It is known that every ovoid of the parabolic quadric Q(4,q), q = ph, p prime, intersects every three-dimensional elliptic quadric in 1 mod p points. We present a new approach which gives us a second proof of this result and, in the case when p = 2, allows us to prove that every ovoid of Q(4,q) either intersects all the three-dimensional elliptic quadrics in 1 mod 4 points or intersects all the...

We discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field. These elliptic curve cryptosystems may be more secure, because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem, especially over GF(2"). We discuss the question of prim...

A homomorphic, or incremental, multiset hash function, associates a hash value to arbitrary collections of objects (with possible repetitions) in such a way that the hash of the union of two collections is easy to compute from the hashes of the two collections themselves: it is simply their sum under a suitable group operation. In particular, hash values of large collections can be computed inc...

In this thesis, we study the software implementation of the NIST-recommended elliptic curves over prime fields. Our implementation goals are to achieve a fast, small, and portable cryptographic library, which supports elliptic curve digital signature generation and verification. The implementation results are presented on a Pentium I1 448.81 MHz. We also consider the sliding window algorithm (S...

New type of elliptic curve point multiplication is proposed, where complex multiplication by 2 − or by 2 ) 7 1 ( − ± is used instead of point duplication. This allows speeding up multiplication about 1.34 times. Using higher radix makes it possible to use one point duplication instead of two and to speed-up computation about 1.6 times. The method takes prime group order factorization: ρ ρ = r a...

There is a re-emerging demand for low-end devices such as 8-bit processors, driven by needs for pervasive applications like sensor networks and RF-ID tags. Security in pervasive applications, however, has been a major concern for their widespread acceptance. Public-key cryptosystems (PKC) like RSA and DSA generally involve computation-intensive arithmetic operations with operand sizes of 1024 −...

The Elliptic Curve Cryptosystem (ECC) provides the highest strength-per-bit of any cryptosystem known today. This paper provides an overview of the three hard mathematical problems which provide the basis for the security of publickey cryptosystems used today: the integer factorization problem (IFP), the discrete logarithm problem (DLP), and the elliptic curve discrete logarithm problem (ECDLP)...

An exploration of the benefits and challenges of a PKI based on elliptic curve cryptography

We consider the problem of finding integer-sided triangles with R/r an integer, where R and r are the radii of the circumcircle and incircle respectively. We show that such triangles are relatively rare.