Periodic sequences, used as keys in cryptosystems, plays an important role in cryptography. Such periodic sequences should possess high linear complexity to resist B-M algorithm. Sequences constructed by cyclotomic cosets have been widely studied in the past few years. In this paper, the linear complexity of n-periodic cyclotomic sequences of order 2 and 4 over 𝔽_p has been calculated, where n and p are two distinct odd primes. The conclusions reveal that the presented sequences have high linear complexity in many cases, which indicates that the sequences can resist the linear attack.