(xn) is irreducible, it has degree mn and order en.

Remark

Observe that the conditions (1) and (2) of Theorem before can be rewritten as

νp(e)≥1 and νp(qm−1) =νp(e) for every prime divisor p of n.

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 6 / 18

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Theorem (Lidl-Niederreiter Theorem 3.35)

Let n be a positive integer and f(x)∈Fq[x]be an irreducible polynomial of degree m and order e. Then the polynomial f(xn) is irreducible overFq

if and only if the following conditions are satisfied:

1 Every prime divisor of n divides e,

2 gcd(n,(qm−1)/e) = 1

3 if4|n then 4|qm−1.

In addition, in the case where the polynomial f(xn)is irreducible, it has degree mn and order en.

Remark

Observe that the conditions (1) and (2) of Theorem before can be rewritten as

νp(e)≥1 and νp(qm−1) =νp(e) for every prime divisor p of n.

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Theorem (Lidl-Niederreiter Theorem 3.35)

Let n be a positive integer and f(x)∈Fq[x]be an irreducible polynomial of degree m and order e. Then the polynomial f(xn) is irreducible overFq

if and only if the following conditions are satisfied:

1 Every prime divisor of n divides e,

2 gcd(n,(qm−1)/e) = 1

3 if4|n then 4|qm−1.

In addition, in the case where the polynomial f(xn)is irreducible, it has degree mn and order en.

Remark

Observe that the conditions (1) and (2) of Theorem before can be rewritten as

νp(e)≥1 and νp(qm−1) =νp(e) for every prime divisor p of n.

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 6 / 18

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Theorem (Butler)

Let f(x)∈Fq[x]be a irreducible polynomial of degree m and order e. Let n be a positive integer such that gcd(n,q) = 1.

1 If rad(n)divides e, then f(xn) splits in exactly ordmn

neq irreducible factors of degree ordneq and order ne.

2 If gcd(n,e) = 1, then for each d divisor of n, f(xn) has in its factorization exactly mordφ(d)

deq irreducible factors of degree orddeq and order de. In addition, every irreducible factor is of this type.

Remark

f(xn) splits into n irreducible factors if ordneq =ordeq. Since m=ordeq, the condition is equivalent to νp(qm−1)≥νp(n) +νp(e) for all p prime divisor of n.

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Theorem (Butler)

Let f(x)∈Fq[x]be a irreducible polynomial of degree m and order e. Let n be a positive integer such that gcd(n,q) = 1.

1 If rad(n)divides e, then f(xn) splits in exactly ordmn

neq irreducible factors of degree ordneq and order ne.

2 If gcd(n,e) = 1, then for each d divisor of n, f(xn) has in its factorization exactly mordφ(d)

deq irreducible factors of degree orddeq and order de. In addition, every irreducible factor is of this type.

Remark

f(xn) splits into n irreducible factors if ordneq =ordeq. Since m=ordeq, the condition is equivalent to νp(qm−1)≥νp(n) +νp(e) for all p prime divisor of n.

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 7 / 18

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Theorem (Butler)

Let f(x)∈Fq[x]be a irreducible polynomial of degree m and order e. Let n be a positive integer such that gcd(n,q) = 1.

1 If rad(n)divides e, then f(xn) splits in exactly ordmn

neq irreducible factors of degree ordneq and order ne.

2 If gcd(n,e) = 1, then for each d divisor of n, f(xn) has in its factorization exactly mordφ(d)

deq irreducible factors of degree orddeq and order de. In addition, every irreducible factor is of this type.

Remark

f(xn) splits into n irreducible factors if ordneq =ordeq. Since m=ordeq, the condition is equivalent to νp(qm−1)≥νp(n) +νp(e) for all p prime divisor of n.

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Lemma

Let f(x) be an irreducible polynomial of degree m and exponent e. Let n >1 be a positive divisor of q−1 such that

νp(n) +νp(e)≤νp(q−1) +νp(ordrpq)

for all prime divisors p of n, where rp is the largest divisor of e prime with p, i.e., rp = e

pvp(e). Then the polynomial f(xn) splits as a product of n irreducible polynomials of degree m. In addition, if g(x)is any monic irreducible factor of f(xn)and c is any element of U(n), then

f(xn) =

n−1

Y

i=0

[c−mjg(cjx)]

is the factorization of f(xn) into irreducible factors.

