Fine-grain decomposition of F q

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1 Fine-grain decomposition of F q n David Thomson with Colin Weir Army Cyber Institute at USMA West Point David.Thomson@usma.edu WCNT 2015 D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

2 Fine-grain decomposition of F q n David Thomson with Dr. Weird Army Cyber Institute at USMA West Point David.Thomson@usma.edu WCNT 2015 D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

3 Motivation Definition. An element α F q n is normal over F q if and only if gcd(x n 1, αx n 1 + α q x n α qn 2 x + α qn 1 ) = 1. This essentially comes by decomposing the trace form T in the discriminant of (α, α q,..., α qn 1 ). Definition. An element α F q n is k-normal (over F q ) if ( ) deg gcd(x n 1, αx n 1 + α q x n α qn 2 x + α qn 1 ) = k. D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

4 Towards a Frobenius module Let R be a commutative ring with 1 R (because every ring has 1 R ), let V be an R-module and let T : V V be an R-module homomorphism on V. Let f R[x]; define the action of f on α V by f α = f(t )(α). Now, we basically turn any extension of R into an R[x]-module. We re most interested when R = F q and T = σ q, the Frobenius q-automorphism. D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

5 Finite fields and normal bases Finite fields and normal bases D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

6 Finite fields and normal bases F q n as a Frobenius module Let R = F q and let T = σ q : F q n F q n be the Frobenius q-automorphism, σ q (α) = α q. Let f(x) = n 1 i=0 a ix i F q [x] and let α F q. Then n 1 f α = a i α qi = F (α), i=0 where F (x) = n 1 i=0 a ix qi is the linearized q-associate of f. We know α F q n if and only if (x n 1) α = α qn α = 0. Remarks. Let F be a q-polynomial. 1 F is linear; that is F (ax + y) = af (x) + F (y) for a F q. 2 The roots of F form a vector space over F q, closed under σ q. D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

7 Finite fields and normal bases Normal bases Definition. Let α F q n and let N = {α, α q,..., α qn 1 }. If N is linearly independent, then N is a normal basis, α N is a normal element of F q n over F q, and N is generated by any of its elements. Proposition. The element α F q n is normal over F q if and only if it is not annihilated by any proper factor of x n 1. D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

8 Finite fields and normal bases Enter k-normality Remark. Let m σq,α F q [x] be the minimal polynomial of α. m σq,α divides x n 1, if deg(m σq,α) = n k, the elements {α, α q,..., α qn k 1 } are linearly independent. D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

9 Finite fields and normal bases Enter k-normality Remark. Let m σq,α F q [x] be the minimal polynomial of α. m σq,α divides x n 1, if deg(m σq,α) = n k, the elements {α, α q,..., α qn k 1 } are linearly independent. deg(gcd(x n 1, αx n α qn 1 )) = k. So the elements we were originally interested in are precisely cyclic vectors of σ q for subspaces of F q n. So let s describe all cyclic σ q -stable subspaces of F q n. D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

10 Finite fields and normal bases Decomposing F q n Remark. Describing normal elements in terms of decompositions has been studied previously; see, for example, Semaev (1989), Blake-Gao-Mullen (1995), Hachenberger (1996), Scheerhorn (1997) and Kyureghyan (2006). We follow the formula as prescribed by Steel (1997): 1 Break F q n into primary σ q -stable factors (primary decomposition of the Frobenius-module), 2 Shatter the factors into irreducible cyclic subspaces, if necessary. 3 Glue the irreducible cyclic subspaces together. D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

11 Finite fields and normal bases Orbits; primary decomposition Definition. The orbit of α under T is the span of {α, T α,...}, denoted Orb T (α). Moreover, α is a cyclic vector of Orb T. Theorem. Let V be a vector space over a field K and let T be a linear transformation of V with m T,V = f e 1 1 f e 2 2 f s es K[x], then 1 where m T,vi = f e i i, 1 i s. V = Orb T (v 1 ) Orb T (v s ), 2 Orb T (v i ) Orb T (v j ) = for i j. 3 Orb T (v i ) Orb T (v j ) = Orb T (v i + v j ). D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

12 Finite fields and normal bases Primary subspaces of F q n Corollary. Let V = F q n, K = F q and T = σ q. Then m T,V (x) = x n 1 = f e 1 1 f s es, and F q n = OrbT (v 1 ) Orb T (v s ), where each v i is a (n deg(f i )e i )-normal element. But we re not done yet: If Orb T (v i ) is not irreducible (e.g., if char(f q ) n), we will miss cyclic vectors of smaller dimensional subspaces. D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

