scholarly journals A NOTE ON THE EQUIVALENCE OF THE PARITY OF CLASS NUMBERS AND THE SIGNATURE RANKS OF UNITS IN CYCLOTOMIC FIELDS

2018 ◽  
Vol 238 ◽  
pp. 206-214 ◽  
Author(s):  
DAVID S. DUMMIT
Keyword(s):  
Prime Power ◽  
Cyclotomic Field ◽  
Class Numbers ◽  
Cyclotomic Fields ◽  
Circular Units

We collect some statements regarding equivalence of the parities of various class numbers and signature ranks of units in prime power cyclotomic fields. We correct some misstatements in the literature regarding these parities by providing an example of a prime cyclotomic field where the signature rank of the units and the signature rank of the circular units are not equal.

scholarly journals The work of K. Ramachandra in algebraic number theory

2013 ◽  
Vol Volume 34-35 ◽  
Author(s):  
M. Ram Murty
Keyword(s):  
Number Theory ◽  
Algebraic Number ◽  
Cyclotomic Field ◽  
Algebraic Number Theory ◽  
Class Numbers ◽  
Cyclotomic Fields ◽  
Relative Class ◽  
International Audience ◽  
Abelian Fields ◽  
Fundamental Units

International audience We give a brief survey of three papers of K. Ramachandra in algebraic number theory. The first paper is based on his thesis and appeared in the Annals of Mathematics and titled, ``Some Applications of Kronecker's Limit Formula.'' The second paper determines a system of fundamental units for the cyclotomic field and is titled, ``On the units of cyclotomic fields.'' This appeared in Acta Arithmetica. The third deals with relative class numbers and is titled, ``The class number of relative abelian fields.'' This appeared in Crelle's Journal.

scholarly journals A GENERALISED KUMMER'S CONJECTURE

2010 ◽  
Vol 52 (3) ◽  
pp. 453-472 ◽  
Author(s):  
M. J. R. MYERS
Keyword(s):  
Natural Number ◽  
Class Numbers ◽  
Cyclotomic Fields ◽  
Rate Of Growth ◽  
Relative Class ◽  
Almost All

AbstractKummer's conjecture predicts the rate of growth of the relative class numbers of cyclotomic fields of prime conductor. We extend Kummer's conjecture to cyclotomic fields of conductor n, where n is any natural number. We show that the Elliott–Halberstam conjecture implies that this generalised Kummer's conjecture is true for almost all n but is false for infinitely many n.

scholarly journals Class numbers of cyclotomic fields

1985 ◽  
Vol 21 (3) ◽  
pp. 260-274 ◽  
Author(s):  
Gary Cornell ◽  
Lawrence C Washington
Keyword(s):  
Class Numbers ◽  
Cyclotomic Fields

scholarly journals Provably Secure Identity-Based Encryption and Signature over Cyclotomic Fields

2019 ◽  
Vol 2019 ◽  
pp. 1-13
Author(s):  
Yang Wang ◽  
Mingqiang Wang ◽  
Jingdan Zou ◽  
Jin Xu ◽  
Jing Wang
Keyword(s):  
Key Management ◽  
Cyclotomic Field ◽  
Cyclotomic Fields ◽  
Secret Key ◽  
Worst Case ◽  
Identity Based Encryption ◽  
Management Procedures ◽  
Identity Based ◽  
Identity Based Cryptography ◽  
Provably Secure

Identity-based cryptography is a type of public key cryptography with simple key management procedures. To our knowledge, till now, the existing identity-based cryptography based on NTRU is all over power-of-2 cyclotomic rings. Whether there is provably secure identity-based cryptography over more general fields is still open. In this paper, with the help of the results of collision resistance preimage sampleable functions (CRPSF) over cyclotomic fields, we give concrete constructions of provably secure identity-based encryption schemes (IBE) and identity-based signature schemes (IBS) based on NTRU over any cyclotomic field. Our IBE schemes are provably secure under adaptive chosen-plaintext and adaptive chosen-identity attacks, meanwhile, our IBS schemes are existentially unforgeable against adaptively chosen message and adaptively chosen identity attacks for any probabilistic polynomial time (PPT) adversary in the random oracle model. The securities of both schemes are based on the worst-case approximate shortest independent vectors problem (SIVPγ) over corresponding ideal lattices. The secret key size of our IBE (IBS) scheme is short—only one (two) ring element(s). The ciphertext (signature) is also short—only two (three) ring elements. Meanwhile, as the case of NTRUEncrypt, our IBE scheme could encrypt n bits in each encryption process. These properties may make our schemes have more advantages for some IoT applications over postquantum world in theory.

scholarly journals The mean values of Dirichlet L-functions at integer points and class numbers of cyclotomic fields

1994 ◽  
Vol 134 ◽  
pp. 151-172 ◽  
Author(s):  
Masanori Katsurada ◽  
Kohji Matsumoto
Keyword(s):  
Positive Integer ◽  
Dirichlet Character ◽  
Class Numbers ◽  
Cyclotomic Fields ◽  
Mean Values ◽  
Integer Points ◽  
Principal Character ◽  
Dirichlet Characters ◽  
The Mean

Let q be a positive integer, and L(s, χ) the Dirichlet L-function corresponding to a Dirichlet character χ mod q. We putwhere χ runs over all Dirichlet characters mod q except for the principal character χ0.

MONOGENEITY IN CYCLOTOMIC FIELDS

2010 ◽  
Vol 06 (07) ◽  
pp. 1589-1607 ◽  
Author(s):  
LEANNE ROBERTSON
Keyword(s):  
Unit Circle ◽  
Number Field ◽  
Complex Plane ◽  
Cyclotomic Field ◽  
Difficult Problem ◽  
Cyclotomic Fields ◽  
Ring Extension ◽  
Integral Basis ◽  
Ring Of Integers ◽  
Integral Bases

A number field is said to be monogenic if its ring of integers is a simple ring extension ℤ[α] of ℤ. It is a classical and usually difficult problem to determine whether a given number field is monogenic and, if it is, to find all numbers α that generate a power integral basis {1, α, α2, …, αk} for the ring. The nth cyclotomic field ℚ(ζn) is known to be monogenic for all n, and recently Ranieri proved that if n is coprime to 6, then up to integer translation all the integral generators for ℚ(ζn) lie on the unit circle or the line Re (z) = 1/2 in the complex plane. We prove that this geometric restriction extends to the cases n = 3k and n = 4k, where k is coprime to 6. We use this result to find all power integral bases for ℚ(ζn) for n = 15, 20, 21, 28. This leads us to a conjectural solution to the problem of finding all integral generators for cyclotomic fields.

On the orders of ideal classes in prime cyclotomic fields

1990 ◽  
Vol 108 (2) ◽  
pp. 197-201 ◽  
Author(s):  
Francisco Thaine
Keyword(s):  
Sylow Subgroup ◽  
Class Group ◽  
Cyclotomic Field ◽  
Cyclotomic Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
The Ideal

In this article we exhibit a method complementary to the method presented in [4], that allows us, at least in some important cases, to obtain exact expressions for the orders of ideal classes of cyclotomic fields in terms of properties of the units of the field. We consider only the particular case in which the classes belong to the p-Sylow subgroup (A)p of the ideal class group of a real p-cyclotomic field, but it appears that the results can be generalized.

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