Theoretically secure cryptosystems, digital signatures may not be secure after being implemented on Internet of Things (IoT) devices and PCs because of side-channel attacks (SCA). Because RSA key generation and ECDSA require GCD computations or modular inversions, which are often computed using the binary Euclidean algorithm (BEA) or binary extended Euclidean algorithm (BEEA), the SCA weaknesses of BEA and BEEA become a serious concern. Constant-time GCD (CT-GCD) and constant-time modular inversion (CTMI) algorithms are effective countermeasures in such situations. Modular inversion based on Fermat's little theorem (FLT) can work in constant time, but it is not efficient for general inputs. Two CTMI algorithms, named BOS and BY in this paper, were proposed by Bos, Bernstein and Yang, respectively. Their algorithms are all based on the concept of BEA. However, one iteration of BOS has complicated computations, and BY requires more iterations. A small number of iterations and simple computations during one iteration are good characteristics of a constant-time algorithm. Based on this view, this study proposes new short-iteration CT-GCD and CTMI algorithms over Fp borrowing a simple concept from BEA. Our algorithms are evaluated from a theoretical perspective. Compared with BOS, BY, and the improved version of BY, our short-iteration algorithms are experimentally demonstrated to be faster.