Vincent Migliore

Maria Méndez Real

Vianney Lapotre

Arnaud Tisserand

Caroline Fontaine

Guy Gogniat

Exploration of Polynomial Multiplication Algorithms for Homomorphic Encryption Schemes

Homomorphic encryption schemes allow performing computations in the ciphertext domain, without the need of the secret key. In most promising schemes based on the ring-learning with errors (R-LWE) problem, polynomial multiplication operation is considered an important bottleneck. In this study, a comparison between the Karatsuba and the fast Fourier transform (FFT) multiplication algorithms in the context of homomorphic encryption is proposed in terms of complexity, flexibility and possible optimizations. A complete hardware architecture to speed up polynomial multiplication is provided and impacts of such an architecture on the Karatsuba and the FFT algorithms is thoroughly studied. The study demonstrates that in a realistic architecture, Karatsuba can be a better alternative than the FFT one.

I. INTRODUCTION

Homomorphic encryption schemes are considered as promising in modern cryptography. The main benefit of such encryption schemes is to process data in the cipher domain. More precisely, one can subcontract a service to a third-party server, possibly unreliable, without compromising privacy and data confidentiality. While classical cryptographic schemes have sometimes homomorphic properties, for addition [START_REF] Paillier | Public-Key Cryptosystems Based on Composite Degree Residuosity Classes[END_REF] or multiplication [START_REF] Rivest | A Method for Obtaining Digital Signatures and Public-key Cryptosystems[END_REF] operations, it has been necessary to wait until 2009 and C. Gentry [START_REF] Gentry | A Fully Homomorphic Encryption Scheme[END_REF] breakthrough to discover a scheme able to perform all types of operations with no restriction. Since then, many schemes have been developed in order to optimize practicability of these schemes, the first one needed a public key of 2.25 GB for a ciphertext of 1.5 GB just for one bit of plaintext [START_REF] Smart | Fully Homomorphic Encryption with Relatively Small Key and Ciphertext Sizes[END_REF]. Now and since adaptation of homomorphic encryption schemes to the R-LWE problem [START_REF] Regev | On lattices, learning with errors, random linear codes, and cryptography[END_REF], one can expect optimized and practical parameters, with a public key and a ciphertext close to 360 Kb [START_REF] Lepoint | A Comparison of the Homomorphic Encryption Schemes FV and YASHE[END_REF]. However, even if recent schemes may be considered as practical compared to the state of the art, these encryption schemes have still important issues. The main problem is the limitation on consecutive multiplications achievable, due to the existence of an inner noise which is growing until making a decryption procedure impossible. This type of encryption schemes are named somewhat-homomorphic encryption (SHE). In order to break this limitation, a procedure named bootstrapping has been prososed in [START_REF] Gentry | A Fully Homomorphic Encryption Scheme[END_REF]. It consists on processing inside the cipher domain an operation of decryption and re-encryption, re-initializing the inner noise, and so allowing to have a fully homomorphic encryption (FHE) scheme. Nonetheless this operation has a very important cost, increasing the size of encryption keys and ciphertexts, and thus increasing calculation time, bandwidth usage and storage cost. Consequently, choosing the best scheme for a given problem, the associated protocol and the security level, must be considered as main scientific and technical concerns. Due to the fact that the bottleneck of R-LWE based encryption schemes is the modular polynomial multiplication operation, various hardware implementations on FPGA are based on the FFT multiplication algorithm [START_REF] Göttert | On the Design of Hardware Building Blocks for Modern Lattice-Based Encryption Schemes[END_REF][8][9] [START_REF] Chen | High-Speed Polynomial Multiplication Architecture for Ring-LWE and SHE Cryptosystems[END_REF], which can be used to speed up the key generation step, the homomorphic multiplication step and the decryption step. However, in this work we do not consider any restriction on the polynomial multiplication algorithm, and we demonstrate that Karatsuba's algorithm [START_REF] Karatsuba | Multiplication of multi-digit numbers on automata (in russian)[END_REF] can be a better alternative compared to the FFT one when dealing with a complete execution scenario. The main contributions of this work are as follows:

• A thoroughly study of the Karatsuba and the FFT multiplication algorithms for a polynomial multiplication is presented. A comparison in the case of SHE in terms of complexity, flexibility and possible optimizations is also provided. • A complete and realistic hardware architecture for SHE is introduced in order to evaluate the impact of such an architecture on the Karatsuba and the FFT multiplication algorithms.

• An optimization of the scheduling of input and output coefficients for Karatsuba is proposed in order to reduce the calculation time by proposing pipeline optimizations.

• Key elements for choosing the multiplication algorithm are proposed with respect to hardware resources limitation.

The paper is organized as follows. Section II recaps some key information on SHE cryptosystems based on a R-LWE problem. It presents and compares the Karatsuba and the FFT multiplication algorithms. The proposed hardware architecture and its impact on the two polynomial multiplication algorithms are presented in Section III. Section IV draws some conclusions on the study. A R-LWE problem is constructed in the ideal lattice Z q [X]/ f (X) , where Z q = Z/qZ and f (X) an irreducible polynomial in Z q of degree n. The modulus q and the degree of the irreducible polynomial n are chosen in order to satisfy a given level of security and multiplicative depth. Table I recaps some practical parameters for SHE schemes based on [START_REF] Lepoint | A Comparison of the Homomorphic Encryption Schemes FV and YASHE[END_REF], considering a security level of 80 bits.

In fast polynomial multiplication algorithms, three steps are required: pre-computation step, a multiplication step and finally a post-computation step. If one tries to reduce the number of sub-products needed for the multiplication step by choosing a specific algorithm, it will be required to adapt, and possibly use larger pre and post computations. For example, for the basic multiplication algorithm, no pre and post computations are needed, but at a cost of O(n 2 ) elementary operations. At the opposite, the FFT multiplication algorithm which requires the smallest number of sub-products, in fact O(n log 2 n), has very complex pre and post computations. For the remainder of the study, we will note with an upper case polynomials of any degree, and with a lower case its coefficients. For a polynomial A, a i will be the i th coefficient of the associated polynomial. In each polynomial multiplication algorithm, input polynomials will be named A and B, where C will be the computation of A × B:

C(x) = 2n-2 i=0 c i x i ; A(x) = n-1 i=0 a i x i ; B(x) = n-1 i=0 b i x i
In what follows, a sub-product will refer to an integer multiplication where a sub-multiplication will refers to a polynomial multiplication (smaller than the operands).

A. Basic Multiplication Algorithm

The basic multiplication algorithm is computed by developing the product of two polynomials. In that case, the coefficients of the output polynomial can be calculated by

c k = k i=0 a i b k-i
For degree n -1 polynomials, the number of elementary operations is O(n 2 ). One can notice here that, as stated before, no pre and post computations are needed. Karatsuba's algorithm [START_REF] Karatsuba | Multiplication of multi-digit numbers on automata (in russian)[END_REF] is an improvement of the basic polynomial multiplication algorithm in order to reduce the number of sub-products in the multiplication step. First, one needs to split input polynomials A and B of degree n -1 into two parts of equivalent size, i.e. n 2 coefficients. Let A H and A L be two polynomials composed respectively by the coefficients of highest degree of A and lowest degree of A. By the same way, one constructs B H and B L . Thus, one therefore obtains

A = A L + A H x n/2 and B = B L + B H x n/2 .
When multiplying classically A × B, one obtains :

A × B = (A L + A H x n/2 )(B L + B H x n/2 ) = A L B L + (A L B H + A H B L )x n/2 + A H B H x n
Karatsuba optimization is to notice that the middle factor (A L B H + A H B L ) can be cleverly computed by

(A L + A H )(B L + B H ) -A L B L -A H B H .
As one can quote, A L B L and A H B H are already computed and so does not require additionnal operations. At the end, Karatsuba requires 3 sub-multiplications instead of 4, at a cost of two pre-computations, namely (A H + A L ) and (B H + B L ), and two post-computations for the reconstruction of the middle factor. However, these pre and post computations are made of additions and subtractions only. To further reduce the number of sub-products, Karatsuba's algorithm can be recursively applied on the sub-polynomials. For degree n -1 polynomials, Karatsuba's complexity is O(n 1.58 ), with a number of sub-products of 3 log 2 n .