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 8 / 18

(26)

Remark Since

νp(qm−1)≥νp(q−1) +νp(ordrpq)≥νp(e) +νp(n) for all prime divisors p of n, and then the condition on Lemma is a sufficient (but not necessary) condition for f(xn)being a reducible polynomial.

Definition

Let f(x)∈Fq[x] be a monic irreducible polynomial of degreemand exponent e. We say that the pair hf(x),nisatisfies thereducible condition if

νp(q−1)≥νp(n) +νp(e) for every prime divisor p of n.

(27)

Remark Since

νp(qm−1)≥νp(q−1) +νp(ordrpq)≥νp(e) +νp(n) for all prime divisors p of n, and then the condition on Lemma is a sufficient (but not necessary) condition for f(xn)being a reducible polynomial.

Definition

Let f(x)∈Fq[x] be a monic irreducible polynomial of degreemand exponent e. We say that the pair hf(x),nisatisfies thereducible condition if

νp(q−1)≥νp(n) +νp(e) for every prime divisor p of n.

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 9 / 18

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Theorem

Let f(x)∈Fq[x]be a monic irreducible polynomial of degree m and exponent e, and let pt be such that hf(x),ptisatisfies the reducible condition. Suppose that k =νp(e) and e =pkr . Then

(a) There exists an unique element c ∈Fq such that f(x) divides xr −c.

(b) Let s be the solution of sr ≡1 (mod pt) with 0<s <pt and let l = sr−1pt . If α∈Fq is a root of f(x), the polynomial

g(x) =Qm

j=1(x−bsα−lqj) is an irreducible factor of f(xpt)overFq. (c) The element a=bpk is in U(pt) and the polynomial f(xpt) has the

following factorization in Fq[x]:

f(xpt) =

pt−1

Y

j=0

[a−mjg(ajx)].

(29)

Theorem

Let f(x)∈Fq[x]be a monic irreducible polynomial of degree m and exponent e, and let pt be such that hf(x),ptisatisfies the reducible condition. Suppose that k =νp(e) and e =pkr . Then

(a) There exists an unique element c ∈Fq such that f(x) divides xr −c.

(b) Let s be the solution of sr ≡1 (mod pt) with0<s <pt and let l = srp−1t . If α∈Fq is a root of f(x), the polynomial

g(x) =Qm

j=1(x−bsα−lqj) is an irreducible factor of f(xpt)overFq.

(c) The element a=bpk is in U(pt) and the polynomial f(xpt) has the following factorization in Fq[x]:

f(xpt) =

pt−1

Y

j=0

[a−mjg(ajx)].

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 10 / 18

(30)

Theorem

Let f(x)∈Fq[x]be a monic irreducible polynomial of degree m and exponent e, and let pt be such that hf(x),ptisatisfies the reducible condition. Suppose that k =νp(e) and e =pkr . Then

(a) There exists an unique element c ∈Fq such that f(x) divides xr −c.

(b) Let s be the solution of sr ≡1 (mod pt) with0<s <pt and let l = srp−1t . If α∈Fq is a root of f(x), the polynomial

g(x) =Qm

j=1(x−bsα−lqj) is an irreducible factor of f(xpt)overFq. (c) The element a=bpk is in U(pt) and the polynomial f(xpt) has the

following factorization in Fq[x]:

f(xpt) =

pt−1

Y[a−mjg(ajx)].

(31)

Remark

Ifhf(x),ni satisfies the reducible condition, where n=Qu

i=1piβi, then iterating the process for each prime divisor we obtain the n irreducible factors of f(xn) overFq.

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 11 / 18

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Example

Consider the irreducible polynomial f(x) =x2−11x+ 1∈F59[x] of degree 2 and order 12

We are going to find the complete factorization of f(x29d+1) for all d ≥0. Case d = 0: Using the notation of Theorem, we have r = 12 and 12s ≡1 (mod 19). Then s = 17 and we setl = rs−129 = 7. Now, by quadratic reciprocity law we can prove that 5∈ U(29)⊂F59.

Since A=

0 1

−1 11

is the companion matrix of f(x) , from Theorem

g(x) = det(xI −bsAl) = det(xI −517A7) is a factor of f(x29).

