Active biological processes like transcription, replication, recombination, DNA repair, and DNA packaging encounter bent DNA. Machineries associated with these processes interact with the DNA at short length (<100 base pair) scale. Thus, the study of elasticity of DNA at such length scale is very important. We use fully atomistic molecular dynamics (MD) simulations along with various theoretical methods to determine elastic properties of dsDNA of different lengths and base sequences. We also study DNA elasticity in nucleosome core particle (NCP) both in the presence and the absence of salt. We determine stretch modulus and persistence length of short dsDNA and nucleosomal DNA from contour length distribution and bend angle distribution, respectively. For short dsDNA, we find that stretch modulus increases with ionic strength while persistence length decreases. Calculated values of stretch modulus and persistence length for DNA are in quantitative agreement with available experimental data. The trend is opposite for NCP DNA. We find that the presence of histone core makes the DNA stiffer and thus making the persistence length 3-4 times higher than the bare DNA. Similarly, we also find an increase in the stretch modulus for the NCP DNA. Our study for the first time reports the elastic properties of DNA when it is wrapped around the histone core in NCP. We further show that the WLC model is inadequate to describe DNA elasticity at short length scale. Our results provide a deeper understanding of DNA mechanics and the methods are applicable to most protein-DNA complexes.