InNCo$_{3}$ and InNNi$_{3}$ with cubic anti-perovskite structure, we carried out the first-principles calculations on these compounds using the pseudopotentials method with plane-wave basis set within the local density approximation and the generalized gradient approximation. Since the elastic properties of a solid are highly associated with various fundamental solid-state properties such as phonon spectra, specific heat, Debye temperature, and so on, we have calculated the independent elastic constant and the elastic moduli of InNCo$_{3}$ and InNNi$_{3}$. II. COMPUTATIONAL DETAILS All calculations on antiperovskite-type InNCo$_{3}$ and InNNi$_{3}$ were performed using the Quantum ESPRESSO code $^{20}$, which is based on the density functional theory (DFT)$^{21}$. The electronic exchange-correlation potential was calculated within the local density ap- proximation (LDA) 22,23 and the generalized gradient approximation using the scheme of Perdew-Burke-Ernzerhof (PBE) $^{24}$. The spin polarization was also considered in the calcula- tion in order to assess the magnetic properties of these compounds. Electron-ion interaction was represented by the norm-conserving optimized 25 designed nonlocal pseudopotentials. The 4 d electrons are explicitly included in the valence of In. The electronic wavefunctions were expanded by the plane waves up to a kinetic energy cutoff of 55 Ry. The k -point sampling in Brillouin zone (BZ) of simple cubic lattice was treated with the Monkhorst- Pack scheme 26 and a 20 × 20 × 20 k -point mesh (i.e., 286 irreducible points in the first BZ) was used. The chosen plane-wave cutoff and number of k points were carefully checked to ensure that the total energy was converged to be better than 1 mRy/cell. The total energies are obtained as a function of volume and they are fitted with the Birch-Murnaghan 3rd- order equation of states (EoS) 27 to give the equilibrium lattice constant and other ground state properties. During the calculation of density of states (DOS), a dense k -point mesh of 30 × 30 × 30 is used, the total DOS is computed by the tetrahedron method $^{28}$, and the atomic-projected DOS is calculated by the L¨ owdin populations $^{29}$. For a cubic crystal, its independent elastic constants are c$_{11}$ , c$_{12}$ , and c$_{44}$ . To determine the elastic constants of InNCo$_{3}$ and InNNi$_{3}$ by means of the curvature of the internal energy versus the strain curves $^{30,31}$, three strain modes32 are adopted and their nonzero strains are as follows: (1) ϵ$_{11}$ = ϵ$_{22}$ = δ , ϵ$_{33}$ = (1 + δ ) $_{−}$2 − 1; (2) ϵ$_{11}$ = ϵ$_{22}$ = ϵ$_{33}$ = δ ; and (3) ϵ$_{12}$ = ϵ$_{21}$ = δ/ 2, ϵ$_{33}$ = δ $^{2}$/ (4 − δ $^{2}$). The deformation magnitudes δ from -0.012 to 0.012 3