13 Finite fields and normal bases Shatter-and-glue Theorem. Let W be a σ q -invariant subspace with m σq,w = f e, f irreducible. Let W i = ker f i and let U i = W i \ W i 1. Then 1 m σq,u i = f i for all u i U i. 2 Orb σq (u i ) is irreducible of dimension i deg(f). 3 If u = u i where u i U i, then Orb σq (u) has dimension k deg(f), where k is the largest index such that u k 0. D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

14 Finite fields and normal bases Corollaries Theorem. Let x n 1 = f e 1 1 f s es, where f i are irreducible (and the factorization is over F q [x]). 1 Any finite field F q n = V i, where V i = Orb σq (v i ) with m σq,v i = f e i i. 2 Furthermore, V i = V i,j, where V i,j = Orb σq (v i,j ) and v i,j ker(f j j 1 i ) \ ker(fi ). Moreover, each V i,j is irreducible. 3 For any α F q n, α = i,j α i,j with α i,j V i,j. Then α is normal over F q if and only if α i,ej 0 for all i. 4 Moreover, k-normal elements are cyclic vectors of co-dimension k subspaces; i.e., take α = α i,j where the sum is over all decompositions of n k into sums of the form j deg(f i ). D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

15 Examples Examples D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

16 Examples Example: V splits into 2 factors Example. Suppose p is a primitive root (mod n). Then x n 1 has irreducible factorization (x 1)(x n x + 1) and F q n = Orb(v ker(x 1)) Orb(w ker(x n x + 1)). There are exactly cyclic vectors of the first factor and exactly cyclic vectors of the second. Hence, there are D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

17 Examples Example: V splits into 2 factors Example. Suppose p is a primitive root (mod n). Then x n 1 has irreducible factorization (x 1)(x n x + 1) and F q n = Orb(v ker(x 1)) Orb(w ker(x n x + 1)). There are exactly q 1 cyclic vectors of the first factor and exactly q n 1 1 cyclic vectors of the second. Hence, there are D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

18 Examples Example: V splits into 2 factors Example. Suppose p is a primitive root (mod n). Then x n 1 has irreducible factorization (x 1)(x n x + 1) and F q n = Orb(v ker(x 1)) Orb(w ker(x n x + 1)). There are exactly q 1 cyclic vectors of the first factor and exactly q n 1 1 cyclic vectors of the second. Hence, there are 1 (q 1) (n 1)-normal elements, D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

19 Examples Example: V splits into 2 factors Example. Suppose p is a primitive root (mod n). Then x n 1 has irreducible factorization (x 1)(x n x + 1) and F q n = Orb(v ker(x 1)) Orb(w ker(x n x + 1)). There are exactly q 1 cyclic vectors of the first factor and exactly q n 1 1 cyclic vectors of the second. Hence, there are 1 (q 1) (n 1)-normal elements, D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

20 Examples Example: V splits into 2 factors Example. Suppose p is a primitive root (mod n). Then x n 1 has irreducible factorization (x 1)(x n x + 1) and F q n = Orb(v ker(x 1)) Orb(w ker(x n x + 1)). There are exactly q 1 cyclic vectors of the first factor and exactly q n 1 1 cyclic vectors of the second. Hence, there are 1 (q 1) (n 1)-normal elements, 2 (q n 1 1) 1-normal elements and D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

21 Examples Example: V splits into 2 factors Example. Suppose p is a primitive root (mod n). Then x n 1 has irreducible factorization (x 1)(x n x + 1) and F q n = Orb(v ker(x 1)) Orb(w ker(x n x + 1)). There are exactly q 1 cyclic vectors of the first factor and exactly q n 1 1 cyclic vectors of the second. Hence, there are 1 (q 1) (n 1)-normal elements, 2 (q n 1 1) 1-normal elements and D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

22 Examples Example: V splits into 2 factors Example. Suppose p is a primitive root (mod n). Then x n 1 has irreducible factorization (x 1)(x n x + 1) and F q n = Orb(v ker(x 1)) Orb(w ker(x n x + 1)). There are exactly q 1 cyclic vectors of the first factor and exactly q n 1 1 cyclic vectors of the second. Hence, there are 1 (q 1) (n 1)-normal elements, 2 (q n 1 1) 1-normal elements and 3 (q 1)(q n 1 1) 0-normal elements. D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

