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Ensuring the integrity of outsourced data is one of several functional requirements in cloud storage services, although designing an effective and secure public auditing scheme remains an ongoing research challenge. For example, a number of existing schemes are designed based on symmetric bilinear pairing, which are computationally expensive. While certificate-based auditing schemes can simplify certificate management and mitigate key escrow concern, such schemes are not popular in the literature. Therefore, we propose a lightweight certificate-based public/private auditing scheme based on asymmetric bilinear pairing for cloud storage. Specifically, we describe a new method of designing lightweight auditing schemes. Additionally, our proposed scheme is proven to be secure in the random oracle model. We then implement our scheme by using type D curve of the pairing-based library. The findings demonstrate that our auditing scheme significantly reduces client's computation cost in the tag-generation (TagGen) phase, particularly in comparison to several other competing schemes. For example, when the block size is 10 KB, our on-line computation cost in the TagGen phase takes only 0.45 seconds, while it requires at least 62.83 seconds in other schemes. Furthermore, the findings show that when the number of blocks is fixed, the on-line computation cost in our TagGen phase is constant despite varying file sizes.