Bitcoin Security with a Twisted Edwards Curve

International audienceThe security of the Bitcoin cryptocurrency system depends on the Koblitz curve secp256k1 combined with the digital signature ECDSA and the hash function SHA-256. In this paper, we show that the security of Bitcoin with ECDSA and secp256k1 is not optimal and present a detailed study of the efficiency of Bitcoin with the digital signature algorithm Ed25519 combined with the twisted Edwards curve CurveEd25519 and the hash function SHA-512. We show that Bitcoin is more secure and more efficient with the digital signature algorithm Ed25519 and the twisted Edwards curve CurveEd25519. Subject Classifications: 94A6

An Efficient Collision Detection Method for Computing Discrete Logarithms with Pollard's Rho

Pollard's rho method and its parallelized variant are at present known as the best generic algorithms for computing discrete logarithms. However, when we compute discrete logarithms in cyclic groups of large orders using Pollard's rho method, collision detection is always a high time and space consumer. In this paper, we present a new efficient collision detection algorithm for Pollard's rho method. The new algorithm is more efficient than the previous distinguished point method and can be easily adapted to other applications. However, the new algorithm does not work with the parallelized rho method, but it can be parallelized with Pollard's lambda method. Besides the theoretical analysis, we also compare the performances of the new algorithm with the distinguished point method in experiments with elliptic curve groups. The experiments show that the new algorithm can reduce the expected number of iterations before reaching a match from 1.309G to 1.295G under the same space requirements for the single rho method

An Internet-Wide Analysis of Diffie-Hellman Key Exchange and X.509 Certificates in TLS

Transport Layer Security (TLS) is a mature cryptographic protocol, but has flexibility during implementation which can introduce exploitable flaws. New vulnerabilities are routinely discovered that affect the security of TLS implementations. We discovered that discrete logarithm implementations have poor parameter validation, and we mathematically constructed a deniable backdoor to exploit this flaw in the finite field Diffie-Hellman key exchange. We described attack vectors an attacker could use to position this backdoor, and outlined a man-in-the-middle attack that exploits the backdoor to force Diffie-Hellman use during the TLS connection. We conducted an Internet-wide survey of ephemeral finite field Diffie-Hellman (DHE) across TLS and STARTTLS, finding hundreds of potentially backdoored DHE parameters and partially recovering the private DHE key in some cases. Disclosures were made to companies using these parameters, resulting in a public security advisory and discussions with the CTO of a billion-dollar company. We conducted a second Internet-wide survey investigating X.509 certificate name mismatch errors, finding approximately 70 million websites invalidated by these errors and additionally discovering over 1000 websites made inaccessible due to a combination of forced HTTPS and mismatch errors. We determined that name mismatch errors occur largely due to certificate mismanagement by web hosting and content delivery network companies. Further research into TLS implementations is necessary to encourage the use of more secure parameters

A framework for World Wide Web client-authentication protocols

Existing client-authentication protocols deployed on the World Wide Web today are based on conventional distributed systems and fail to address the problems specific to the application domain. Some of the protocols restrict the mobility of the client by equating user identity to a machine or network address, others depend on sound password management strategies, and yet others compromise the privacy of the user by transmitting personal information for authentication. We introduce a new framework for client-authentication by separating two goals that current protocols achieve simultaneously: 1. Maintain persistent sense of identity across different sessions. 2. Prove facts about the user to the site. These problems are independent, in the sense that any protocol for solving the first problem can be combined with any protocol for solving the second. Separation of the two purposes opens up the possibility of designing systems which balance two conflicting goals, authentication and anonymity. We propose a solution to the first problem, based on the Digital Signature Standard. The implications of this framework from the point of view of user privacy are examined. The paper is concluded with suggestions for integrating the proposed scheme into the existing WWW architecture

Computing Discrete Logarithms in an Interval

The discrete logarithm problem in an interval of size $N$ in a group $G$ is: Given $g, h \in G$ and an integer $N$ to find an integer $0 \le n \le N$, if it exists, such that $h = g^n$. Previously the best low-storage algorithm to solve this problem was the van Oorschot and Wiener version of the Pollard kangaroo method. The heuristic average case running time of this method is $(2 + o(1)) \sqrt{N}$ group operations. We present two new low-storage algorithms for the discrete logarithm problem in an interval of size $N$. The first algorithm is based on the Pollard kangaroo method, but uses 4 kangaroos instead of the usual two. We explain why this algorithm has heuristic average case expected running time of $(1.715 + o(1)) \sqrt{N}$ group operations. The second algorithm is based on the Gaudry-Schost algorithm and the ideas of our first algorithm. We explain why this algorithm has heuristic average case expected running time of $(1.661 + o(1)) \sqrt{N}$ group operations. We give experimental results that show that the methods do work close to that predicted by the theoretical analysis. This is a revised version since the published paper that contains a corrected proof of Theorem 6 (the statement of Theorem 6 is unchanged). We thank Ravi Montenegro for pointing out the errors

Reconfigurable elliptic curve cryptography

Elliptic Curve Cryptosystems (ECC) have been proposed as an alternative to other established public key cryptosystems such as RSA (Rivest Shamir Adleman). ECC provide more security per bit than other known public key schemes based on the discrete logarithm problem. Smaller key sizes result in faster computations, lower power consumption and memory and bandwidth savings, thus making ECC a fast, flexible and cost-effective solution for providing security in constrained environments. Implementing ECC on reconfigurable platform combines the speed, security and concurrency of hardware along with the flexibility of the software approach. This work proposes a generic architecture for elliptic curve cryptosystem on a Field Programmable Gate Array (FPGA) that performs an elliptic curve scalar multiplication in 1.16milliseconds for GF (2163), which is considerably faster than most other documented implementations. One of the benefits of the proposed processor architecture is that it is easily reprogrammable to use different algorithms and is adaptable to any field order. Also through reconfiguration the arithmetic unit can be optimized for different area/speed requirements. The mathematics involved uses binary extension field of the form GF (2n) as the underlying field and polynomial basis for the representation of the elements in the field. A significant gain in performance is obtained by using projective coordinates for the points on the curve during the computation process