NowA7 =

0 −1 1 −11

=−Aand 517≡36 (mod 59),therefore

g(x) = det(xI + 23A) =

x 36

23 x−17

=x2−17x−2 =x2+ 42x+ 57.

(33)

Example

Consider the irreducible polynomial f(x) =x2−11x+ 1∈F59[x] of degree 2 and order 12

We are going to find the complete factorization of f(x29d+1) for all d ≥0.

Case d = 0: Using the notation of Theorem, we have r = 12 and 12s ≡1 (mod 19). Then s = 17 and we setl = rs−129 = 7. Now, by quadratic reciprocity law we can prove that 5∈ U(29)⊂F59.

Since A=

0 1

−1 11

is the companion matrix of f(x) , from Theorem

g(x) = det(xI −bsAl) = det(xI −517A7) is a factor of f(x29).

NowA7 =

0 −1 1 −11

=−Aand 517≡36 (mod 59),therefore

g(x) = det(xI + 23A) =

x 36

23 x−17

=x2−17x−2 =x2+ 42x+ 57.

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 12 / 18

(34)

Example

Consider the irreducible polynomial f(x) =x2−11x+ 1∈F59[x] of degree 2 and order 12

We are going to find the complete factorization of f(x29d+1) for all d ≥0.

Case d = 0: Using the notation of Theorem, we have r = 12 and 12s ≡1 (mod 19). Thens = 17 and we setl = rs−129 = 7.

Now, by quadratic reciprocity law we can prove that 5∈ U(29)⊂F59.

Since A=

0 1

−1 11

is the companion matrix of f(x) , from Theorem

g(x) = det(xI −bsAl) = det(xI −517A7) is a factor of f(x29).

NowA7 =

0 −1 1 −11

=−Aand 517≡36 (mod 59),therefore

g(x) = det(xI + 23A) =

x 36

23 x−17

=x2−17x−2 =x2+ 42x+ 57.

(35)

Example

Consider the irreducible polynomial f(x) =x2−11x+ 1∈F59[x] of degree 2 and order 12

We are going to find the complete factorization of f(x29d+1) for all d ≥0.

Case d = 0: Using the notation of Theorem, we have r = 12 and 12s ≡1 (mod 19). Thens = 17 and we setl = rs−129 = 7. Now, by quadratic reciprocity law we can prove that 5∈ U(29)⊂F59.

Since A=

0 1

−1 11

is the companion matrix of f(x) , from Theorem

g(x) = det(xI −bsAl) = det(xI −517A7) is a factor of f(x29).

NowA7 =

0 −1 1 −11

=−Aand 517≡36 (mod 59),therefore

g(x) = det(xI + 23A) =

x 36

23 x−17

=x2−17x−2 =x2+ 42x+ 57.

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 12 / 18

(36)

Example

Consider the irreducible polynomial f(x) =x2−11x+ 1∈F59[x] of degree 2 and order 12

We are going to find the complete factorization of f(x29d+1) for all d ≥0.

Case d = 0: Using the notation of Theorem, we have r = 12 and 12s ≡1 (mod 19). Thens = 17 and we setl = rs−129 = 7. Now, by quadratic reciprocity law we can prove that 5∈ U(29)⊂F59.

Since A=

0 1

−1 11

is the companion matrix of f(x) , from Theorem

g(x) = det(xI −bsAl) = det(xI−517A7) is a factor of f(x29).

NowA7 =

0 −1 1 −11

=−Aand 517≡36 (mod 59),therefore

g(x) = det(xI + 23A) =

x 36

23 x−17

=x2−17x−2 =x2+ 42x+ 57.

(37)

Example

Consider the irreducible polynomial f(x) =x2−11x+ 1∈F59[x] of degree 2 and order 12

We are going to find the complete factorization of f(x29d+1) for all d ≥0.

Case d = 0: Using the notation of Theorem, we have r = 12 and 12s ≡1 (mod 19). Thens = 17 and we setl = rs−129 = 7. Now, by quadratic reciprocity law we can prove that 5∈ U(29)⊂F59.

Since A=

0 1

−1 11

is the companion matrix of f(x) , from Theorem

g(x) = det(xI −bsAl) = det(xI−517A7) is a factor of f(x29).

