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This paper proposes a novel hybrid cryptographic scheme for the generation of pair-wise network topology authenticated (TAK) keys in a Wireless Sensor Network (WSN) using vector algebra in GF(q). The proposed scheme is deterministic, pair-wise keys are not pre-distributed but generated starting from partial key components, keys management exploits benefits from both symmetric and asymmetric schemes (hybrid cryptography) and each key in a pair node can be generated only if nodes have been authenticated (key authentication). Network topology authentication, and hybrid key cryptography are the building blocks for this proposal: the former means that a cryptographic key can be generated if and only if the current network topology is compliant to the ldquoplanned network topologyrdquo, which acts as the authenticated reference; the latter means that the proposed scheme is a combination of features from symmetric (for the ciphering and authentication model) and asymmetric cryptography (for the key generation model). The proposal fits the security requirement of a cryptographic scheme for WSN in a limited computing resource. A deep quantitative security analysis has been carried out. Moreover the cost analysis of the scheme in terms of computational time and memory usage for each node has been carried on and reported for the case of a 128-bit key.