Based on residue arithmetic

a new form of polynomial named PAPB in Fp[x] had been constructed.The amount and distribution of such irreducible polynomials had also been investigated.Then

an efficient algorithm for multiplication modulo PAPB had been proposed using residue arithmetic and Chinese remainder theorem.In the end

the result analysis was given.The theoretic analysis and simulation experiment show that for some finite field of medium prime characteristic

the given algorithm only costs O（k1.5） operations in Fp while the multiplication modulo irreducible binomial cost O（k2） operations

which shows a better performance.As a result

the work presents an interesting perspective for optimal extension field and elliptic curve cryptography.