A strategy to implement Karatsuba in hardware consists in designing a small polynomial multiplier of degree p and recursively reconstruct the output polynomial. Because pre and post computations are only composed by additions and subtrations, one can at a first approximation calculates the number of operations required to perform a polynomial multiplication by considering the number of sub-products only. In that case, the number of operations required can be easily calculated by

3 log 2 n 3 log 2 p = 3 log 2 n 3 log 2 t/2
, where t is the number of coefficients processed at each operation. Fig. 1a shows an high level architecture for Karatsuba. Depending on the degree of the deepest polynomial multiplication, Karatsuba can achieve a polynomial multiplication with various degrees. For example, if the deepest polynomial multiplication is of degree 0 (i.e. integers), Karatsuba can perform a polynomial multiplication of degree 2 i-1 , where i is the number of times Karatsuba was applied. In general, if the deepest polynomial multiplication is of degree p, Karatsuba can perform a polynomial multiplication of degree p × 2 i (p > 0). Fig. 2 shows different polynomial multiplications achievable for p ∈ [START_REF] Paillier | Public-Key Cryptosystems Based on Composite Degree Residuosity Classes[END_REF][START_REF] Pollard | The Fast Fourrier Transform in a Finite Field[END_REF] and n ∈ [1024, 8192].

C. The FFT Multiplication Algorithm

The FFT multiplication [START_REF] Pollard | The Fast Fourrier Transform in a Finite Field[END_REF] is the algorithm which requires the smallest number of sub-products, but at a cost of complex pre and post computations. The FFT multiplication in the context of R-LWE has been partially studied, we will recall here the main advances so far. The FFT multiplication algorithm performs a convolution operation between 2 vectors. Applied to 2 polynomials, it performs a polynomial multiplication as it can be seen in Section II-A. Because applying directly a FFT multiplication on two polynomials will led to a wrong convolution, it is necessary to resize the input polynomials. Let n -1 be the degree of the input polynomials and k be the smallest integer which satisfy 2 k > 2n -2. The input polynomials must be filled with 0 in order to be degree 2 k . To perform the polynomial multiplication, one must first apply a 2n-FFT operation on each polynomial. Then, a point-wise multiplication is performed on the resulting vectors. The output polynomial is finally reconstructed by applying the inverted FFT algorithm, which is quite similar to the FFT but scaled by n -1 . In short, the complete equation is given by:

A × B = IFFT(FFT(A) FFT(B)),
Algorithm 1 FFT algorithm Let ω be a primitive n-th root of unity in Z q . Let A and A be a polynomial of a degree less than n, A

(x) = a 0 + • • • + a n-1 x n-1 and A (x) = a 0 + • • • + a n-1 x n-1 . A = reverse order(A) for i = 0 to log 2 (n) -1 do for j = 0 to n/2 -1 do P i,j = j 2 log 2 (n-1-i) .2 log 2 (n-1-i) a j = a 2j + a 2j+1 ω Pi,j mod q a j+ n 2 = a 2j -a 2j+1 ω Pi,j mod q end for if i = log 2 (n) -1 then A = A end if end for return A
where denotes a point-wise multiplication. The FFT algorithm is shown in Algorithm 1. It consists of two nested loop, an inner loop where coefficients of a polynomial A are recombined to produce a polynomial A , and an outer loop which role is to iterate the inner loop log 2 d times, where d is the size of the FFT. Fig. 1b shows an high level architecture for the FFT polynomial multiplication algorithm. In the followings, the Butterfly unit will refer to the inner loop. As one can see, pre and post computations are quite complex and require a sub-product in each operation. For a polynomial multiplication of degree n -1, the number of sub-products for the FFT is given by

2n
+ 2n t log 2 2n IFFT .
Table II compares Karatsuba to the FFT in terms of operations count regarding the number of coefficients processed in parallel. The negative wrapped convolution (NWC) is also compared and will be presented in the next section.

D. Application in the Context of Homomorphic Encryption

As stated before, the R-LWE problem requires a reduction by an irreducble polynomial of degree n. As it can be noticed in Table I, if the irreducible polynomial degree n is a power of two, some specific multiplicative depths L are not reachable and thus can lead to unoptimized parameters for a specific application. Because Karatsuba can achieve multiplications with degrees of non power of two, parameters of the R-LWE problem can be optimized. At the oposite, FFT polynomial multiplication algorithm which requires such a n may lead to oversize the solution, and so on as n increases. However, at the same time, asymptotic complexity of the FFT compaired to Karatsuba counterbalance this phenomenon. Karatsuba can also be adapted in order to reduce the complexity of the reduction step required by the R-LWE problem. Nevertheless, in the special case when n is a power of two, the irreducible polynomial can be fixed to x n + 1, and optimizations can be made. First, computation of the reduction is very simple, composed by a simple polynomial addition. Second, one can perform a special form of the FFT, the NWC which requires a n-FFT instead of 2n. The NWC requires specific parameters to be functional, and in particular n and q must satisfy 4 conditions:

• The polynomial modular must be fixed to x n + 1 • q must be prime

• n must be a power of 2

• q ≡ 1 (mod 2n) These parameters bring important issues, especially parameter q. Because the integer modular reduction complexity is inversely proportional to the hamming weight of the modulus, finding a modulus close to the modulus required for security purposes, respecting q ≡ 1 (mod 2n) and with a small hamming weight is a very hard work. Moreover, due to the fact that the modular reduction by q is performed in each pre and post computations, this operation is an important bottleneck. In addition, the choice of x n + 1 for the irreducible polynomial can be considered under-optimized. Indeed, because the polynomial x n + 1 is also irreducible in Z 2 , the batching technique [START_REF] Brakerski | leveled) fully homomorphic encryption without bootstrapping[END_REF] cannot be used. This technique is very promising for SHE schemes because it "packs" several ciphertexts into a bigger one, where the resulting ciphertext is much smaller than the sum the separated ones. A summary of pros and cons on Karatsuba and the NWC algorithm is shown in Table III.

III. COMPARISON OF THE MULTIPLICATION ALGORITHMS IN A COMPLETE ARCHITECTURE

A. Proposed Architecture

The proposed architecture is shown in Fig. 3. Due to the complexity of SHE schemes, implementing a complete crypto-processor is a hard work. To accelerate specific steps of homomorphic cryptosystems, an hybrid and flexible structure is needed. This is why the proposed architecture is composed by an ARM which task is to run a SHE software library on a Linux OS, and a FPGA which runs hardware accelerators like a polynomial multiplier. The communication between the ARM and the FPGA is performed through an AXI bus with a 128-bit width. However, the flexibility of the architecture brings some technological issues. The hardware is penalized by the bandwidth between the ARM and the FPGA. For a polynomial multiplication of degree n -1, the transfer cost can be estimated by 4n log 2 q/axi bus width.

B. Impact of the Architecture on the FFT Algorithm

The FFT algorithm is much impacted by the transfer. Indeed, in each step of the FFT, the entire polynomial is needed in order to complete a Butterfly stage. Thus, the coefficients of the output polynomial cannot be calculated where m is the number of parallel multipliers. In the case of the NWC, the equation can be reduced to Due to lower data dependencies compared to the FFT, Karatsuba algorithm is able to produce output coefficients even if the pre-computations are not already done. Thus, Karatsuba can pipeline the calculation processes more efficiently than the FFT. Moreover, the scheduling of input and output coefficients can be optimized to further reduce the total calculation time. We investigate different scheduling strategies in order to measure the possibility of speedup. If one carefully examines Karatsuba's algorithm, coefficients with the most important data dependencies are the coefficients closed to the middle, because pre-computations and post-computations are performed in the middle factor. That is why one can optimize the pipelining by sending first the coefficients of low degree and high degree, and finishing by the coefficients in the middle. Due to the penalty of the transfer, we implemented a scheduling algorithm to minimize at the same time the number of operations needed before producing the first output coefficient, and the pre-computations needed between each output coefficient. Fig. 4 compares the naive sequence with the optimized one. The columns correspond to input coefficients (left) and output coefficients (right). For a given line which corresponds to a sub-product, a red square refers to an addition of the related coefficient, where a green one refers to a subtration. In the optimized strategy, one can observe that output coefficients are gradually ready compared to the naive strategy, and the number of sub-products required between each output coefficients has been reduced. As an optimization, one can exploit the When the number of multipliers tends towards zeros, the two FFT algorithms outperform Karatsuba due to the fact that Karatsuba has much more sub-products to achieve. When the number of multipliers tends towards the infinity, the two FFT algorithms improve their calculation time because Butterfly units can be performed fastly and balance the penalty of pre and post computations.

