This paper by Victor Shoup, published in IBM Research, focuses on the computational complexity of the discrete logarithm problem and related cryptographic problems, particularly in the context of "generic algorithms." These algorithms do not exploit any special properties of the encodings of group elements beyond their unique binary string representation. The main results include: 1. **Lower Bounds for Discrete Logarithm and Diffie-Hellman Problems**: Any generic algorithm solving the discrete logarithm problem or the Diffie-Hellman problem must perform at least \(\Omega(p^{1/2})\) group operations, where \(p\) is the largest prime dividing the order of the group. This bound matches known upper bounds. 2. **Noncyclic Groups**: For the product of \(r\) cyclic groups of prime order \(p\), any generic algorithm must perform at least \(\Omega(p^{r/2})\) group operations. 3. **Diffie-Hellman Problem**: The paper proves that determining which of two possible solutions is correct is hard when the group order is divisible by only large primes. 4. **Subgroups**: If a Diffie-Hellman oracle for a group \(G\) is available, solving the Diffie-Hellman problem in a proper subgroup \(H\) is not significantly easier, even if \(H\) is a prime-order subgroup. 5. **Security of Schnorr's Identification Scheme**: The scheme is secure against active attacks by generic algorithms, provided the discrete logarithm problem is hard. 6. **Correcting a Faulty Diffie-Hellman Oracle**: The paper presents a method to efficiently turn a faulty Diffie-Hellman oracle into one that is almost always correct, with a running time that depends on the success probability of the faulty oracle. The paper also discusses related work, including lower bounds in a "black box" model and earlier results by Babai, Szemerédi, and Nechaev, and provides a detailed proof of the main theorems using game-theoretic arguments.This paper by Victor Shoup, published in IBM Research, focuses on the computational complexity of the discrete logarithm problem and related cryptographic problems, particularly in the context of "generic algorithms." These algorithms do not exploit any special properties of the encodings of group elements beyond their unique binary string representation. The main results include: 1. **Lower Bounds for Discrete Logarithm and Diffie-Hellman Problems**: Any generic algorithm solving the discrete logarithm problem or the Diffie-Hellman problem must perform at least \(\Omega(p^{1/2})\) group operations, where \(p\) is the largest prime dividing the order of the group. This bound matches known upper bounds. 2. **Noncyclic Groups**: For the product of \(r\) cyclic groups of prime order \(p\), any generic algorithm must perform at least \(\Omega(p^{r/2})\) group operations. 3. **Diffie-Hellman Problem**: The paper proves that determining which of two possible solutions is correct is hard when the group order is divisible by only large primes. 4. **Subgroups**: If a Diffie-Hellman oracle for a group \(G\) is available, solving the Diffie-Hellman problem in a proper subgroup \(H\) is not significantly easier, even if \(H\) is a prime-order subgroup. 5. **Security of Schnorr's Identification Scheme**: The scheme is secure against active attacks by generic algorithms, provided the discrete logarithm problem is hard. 6. **Correcting a Faulty Diffie-Hellman Oracle**: The paper presents a method to efficiently turn a faulty Diffie-Hellman oracle into one that is almost always correct, with a running time that depends on the success probability of the faulty oracle. The paper also discusses related work, including lower bounds in a "black box" model and earlier results by Babai, Szemerédi, and Nechaev, and provides a detailed proof of the main theorems using game-theoretic arguments.