A Lower Bound for the Class Number of a Real Quadratic Field of ERD-Type

Mollin, R. A.; Zhang, L.-C.; Kemp, Paula

doi:10.4153/cmb-1994-014-8

Cited by 3 publications

(1 citation statement)

References 3 publications

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“…Most of the material in this section can be found in [11] and Chapter 2, §7, of [4]; while the latter source considers mostly imaginary quadratic fields, much of the results are easily extendable to real quadratic fields as was done in [19].…”

Section: Overview Of Quadratic Ordersmentioning

confidence: 99%

An Adaptation of the NICE Cryptosystem to Real Quadratic Orders

Jacobson

Scheidler

Weimer³

2008

Progress in Cryptology – AFRICACRYPT 2008

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Abstract. In 2000, Paulus and Takagi introduced a public key cryptosystem called NICE that exploits the relationship between maximal and non-maximal orders in imaginary quadratic number fields. Relying on the intractability of integer factorization, NICE provides a similar level of security as RSA, but has faster decryption. This paper presents REAL-NICE, an adaptation of NICE to orders in real quadratic fields. REAL-NICE supports smaller public keys than NICE, and while preliminary computations suggest that it is somewhat slower than NICE, it still significantly outperforms RSA in decryption.

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Section: Overview Of Quadratic Ordersmentioning

confidence: 99%

An Adaptation of the NICE Cryptosystem to Real Quadratic Orders

Jacobson

Scheidler

Weimer³

2008

Progress in Cryptology – AFRICACRYPT 2008

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Effective lower bound for the class number of a certain family of real quadratic fields

Lapkova¹

2012

Journal of Number Theory

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In this work we establish an effective lower bound for the class number of the family of real quadratic fields Q( √ d), where d = n 2 + 4 is a square-free positive integer with n = m(m 2 − 306) for some odd m, with the extra condition ( d N ) = −1 for N = 2 3 · 3 3 · 103 · 10 303. This result can be regarded as a corollary of a theorem of Goldfeld and some calculations involving elliptic curves and local heights. The lower bound tending to infinity for a subfamily of the real quadratic fields with discriminant d = n 2 + 4 could be interesting having in mind that even the class number two problem for these discriminants is not yet solved unconditionally.

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Proof, Disproof and Advances Concerning Certain Conjectures on Real Quadratic Fields

Mollin¹,

Williams²

1995

Can. j. math.

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The purpose of this paper is to address conjectures raised in [2]. We show that one of the conjectures is false and we advance the proof of another by proving it for an infinite set of cases. Furthermore, we give hard evidence as to why the conjecture is true and show what remains to be done to complete the proof. Finally, we prove a conjecture given by S. Louboutin, for Mathematical Reviews, in his discussion of the aforementioned paper.

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A Lower Bound for the Class Number of a Real Quadratic Field of ERD-Type

Abstract: In this paper, we use the Lagrange neighbour and our equivalence theorem for primitive ideals to obtain lower bounds which are sharper than those given in the literature for class numbers of real quadratic fields in general, but applied to greatest advantage when d is of ERD type.

Cited by 3 publications

References 3 publications

An Adaptation of the NICE Cryptosystem to Real Quadratic Orders

An Adaptation of the NICE Cryptosystem to Real Quadratic Orders

Effective lower bound for the class number of a certain family of real quadratic fields

Proof, Disproof and Advances Concerning Certain Conjectures on Real Quadratic Fields

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