Multi-Key Fully Homomorphic Encryption (MKFHE) schemes represent a significant advancement in cryptographic technology, enabling multiple parties to perform arbitrary computations on data encrypted under different keys.These schemes utilize the complex ciphertext extension algorithms that extend the ciphertexts encrypted by the public key of a single party to the ciphertexts encrypted under all the participants' public keys, enabling collaborative evaluation.However, this process often compromises efficiency by increasing ciphertext size and requiring intricate operations.Moreover, the hardness of the majority of existing MKFHE schemes based on lattice problems such as Learning with Errors (LWE) or Ring Learning with Errors (RLWE) necessitates extensive sampling from error distributions.This approach leads to substantial noise propagation after only a few operations, limiting the scheme's practical utility.By addressing these key challenges, our work proposes a more streamlined and efficient Key Extension Multi-key Fully Homomorphic Encryption (MKEFHE) scheme based on the Learning with Rounding (LWR) problem, which eliminates the need for a complex ciphertext extension algorithm and an error distribution.

Parkerconsidered a new type of discrete Fourier transform, called nega-Hadamard transform. We prove several results regarding its behavior on combinations of Boolean functions and use this theory to derive several results on negabentness (that is, flat nega-spectrum) of concatenations, and partially symmetric functions. We derive the upper bound$\lceil {{ n}\over { 2}} \rceil $for the algebraic degree of a negabent function on$n$variables. Further, a characterization of bent–negabent functions is obtained within a subclass of the Maiorana–McFarland set. We develop a technique to construct bent–negabent Boolean functions by using complete mapping polynomials. Using this technique, we demonstrate that for each$\ell \geq 2$, there exist bent–negabent functions on$n = 12\ell $variables with algebraic degree$ {{ n}\over { 4}}+1 = 3\ell + 1$. It is also demonstrated that there exist bent–negabent functions on eight variables with algebraic degrees 2, 3, and 4. Simple proofs of several previously known facts are obtained as immediate consequences of our work.