The eiDAS 2.0 regulation (electronic identification and trust services) that defines the new EU Digital Identity Wallet (EUDIW) is an important step towards developing interoperable digital identities in Europe for the public and private sectors. The regulation, if realized with the right technology, can make Europe the front runner in private and secure identification mechanisms in the digital space, and act as a template for future digital identity systems in other regions. Unfortunately, we believe that some of the currently suggested design aspects of the EUDI and its credential mechanism fall short of the privacy requirements that were explicitly defined after extensive debate in the Digital Identity regulation. The main reason for this shortcoming in the current proposal is that it relies on cryptographic methods that were never designed for such requirements. We do not see a way to fix the proposed solution to meet all the privacy features as required by the regulation; we believe that a larger redesign is in order. In this document, we propose to use a different cryptographic mechanism instead; namely, anonymous credentials. Anonymous credentials were designed specifically to achieve authentication and identification that are both secure and privacy-preserving. As a result, they fully meet the requirements put forth in the eiDAS 2.0 regulation. Moreover, they are by now a mature technology. This technology was developed more than twenty years ago, and extensive efforts have been expended to analyze, improve, implement, standardize, test, and deploy it. Anonymous credentials are well understood by the scientific community. Our specific recommendation is to use the BBS family of anonymous credentials. For BBS, thanks to prior work by the W3C, the Decentralized Identity Foundation, IETF/IRTF, ISO, and other standardization bodies, as well as the availability of open-source software libraries, the EC can develop a standard and reference implementation with only a modest effort. We additionally recommend that the EUDI be designed following the principle of crypto-agility, meaning that its underlying technologies can be upgraded quickly in the future if the need arises. We thank the EC for this opportunity to weigh in. We would be excited to continue to provide feedback on this important endeavor, and see this document as the beginning of a longer dialogue.