Lattices associated with Octonion Algebras

In this paper we propose a construction of lattices in dimension 8n via octonion orders over a totally real number field. To show the potential of this construction, we present rotated versions of E8, Λ16, the densest known lattices in these dimensions and a lattice with the same density of the Barnes-Wall lattice in dimension 32. Perspectives applications of these lattices based on octonion algebras are in lattice based cryptography and lattice coding for MIMO and MISO channels.


I. INTRODUCTION
Signal constellations having a lattice structure have been studied as meaningful tools for transmitting data over both Gaussian and single-antenna Rayleigh fading channels [1].The problem of finding good signal constellations for a Gaussian channel is associated to the search for lattices with high packing density [2].
Algebraic number theory has been used as mathematical tool that enables the design of good coding schemes also due to coding/decoding properties.It has been shown that algebraic lattices, i.e., lattices constructed via the canonical embedding of an algebraic number field, provide an efficient tool for designing lattice codes for transmission over the single-antenna Rayleigh fading channel [3].The reason is that the two main design parameters, namely the modulation diversity and the minimum product distance, can be related to properties of the underlying number field: the maximal diversity is guaranteed when using totally real number fields and the minimum product distance can be related to the field discriminant [1].In the search for lattices which can be the support for design codes for both Gaussian and Rayleigh channels, algebraic rotated lattices can be considered, by requiring both bigger product distance (smaller error probability) and good density (coding gain) [4].
Space-time block codes provide the bridge between the practice of wireless communication with the mathematics of quadratic forms developed by Radon and Hurwitz.The columns of the code represent different time slots and the rows the transmit antennas, and the entries are the symbols to be transmitted [7].This connection was explored in the Alamouti space time block code for two transmit antennas in the pionner paper [8] where multiplication in the ring of quaternions was used.Maximal quaternion orders have been proposed in the context of space-time block codes in [9] and complex codes constructions based on cyclic division algebras are proposed in [10].More recently, the E 8 -lattice was constructed over an imaginary quadratic field [11].Codewords are usually built over the complex field.However for ultra wideband communication, one needs to design them over the real field [12].Thus, having the construction of rotated lattices as our goal, we are interested in constructing dense lattices from octonion orders over a totally real number field.In this construction we use algebraic lattice theory.
Algebraic lattices which are obtained via ideals of number fields [13] are also called ideal lattices.In [14], [15] such lattices were constructed in dimension 4n, via maximal orders in quaternion algebras.We propose here the construction of ideal lattices in dimension 8n using octonion orders.To exemplify the potential of this construction we present, for n = 1 and 2, rotated versions of E 8 and Λ 16 lattices (the densest known lattices in these dimensions) and for n = 3 a lattice with the same center density of the Barnes-Wall lattice in dimension 32. it is also meaningful to note that the division algebra of the octonions can be used in coding and decoding processes.Lattices via octonion algebras over finite fields and polynomial rings have been proposed in [16] to construct lattice-based public key cryptossystems and it was proved that such encoding based on a non-associative algebra is feasible and also more secure than the lattice-based cryptosystem NTRU.
This work is organized as follows: In Section II we review some basics concepts on lattices and algebraic number theory.In Section III some definitions and results on octonion algebras are presented.In Section IV we propose an ideal lattice construction via octonion algebras over totally positive algebraic number fields and characterize its Gram matrix.In Section V, two families of lattices in dimension 2 n , n ≥ 3 are derived and examples of lattices in dimenson 8, 16 and 32 are presented.Conclusions and perspectives are drawn in Section VI.