The speedup of Karatsuba compared to the NWC one when one has limited embedded multipliers must also be put into perspective because the polynomial modular reduction remains in the case of Karatsuba, and the optimized scheduling is not fitted to perform the polynomial reduction by x n + 1.

At the end, Karatsuba remains competitive in the case of limited hardware resources, but not in ultra-compact architectures, and when the reduction polynomial is not fixed to x n + 1 because one wants to perform the batching technique.

A deeper study on the complexity of calculating the NWC parameters may be interesting, because as stated before, the bottleneck of the NWC is the integer modular reduction. In that case, Karatsuba may renewed interest compared to the NWC especially because in the NWC implementation in [START_REF] Chen | High-Speed Polynomial Multiplication Architecture for Ring-LWE and SHE Cryptosystems[END_REF], the modulus chosen is under estimated in the case of SHE and the complexity to find such a modulus of higher length has not been presented.

IV. CONCLUSIONS AND FUTURE WORK A comparison between Karatsuba and the FFT polynomial multiplication was proposed in the context of SHE schemes based on the R-LWE problem, in terms of complexity, flexibility and possible optimizations. A practical architecture for a complete hardware acceleration of SHE schemes has been proposed and impacts on polynomial multiplication algorithms when dealing with this architecture has been studied. The speedup of Karatsuba compared to the FFT one has been studied for polynomials of few thousands coefficients and for different hardware resources limitations. Future work will focus on implementing different solutions of the polynomial multiplication in order to have a deeper analysis between each polynomial multiplication algorithm. A study of Karatsuba's algorithm for higher n will also be led, especially in the case of memory limited hardware. Investigations on the way how one can optimize Karatsuba's algorithm in the case of consecutive homomorphic operations will also be led. Toom-Cook algorithm will also be considered for the polynomial multiplication, in particular when algorithms require a division by 3 like an average operation. A second step of the future work will investigate the acceleration of other primitives of SHE schemes, for example the generation of secret keys which needs the generation of a discrete Gaussian.

Fig. 1 :

 1 Fig. 1: High level architectures for Karatsuba and the FFT

Fig. 2 :

 2 Fig. 2: Polynomial multiplications achievable for Karatsuba and the FFT multiplication algorithms. Karatsuba multiplier has a root polynomial of degree less or equal to 12.

Fig. 3 :

 3 Fig. 3: Proposed architecture for SHE schemes hardware acceleration

  of the Architecture on Karatsuba's Algorithm

Fig. 4 :FFTFig. 5 :

 45 Fig. 4: Comparison between two scheduling strategies for the input and output coefficients in Karatsuba's algorithm, for a multiplication of polynomials of degree 63

TABLE I :

 I Practical parameters for SHE schemes[START_REF] Lepoint | A Comparison of the Homomorphic Encryption Schemes FV and YASHE[END_REF] based on a R-LWE problem for a security level of 80 bits, where L is the multiplicative depth, and (q, n) are the R-LWE settings

	L	log 2 q	n
	0	20	512
	1	40	1024
	3	80	2048
	5	128	4096
	10	392	8192
	50	1225	32768
	II. STUDY OF KARATSUBA AND THE FFT
	MULTIPLICATION ALGORITHMS FOR SHE

  log 2 2n

			+	2n	+ nlog 2 2n	.
		FFT		point-wise mult	IFFT
	By following the same reasoning than for Karatsuba, one can
	calculate the number of operations required by neglecting the
	additions and subtractions. Because the FFT algorithm is a
	simple nested loop, the number of operations can easily be
	calculated by 2n t log 2 (2n), where t is the number coefficients
	processed in parallel. The total number of operations required
	can now be calculated by
	4n t	log 2 2n	+	4n t
		FFT	point-wise mult

TABLE II :

 II Comparison of the number of operations required for Karatsuba and the two FFTs multipliers, for polynomials of degree 1023

	Coefficients		Number of operations
	per operation
		Karatsuba	FFT	NWC
	4	19683	17920	8192
	8	6561	8960	4096
	16	2187	4480	2048
	32	729	2240	1024
	64	243	1120	512
	128	81	560	256
	256	27	280	128
	512	9	140	64
	1024	3	70	32

TABLE III :

 III Summary of pros and cons on Karatsuba and the NWC

	constraint on n	∅	power of 2
	constraint on q	∅	q ≡ 1 (mod 2n) and prime
	batching technique	yes	no

Karatsuba NWC multiplications O(n 1.58 ) O(n log 2 n)

TABLE IV :

 IV Comparaison of Karatsuba, FFT and NWC in terms of operations required, for polynomials of degree n -1, in the proposed architecture scenario

	n	Karatsuba (basic scheduling)	Karatsuba (opti-mized scheduling)	FFT	NWC
	256	895			767		1074	812
	512	1791			1535		2104 1586
	1024	3583			3071		4158 3128
	during the downloading process. The updated version of the
	number of operations required is described by
	2n	+ 3 (log 2 (2n) -1)	2n m	+	2n m	+	2n
	transfer in						transfer out
	+ first butterfly	FFT, IFFT				point-wise mult

+ last butterfly

ACKNOWLEDGMENT

This study has been financially supported by the french Direction General de l'Armement (DGA).


Cecilia García

-Campos María

Martinón-Torres Marina Martínez De Pinillos

Mario Modesto- Mata

Laura Martín-Francés

Bernardo Perea-Pérez

Clément Zanolli

José María Bermúdez De Castro

Modern humans sex estimation through dental tissue patterns of maxillary canines

Objectives

Dental tissue proportions of human permanent canines is one of only a few sexually dimorphic features that is present in childhood and maintained in adults, offering the opportunity for this to be used in sex determination. This study assesses dental tissue volumes and surface areas of maxillary permanent canines in a sample of known sex to provide new data and to explore the potential of these variables as reliable sexual estimators.

Materials and methods

The teeth studied here derive from 56 individuals (27 females and 29 males) of known sex and age, and of different geographic origins. The teeth were scanned and three-dimensional (3D) measurements (volumes and surface areas) were obtained. In addition, a discriminant function analysis was applied.

Results

The results presented here concur with those previously published in relation to both size and dental tissue patterns. Male maxillary canines have a greater dentine component, whereas female enamel is thicker, leading to a difference in dental size in favor of males. Discriminant functions were calculated using these histological variables successfully identifying sex in between 87.5% and 93.75% of the known-sex hold-out sample, with 92.3% correctly assigned when all functions were applied together.

Discussion

The present study supports that methods for sex determination based on dental tissue measurements can achieve high allocation accuracies, being especially useful in the case of subadults or when no other appropriate method is available.

INTRODUCTION

Sexual variation in the human skeleton is of great concern for anthropologists. Sexual dimorphism, the differences in size and form between males and females of the same species, is also observed in dentition [START_REF] Feeney | Enamel thickness in Asian human canines and premolars[END_REF][START_REF] Harris | Tissue contributions to sex and race: Differences in tooth crown size of deciduous molars[END_REF]Hillson, 1996a;[START_REF] Saunders | Sexual dimorphism of the dental tissues in human permanent mandibular canines and third premolars[END_REF][START_REF] Schwartz | Sexual dimorphism in modern human permanent teeth[END_REF][START_REF] Stroud | Sexual dimorphism in mesiodistal dentin and enamel thickness[END_REF][START_REF] Zilberman | Sex-and age-related differences in primary and secondary dentin formation[END_REF]. In modern human populations, males have larger tooth crowns than females, which has allowed the development of different sexual assignment methodologies based on linear measurements of coronal diameters [START_REF] Cardoso | Testing discriminant functions for sex determination from deciduous teeth[END_REF][START_REF] Karaman | Use of diagonal teeth measurements in predicting gender in a Turkish population[END_REF][START_REF] Peckmann | Sex determination using the mesio-distal dimension of permanent maxillary incisors and canines in a modern Chilean population[END_REF][START_REF] Zorba | Sex determination in modern Greeks using diagonal measurements of molar teeth[END_REF].

