Gaussian SF estimates of the Hessian are derived by taking the convolution of the Hessian of the objective function with a multi-variate Gaussian density functional. Through an integration-by-parts argument applied twice, the same is seen to be the convolution of the function itself with a scaled multi-variate Gaussian density. This results in a one-simulation estimate of the Hessian. The same simulation also helps in obtaining a one-simulation gradient estimate (see Chapter 6). Thus, one obtains a one-simulation Newton-based SF algorithm. A two-simulation estimate of the Hessian is also derived that incorporates the same two simulations as for the two-simulation gradient estimate, also derived in Chapter 6. This results in a two-simulation Newton SF algorithm. We limit the discussion in this chapter to Gaussian-based SF estimates only.