NowA7 =

0 −1 1 −11

=−Aand 517≡36 (mod 59),therefore

g(x) = det(xI + 23A) =

x 36

23 x−17

=x2−17x−2 =x2+ 42x+ 57.

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 12 / 18

(38)

Example

Moreover, every monic irreducible factors of f(x29) have the form

gj(x) = 5−2jg(5jx) = 5−2j(25jx2+42·5jx+57) =x2+(42·5−j)x+57·5−2j where j = 0,· · ·,28. i.e

x58−11x29+ 1 =

28

Y

i=0

(x2+ 42·12jx+ 57·26j).

Each factor gj(x) has degree 2 and exponent 12·29. Hence the polynomials gj(x29d) are irreducible. Therefore

f(x29d+1) =

28

Y

i=0

(x2·29d + 42·12jx29d + 57·26j).

(39)

Example

Moreover, every monic irreducible factors of f(x29) have the form

gj(x) = 5−2jg(5jx) = 5−2j(25jx2+42·5jx+57) =x2+(42·5−j)x+57·5−2j where j = 0,· · ·,28. i.e

x58−11x29+ 1 =

28

Y

i=0

(x2+ 42·12jx+ 57·26j).

Each factor gj(x) has degree 2 and exponent 12·29. Hence the polynomials gj(x29d) are irreducible. Therefore

f(x29d+1) =

28

Y

i=0

(x2·29d + 42·12jx29d + 57·26j).

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 13 / 18

(40)

Algorithm A.

This algorithm takes as input an irreducible polynomialf ∈Fq[x] of degree m and ordere, andpt a power of a prime.

Step A1. Computeνp(e), νp(q−1) andr := pνe

p(e) and verify that νp(q−1)≥t+ν(e)

Step A2. Computec :=xr (mod f(x)).

Step A3. Compute an elementb such that bpt =c.

Step A4. Computesandl such thatrs≡1 (mod pt) andl := sr−1pt . Step A5. Computeβ=x−lbs modf(x).

Step A6. Compute one factor of f(y) as g0(y) = (y −β)(y − βq)· · ·(y−βqm−1)∈ (fFq(x))[x][y].

Step A7. Pick random elementsα ∈Fq until α(q−1)/p 6= 1. Then a:=α(q−1)/pt is an element of orderpt.

Step A8. Compute the other factors off(y) asgj(y) =a−jmg(ajy) for j = 1, . . . ,pt−1.

(41)

Algorithm A.

This algorithm takes as input an irreducible polynomialf ∈Fq[x] of degree m and ordere, andpt a power of a prime.

Step A1. Computeνp(e), νp(q−1) andr := pνe

p(e) and verify that νp(q−1)≥t+ν(e)

Step A2. Computec :=xr (mod f(x)).

Step A3. Compute an elementb such that bpt =c.

Step A4. Computesandl such thatrs≡1 (mod pt) andl := sr−1pt . Step A5. Computeβ=x−lbs modf(x).

Step A6. Compute one factor of f(y) as g0(y) = (y −β)(y − βq)· · ·(y−βqm−1)∈ (fFq(x))[x][y].

Step A7. Pick random elementsα ∈Fq until α(q−1)/p 6= 1. Then a:=α(q−1)/pt is an element of orderpt.

Step A8. Compute the other factors off(y) asgj(y) =a−jmg(ajy) for j = 1, . . . ,pt−1.

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 14 / 18

(42)

Algorithm A.

This algorithm takes as input an irreducible polynomialf ∈Fq[x] of degree m and ordere, andpt a power of a prime.

Step A1. Computeνp(e), νp(q−1) andr := pνe

p(e) and verify that νp(q−1)≥t+ν(e)

Step A2. Computec :=xr (mod f(x)).

Step A3. Compute an elementb such that bpt =c.

Step A4. Computesandl such thatrs≡1 (mod pt) andl := sr−1pt . Step A5. Computeβ=x−lbs modf(x).

Step A6. Compute one factor of f(y) as g0(y) = (y −β)(y − βq)· · ·(y−βqm−1)∈ (fFq(x))[x][y].

Step A7. Pick random elementsα ∈Fq until α(q−1)/p 6= 1. Then a:=α(q−1)/pt is an element of orderpt.

Step A8. Compute the other factors off(y) asgj(y) =a−jmg(ajy) for j = 1, . . . ,pt−1.