Among the teeth, mandibular canines have been widely used in sexual identification [START_REF] Acharya | Univariate sex dimorphism in the Nepalese dentition and the use of discriminant functions in gender assessment[END_REF][START_REF] Garn | Shape similarities throughout the dentition[END_REF]Hillson, 1996a;[START_REF] Lund | Gender determination by odontometrics in a Swedish population[END_REF][START_REF] Rao | Mandibular canine index-A clue for establishing sex identity[END_REF][START_REF] Schwartz | Sexual dimorphism in modern human permanent teeth[END_REF]. The measurement of the mesiodistal (MD) diameter and mandibular intercanine distance consists of a simple method that has been used to establish sex identity, particularly in adults [START_REF] Rao | Mandibular canine index-A clue for establishing sex identity[END_REF]. In recent years, the assessment of coronal tissue proportions has aroused great interest due to the influence that sexual chromosomes and hormones seem to have on them [START_REF] Alvesalo | Sex chromosomes and human growth. A dental approach[END_REF][START_REF] Alvesalo | 47, XXY males: Sex chromosomes and tooth size[END_REF][START_REF] Alvesalo | Enamel thickness in 47,XYY males' permanent teeth[END_REF][START_REF] Guatelli-Steinberg | Dental crown size and sex hormone concentrations: Another look at the development of sexual dimorphism[END_REF][START_REF] Pentinpuro | Crown heights in the permanent teeth of 47,XYY males[END_REF][START_REF] Zilberman | Sex-and age-related differences in primary and secondary dentin formation[END_REF]. Studies of enamel and dentine surface areas, measured on buccolingual (BL) planes of the mandibular canine crowns, have suggested that males have more dentine than their female counterparts, with the latter exhibiting greater enamel thicknesses [START_REF] Feeney | Enamel thickness in Asian human canines and premolars[END_REF][START_REF] Saunders | Sexual dimorphism of the dental tissues in human permanent mandibular canines and third premolars[END_REF][START_REF] Schwartz | Sexual dimorphism in modern human permanent teeth[END_REF]. A recent study using three-dimensional (3D) measurements (volumes and surface areas) of hard dental tissues supports results previously obtained: "Sexual dimorphism of mandibular canine size is mainly due to males having a greater dentine volume, whereas sex differences in enamel dimensions do not make a large contribution to overall tooth size" [START_REF] García-Campos | Contribution of dental tissues to sex determination in modern human populations[END_REF]. These differences were large enough to allow the authors to determine sex with 92.3% accuracy, through the use of a combination of seven discriminant functions. These results demonstrate that mandibular canines should continue to be considered as "key teeth" for personal identification, especially when other skeletal elements are fragmented and/or DNA is unavailable.

Sexual dimorphism of maxillary canines has been also described, although in contrast with its mandibular counterpart, studies focused on this tooth class are scarce. As in the case of lower canines, upper canine crown dimensions have been shown to be larger in males than in females, which has once again been used in the development of different multivariate methodologies to explore sex differentiation [START_REF] Ateş | Sexual differences in Turkish dentition[END_REF][START_REF] Ditch | A multivariate dental sexing technique[END_REF][START_REF] Işcan | Sexual variation in bucco-lingual dimensions in Turkish dentition[END_REF][START_REF] Pereira | Contribution of teeth in human forensic identification-Discriminant function sexing odontometrical techniques in Portuguese population[END_REF]. However, only a few studies have quantified the dental tissue proportions of this tooth class in the modern human population. In the majority of these studies, two-dimensional (2D) quantification of the amount of enamel and dentine was performed using physical cross sections [START_REF] Smith | Variation in enamel thickness within the genus homo[END_REF][START_REF] Smith | Brief communication: Enamel thickness trends in the dental arcade of humans and chimpanzees[END_REF] and virtual sections obtained from 3D rendered models [START_REF] Feeney | Enamel thickness in Asian human canines and premolars[END_REF]. Nevertheless, despite its potential for assessing sexual dimorphism, a comparison between male and female maxillary canines was performed only in the latter study, in which it was shown that males have significantly greater values for dentine area and enamel-dentine junction (EDJ) length, whereas no significant differences were recorded in the enamel area and thickness. Although the published data is unique, the study carried out by [START_REF] Feeney | Enamel thickness in Asian human canines and premolars[END_REF] has two main limitations: the small sample size, which was no more than 20 individuals per tooth class; and the methodology employed. Given that enamel thickness is not homogenously distributed in the crown, conventional 2D measurements of ideal BL section planes may not be a reliable estimator of 3D whole-crown variability. This means that some measurements may be overestimated or underestimated when only 2D variables from the crown are considered [START_REF] Benazzi | Technical note: Guidelines for the digital computation of 2D and 3D enamel thickness in hominoid teeth[END_REF][START_REF] García-Campos | Contribution of dental tissues to sex determination in modern human populations[END_REF][START_REF] Olejniczak | Dental tissue proportions and enamel thickness in Neandertal and modern human molars[END_REF]. 3D measurements have been considered as a more precise method of evaluating tissue proportions as they counteract the effect of dimensional loss suffered by classic 2D estimations [START_REF] Benazzi | Technical note: Guidelines for the digital computation of 2D and 3D enamel thickness in hominoid teeth[END_REF][START_REF] Feeney | Microtomographic analysis of sexual dimorphism and dental tissue distribution in human molars[END_REF][START_REF] García-Campos | Contribution of dental tissues to sex determination in modern human populations[END_REF][START_REF] Molnar | Hominid enamel thickness: I. The Krapina Neandertals[END_REF][START_REF] Shellis | Variations in molar enamel thickness among primates[END_REF].

The current study is the first to apply microtomographic imaging to a broad forensic sample of known sex, to determine the degree of sexual dimorphism in maxillary canine 3D dental tissue proportions. The objectives of this work are to: (a) assess the variability within sexes in the volumes and surface areas of the permanent upper canine dental tissues; (b) explore its potential for reliable sex determination through the application of discriminant function analysis; and (c) contrast our results against those obtained in lower canines by [START_REF] García-Campos | Contribution of dental tissues to sex determination in modern human populations[END_REF].

MATERIALS AND METHODS

Details of the sample studied, the scanning parameters, and the methodology employed here can be found below and in [START_REF] García-Campos | Contribution of dental tissues to sex determination in modern human populations[END_REF].

The teeth studied here derive from 56 individuals (27 females and 29 males) of known sex and age, and of different geographic origins. The sample was selected from the anthropological collections housed at the Escuela de Medicina Legal of Madrid (Spain) and the University of Pretoria (South Africa), as well as from a sample of dental extractions carried out in different clinics in Sudan [START_REF] Elamin | Malnutrition has no effect on the timing of human tooth formation[END_REF]. The sample was designed to obtain a similar representation of individuals with African and European origin in the female and male subsamples. In the present study, only one antimere per individual was included in the analysis. The studied upper canines showed a degree of wear equal to or smaller than three [START_REF] Molnar | Human tooth wear, tooth function and cultural variability[END_REF], which is characterized by the obliteration of the apex at the incisal border and the presence of a dentine point instead.

The specimens were scanned in three facilities: the Phoenix v/tome/x s (GE Measurement & Control) microtomographic system housed in the Microscopy Laboratory of the Centro Nacional de Investigación sobre la Evolución Humana (CENIEH) in Burgos (Spain); the CTP-Mlab micro-CT located in the Multidisciplinary Laboratory of the International Centre for Theoretical Physics (ICTP) in Trieste (Italy); and in the South African Nuclear Energy Corporation (Necsa), Pelindaba, using Nikon XTH 225 ST equipment. The output images had a voxels size ranging between 17 and 50.8 μm. The subsequent image processing of each of the teeth was performed using Amira 6.0.0 software (Visage Imaging, Inc., San Diego, CA). Dental tissues (enamel and dentine-pulp complex) were semiautomatically segmented using the Watershed Segmentation Tool and through manual editing. A Non-Local Means filter was also applied. Small fractures and cracks were virtually filled in.

