## Key-exchange in real quadratic congruence function fields (1996)

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Venue: | Designs, Codes and Cryptography 7 |

Citations: | 31 - 20 self |

### BibTeX

@INPROCEEDINGS{Scheidler96key-exchangein,

author = {R. Scheidler and A. Stein and H. C. Williams},

title = {Key-exchange in real quadratic congruence function fields},

booktitle = {Designs, Codes and Cryptography 7},

year = {1996},

pages = {153--174}

}

### Years of Citing Articles

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### Abstract

### Citations

3071 | New directions in cryptography
- Diffie, Hellman
- 1976
(Show Context)
Citation Context ...st and sufficiently secure for most applications. The real difficulty in employing such cryptosystems is the problem of securely transmitting the key between communicants. In 1976, Diffie and Hellman =-=[8]-=- described a possible solution to this problem by making use of the multiplicative group F p of integers relatively prime to a large prime p. More generally, we can let G be any group such that jGj (=... |

1025 |
A course in computational algebraic number theory
- Cohen
- 1993
(Show Context)
Citation Context ... closest to the left of nffi(r). The number of iterations required is approximately n. By adapting a well-known exponentiation technique based on repeated squaring (see for example Algorithm 1:2:3 in =-=[4]-=-), this can be reduced to O(log n). We will now describe the ideal multiplication and reduction process in more detail. Algorithm MULT Input: a = (Q a ; P a ), b = (Q b ; P b ) 2 R Output: c = (Q c ; ... |

105 |
An Implementation of Elliptic Curve Cryptosystems over F 155
- Agnew, Mullin, et al.
- 1993
(Show Context)
Citation Context ...ptic curves, performed best. Since our implementation is algorithmic and computes keys for arbitrary parameters q and D, it is slower than some implementations of elliptic curve cryptosystems such as =-=[2]-=-. However, we believe that a significant speed-up could be achieved by making use of features such as special-purpose arithmetic, hardware implementation, and code optimization. For our procedures, we... |

61 |
Quadratische Körper im Gebiete der höheren
- Artin
- 1924
(Show Context)
Citation Context ...set of examples. 2 Real Quadratic Congruence Function Fields 2.1 Basic Definitions In this section, we present the situation as described in [18], [16] and [19]. Basic references for this subject are =-=[3]-=-, [7] and [20]. Let K=F q be a quadratic congruence function field over a finite field F q of constants of odd characteristic with q elements. Then K is a quadratic extension of the rational function ... |

60 | Heuristics on class groups of number fields
- Cohen, Lenstra
- 1984
(Show Context)
Citation Context ...me or a product of two odd degree prime polynomials, then 2 6 j h 0 . In order to examine the odd part of the class group G of K, we can apply the same heuristic arguments that Cohen and Lenstra [5], =-=[6]-=- used for real quadratic number fields. This is possible because of the complete analogy that exists between the infrastructures of the ideal classes in K and in a real quadratic number field. For exa... |

49 |
The infrastructure of a real quadratic field and its applications
- SHANKS
- 1972
(Show Context)
Citation Context ...it a secure key exchange protocol, similar in concept to that of Diffie-Hellman, which does not make use of a group as the underlying structure. This scheme is based on the infrastructure (see Shanks =-=[15]-=-) of the principal ideal class of a real quadratic number field. Unfortunately, this technique possesses a number of disadvantages not shared by the standard Diffie-Hellman protocol: increased bandwid... |

46 |
Algebraic Number Theory
- Weiss
- 1963
(Show Context)
Citation Context ...es. 2 Real Quadratic Congruence Function Fields 2.1 Basic Definitions In this section, we present the situation as described in [18], [16] and [19]. Basic references for this subject are [3], [7] and =-=[20]-=-. Let K=F q be a quadratic congruence function field over a finite field F q of constants of odd characteristic with q elements. Then K is a quadratic extension of the rational function field F q (x) ... |

34 |
Introduction to the theory of algebraic numbers and functions
- Eichler
- 1966
(Show Context)
Citation Context ... by Lemma 3.2 c) and (3.7), or equivalently ms2 R deg(D) : Thus, to get a lower bound on m, we require a lower bound on R. By using standard results on zeta functions for function fields (see Eichler =-=[9]-=-, p. 299-307), we can bound the value of the divisor class number h by ( p q \Gamma 1 ) 2gshs( p q + 1 ) 2g ; where, in this case, g = 1 2 deg(D) \Gamma 1. Since h = Rh 0 , we see that for fixed h, R ... |