23 Examples Example: V splits into 2 factors Example. Suppose p is a primitive root (mod n). Then x n 1 has irreducible factorization (x 1)(x n x + 1) and F q n = Orb(v ker(x 1)) Orb(w ker(x n x + 1)). There are exactly q 1 cyclic vectors of the first factor and exactly q n 1 1 cyclic vectors of the second. Hence, there are 1 (q 1) (n 1)-normal elements, 2 (q n 1 1) 1-normal elements and 3 (q 1)(q n 1 1) 0-normal elements. Corollary. An element α is normal if and only if α = β + γ, where β F q, Tr(γ) = 0 and γβ = 0. D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

24 Examples Example: degree a power of the characteristic Example. Let q = 2, n = 64, then m σq,f q n (x) = (x 1) 64. Let U i = ker(x 1) i \ ker(x 1) i 1, then U 1 = F 2 \ {0} U 2 = F 4 \ F 2 U 3 = ker(x 3 + x 2 + x + 1) \ F 4. U 64 = F 2 64 \ ker(x n x + 1). Hence, the number of k-normal elements is 2 64 k 1, k = 0, 1,..., 63. D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

25 Examples Example: degree a power of the characteristic Example. Let q = 2, n = 64, then m σq,f q n (x) = (x 1) 64. Let U i = ker(x 1) i \ ker(x 1) i 1, then U 1 = F 2 \ {0} U 2 = F 4 \ F 2 U 3 = ker(x 3 + x 2 + x + 1) \ F 4. U 64 = F 2 64 \ ker(x n x + 1). Hence, the number of k-normal elements is 2 64 k 1, k = 0, 1,..., 63. Corollary. An element α F q p n is normal if and only if Tr(α) 0. D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

26 Examples Another example: Cyclic codes Definition. A linear code C of length n and dimension k is a k-dimensional subspace of F q n over F q. Let (a 0,..., a n 1 ) F n q, and let E be the cyclic left-shift operator; hence, E(a 0,..., a n 1 ) = (a 1,..., a n 1 ). Certainly, E n E = 0; that is, every α F n q is annihilated by x n 1. Definition. If E(C) = C, then C is a cyclic code. D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

27 Examples Classifying all quasi-cyclic codes Definition. A quasi-cyclic code C is a code such that x k C = C for some k coprime to n. In fact, by repeating the above process on k-normals, we have a correspondence between k-normal elements and quasi-cyclic codes of dimension n k. Other codes of interest: Negacyclic: E n (α) = α, so every α is annihilated by x n ± 1. Consta-cyclic: E b (α) = bα, so every α is annihilated by x n b n. D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

28 For the problem session: Completely normal elements For the problem session: Completely normal elements D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

29 For the problem session: Completely normal elements Completely normal Definition. An element α F q n is completely normal if it is normal over every intermediate extension F q d, where d n. Theorem. An element α F q n is completely normal if and only if it is annihilated by x n/d 1 and by no smaller factor when the factorization is over F q d for all d n. Question. Characterize completely normal elements in a similar fashion as normality: Seems easy if x n 1 splits over F q. Simple characterization for n = p e, p = char(f q ). Constructions for n = r e, r any prime, plus a product construction, gives existence. We (me and Colin and you) should be able to give characterizations. D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

30 For the problem session: Completely normal elements Bonus Let α be k-normal and β be l-normal. There are many other properties we may want to know about, for example: Discuss the normality of αβ (if α 0-normal in F q n and β 0-normal in F q m with gcd(n, m) = 1, then αβ normal in F q nm). β is dual to α if Tr(α qi β qj ) = δ(i, j). If α is k-normal with dual β, what is β? When is Tr(α) k-normal of the subfield? Primitive, k-normal elements. Character sum arguments for primitive 1-normality in [HMPT 2013], but room to improve. And the big question: Express α = i,j α i,j and β = i,j α i,j. What can we say about αβ? Can we compute αβ (more) efficiently? D. Thomson (ACI at West Point) Linear algebra over finite fields WCNT / 15

Something about normal bases over finite fields

Something about normal bases over finite fields David Thomson Carleton University dthomson@math.carleton.ca WCNT, December 17, 2013 Existence and properties of k-normal elements over finite fields David