(43)

Algorithm A.

This algorithm takes as input an irreducible polynomialf ∈Fq[x] of degree m and ordere, andpt a power of a prime.

Step A1. Computeνp(e), νp(q−1) andr := pνe

p(e) and verify that νp(q−1)≥t+ν(e)

Step A2. Computec :=xr (mod f(x)).

Step A3. Compute an elementb such that bpt =c.

Step A4. Computesandl such thatrs≡1 (mod pt) andl := sr−1pt . Step A5. Computeβ=x−lbs modf(x).

Step A6. Compute one factor of f(y) as g0(y) = (y −β)(y − βq)· · ·(y−βqm−1)∈ (fFq(x))[x][y].

Step A7. Pick random elementsα ∈Fq until α(q−1)/p 6= 1. Then a:=α(q−1)/pt is an element of orderpt.

Step A8. Compute the other factors off(y) asgj(y) =a−jmg(ajy) for j = 1, . . . ,pt−1.

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 14 / 18

(44)

Algorithm A.

This algorithm takes as input an irreducible polynomialf ∈Fq[x] of degree m and ordere, andpt a power of a prime.

Step A1. Computeνp(e), νp(q−1) andr := pνe

p(e) and verify that νp(q−1)≥t+ν(e)

Step A2. Computec :=xr (mod f(x)).

Step A3. Compute an elementb such that bpt =c.

Step A4. Computesandl such thatrs ≡1 (mod pt) andl := sr−1pt .

Step A5. Computeβ=x−lbs modf(x).

Step A6. Compute one factor of f(y) as g0(y) = (y −β)(y − βq)· · ·(y−βqm−1)∈ (fFq(x))[x][y].

Step A7. Pick random elementsα ∈Fq until α(q−1)/p 6= 1. Then a:=α(q−1)/pt is an element of orderpt.

Step A8. Compute the other factors off(y) asgj(y) =a−jmg(ajy) for j = 1, . . . ,pt−1.

(45)

Algorithm A.

This algorithm takes as input an irreducible polynomialf ∈Fq[x] of degree m and ordere, andpt a power of a prime.

Step A1. Computeνp(e), νp(q−1) andr := pνe

p(e) and verify that νp(q−1)≥t+ν(e)

Step A2. Computec :=xr (mod f(x)).

Step A3. Compute an elementb such that bpt =c.

Step A4. Computesandl such thatrs ≡1 (mod pt) andl := sr−1pt . Step A5. Computeβ=x−lbs modf(x).

Step A6. Compute one factor of f(y) as g0(y) = (y −β)(y − βq)· · ·(y−βqm−1)∈ (fFq(x))[x][y].

Step A7. Pick random elementsα ∈Fq until α(q−1)/p 6= 1. Then a:=α(q−1)/pt is an element of orderpt.

Step A8. Compute the other factors off(y) asgj(y) =a−jmg(ajy) for j = 1, . . . ,pt−1.

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 14 / 18

(46)

Algorithm A.

This algorithm takes as input an irreducible polynomialf ∈Fq[x] of degree m and ordere, andpt a power of a prime.

Step A1. Computeνp(e), νp(q−1) andr := pνe

p(e) and verify that νp(q−1)≥t+ν(e)

Step A2. Computec :=xr (mod f(x)).

Step A3. Compute an elementb such that bpt =c.

Step A4. Computesandl such thatrs ≡1 (mod pt) andl := sr−1pt . Step A5. Computeβ=x−lbs modf(x).

Step A6. Compute one factor of f(y) as g0(y) = (y −β)(y − βq)· · ·(y−βqm−1)∈ (fFq(x))[x][y].

Step A7. Pick random elementsα ∈Fq until α(q−1)/p 6= 1. Then a:=α(q−1)/pt is an element of orderpt.

Step A8. Compute the other factors off(y) asgj(y) =a−jmg(ajy) for j = 1, . . . ,pt−1.

(47)

Algorithm A.

This algorithm takes as input an irreducible polynomialf ∈Fq[x] of degree m and ordere, andpt a power of a prime.

Step A1. Computeνp(e), νp(q−1) andr := pνe

p(e) and verify that νp(q−1)≥t+ν(e)

Step A2. Computec :=xr (mod f(x)).

Step A3. Compute an elementb such that bpt =c.