II. LATTICES
A full-rank lattice Λ is a set of R n composed by all integer linear combination of n linearly independent vectors The matrix M a whose rows are the vectors of a is called a generator matrix to Λ and G a = M a M a is the Gram matrix of Λ associated with M a .Another set b = {w 1 , • • • , w n } of linearly independent vectors is also a basis of Λ if and only if M b = U M a , where U is an unimodular matrix [2] (U has integers entries and det (U) = ±1).The determinant of Λ is defined by det (Λ) = det (G a ), where G a is any Gram matrix of Λ.The determinant of Λ is invariant under the change of basis and it measures the squared volume of the n-dimensional parallelotope generated by any lattice basis.If there is a Gram matrix G which is unimodular we say that Λ is a unimodular lattice.
If d Λ is the minimum distance between two points of a lattice Λ, the packing density, ∆(Λ) is the proportion of R n covered by the union of congruent and disjoint spheres with the biggest possible radius and centered at points of Λ. Due the lattice homogeneity, its packing density can be expressed as , where Vol B n (r) is the volume of a n-dimensional ball of radius r.Another expression to be used here is the so called center density of a lattice, defined as .
Algebraic constructions of lattices allow explicit calculation of parameters and figures of merit of a lattice.The algebraic construction considered here is also called ideal lattice [13] and it is based on concepts and properties recalled next.
A group is a set G with an operation " + ", (G, +) such that for all a, b, c, ∈ G: An ideal I of a commutative ring R is an additive subgroup of R which is stable under multiplication by R, i.e., aI ⊂ I for all a ∈ R. If the ring operation "•" is also commutative then the ring R is a commutative ring and if there is a multiplicative identity in R (1 R ) such that any nonzero element of F has an inverse, i.e, if for all a ∈ F\{0} there is b ∈ F such that ab = 1 R then R = F is a field.Standard examples of these concepts are the ring of integers Z and the fields of the rational numbers Q, of the real numbers R and of the complex numbers C.
The structure of vector spaces which are defined over fields can be generalized to modules.Given a ring R, a commutative group (M, ) is an R-module if there is an operation ϕ : R × M → M defined as ϕ(a, m) = am such that (1) Also, M is said to be a finitely generated module if it has a finite set of generators.A vector space of dimension n is a finitely generated module over a field F.
A field K is called an algebraic number field if it contains Q and has finite dimension n when considered as a vector space over Q (n is called the degree of K).For all y ∈ K there is a monic polynomial g with coefficients in Q such that g(y) = 0 and the polynomial with this property and smallest degree is called minimal polynomial of y.The minimal polynomial is unique and has degree less or equal to the degree of K.The set of all y ∈ K such that the minimal polynomial of y has integer coefficients is called the ring of integers of K and denoted by o K .
) is an algebraic number field of degree 2. The minimal polynomial of any element As another example, if we consider L = Q( √ 5) an algebraic number field of degree 2, the minimal polynomial of any A homomorphism σ : K 1 → K 2 between two fields is a map which preserves both field operations: (1) σ(a + b) = σ(a) + σ(b) and ( 2) Given an algebraic number field K there are exactly n field homomorphisms [17].The mapping σ(x) = (σ 1 (x), σ 2 (x), . . ., σ n (x)) is called the canonical (or Minkowski) embedding of K in C n and the signature (r 1 , r 2 ) of K is defined by the number of real (r 1 ) and pairs of complex (r 2 ) embeddings (n = r 1 + 2r 2 ).If r 2 = 0 we say K is a totally real number field.
Let α ∈ K totally positive (i.e., is called a twisted embedding.When α = 1, we have the canonical embedding.
. We remark that the discriminant is independent of the choice of basis [18].
It can be shown [13] that, given an ideal is called an ideal lattice associated to α and and we will denote such lattice as Λ = (I, α).

