  The Johnson-Lindenstrauss property ({\sf JLP}) of random matrices has immense
application in computer science ranging from compressed sensing, learning
theory, numerical linear algebra, to privacy. This paper explores the
properties and applications of a distribution of random matrices. Our
distribution satisfies {\sf JLP} with desirable properties like fast
matrix-vector multiplication, sparsity, and optimal subspace embedding. We can
sample a random matrix from this distribution using exactly $2n+n \log n$
random bits. We show that a random matrix picked from this distribution
preserves differential privacy under the condition that the input private
matrix satisfies certain spectral property. This improves the run-time of
various differentially private mechanisms like Blocki {\it et al.} (FOCS 2012)
and Upadhyay (ASIACRYPT 13). Our final construction has a bounded column
sparsity. Therefore, this answers an open problem stated in Blocki {\it et al.}
(FOCS 2012). Using the results of Baranuik {\it et al.} (Constructive
Approximation: 28(3)), our result implies a randomness efficient matrices that
satisfies the Restricted-Isometry Property of optimal order for small sparsity
with exactly linear random bits.
  We also show that other known distributions of sparse random matrices with
the {\sf JLP} does not preserves differential privacy; thereby, answering one
of the open problem posed by Blocki {\it et al.} (FOCS 2012). Extending on the
works of Kane and Nelson (JACM: 61(1)), we also give unified analysis of some
of the known Johnson-Lindenstrauss transform. We also present a self-contained
simplified proof of an inequality on quadratic form of Gaussian variables that
we use in all our proofs.