Step A4. Computesandl such thatrs ≡1 (mod pt) andl := sr−1pt . Step A5. Computeβ=x−lbs modf(x).

Step A6. Compute one factor of f(y) as g0(y) = (y −β)(y − βq)· · ·(y−βqm−1)∈ (fFq(x))[x][y].

Step A7. Pick random elements α ∈Fq until α(q−1)/p 6= 1. Then a:=α(q−1)/pt is an element of orderpt.

Step A8. Compute the other factors off(y) asgj(y) =a−jmg(ajy) for j = 1, . . . ,pt−1.

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 14 / 18

(48)

Algorithm A.

This algorithm takes as input an irreducible polynomialf ∈Fq[x] of degree m and ordere, andpt a power of a prime.

Step A1. Computeνp(e), νp(q−1) andr := pνe

p(e) and verify that νp(q−1)≥t+ν(e)

Step A2. Computec :=xr (mod f(x)).

Step A3. Compute an elementb such that bpt =c.

Step A4. Computesandl such thatrs ≡1 (mod pt) andl := sr−1pt . Step A5. Computeβ=x−lbs modf(x).

Step A6. Compute one factor of f(y) as g0(y) = (y −β)(y − βq)· · ·(y−βqm−1)∈ (fFq(x))[x][y].

Step A7. Pick random elements α ∈Fq until α(q−1)/p 6= 1. Then

(49)

Computational Complexity

Taking powers in Fq and calculatingxd (mod f(x)) (Steps A2 and A5)

Ifa∈Fq, taking squares successively is a well-known fast process for finding an in essentially 2 log2(n) products of elements inFq.

The product of two polynomials and reduction modulof(x) can be done with

O(mlogmlog logm)

products in Fq using the fast Euclidean algorithm and the Cantor-Kaltofen Algorithm.

Thus the computation of xd (mod f(x)) whend >m requires O(mlog d

mlogmlog logm) products in Fq.

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 15 / 18

(50)

Computational Complexity

Taking powers in Fq and calculatingxd (mod f(x)) (Steps A2 and A5)

Ifa∈Fq, taking squares successively is a well-known fast process for finding an in essentially 2 log2(n) products of elements inFq.

The product of two polynomials and reduction modulof(x) can be done with

O(mlogmlog logm)

products in Fq using the fast Euclidean algorithm and the Cantor-Kaltofen Algorithm.

Thus the computation of xd (mod f(x)) whend >m requires O(mlog d

mlogmlog logm) products in Fq.

(51)

Computational Complexity

Taking powers in Fq and calculatingxd (mod f(x)) (Steps A2 and A5)

Ifa∈Fq, taking squares successively is a well-known fast process for finding an in essentially 2 log2(n) products of elements inFq.

The product of two polynomials and reduction modulof(x) can be done with

O(mlogmlog logm)

products in Fq using the fast Euclidean algorithm and the Cantor-Kaltofen Algorithm.

Thus the computation of xd (mod f(x)) whend >m requires O(mlog d

mlogmlog logm) products in Fq.

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 15 / 18

(52)

Taking roots in Fq (Step A3)

Taking p-root in a finite field can be computed by means of the Adleman Manders Miller algorithm in

O(pνp(q−1) log3q) steps.

Iterating this algorithm, we can solve the equation xpt −c = 0 (or find a primitive pt−th root of unity whenc = 1) and the algorithm has

complexity

O(ptlog3q).

In the special case when t=νp(q−1), i.e. gcd(pt,(q−1)/pt) = 1, we can use Barreto Voloch algorithm, which has complexity

O(ptlog logqlogq).

(53)

Taking roots in Fq (Step A3)

Taking p-root in a finite field can be computed by means of the Adleman Manders Miller algorithm in

O(pνp(q−1) log3q) steps.

Iterating this algorithm, we can solve the equation xpt −c = 0 (or find a primitive pt−th root of unity whenc = 1) and the algorithm has

complexity

O(ptlog3q).

In the special case when t=νp(q−1), i.e. gcd(pt,(q−1)/pt) = 1, we can use Barreto Voloch algorithm, which has complexity

O(ptlog logqlogq).

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 16 / 18

(54)

Taking roots in Fq (Step A3)

Taking p-root in a finite field can be computed by means of the Adleman Manders Miller algorithm in

O(pνp(q−1) log3q) steps.