29 |
Lectures on the Theory of Algebraic Functions of One Variable, Lecture
- Deuring
(Show Context)
Citation Context ...f examples. 2 Real Quadratic Congruence Function Fields 2.1 Basic Definitions In this section, we present the situation as described in [18], [16] and [19]. Basic references for this subject are [3], =-=[7]-=- and [20]. Let K=F q be a quadratic congruence function field over a finite field F q of constants of odd characteristic with q elements. Then K is a quadratic extension of the rational function field... |

27 | Analytische Zahlentheorie in Körpern der Charakteristik p - Schmidt - 1931 |

24 |
On the distribution of divisor class groups of curves over a finite field
- Friedman, Washington
- 1989
(Show Context)
Citation Context ...hat the characteristic of F q behaves like any other odd prime p with respect to the p-component of G. In their investigation of the structure of the divisor class group of K, Friedman and Washington =-=[10]-=- excluded this prime because the p-rank of the divisor class group for this prime cannot be as large as that for other primes; however, there seems to be no a priori reason for excluding it in an inve... |

16 |
Fin Algorithmus zur Berechnung der Klassenzahl und des Regulators reellquadratischer Ord- nungen, Dissertation, Universit~t des Saarlandes, Saarbriicken
- Abel
- 1994
(Show Context)
Citation Context ...known whether the DLP is in fact equivalent to the difficulty of breaking thesprotocol in the sense that any fast method for breaking the scheme gives rise to a fastsalgorithm for solving the DLP.sIn =-=[1]-=-, Abel shows that the DLP in a real quadratic number field Q(v~)scan be solvedsin time subexponential n log A. Also, any algorithm for solving the DLP can be used tos170 R. SCHEIDLER, A. STEIN, AND H.... |

15 | Equivalences between elliptic curves and real quadratic congruence function
- Stein
- 1997
(Show Context)
Citation Context ... in the case of a real quadratic number field, Shanks' Baby step-Giant step technique [11] can be used to compute the distance of a reduced ideal. This method has complexity O i q 1 4 deg( D ) j . In =-=[17]-=-, it is shown that the DLP in real quadratic congruence function fields F q (x)( p D) where deg(D) = 4, (the simplest non-trivial case, since it is known that R = 1 if deg(D) = 2) is equivalent to the... |

13 |
Heuristics on class groups
- Cohen, Lenstra
- 1984
(Show Context)
Citation Context ...s prime or a product of two odd degree prime polynomials, then 2 6 j h 0 . In order to examine the odd part of the class group G of K, we can apply the same heuristic arguments that Cohen and Lenstra =-=[5]-=-, [6] used for real quadratic number fields. This is possible because of the complete analogy that exists between the infrastructures of the ideal classes in K and in a real quadratic number field. Fo... |

13 |
Baby step-Giant step-Verfahren i reell-quadratischen Kongruenzfunktionenk6rpern mit Charak- teristik ungleich 2
- Stein
- 1992
(Show Context)
Citation Context ...real quadratic number field. This appears to be the first time that the theory of algebraic function fields has been applied to cryptography. Research supported by NSERC of Canada Grant #A7649. Stein =-=[16]-=- has shown that Shanks' infrastructure idea also applies to the set of reduced principal ideals in a real quadratic congruence function field. Thus, many of the techniques needed to produce the scheme... |

12 |
Quadratic fields and factorization, Computational methods in number theory
- Schoof
- 1983
(Show Context)
Citation Context ...olved in time subexponential in log \Delta. Also, any algorithm for solving the DLP can be used to find the regulator of this field. Knowledge of the regulator together with a technique due to Schoof =-=[14]-=- can then in turn be used to factor \Delta. Hence the DLP for real quadratic number fields is at least as difficult as the problem of factoring the integer \Delta. The situation in real quadratic cong... |

12 |
Artins Theorie der quadratischen Kongruenzfunktionenk orper und ihre Anwendung auf die Berechnung der Einheiten- und Klassengruppen
- Weis, Zimmer
(Show Context)
Citation Context ...ation issues and some timings for a certain set of examples. 2 Real Quadratic Congruence Function Fields 2.1 Basic Definitions In this section, we present the situation as described in [18], [16] and =-=[19]-=-. Basic references for this subject are [3], [7] and [20]. Let K=F q be a quadratic congruence function field over a finite field F q of constants of odd characteristic with q elements. Then K is a qu... |