III. OCTONION ALGEBRAS
In this section we recall some concepts on octonion algebras and present results to be used in the following sections.We start with the most general definition of an octonion algebra.
Let K be a field, and V be a vector space over K with the usual sum and an additional operation from V × V to V called multiplication and denoted by x • y or simply x y.Then V is an algebra over K if the multiplication is bilinear.Classical examples of algebras are the rational numbers, the real and the complex numbers and also the set of n × n matrices with entries in some field, for some n > 1.Also, we say that an algebra is a division algebra if for any element a ∈ V and a nonzero b in V there exist unique elements x, y ∈ V such that a = bx and a = yb.Now, let K be a totally real algebraic number field, a, b, c nonzeros elements of K and C be a vector space of dimension 8 over K with basis {e 0 , . . ., e 7 }.A product in C can be defined by setting e 0 = 1, the identity element, e 2 1 = ae 0 = a, e 2  2 = be 0 = b, e 2 4 = ce 0 = c as described in following table: • Definition 3. The set C with the usual sum of vector spaces and the multiplication as described above is called an octonion algebra and denoted by C = (a, b, c) K .
Note that, if we choose K = R e a = b = c = −1, we will obtain the Cayley numbers, the most known octonion algebra [19].Along this paper we consider a = b = c = −1, but different fields starting with Q (Example 2) and number fields as Q( √ 2) and others as we will see in Section V. We define the conjugate of x as x = x 0 − 7 i=1 x i e i , the reduced norm of x by N(x) = x x and the reduced trace of x by Tr d(x) = x + x.In particular, we have N(x) ∈ K and N(x) = N(x) = x x.It can be shown that the algebra (a, b, c) K in the Definition 3 is a nonassociative division algebra [19].Given an octonion algebra, we can obtain an orthogonal basis to this algebra regarding to a natural inner product.Proposition 4. [20] In any octonion algebra C = (a, b, c) K over a number field K such that 1 + 1 0 there are elements x, y, z ∈ C with nonzero norm such that is an orthogonal basis of C with respect to the associate bilinear form We introduce next the concept of R-order in the context of the octonion algebras.In particular, for the results of the next section we will assume R = o K for some number field K.
An R-order (or simply order when the ring R is clear in the context) O in the octonion algebra An order M ⊂ C is called a maximal order if it is not properly contained in any other order O. Every R-order is contained in a maximal R-order of C [21].
Note that this algebra can be seen as a vector space of dimension 16 over the field Q since we can write any x ∈ C as In this section we propose an algebraic construction of lattices of dimension 8n via orders in octonion algebras, identifying their Gram matrix.We can define ideal lattices from octonion orders in the same way that we define ideal lattices from number fields [6].
Let K be a totally real algebraic number field of degree n and C an octonion algebra over K.If I ⊂ O is an ideal of an order in C, α a totally positive element in K and Q α : I × I → Q a positive definite quadratic form given by Q α (x, y) = tr K/Q (αTr d(x y)).Considering I an ideal on C with basis B = {v 1 , . . ., v 8 }, assuming that K has degree n and o K with Z-basis {u 1 , . . ., u n }, we can write a basis of the lattice Λ = (I, α) ⊂ R 8n as B = {σ(v i u j )} = {w 11 , w 12 , . . ., w 1n , . . ., w 81 , . . ., w 8n }, where Moreover since K is a totally real number field, the Gram matrix associated to the basis w i j of Λ is given by Then, Λ = (I, α) is the ideal lattice associated to the quadratic form Q α .
The following result provides the determinant of the matrix G of the lattice Λ = (I, α).The proof of this result follows similar steps as ones given in [14] for the quaternionic case.
Theorem 7. Let Λ = (I, α) be the ideal lattice with a Gram matrix G as in (5).Then the determinant of Λ can be written as where B = Tr d(v v ) 8 , =1 and v , v ∈ B basis of I. Proof.Let σ 1 , . . ., σ n be the n-homomorphism from K to R. The elements of the Gram matrix of Λ, G = (g i j ) 8  i, j=1 , can be written as with tr(x) = n i=1 σ i (x).We can then expand the trace form on the number field as the sum of the homomorphisms Hence, the matrix G can be factorized as the product of three matrices, The operation A ⊗ C means that if A is a matrix m × n and C a matrix r × s, we have Therefore, the determinant of G is To calculate the determinant of ϕ, we can permute rows and columns of this matrix in order to obtain block matrices as follows ϕ = σ αTr d(v i v j ) n i, j=1 , for = 1, . . ., n, and then ϕ = diag(ϕ 1 , . . ., ϕ n ). Thus, with B = Tr d(v i v j ) 8 i, j=1 .Combining ( 7) and ( 8) we have, then V. FAMILIES OF LATTICES IN DIMENSION 2 n , n ≥ 3 In this section, we will present two families of lattices in dimension 2 n , n ≥ 3 by using fixed orders in octonion algebras Note that K n has dimension 2 n−3 over Q.
Using the construction proposed in Section IV and choosing I = O it is possible to obtain lattices from these fields.We remark that other choices of I can be taken in order to obtain different lattices in the same dimensions.In the next constructions, for n = 3 we obtain the lattice with optimal center density in dimension 8 and for n = 4 and 5 we get lattices with the same center density of the Barnes-Wall lattices, a well known family that includes the densest lattices in dimension 1, 2, 4, 8 and 16 [2], [22].
First we consider the field K n for some n ≥ 3 and construct a fixed basis of C n as in Proposition 4 by choosing the elements and applying (2) x = e 1 ; y = e 2 ; z = (e 1 + e 2 + e 3 + e 4 )/2;  [23].By using the basis A and choosing the totally positive element α = 1 for n = 3 and α = 2 − η n , n ≥ 4 (9), which has norm N(α) = 2, we can construct a family of lattices in dimension 2 n , n ≥ 3 with determinant We can then consider another Gram matrix G for a scaled version of Λ by a factor 2 n−2 2 , related to G as follows where U is an unimodular matrix.Therefore, det (G ) = 1 for all n ≥ 3. We can also remark that since in this case every element of G described in ( 5) is a multiple of 2 (n−2) , the scaled version of Λ is an unimodular lattice.Then, we have a family of lattices in dimension 2 n associated to the octonion algebra C n , basis A and α = 1 if n = 3 and α = 2 − η n , n ≥ 4.
To exemplify and show an element of this family, we will construct a lattice in dimension 8.
Example 8. To construct a lattice congruent to E 8 , we consider C 3 = (−1, −1, −1) K 3 , K 3 = Q and the order O generated by the basis A. For α = 1, applying (5) we get the following Gram matrix , which is a unimodular matrix which main diagonal is even.Therefore, the lattice Λ = (O, α) is an ideal lattice congruent to the E 8 lattice.
Remark 9.If we use the same basis A to generate an octonion order in C n for n = 4, 5, 6 we obtain after scaling unimodular lattices in dimension 2 n with squared minimum distance equal to 2.
Now, instead to use the basis defined before, we can choose other triple x, y, z as Now, we consider again the field K n for some n ≥ 5 and the basis C of C n as in (11).Associated to this basis we have det (B) = 2 12 and we want to choose an appropriate 3) , and m = 2 n−4 + 1.This imply that we can consider an integer Gram matrix G of a scaled version of this lattice.
where U is an unimodular matrix and det (G ) = 1 for all n ≥ 4. We have then constructed another family of unimodular lattices in dimension ) and the field K n .
In the next examples we have implemented algorithms following the ideas of [24], [25].We have used thein the Mathematica software for the Gram matrix LLL reduction and the SAGE [26] for the computation of minimum distance.In the minimum distance search we took account the Minkowski bound for this parameter .
Using such basis we get in dimensions 32 and 64, unimodular lattices with minimum distance equal to 4. To exemplify and show an element of this family, we will present the construction of a lattice with the same center density of the Barnes-Wall lattice in dimension 32.
Example 11.To construct a lattice in dimension 32 with the same center density of the Barnes-Wall lattice we choose where and The squared minimum distance, computed with Sage-Math [26], associated to the lattice with Gram matrix G 2 is 4, and the obtained lattice has then the same center density, determinant and minimum distance of the Barnes-Wall lattice in dimension 32.Now by using the same parameters, we will construct a lattice in dimension 16 that has the same center density and minimum distance of the Barnes-Wall lattice which is not unimodular.Then, the lattice Λ has the same center density of the Barnes-Wall lattice Λ 16 , and this is the biggest known center density in this dimension.
Remark 13.It is interesting to note that in the cases analyzed above the search for the square of minimum distance found using the mentioned algorithm is an element of the diagonal of the reduced Gram matrix.This means that the LLL algorithm used in the matrix reduction provided a lattice basis which contains minimal norm vectors.

