This paper attempts to disprove the asymptotic in time O(n log n) prediction of Schönhage-Strassen, claiming its ‘best possible’ result and remarking that no one will ever find a faster multiplication algorithm. Accordingly, this paper postulates that, the most desired complexity in time, i.e. O(n) is as achievable using only two basic arithmetic operations. We present four algorithms for large integer multiplications. First algorithm is based on the place value approach and achieves the much desired complexity of O(n), based on Nearest Place Values (NPV) approach. Second algorithm extends and improves Karatsuba algorithm for any ordered pairs greater than 2. It is important to remind that the present version of Karatsuba algorithm works only for ordered pairs of 2. Third algorithm called Addition and Subtraction (AnS) achieves time complexity of O(n) for very large integer multiplications using only repeated additions and subtractions. The fourth algorithm is called the Repeated Doubling Method (RDM), which is an improvised version of AnS algorithm and achieves time complexity of O(n).