Iterating this algorithm, we can solve the equation xpt −c = 0 (or find a primitive pt−th root of unity whenc = 1) and the algorithm has

complexity

O(ptlog3q).

In the special case when t=νp(q−1), i.e. gcd(pt,(q−1)/pt) = 1, we can use Barreto Voloch algorithm, which has complexity

(55)

Computation of the minimal polynomial of β∈Fq[x]/(f(x)) (Step A6) Using an algorithm of Shoup, the minimal polynomial ofβ can be computed in

O(m1.688) operations in Fq.

Note that if n=pt11· · ·piti, we can iterate the algorithmi times, where i is at mostO(logn), hence at mostO(logq).

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 17 / 18

(56)

In conclusion, ifhf(x),ni satisfies the reducible condition, we find the complete factorization of f(xn) over Fq with complexity bounded by

O(mlog(M/m) logmlog logmlogq+m1.688logq+nlog3q), where M := max{r,l}<qm.

In the worst case, we have logM =O(mlogq), and the complexity is bounded by

O(m˜ 2log2q+nlog3q).

On other hand,f(xn) is a polynomial of degreemn such that each of its irreducible factors has degree m, using the probabilistic algorithm of von zur Gathen and Shoup the expected number of operations is

O((nm)1.688+ (nm)1+o(1)logq).

Therefore, our algorithm is faster than the one of von zur Gathen and Shoup in the case where q is not very big (q <exp (mn)0.5626

) and the order of growth of n is greater than

O(m˜ 0.185(logq)1.185).

(57)

In conclusion, ifhf(x),ni satisfies the reducible condition, we find the complete factorization of f(xn) over Fq with complexity bounded by

O(mlog(M/m) logmlog logmlogq+m1.688logq+nlog3q), where M := max{r,l}<qm. In the worst case, we have

logM =O(mlogq), and the complexity is bounded by O(m˜ 2log2q+nlog3q).

On other hand,f(xn) is a polynomial of degreemn such that each of its irreducible factors has degree m, using the probabilistic algorithm of von zur Gathen and Shoup the expected number of operations is

O((nm)1.688+ (nm)1+o(1)logq).

Therefore, our algorithm is faster than the one of von zur Gathen and Shoup in the case where q is not very big (q <exp (mn)0.5626

) and the order of growth of n is greater than

O(m˜ 0.185(logq)1.185).

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 18 / 18

(58)

In conclusion, ifhf(x),ni satisfies the reducible condition, we find the complete factorization of f(xn) over Fq with complexity bounded by

O(mlog(M/m) logmlog logmlogq+m1.688logq+nlog3q), where M := max{r,l}<qm. In the worst case, we have

logM =O(mlogq), and the complexity is bounded by O(m˜ 2log2q+nlog3q).

On other hand,f(xn) is a polynomial of degreemn such that each of its irreducible factors has degree m, using the probabilistic algorithm of von zur Gathen and Shoup the expected number of operations is

O((nm)1.688+ (nm)1+o(1)logq).

Therefore, our algorithm is faster than the one of von zur Gathen and Shoup in the case where q is not very big (q <exp (mn)0.5626

) and the order of growth of n is greater than

O(m˜ 0.185(logq)1.185).

(59)

In conclusion, ifhf(x),ni satisfies the reducible condition, we find the complete factorization of f(xn) over Fq with complexity bounded by

O(mlog(M/m) logmlog logmlogq+m1.688logq+nlog3q), where M := max{r,l}<qm. In the worst case, we have

logM =O(mlogq), and the complexity is bounded by O(m˜ 2log2q+nlog3q).

On other hand,f(xn) is a polynomial of degreemn such that each of its irreducible factors has degree m, using the probabilistic algorithm of von zur Gathen and Shoup the expected number of operations is

O((nm)1.688+ (nm)1+o(1)logq).

Therefore, our algorithm is faster than the one of von zur Gathen and Shoup in the case where q is not very big (q <exp (mn)0.5626

) and the order of growth of n is greater than

O(m˜ 0.185(logq)1.185).

F.E. Brochero Mart´ınez (UFMG) Factoring Polynomials of the formf(xn)Fq[x] July 11, 2017 18 / 18

Referências

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