A kilobit hidden SNFS discrete logarithm computation

We perform a special number field sieve discrete logarithm computation in a 1024-bit prime field. To our knowledge, this is the first kilobit-sized discrete logarithm computation ever reported for prime fields. This computation took a little over two months of calendar time on an academic cluster using the open-source CADO-NFS software. Our chosen prime $p$ looks random, and $p--1$ has a 160-bit prime factor, in line with recommended parameters for the Digital Signature Algorithm. However, our p has been trapdoored in such a way that the special number field sieve can be used to compute discrete logarithms in $\mathbb{F}\_p^*$ , yet detecting that p has this trapdoor seems out of reach. Twenty-five years ago, there was considerable controversy around the possibility of back-doored parameters for DSA. Our computations show that trapdoored primes are entirely feasible with current computing technology. We also describe special number field sieve discrete log computations carried out for multiple weak primes found in use in the wild. As can be expected from a trapdoor mechanism which we say is hard to detect, our research did not reveal any trapdoored prime in wide use. The only way for a user to defend against a hypothetical trapdoor of this kind is to require verifiably random primes

Security Estimates for Quadratic Field Based Cryptosystems

We describe implementations for solving the discrete logarithm problem in the class group of an imaginary quadratic field and in the infrastructure of a real quadratic field. The algorithms used incorporate improvements over previously-used algorithms, and extensive numerical results are presented demonstrating their efficiency. This data is used as the basis for extrapolations, used to provide recommendations for parameter sizes providing approximately the same level of security as block ciphers with $80,$ $112,$ $128,$ $192,$ and $256$-bit symmetric keys

Certifying RSA public keys with an efficient NIZK

In many applications, it is important to verify that an RSA public key ( N,e ) specifies a permutation, in order to prevent attacks due to adversarially-generated public keys. We design and implement a simple and efficient noninteractive zero-knowledge protocol (in the random oracle model) for this task. The key feature of our protocol is compatibility with existing RSA implementations and standards. The protocol works for any choice of e. Applications concerned about adversarial key generation can just append our proof to the RSA public key without any other modifications to existing code or cryptographic libraries. Users need only perform a one- time verification of the proof to ensure that raising to the power e is a permutation of the integers modulo N . For typical parameter settings, the proof consists of nine integers modulo N; generating the proof and verifying it both require about nine modular exponentiations.https://eprint.iacr.org/2018/057.pdfFirst author draf

The Elgamal Cryptosystem is better than Th RSA Cryptosystem for Mental Poker

Cryptosystems are one of the most important parts of secure online poker card games. However, there is no research comparing the RSA Cryptosystem (RC) and Elgamal Cryptosystem (EC) for mental poker card games. This paper compares the RSA Cryptosystem and Elgamal Cryptosystem implementations of mental poker card games using distributed key generation schemes. Each implementation is based on a joint encryption/decryption of individual cards. Both implementations use shared private key encryption/decryption schemes and neither uses a trusted third party (TTP). The comparison criteria will be concentrated on the security and computational complexity of the game, collusions among the players and the debate between the discrete logarithm problem (DLP) and the factoring problem (FP) for the encryption/decryption schemes. Under these criteria, the comparison results demonstrate that the Elgamal Cryptosystem has better efficiency and effectiveness than RSA for mental poker card games