VI. CONCLUSION
In this paper we have presented a construction of ideal lattices via octonion orders extending the construction using quaternion algebras [14], [15] to get lattices in dimension 8n, n ≥ 1.For n = 1, 2, 4 the obtained lattices are rotated versions of well known dense lattices.We also provide two families in dimension 2 n for n ≥ 3 using specific basis of the octonion orders involved.A future work is to construct dense lattices in higher dimensions and also discuss advantages of this construction regarding coding/decoding processes by using octonion operations.Perspective applications of these results include lattice-based cryptography (which requires higher dimensions) and coding for MIMO and MISO channels.

( 1 )
(a + b) + c = a + (b + c); (2) ∃ e ∈ G; a + e = e + a = a, (e ≡ 0); (3) for any a ∈ G there is b ∈ G; a + b = b + a = e.If a + b = b + a for all a, b ∈ G then the group G is called a commutative group.A ring is a set R with two operations (R, +, •) such that (R, +) is a commutative group and for all a, b, c ∈ 2b 2 for b 0. Therefore y belongs to the ring of integers if and only if a is integer and b 5b 2 for b 0. Since this last polynomial has integers coefficients we can assert that the the ring of integers of L is Z[(1 + √ 5)/2].

Proposition 5 .
[21] Let C be an octonion algebra and O ⊂ C. If O is an order, then every element x ∈ O is an integer over R i.e.Tr d(x) ∈ R, N(x) ∈ R.