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Weber class invariants with 'Thetanullwerte'

101 pages
Weber class invariants with 'Thetanullwerte' Weber class invariants with 'Thetanullwerte' Osmanbey Uzunkol Fakultat fur Mathematik Carl von Ossietzky Universitat Oldenburg GeoCrypt 2011, Bastia 23. June 2011

  • roots

  • retrieving original

  • having significiantly smaller

  • mathematik carl von

  • original roots

  • weber class invariants


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Weber class invariants with ’Thetanullwerte’
Weber class invariants with ’Thetanullwerte’
Osmanbey Uzunkol
Fakultat fur Mathematik
Carl von Ossietzky Universitat Oldenburg
GeoCrypt 2011, Bastia
23. June 2011Weber class invariants with ’Thetanullwerte’
Weber class invariants with ’Thetanullwerte’
Osmanbey Uzunkol
Fakultat fur Mathematik
Carl von Ossietzky Universitat Oldenburg
GeoCrypt 2011, Bastia
23. June 2011Weber class invariants with ’Thetanullwerte’
Outline
Motivation and applications
Problem
Primality, DLP, Hilbert’s 12. problem
Class polynomials
Class invariants
Class invariants as quotients of ’Thetanullwerten’
Class units
Unit group
Ongoing workwith the possibility of retrieving original roots?
Weber class invariants with ’Thetanullwerte’
Motivation and applications
Problem
The problem with an example
I (Minimal polynomial of j()) :
6 5H (x) = x 30703802307926880672x +204
495864841637996112067555072x +
3775121756231241041610849730560x +
2534484930703209896960446929872814080x +
6020337293681148983229932704488367325184x+
28508041377034538166862450172153093456658432:
I Problem: Is it possible to construct alternative class
polynomials having signi ciantly smaller coe cients than the
above oneWeber class invariants with ’Thetanullwerte’
Motivation and applications
Problem
The problem with an example
I (Minimal polynomial of j()) :
6 5H (x) = x 30703802307926880672x +204
495864841637996112067555072x +
3775121756231241041610849730560x +
2534484930703209896960446929872814080x +
6020337293681148983229932704488367325184x+
28508041377034538166862450172153093456658432:
I Problem: Is it possible to construct alternative class
polynomials having signi ciantly smaller coe cients than the
above one with the possibility of retrieving original roots?Weber class invariants with ’Thetanullwerte’
Motivation and applications
Problem
The problem with an example
I (Minimal polynomial of j()) :
6 5H (x) = x 30703802307926880672x +204
495864841637996112067555072x +
3775121756231241041610849730560x +
2534484930703209896960446929872814080x +
6020337293681148983229932704488367325184x+
28508041377034538166862450172153093456658432:
I Problem: Is it possible to construct alternative class
polynomials having signi ciantly smaller coe cients than the
above one with the possibility of retrieving original roots?I Thm, Pocklington: Let a;N2N with gcd(a;N) = 1. If
N 1
N 1
da 1 mod N and a 6 1 mod N for all divisors d of
N 1, then N2P.
Weber class invariants with ’Thetanullwerte’
Motivation and applications
Primality, DLP, Hilbert’s 12. problem
Inverting Fermat’s little theorem
p 1Fermat: Let p2P, then we have a 1 mod p for
gcd(a;p) = 1.
Can we invert the theorem? Are there composite numbers N, for
N 1which a 1 mod N for all a with gcd(a;N) = 1?
I Carmichael numbers! They are in nitely many!Weber class invariants with ’Thetanullwerte’
Motivation and applications
Primality, DLP, Hilbert’s 12. problem
Inverting Fermat’s little theorem
p 1Fermat: Let p2P, then we have a 1 mod p for
gcd(a;p) = 1.
Can we invert the theorem? Are there composite numbers N, for
N 1which a 1 mod N for all a with gcd(a;N) = 1?
I Carmichael numbers! They are in nitely many!
I Thm, Pocklington: Let a;N2N with gcd(a;N) = 1. If
N 1
N 1
da 1 mod N and a 6 1 mod N for all divisors d of
N 1, then N2P.Weber class invariants with ’Thetanullwerte’
Motivation and applications
Primality, DLP, Hilbert’s 12. problem
Inverting Fermat’s little theorem
p 1Fermat: Let p2P, then we have a 1 mod p for
gcd(a;p) = 1.
Can we invert the theorem? Are there composite numbers N, for
N 1which a 1 mod N for all a with gcd(a;N) = 1?
I Carmichael numbers! They are in nitely many!
I Thm, Pocklington: Let a;N2N with gcd(a;N) = 1. If
N 1
N 1
da 1 mod N and a 6 1 mod N for all divisors d of
N 1, then N2P.Weber class invariants with ’Thetanullwerte’
Motivation and applications
Primality, DLP, Hilbert’s 12. problem
ECPP
Determine, whether a given number N > 1 with gcd(N; 6) = 1 is a
prime.
I Goldwasser: Let E be an elliptic curve overZ=NZ with a
point P2 E and m be a number, for which:
1. there exists a prime factor q of m with
p
4 2q> ( N + 1) ;
2. mP =O = (0 : 1 : 0),E
m 3. P = (x : y : t) with t2 (Z=NZ) .q
Then N is a prime.
I Down-Run: N = N > q = N >> N .1 2 r
N prime) N prime!r
I Record (Morain): 2580 decimal digits!