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We consider the operator,$f$(Δ) for Δ the Laplacian, on spaces of measures on the sphere in$R$^{$d$}, show how to determine a family of approximating kernels for this operator assuming that certain technical conditions are satisfied, and give estimates for the$L$^{2}-norm of$f$(Δ)μ in terms of the energy of the measure μ. We derive a formula, analogous to the classical formula relating the energy of a measure on$R$^{$d$}with its Fourier transform, comparing the energy of a measure on the sphere with the size of its spherical harmonics. An application is given to pluriharmonic measures.

In this paper, we show that the signal function used in Ring-Learning with Errors (RLWE) key exchange could leak information to find the secret s of a reused public key p = as+2e. This work is motivated by an attack proposed in [1] and gives an insight into how public keys reused for long term in RLWE key exchange protocols can be exploited. This work specifically focuses on the attack on the KE protocol in [2] by initiating multiple sessions with the honest party and analyze the output of the signal function. Experiments have confirmed the success of our attack in recovering the secret.

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