This paper presents a new provably secure group signature scheme and a companion identity escrow scheme that are significantly more efficient than existing ones. The group signature allows a group member to sign messages anonymously, with the ability to reveal the signer's identity in case of a dispute. The scheme is proven secure under the strong RSA and decisional Diffie-Hellman assumptions, and is also coalition-resistant, meaning that a group of members cannot generate a signature that cannot be traced back to one of them. The identity escrow scheme is a variant of the group signature that allows for revocable anonymity. The new scheme improves upon previous work by being more efficient, statistically zero-knowledge, and secure against adaptive adversaries. The group signature is based on a membership certificate and a secret key, with the ability to generate anonymous signatures. The scheme is proven secure in the random oracle model and is efficient in both the interactive and non-interactive forms. The paper also discusses the security properties of the scheme, including correctness, unforgeability, anonymity, unlinkability, exculpability, and traceability. The scheme is shown to be secure under the strong RSA and decisional Diffie-Hellman assumptions, and the identity escrow scheme is also proven secure under these assumptions. The paper concludes that the new scheme offers significant improvements over previous work in terms of efficiency and security.This paper presents a new provably secure group signature scheme and a companion identity escrow scheme that are significantly more efficient than existing ones. The group signature allows a group member to sign messages anonymously, with the ability to reveal the signer's identity in case of a dispute. The scheme is proven secure under the strong RSA and decisional Diffie-Hellman assumptions, and is also coalition-resistant, meaning that a group of members cannot generate a signature that cannot be traced back to one of them. The identity escrow scheme is a variant of the group signature that allows for revocable anonymity. The new scheme improves upon previous work by being more efficient, statistically zero-knowledge, and secure against adaptive adversaries. The group signature is based on a membership certificate and a secret key, with the ability to generate anonymous signatures. The scheme is proven secure in the random oracle model and is efficient in both the interactive and non-interactive forms. The paper also discusses the security properties of the scheme, including correctness, unforgeability, anonymity, unlinkability, exculpability, and traceability. The scheme is shown to be secure under the strong RSA and decisional Diffie-Hellman assumptions, and the identity escrow scheme is also proven secure under these assumptions. The paper concludes that the new scheme offers significant improvements over previous work in terms of efficiency and security.