In today's world, wireless communication pervades every part of life and is currently advancing to include transportation. Vehicles can communicate with one another, with pedestrians, with roadside infrastructure, and other moving objects using the basic type of vehicular communication known as vehicle-to-everything (V2X) communication. In addition to road safety, security and privacy issues must be taken into consideration in V2X activities. The objective of this research is to ensure a high level of security in various forms of vehicular communication (V2V, V2I, and V2N) and to support vehicles in safely receiving all keys and messages from Road side Unit (RSU), other vehicles, or the network with the support of simple cryptographic techniques. Elliptic Curve Cryptography (ECC) is one of the many cryptographic techniques that offer a desirable response to this requirement. The Koblitz curve secp256k1 is supported by the designed ECC processor for 256-bit point multiplication and point addition. A powerful illustration of the "Divide and Conquer" strategy's ability to accelerate multiplication asymptotically is given by the Karatsuba algorithm. One of the algorithms made to increase effectiveness and decrease cost in order to simplify multiplication is the Karatsuba algorithm. The ECC processor's multiplication operation is accelerated further by the incorporation of pipelining. With a maximum clock frequency of 238.40MHz, the proposed Karatsuba-based ECC processor executes 256-bit single point multiplication in 0.937ms, providing 273.21kbps throughput, and taking up 8.42k slices in a Virtex-7 FPGA. Scalar multiplication is extended by incorporating a pipeline by increasing the highest clock frequency by up to 7.97%, which decreases time consumption by 9.90% and boosts throughput by 10.99%. In terms of area, operating frequency, area-delay product, and throughput, the suggested pipelined Karatsuba multiplier based ECC processor performs better than the existing designs. As a result, the suggested method's implementation provides fast and time-efficient scalar multiplication with effective hardware usage, all without sacrificing performance.