Dental tissue volumes and surface areas described in [START_REF] Olejniczak | Dental tissue proportions and enamel thickness in Neandertal and modern human molars[END_REF], [START_REF] Olejniczak | Three-dimensional primate molar enamel thickness[END_REF], [START_REF] Skinner | Enamel-dentine junction (EDJ) morphology distinguishes the lower molars of Australopithecus africanus and Paranthropus robustus[END_REF]Hublin (2008), andSkinner, Wood, et al. (2008) were quantified following the protocol put forward by García-Campos et al.

(2018) (Figure 1), which was based on a previous study by [START_REF] Benazzi | Technical note: Guidelines for the digital computation of 2D and 3D enamel thickness in hominoid teeth[END_REF]. The variables and indices included were: volume of the enamel cap (Ve, mm 3 ); volume of the coronal dentine including the coronal pulp (Vcdp, mm 3 ); the surface area of the enamel-dentine junction (EDJS, in mm 2 ); the outer surface of the enamel cap (OES, mm 2 ); 3D average enamel thickness index (3DAET = Ve/EDJS, mm); the 3D relative enamel thickness index (3DRET = 3DAET/ ) × 100, free scale); and the relative dentine volume defined as the percentage of coronal volume that is dentine and pulp (Vcdp/Vc = Vcdp/Vc × 100, percentage scale). In addition, the volume of the root dentine including the pulp (Vr, mm 3 ), the coronal volume (Vc, mm 3 ) and the basal surface of the crown (BS, mm 2 ) were measured and subsequently used to compute the total tooth volume (Vt = Vc + Vr, mm 3 ) and the relative coronal volume (Vc/Vt = Vc/Vt × 100, percentage scale). A detailed explanation of each of the variable and indices can be found in [START_REF] García-Campos | Contribution of dental tissues to sex determination in modern human populations[END_REF]. and (c) mesial view. Then, a straight line between the maximum enamel extensions is drawn in the mesiodistal stack of slices (b) defining the lower limit of the crown and giving rise to concave curve in the buccolingual slices (d). In that way, the crown is limited on the base by a curve with a smooth surface which isolates it from the root volume.

Statistical analyses were performed using SPSS software (v. 18.0, SPSS Science, Inc.). First, standard descriptive statistics were calculated for each variable. The normal distribution and the equal variance assumption were assessed using the Kolmogorov-Smirnov one-sample test and Levene's test respectively. Analysis of variance (ANOVA) and covariance (ANCOVA) were employed to examine the possible differences between males and females. When equal variance could not be assumed a Student's t test was applied instead. Means were considered to be significantly different at α = 0.05 level. In addition, a discriminant function analysis (DFA) was performed to generate a set of univariate and multivariate functions from the 3D variables. The whole sample was divided into two sub-samples for statistical purposes. A total of 40 slightly worn teeth (19 females, 21 males) were used to create the discriminant functions, whereas 16 teeth (8 females, 8 males) were used as a hold-out sample. The functions created were tested in the original sample and the leave-one-out method was chosen to calculate the cross-validation error index. The percentage of correct allocation accuracy in determining sex was also calculated for the hold-out sample: females and males separately, and for the whole of the sample. Lastly, only the equations with a high percentage of correct assignments were selected and included in this study.

RESULTS

Table 1 and Figure 2 provide the descriptive statistics for the 3D dental tissue measurements (both absolute data and associated indices) for males and females, together with a summary of the results of statistical comparisons of means (ANOVA, Student's t test and ANCOVA).

Tables 2 and3 present, respectively, the discriminant functions obtained in this study and their accuracy at correctly assigning sex.

Intra-population variability

The Kolmogorov-Smirnov test showed that all dimensions were normally distributed within each sex category. The results of the homogeneity of variance test indicate that the sample is statistically homogeneous for all dimensions and indices, except for the enamel volume (Ve), the EDJ surface area (EDJS), the outer enamel surface area (OES), and the index Vc/Vt. In these cases, a Student's t test was applied (indicated with an asterisk in Table 1); an ANCOVA test could not be applied to compare the means for these variables.

All absolute tooth dimensions were larger in males and the mean comparison analysis showed that these differences were statistically significant (Table 1). The relative measurements associated with these absolute variables also showed statistically significant differences, except for average enamel thickness (3DAET). The relative enamel thickness (3DRET) lets us observe that females have significantly thicker enamel than males. The same happens with the OES, which is significantly greater in females in relation to their EDJ surface area, indicated by the OES/EDJS index. Therefore, although the ANCOVA test could not be applied to enamel volume (Ve), these indices (3DRET and OES/EDJS) show that the enamel seems to have relatively larger dimensions in females than in males. ANCOVA tests could be applied to dentine volume measurements (Vcdp, Vtdp), as well as to the basal surface area of the crown (BS). The results of this test shows that these three variables were significantly larger in males even after controlling for size (Vt). This is corroborated by the relative dentine volume (Vcdp/Vc) results, supporting that males have absolutely, but also relatively, more dentine than females.

Discriminant function analysis

Table 2 includes the unstandardized discriminant function coefficients, constants, sectioning points and main statistical values obtained from DFA. This analysis produced a set of univariate and multivariate functions, seven of which have high percentages of correct determinations and were selected consequently. Four univariate functions were included, which were based on the following variables: the coronal dentine and pulp volume (D1), the total dentine and pulp volume (D2), the EDJ surface area (D3), and the basal surface area (D4). Furthermore, a multivariate analysis that used all the measured variables and associated indices was carried out (D5), as well as a second one that employed the variables and indices from the crown (D7). We also created multivariate equations using all variables and indices through a stepwise DFA using the Wilks' lambda method (D6). All functions were statistically significant in discriminating between groups (F value, p < .01).

Table 3 shows the sex allocation accuracy (%) for each discriminant function, tested on the original sample used for the creation of the functions (n = 40) and via the cross-validation technique, as well as on a hold-out sample of slightly worn teeth (n = 16). The percentage of correct determinations for the univariate functions ranged between 82.50% and 90.00% for the original sample and the cross-validated analysis, and between 87.50% and 93.75% for the holdout sample validation. On the other hand, multivariate equations had a classification accuracy that ranged between 87.50% and 92.50% for the original sample, between 80.00% and 87.50% for the cross-validated analysis, and was 93.75% for the hold-out sample validation. The function with the greatest correct sex allocation was D4, generated from the basal surface area of the crown.

We also used a combination of the seven functions (D1-D7) in the hold-out sample, assigning each individual to a specific sex when at least four of the outcomes indicated either "male" or "female". Using this methodology, we obtained 92.30% correct classifications.

Both sexes were not classified with the same accuracy by all discriminant functions. Overall, the percentage of correct classifications was higher in females than in males, except for functions D2

(Vtdp) and D4 (BS). The percentage of correct determinations in the original sample ranges between 76.20% and 90.50% in the case of males and between 78.90% and 100% in the case of females. The cross-validation test results ranged between 76.20% and 90.50% in the case of males and between 73.70% and 94.70% in the case of females. Validation of the hold-out sample ranged between 87.50% and 88.00% in the case of males and between 87.50% and 100.00% in the case of the females. The combination of the results from the seven functions correctly classified 100% of the females and 87.5% of the males.

DISCUSSION

Sexual dimorphism of maxillary canines

Teeth are the most resistant tissue of the human body, which is why they have been the focus of anthropological research. In forensic sciences, teeth have been used mostly for age estimation [START_REF] Cameriere | Reliability in age determination by pulp/tooth ratio in upper canines in skeletal remains[END_REF][START_REF] Someda | Age estimation based on three-dimensional measurement of mandibular central incisors in Japanese[END_REF][START_REF] Stavrianos | Dental age estimation of adults: A review of methods and principals[END_REF] and sex determination [START_REF] Acharya | Univariate sex dimorphism in the Nepalese dentition and the use of discriminant functions in gender assessment[END_REF][START_REF] Ateş | Sexual differences in Turkish dentition[END_REF][START_REF] Pereira | Contribution of teeth in human forensic identification-Discriminant function sexing odontometrical techniques in Portuguese population[END_REF][START_REF] Zorba | Sex determination in modern Greeks using diagonal measurements of molar teeth[END_REF], with most of these odontometric studies using discriminant function statistical models to assess sexual dimorphism in human dentition.

