6,471 research outputs found
Elliptic Curve Cryptography in Practice
In this paper, we perform a review of elliptic curve cryptography (ECC), as it is used in practice today, in order to reveal unique mistakes and vulnerabilities that arise in implementations of ECC. We study four popular protocols that make use of this type of public-key cryptography: Bitcoin, secure shell (SSH), transport layer security (TLS), and the Austrian e-ID card. We are pleased to observe that about 1 in 10 systems support ECC across the TLS and SSH protocols. However, we find that despite the high stakes of money, access and resources protected by ECC, implementations suffer from vulnerabilities similar to those that plague previous cryptographic systems
L’algoritme de Schoof i la criptografia basada en corbes el·lÃptiques
Treballs Finals de Grau de Matemà tiques, Facultat de Matemà tiques, Universitat de Barcelona, Any: 2018, Director: Xavier Guitart Morales[en] In this project we study elliptic curve cryptography, which uses groups of points of elliptic curves over finite fields, and Schoof’s algorithm for computing the cardinality of such groups. We introduce the mathematical concepts related to elliptic curves and their group of points, and we describe some elliptic curve based public
key cryptosystems. In addition, we provide the necessary background for Schoof’s algorithm, which we describe and implement using the software Sage. We also discuss some criteria used in practice to choose suitable elliptic curves for cryptography, which show the importance of counting points algorithms in cryptographic applications
Elliptic Curves of Nearly Prime Order
Constructing an elliptic curve of prime order has a significant role in
elliptic curve cryptography. For security purposes, we need an elliptic curve of
almost prime order. In this paper, we propose an efficient technique to generate
an elliptic curve of nearly prime order. In practice, this algorithm produces an
elliptic curve of order 2 times a prime number. Therefore, these elliptic curves are
appropriate for practical uses. Presently, the most known working algorithms for
generating elliptic curves of prime order are based on complex multiplication.
The advantages of the proposed technique are: it does not require a deep mathe-
matical theory, it is easy to implement in any programming language and produces
an elliptic curve with a remarkably simple expression
Still Wrong Use of Pairings in Cryptography
Several pairing-based cryptographic protocols are recently proposed with a
wide variety of new novel applications including the ones in emerging
technologies like cloud computing, internet of things (IoT), e-health systems
and wearable technologies. There have been however a wide range of incorrect
use of these primitives. The paper of Galbraith, Paterson, and Smart (2006)
pointed out most of the issues related to the incorrect use of pairing-based
cryptography. However, we noticed that some recently proposed applications
still do not use these primitives correctly. This leads to unrealizable,
insecure or too inefficient designs of pairing-based protocols. We observed
that one reason is not being aware of the recent advancements on solving the
discrete logarithm problems in some groups. The main purpose of this article is
to give an understandable, informative, and the most up-to-date criteria for
the correct use of pairing-based cryptography. We thereby deliberately avoid
most of the technical details and rather give special emphasis on the
importance of the correct use of bilinear maps by realizing secure
cryptographic protocols. We list a collection of some recent papers having
wrong security assumptions or realizability/efficiency issues. Finally, we give
a compact and an up-to-date recipe of the correct use of pairings.Comment: 25 page
Cryptography: Mathematical Advancements on Cyber Security
The origin of cryptography, the study of encoding and decoding messages, dates back to ancient times around 1900 BC. The ancient Egyptians enlisted the use of basic encryption techniques to conceal personal information. Eventually, the realm of cryptography grew to include the concealment of more important information, and cryptography quickly became the backbone of cyber security. Many companies today use encryption to protect online data, and the government even uses encryption to conceal confidential information. Mathematics played a huge role in advancing the methods of cryptography. By looking at the math behind the most basic methods to the newest methods of cryptography, one can learn how cryptography has advanced and will continue to advance
Analysis of Parallel Montgomery Multiplication in CUDA
For a given level of security, elliptic curve cryptography (ECC) offers improved efficiency over classic public key implementations. Point multiplication is the most common operation in ECC and, consequently, any significant improvement in perfor- mance will likely require accelerating point multiplication. In ECC, the Montgomery algorithm is widely used for point multiplication. The primary purpose of this project is to implement and analyze a parallel implementation of the Montgomery algorithm as it is used in ECC. Specifically, the performance of CPU-based Montgomery multiplication and a GPU-based implementation in CUDA are compared
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