This chapter discusses the algorithms that perform several basic arithmetic operations over the field Fp, sometimes referred to as base field arithmetic. A careful implementation of elementary field arithmetic operations is a crucial step to achieving efficient implementations of pairing-based cryptography. Base field arithmetic over a prime field Fp, is the most important layer for achieving high-performance pairing-based implementations. Efficient field arithmetic implementations must consider the processor’s word size and the bitsize of the prime p used by the cryptographic application. In contemporary microprocessor architectures, the processor’s word size ranges from 8 bits, for the smallest mobile processors, up to 64 bits, for high-end Intel or ARM microprocessors. The chapter also discusses the computation of squaring, exponentiation, multiplicative inverse, and square root over the field Fp.