The sexual dimorphism of permanent teeth has been established by many researchers, being the greatest in canines [START_REF] Harris | A principal components analysis of human odontometrics[END_REF][START_REF] Hillson | Dental anthropology[END_REF][START_REF] Işcan | Sexual variation in bucco-lingual dimensions in Turkish dentition[END_REF][START_REF] Peckmann | Sex determination using the mesio-distal dimension of permanent maxillary incisors and canines in a modern Chilean population[END_REF]. These differences take the form of a marked difference in size, with male canines being more volumetric than those of females [START_REF] De Angelis | Sexual dimorphism of canine volume: A pilot study[END_REF][START_REF] García-Campos | Contribution of dental tissues to sex determination in modern human populations[END_REF]. Most of the studies in which the sexual dimorphism of canines has been assessed are based on traditional MD and BL crown diameters [START_REF] Acharya | Univariate sex dimorphism in the Nepalese dentition and the use of discriminant functions in gender assessment[END_REF][START_REF] Ateş | Sexual differences in Turkish dentition[END_REF][START_REF] Ditch | A multivariate dental sexing technique[END_REF][START_REF] Işcan | Sexual variation in bucco-lingual dimensions in Turkish dentition[END_REF][START_REF] Moorrees | Mesiodistal crown diameters of the deciduous and permanent teeth in individuals[END_REF][START_REF] Sabóia | Sexual dimorphism involved in the mesiodistal and buccolingual dimensions of permanent teeth[END_REF]. However, these measurements usually are modified by wear and pathologies, so some authors have explored alternative variables like the cervical MD and BL diameters as well as diagonal diameters of the crown [START_REF] Hillson | Alternative dental measurements: Proposals and relationships with other measurements[END_REF][START_REF] Viciano | Odontometric sex discrimination in the Herculaneum sample (79 AD, Naples, Italy), with application to juveniles[END_REF][START_REF] Viciano | Sex estimation based on deciduous and permanent dentition in a contemporary Spanish population[END_REF][START_REF] Zorba | Sexual dimorphism in permanent teeth of modern Greeks[END_REF]. All such dimensions have been found to be sexually dimorphic, reaching allocation accuracies in the original sample ranging from 67% to 88%, approximately.

The results of the present study corroborate the differences in size observed in previous studies (Table 1). The male maxillary canines showed a significantly greater volume (Vt) than those of their female counterparts. Their crowns are more volumetric (Vc), showing very significant differences to those of the females. These differences in crown size are also reflected in their coronal basal surface (BS), showing absolutely and relatively larger areas in males. All these agree with the results obtained from coronal linear measurements [START_REF] Viciano | Odontometric sex discrimination in the Herculaneum sample (79 AD, Naples, Italy), with application to juveniles[END_REF][START_REF] Viciano | Sex estimation based on deciduous and permanent dentition in a contemporary Spanish population[END_REF][START_REF] Zorba | Sexual dimorphism in permanent teeth of modern Greeks[END_REF] as well as those from mandibular canine volumes [START_REF] García-Campos | Contribution of dental tissues to sex determination in modern human populations[END_REF]. Not only are the crowns of female maxillary canines smaller, but so are their roots (Vr). The study of root measurements in the context of sexual dimorphism has greatly expanded in recent years [START_REF] Garn | Relationship between root lengths and crown diameters of corresponding teeth[END_REF][START_REF] Zorba | Sexual dimorphism of root length on a Greek population sample[END_REF]. Roots tend to be better preserved compared to crowns, as they are protected by the alveolar bone; in addition, they are far less frequently affected by pathological conditions or wear. Zorba et al.

(2013) tested the existence of sexual dimorphism in the root length of single-rooted teeth and showed that root length was greater in males than in females (allocation accuracy in the original sample: 77%-88%). In the light of the results of the present and previous studies [START_REF] García-Campos | Contribution of dental tissues to sex determination in modern human populations[END_REF], it can be confirmed that canine roots are not only longer in males but also more volumetric, representing a greater percentage of their total volume (Vc/Vt) in contrast to female canines.

Hypotheses proposed for tooth-size dimorphism between males and females include: a greater enamel thickness owing to a longer period of amelogenesis in males [START_REF] Acharya | Are dental indexes useful in sex assessment[END_REF]; a relatively higher quantity of dentine in male teeth [START_REF] García-Campos | Contribution of dental tissues to sex determination in modern human populations[END_REF][START_REF] Schwartz | Sexual dimorphism in modern human permanent teeth[END_REF][START_REF] Smith | Modern human molar enamel thickness and enamel-dentine junction shape[END_REF]; or, differential effects of sex chromosomes and hormonal influences in promoting dental tissue growth [START_REF] Alvesalo | Human sex chromosomes in oral and craniofacial growth[END_REF][START_REF] Alvesalo | Upper central incisor and canine tooth crown size in 47,XXY males[END_REF][START_REF] Guatelli-Steinberg | Dental crown size and sex hormone concentrations: Another look at the development of sexual dimorphism[END_REF][START_REF] Pentinpuro | Crown heights in the permanent teeth of 47,XYY males[END_REF][START_REF] Pentinpuro | Crown heights in the permanent teeth of 45,X and 45,X/46,XX females[END_REF][START_REF] Ribeiro | Intrauterine hormone effects on tooth dimensions[END_REF][START_REF] Ribeiro | Sexual dimorphism in the primary and permanent dentitions of twins: An approach to clarifying the role of hormonal factors[END_REF][START_REF] Smith | Modern human molar enamel thickness and enamel-dentine junction shape[END_REF]. That is the reason why, whereas most odontometric studies were focused on crown size differences, some researchers have attempted to quantify tissue differences between sexes [START_REF] Feeney | Microtomographic analysis of sexual dimorphism and dental tissue distribution in human molars[END_REF][START_REF] Harris | A radiographic assessment of enamel thickness in human maxillary incisors[END_REF][START_REF] Saunders | Sexual dimorphism of the dental tissues in human permanent mandibular canines and third premolars[END_REF][START_REF] Schwartz | Sexual dimorphism in modern human permanent teeth[END_REF][START_REF] Smith | Modern human molar enamel thickness and enamel-dentine junction shape[END_REF][START_REF] Stroud | Sexual dimorphism in mesiodistal dentin and enamel thickness[END_REF]. [START_REF] Schwartz | Sexual dimorphism in modern human permanent teeth[END_REF] and [START_REF] Saunders | Sexual dimorphism of the dental tissues in human permanent mandibular canines and third premolars[END_REF] provided enamel thickness values obtained from physical sections of mandibular canines, including the first study also third molars. Variation in lateral enamel thickness has also been examined through clinical radiographs of maxillary incisors [START_REF] Harris | A radiographic assessment of enamel thickness in human maxillary incisors[END_REF] and mandibular premolars [START_REF] Stroud | Sexual dimorphism in mesiodistal dentin and enamel thickness[END_REF]. Two more recent studies employed 3D models obtained from micro-CT scanning. In the first of these, [START_REF] Feeney | Enamel thickness in Asian human canines and premolars[END_REF] employed virtual sections to quantify the average enamel thickness of a sample of clinically extracted Indonesian canine and premolar teeth. In the second, García-Campos et al. ( 2018), using the same methodology applied in the present study, determined the degree of sexual dimorphism in the dental tissue volumes and surface areas of mandibular canines. Although all these studies vary in terms of the tooth class assessed, overall it is observed to a greater or lesser extent similar patterns of sexual dimorphism in dental tissue proportions: male teeth have greater dentine component, whereas in females the enamel is thicker [START_REF] Feeney | Enamel thickness in Asian human canines and premolars[END_REF][START_REF] García-Campos | Contribution of dental tissues to sex determination in modern human populations[END_REF][START_REF] Saunders | Sexual dimorphism of the dental tissues in human permanent mandibular canines and third premolars[END_REF][START_REF] Schwartz | Sexual dimorphism in modern human permanent teeth[END_REF][START_REF] Smith | Modern human molar enamel thickness and enamel-dentine junction shape[END_REF][START_REF] Stroud | Sexual dimorphism in mesiodistal dentin and enamel thickness[END_REF].