10 |
Ambiguous classes and 2-rank of class group of quadratic function
- Zhang
- 1987
(Show Context)
Citation Context ...by ( p q \Gamma 1 ) 2gshs( p q + 1 ) 2g ; where, in this case, g = 1 2 deg(D) \Gamma 1. Since h = Rh 0 , we see that for fixed h, R will be large as long as h 0 is small. We can use a result of Zhang =-=[21]-=- to ensure that h 0 is odd. Namely, if D is prime or a product of two odd degree prime polynomials, then 2 6 j h 0 . In order to examine the odd part of the class group G of K, we can apply the same h... |

9 |
An algorithm for determining the regulator and the fundamental unit of a hyperelliptic congruence function field
- Stein, Zimmer
(Show Context)
Citation Context ...puter implementation issues and some timings for a certain set of examples. 2 Real Quadratic Congruence Function Fields 2.1 Basic Definitions In this section, we present the situation as described in =-=[18]-=-, [16] and [19]. Basic references for this subject are [3], [7] and [20]. Let K=F q be a quadratic congruence function field over a finite field F q of constants of odd characteristic with q elements.... |

8 |
A key exchange protocol using real quadratic
- Scheidler, Buchmann, et al.
- 1994
(Show Context)
Citation Context ...s extensions make use of this idea, only the choice of G varies. Of course, G here should be selected such that the DLP in this structure is a hard problem. Recently, Scheidler, Buchmann and Williams =-=[12]-=- were able, for the first time, to exhibit a secure key exchange protocol, similar in concept to that of Diffie-Hellman, which does not make use of a group as the underlying structure. This scheme is ... |

4 |
Quadratische K"orper im Gebiete der h"oheren Kongruenzen
- Artin
- 1924
(Show Context)
Citation Context ...set of examples. 2 Real Quadratic Congruence Function Fields 2.1 Basic Definitions In this section, we present the situation as described in [18], [16] and [19]. Basic references for this subject are =-=[3]-=-, [7] and [20]. Let K=Fq be a quadratic congruence function field over a finite field Fq of constants of odd characteristic with q elements. Then K is a quadratic extension of the rational function fi... |

4 | On the calculation of regulators and class numbers of quadratic fields - Jr - 1982 |

2 |
Abel Ein Algorithmus zur Berechnung der Klassenzahl und des Regulators reellquadratischer Ordnungen
- S
- 1994
(Show Context)
Citation Context ...known whether the DLP is in fact equivalent to the difficulty of breaking the protocol in the sense that any fast method for breaking the scheme gives rise to a fast algorithm for solving the DLP. In =-=[1]-=-, Abel shows that the DLP in a real quadratic number field Q ( p \Delta) can be solved in time subexponential in log \Delta. Also, any algorithm for solving the DLP can be used to find the regulator o... |

1 | Analytische Zahlentheorie in K"orpern der Charakteristik p - Schmidt - 1931 |

1 |
Quadratische KGrper im Gebiete der hGheren Kongruenzen
- Artin
- 1924
(Show Context)
Citation Context ...t of examples.s2. Real Quadratic Congruence Function Fieldss2.1. Basic DefinitionssIn this section, we present the situation as described in [18], [16] and [19]. Basic referencessfor this subject are =-=[3]-=-, [7] and [20].sLet K/Fq be a quadratic ongruence function field over a finite field Fq of constants of oddscharacteristic with q elements. Then K is a quadratic extension of the rational function fie... |

1 | Analytische Zahlentheorie in KGrpern der Charakteristik p - Schmidt - 1931 |

1 |
Artin's Theode der quadratischen Kongruenzfunktionenk6rper und ihre Anwan- dung auf die Berechnung der Einheiten-und
- Weis, Zimmer
- 1991
(Show Context)
Citation Context ...ion issues and some timings for a certain set of examples.s2. Real Quadratic Congruence Function Fieldss2.1. Basic DefinitionssIn this section, we present the situation as described in [18], [16] and =-=[19]-=-. Basic referencessfor this subject are [3], [7] and [20].sLet K/Fq be a quadratic ongruence function field over a finite field Fq of constants of oddscharacteristic with q elements. Then K is a quadr... |