11 research outputs found
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
Constructing elliptic curves of prime order
We present a very efficient algorithm to construct an elliptic curve E and a
finite field F such that the order of the point group E(F) is a given prime
number N. Heuristically, this algorithm only takes polynomial time Otilde((\log
N)^3), and it is so fast that it may profitably be used to tackle the related
problem of finding elliptic curves with point groups of prime order of
prescribed size. We also discuss the impact of the use of high level modular
functions to reduce the run time by large constant factors and show that recent
gonality bounds for modular curves imply limits on the time reduction that can
be obtained.Comment: 13 page
Elliptic Reciprocity
The paper introduces the notions of an elliptic pair, an elliptic cycle and
an elliptic list over a square free positive integer d. These concepts are
related to the notions of amicable pairs of primes and aliquot cycles that were
introduced by Silverman and Stange. Settling a matter left open by Silverman
and Stange it is shown that for d=3 there are elliptic cycles of length 6. For
d not equal to 3 the question of the existence of proper elliptic lists of
length n over d is reduced to the the theory of prime producing quadratic
polynomials. For d=163 a proper elliptic list of length 40 is exhibited. It is
shown that for each d there is an upper bound on the length of a proper
elliptic list over d. The final section of the paper contains heuristic
arguments supporting conjectured asymptotics for the number of elliptic pairs
below integer X. Finally, for d congruent to 3 modulo 8 the existence of
infinitely many anomalous prime numbers is derived from Bunyakowski's
Conjecture for quadratic polynomials.Comment: 17 pages, including one figure and two table
Computation of Trusted Short Weierstrass Elliptic Curves for Cryptography
Short Weierstrass's elliptic curves with underlying hard Elliptic Curve
Discrete Logarithm Problems was widely used in Cryptographic applications. This
paper introduces a new security notation 'trusted security' for computation
methods of elliptic curves for cryptography. Three additional "trusted security
acceptance criteria" is proposed to be met by the elliptic curves aimed for
cryptography. Further, two cryptographically secure elliptic curves over 256
bit and 384 bit prime fields are demonstrated which are secure from ECDLP, ECC
as well as trust perspectives. The proposed elliptic curves are successfully
subjected to thorough security analysis and performance evaluation with respect
to key generation and signing/verification and hence, proven for their
cryptographic suitability and great feasibility for acceptance by the
community.Comment: CYBERNETICS AND INFORMATION TECHNOLOGIES, Volume 21, No
Aliquot Cycles for Elliptic Curves with Complex Multiplication
We review the history of elliptic curves and show that it is possible to form a group law using the points on an elliptic curve over some field L. We review various methods for computing the order of this group when L is finite, including the complex multiplication method. We then define and examine the properties of elliptic pairs, lists, and cycles, which are related to the notions of amicable pairs and aliquot cycles for elliptic curves, defined by Silverman and Stange. We then use the properties of elliptic pairs to prove that aliquot cycles of length greater than two exist for elliptic curves with complex multiplication, contrary to an assertion of Silverman and Stange, proving that such cycles only occur for elliptic curves of j-invariant equal to zero, and they always have length six. We explore the connection between elliptic pairs and several other conjectures, and propose limitations on the lengths of elliptic lists
Batch Verification of Elliptic Curve Digital Signatures
This thesis investigates the efficiency of batching the verification of elliptic curve signatures. The first signature scheme considered is a modification of ECDSA proposed by Antipa et al.\ along with a batch verification algorithm by Cheon and Yi. Next, Bernstein's EdDSA signature scheme and the Bos-Coster multi-exponentiation algorithm are presented and the asymptotic runtime is examined. Following background on bilinear pairings, the Camenisch-Hohenberger-Pedersen (CHP) pairing-based signature scheme is presented in the Type 3 setting, along with the derivative BN-IBV due to Zhang, Lu, Lin, Ho and Shen. We proceed to count field operations for each signature scheme and an exact analysis of the results is given. When considered in the context of batch verification, we find that the Cheon-Yi and Bos-Coster methods have similar costs in practice (assuming the same curve model). We also find that when batch verifying signatures, CHP is only 11\% slower than EdDSA with Bos-Coster, a significant improvement over the gap in single verification cost between the two schemes