The results of the present study confirm this pattern. As in the case of lower canines [START_REF] García-Campos | Contribution of dental tissues to sex determination in modern human populations[END_REF], male maxillary canines have absolutely (Vcdp, Vtdp) and relatively (Vcdp/Vc, ANCOVA results) more dentine than those of females, whereas females have been found to have relatively thicker enamel (3DRET). In contrast to the lower canines, the enamel cap volume was smaller in female upper canines when compared to that of males. However, although the ANCOVA test could not be applied to this variable, the 3DRET index results lend support to the idea that this may be due to the differences in size between sexes, and not because males have a proportionally greater amount of enamel. As with tissue volumes, the surface areas of enamel (OES) and dentine (EDJS) are smaller in females; nevertheless, in both upper and lower canines their OES have a significantly larger area regarding the EDJ surface (OES/EDJS index) (García-Campos et al., 2018, present study), which has been related to greater enamel thickness by some authors [START_REF] Smith | Modern human molar enamel thickness and enamel-dentine junction shape[END_REF].

4.2 Sex allocation accuracy of the discriminant functions based on dental tissue volumes and surface areas of maxillary canines DFA has become a method extensively used by archaeologists and forensic anthropologists to determine sex [START_REF] Acharya | Univariate sex dimorphism in the Nepalese dentition and the use of discriminant functions in gender assessment[END_REF][START_REF] García-Campos | Contribution of dental tissues to sex determination in modern human populations[END_REF][START_REF] Hassett | Technical note: Estimating sex using cervical canine odontometrics: A test using a known sex sample[END_REF][START_REF] Işcan | Sexual variation in bucco-lingual dimensions in Turkish dentition[END_REF][START_REF] Viciano | Odontometric sex discrimination in the Herculaneum sample (79 AD, Naples, Italy), with application to juveniles[END_REF][START_REF] Viciano | Sex estimation based on deciduous and permanent dentition in a contemporary Spanish population[END_REF][START_REF] Zorba | Sexual dimorphism of root length on a Greek population sample[END_REF], as it calculates the optimal combination of variables and indices to reflect their contribution to sex diagnosis.

Univariate functions selected in this study are defined by the following variables: the coronal dentine and pulp volume (Vcdp, D1); the total dentine and pulp volume (Vtdp, D2); the EDJ surface area (EDJS, D3); and the basal surface area of the crown (BS, D4). Most of these variables are related to the dentine component, highlighting its importance in sex determination.

Overall, multivariate functions (D5, D6, and D7) have higher accuracy than univariate ones (original: 87.5%-92.50%; cross-validation: 80.00%-82.50%; hold-out sample: 93.75%), although the univariate function D4 was the most effective in sex diagnosis. The percentage of correct determinations when we combined all functions was 92.30%, which, together with the previous results, shows the effectiveness of the methodology introduced in this study, which overtakes that of methods based on linear measurements. The discriminant function results obtained for upper canines were similar to those obtained for lower canines [START_REF] García-Campos | Contribution of dental tissues to sex determination in modern human populations[END_REF]. It should be pointed out that in both tooth classes the functions based on BS and Vtdp were selected due to their high accuracy rates. As in the case of mandibular canines [START_REF] García-Campos | Contribution of dental tissues to sex determination in modern human populations[END_REF], as well as the results of previous researches [START_REF] Acharya | Univariate sex dimorphism in the Nepalese dentition and the use of discriminant functions in gender assessment[END_REF][START_REF] Ateş | Sexual differences in Turkish dentition[END_REF][START_REF] Işcan | Sexual variation in bucco-lingual dimensions in Turkish dentition[END_REF], in general terms females were correctly assigned more often than males, which may indicate greater variability in male maxillary canines.

4.3

The potential of the methodology exposed for sex determination and its limitations Sex determination in fragmented human remains constitutes the foremost step for identification in physical anthropology. The pelvis and skull have been shown to be the most accurate skeletal elements for determining sex [START_REF] Ferembach | Recommendation for age and sex diagnoses of skeletons[END_REF][START_REF] İşcan | Forensic anthropology of sex and body size[END_REF][START_REF] Iscan | Determination of sex from the sacroiliac joint: A visual assessment technique[END_REF][START_REF] Phenice | A newly developed visual method of sexing the os pubis[END_REF]. However, in both forensic practice and an archaeological context it is common to recover human remains where most skeletal structures appear fractured or are

absent. An alternative is to use teeth. Considering the fact that teeth are the most resistant mineral components of the human skeleton [START_REF] Hillson | Dental anthropology[END_REF], a method based on dental measurements may help to fill this gap. The results of the present study and previous research show that there are several dental variables that can be used to develop sample-specific methods for sex determination, and that the correct allocation accuracies that they achieve are comparable to other metric methods commonly applied to other skeletal elements [START_REF] Acharya | Univariate sex dimorphism in the Nepalese dentition and the use of discriminant functions in gender assessment[END_REF][START_REF] Ateş | Sexual differences in Turkish dentition[END_REF][START_REF] García-Campos | Contribution of dental tissues to sex determination in modern human populations[END_REF][START_REF] Hassett | Technical note: Estimating sex using cervical canine odontometrics: A test using a known sex sample[END_REF][START_REF] Zorba | Sexual dimorphism of root length on a Greek population sample[END_REF]. Their application might be especially useful in sex determination of subadult individuals. This is because the crowns of the permanent dentition develop early, and once formed, they hardly change during growth. This means that any discriminant function created from their measurement, and which is powerful at discriminating between sexes, could also be useful in subadults for whom secondary sex characteristics of the skeleton are not yet discernible. In particular, techniques based on permanent canine traits might allow us to determine the sex of individuals older than 6 years of age, which is the age around which the crown finishes its formation [START_REF] Moorrees | Age variation of formation stages for ten permanent teeth[END_REF]. One limitation is that these functions cannot be employed to determine the sex of very young individuals whose permanent teeth have not yet formed. Although deciduous crown dimensions seem to show considerably less sexual dimorphism than permanent teeth [START_REF] Black | Sexual dimorphism in the tooth-crown diameters of the deciduous teeth[END_REF][START_REF] De Vito | A discriminant function analysis of deciduous teeth to determine sex[END_REF], future studies of dental tissue proportions of primary dentition in general, and of deciduous canines in particular, could help to shed light on this issue.

The usefulness in sex estimation studies of the multivariate methods based on linear dimensions of the canines has been deeply explored, which have obtained an accuracy which ranges from 67% to 88.2% (e.g., [START_REF] Ateş | Sexual differences in Turkish dentition[END_REF][START_REF] Işcan | Sexual variation in bucco-lingual dimensions in Turkish dentition[END_REF][START_REF] Karaman | Use of diagonal teeth measurements in predicting gender in a Turkish population[END_REF][START_REF] Peckmann | Sex determination using the mesio-distal dimension of permanent maxillary incisors and canines in a modern Chilean population[END_REF][START_REF] Zorba | Sexual dimorphism of root length on a Greek population sample[END_REF]. In contrast to these studies, the methodology based on 3D measurements proposed here confidently offers greater percentages of correct assignment, which has been assessed employing geographical mixed samples and tested in a hold-out sample. On the other hand, a limitation of sex determination methodologies based on linear odontometric measurements is that teeth should not be extremely worn down nor should their dimensions be affected by dental anomalies and disease. In many dental samples, adult teeth show a severe interproximal attrition and/or crown reduction, making it impossible to collect the traditional linear measurements of the crown (MD, BL); yet, the cervical dimensions of the crown can be used as an alternative to solve this problem [START_REF] Hillson | Alternative dental measurements: Proposals and relationships with other measurements[END_REF][START_REF] Viciano | Odontometric sex discrimination in the Herculaneum sample (79 AD, Naples, Italy), with application to juveniles[END_REF][START_REF] Viciano | Sex estimation based on deciduous and permanent dentition in a contemporary Spanish population[END_REF][START_REF] Zorba | Sexual dimorphism in permanent teeth of modern Greeks[END_REF]. The variables considered in this study might be also be affected by wear.

García-Campos et al. ( 2018) included extensively worn teeth (4-5 wear degree, [START_REF] Molnar | Human tooth wear, tooth function and cultural variability[END_REF]) as a hold-out sample, showing that although the percentage of correct assignment only decreased to 9.64% compared to the slightly worn sample, the differences between males and females increased, which made them doubt the precision of this methodology as wear increases. A solution to this problem may be the discriminant function based on the basal surface area of the crown (BS), as in both maxillary and mandibular permanent canines the univariate functions created using this variable have high accuracy rates. As with cervical margins diameters, the basal surface is not affected by wear and could be assessed even when the root is either absent, which could be common in single rooted teeth, or is not completed (as with subadult individuals).

In light of the findings of the present study, as well as those in literature, it can be affirmed that canine dental tissue proportions provide several variables that can be used to develop methods for sex determination, whose correct allocation accuracies are comparable to other metric methods commonly applied to other skeletal structures. The results of this study show that male maxillary canines have an absolutely and relatively greater dentine component, whereas female enamel is thicker, leading to a difference in dental size in favor of males. These differences have permitted the development of a discriminant function model that achieves correct assignation rate of 92.30%. This represents an alternative methodology that might become of interest when DNA and other skeletal elements are not available, or in the case of subadults, for whom secondary sexual characteristics cannot always be discerned accurately using techniques based on postcranial measurements or morphological features.

Figure 1 .

 1 Figure 1. Crown isolation. The canine is orientated in its anatomical position: (a) lingual view

Table 1 .

 1 Descriptive statistics, analysis of variance (ANOVA) and analysis of covariance (ANCOVA) results for the three-dimensional variables measured on slightly worn maxillary canines (mean and standard deviation, SD). When equal variance could not be assumed, a Standard box and whisker plot of the upper canine tissue proportions (wear stages 1-3).

	Student's t test was applied (*)
		Female		Male			ANOVA ANCOVA
	Measurement n	Mean	SD	n	Mean	SD	Sig.	Sig. Female	Male
										mean	mean
	Vc	27 223.94 35.93 29	286.86 57.05	.00	-	-	-
	Vr	27 299.37 59.93 29	420.98 81.11	.00	-	-	-
	Vt	27 523.31 93.41 29	707.84 121.52 .00	-	-	-
	Ve	27 109.96 20.04 29	128.80 34.77	.016*	-	-	-
	Vcdp	27 113.98 18.57 29	158.07 26.95	.00	.02 130.61	142.58
	EDJS	27 115.25 11.32 29	142.43 17.81	.00*	-	-	-
	OES	27 174.43 18.64 29	204.31 29.61	.00*	-	-	-
	BS	27	33.08 3.72	29	43.82 5.61	.00	.00 36.33	40.79
	Vtdp	27 413.34 76.35 29	579.05 97.92	.00	.00 489.16	508.45
	Index
	Vc/Vt	27 43.016 2.53	29	40.58 4.46	.00*	-	-	-
	3DAET	27	0.95 0.12	29	0.90 0.17	.17	-	-	-
	3DRET	27	19.71 2.52	29	16.61 2.97	.00	-	-	-
	Vcdp/Vc	27	50.94 3.36	29	55.54 4.64	.00	-	-	-
	OES/EDJS	27	1.51 0.07	29	1.43 0.08	.00	-	-	-

Table 2 .

 2 Discriminant function analysis results a obtained from sub-sample of slightly worn

	maxillary canines (n = 40)
	Variables	Unstandardized coefficients
		D1	D2	D3	D4	D5	D6	D7
	Vc	-	-	-	-	0.025	-	-
	Vr	-	-	-	-	0.005	-	-
	Vt	-	-	-	-	-	-	-
	Ve	-	-	-	-	-	-	0.142
	Vcdp	0.041	-	-	-	-	-	-0.044
	EDJS	-	-	0.062	-	-0.050	-	0.073
	OES	-	-	-	-	-0.035	-	-0.033
	BS	-	-	-	0.195	0.128	0.164	0.142
	Vtdp	-	0.011	-	-	0.004	-	-
	Vc/Vt	-	-	-	-	0.096	-	-
	3DAET	-	-	-	-	-	-	-39.705
	3DRET	-	-	-	-	-	-	2.507
	Vcdp/Vc	-	-	-	-	-	0.120	1.074
	OES/EDJS	-	-	-	-	-9.671	-	-

Table 3 .

 3 Sex

allocation accuracy (%) for each discriminant function generated from data collected in this study, tested on the original sample and using cross-validation. The functions were also tested on a hold-out sample of slightly worn teeth. Discriminant function scores were calculated using the raw coefficients multiplied by the corresponding dental dimensions plus the constant. The resulting score was compared with the sectioning point. If the score was greater than the sectioning point the sex was determined to be male

Tested on equations sample Tested on hold-out sample Original Cross-validation Slightly worn teeth Discriminant functions

		Tested on equations sample				Tested on hold-out sample
		Original			Cross-validation		Slightly worn teeth
	Discriminant	Female	Male	Total	Female Male	Total	Female	Male	Total
	functions
	D4	89.5%	90.5%	90.00% 89.5%	90.50%	90.00% 100.00%	88.00% 93.75%
	D5	84.2%	90.50%	87.50% 84.20% 76.2%	80.00% 100.00%	88.00% 93.75%
	D6	94.7%	85.70%	90.00% 89.50% 85.70%	87.50% 100.00%	88.00% 93.75%
	D7	100.00%	85.70%	92.5%	89.50% 76.20%	82.50% 100.00%	88.00% 93.75%
	All	-	-	-	-	-	-	100.00%	87.50% 92.30%
	combined
		Female	Male	Total	Female Male	Total	Female	Male	Total
	D1	94.70%	81.00%	87.50% 94.70% 81.00%	87.50% 87.50%	87.50% 87.50%
	D2	78.90%	90.50%	85.00% 73.70% 85.70%	80.00% 87.50%	87.50% 87.50%
	D3	89.50%	76.20%	82.50% 89.5%	76.20%	82.50% 87.50%	87.50% 87.50%

 Dimensions in millimeters. Significant p values in bold.

ACKNOWLEDGMENTS

This study has been carried out thanks to the Dirección General de Investigación of the Spanish Ministerio de Economía y Competitividad (MINECO/FEDER) support, grant number: CGL2015-65387-C3-1, 3-P. Likewise, we also wish to acknowledge The Leakey Foundation for the funds provided by Gordon Getty and Dub Crook to one of the authors (MM-T). CG-C and MM-M are funded by a doctoral grant from the Junta de Castilla y León, financed in turn by the European Social Funds through the Consejería de Educacion. MMP is the recipient of a post-doctoral research grant at the Atapuerca Foundation. LM-F received financial support from the French State as part of the 'Investments for the future' Programme IdEx Bordeaux, reference ANR-10-IDEX-03-02. Some of the micro-CT images were obtained in the Laboratory of Microscopy of the CENIEH-ICTS (Spain) and in collaboration with CENIEH staff. The remaining dental samples were scanned in the Multidisciplinary Laboratory of the International Centre for Theoretical Physics (ICTP) in Trieste in collaboration with Federico Bernardini and Claudio Tuniz. The African sample from Sudan was provided by Dr. Christopher Dean from the Anatomy Department at University College London. Part of the sample was analyzed in the Anthropology Department at University College London in collaboration with Dr. Christopher Soligo. We are indebted to A. Oettlé, G. Krüger and E.N. L'Abbé for kindly authorizing access to the Pretoria Bone Collection (PBC) of the University of Pretoria. We also deeply thank F. de Beer and J.

Hoffman for carrying out the X-ray micro-CT imaging of the human specimens from the PBC included in this study.

Dirección General de Investigación of the Spanish Ministerio de Economía y Competitividad (MINECO/FEDER), Grant/Award Number: CGL2015-65387-C3-1, 3-P; The European Social Funds; Consejería de Educacion; of the Junta de Castilla y León; Atapuerca Foundation; Leakey Foundation