Precisely, let

  p = \lfloor 10^{119} \pi \rfloor+ 207819,
    = 314159265358979323846264338327950288419716939937510582097494\
      459230781640628620899862803482534211706798214808651328438483,

  g = 2,

and

  y = \lfloor 10^{119} e \rfloor,
    = 271828182845904523536028747135266249775724709369995957496696\
      762772407663035354759457138217852516642742746639193200305992.

Then

  y   = g^262112280685811387636008622038191827370390768520656974243035\
          380382193478767436018681449804940840373741641452864730765082,

and, similarly,

  y+1 = g^39657965519539238631090956325038481900751981791165229696297\
          421520645832904710912189562251329527994908449750607046857937.

Sieving:
--------

                        t^3-9*t^2-9*t+9

                        s^2
                        -7374389167922711279538633461199308033087*s
                        -333238556260219119547406855509826713348*c

                        c = 1201639291188427271122019272295979872125.


Linear algebra:
---------------

For instance,
  3  = g ^ 28812588093314776509699010256332271205911219293533606948309\
           244629961378102894412283315317373685769257402738003506902138,

  5  = g ^ 21755718305811583829459340786707488931552080300620288049491\
           6079418842612898727245046247623746814003335266081854116641401,

  7  = g ^ 30620343436458977106289725314901617172209074240481960209355\
           5007057950445875245076830599652008193628520847056151254242513,

  11 = g ^ 26369065546060570062275123759054389628546695463387446804852\
           2904503050215021182824543943461120732653168766053846377967922,

etc ...

Individual Logarithms:
----------------------

We take advantage of the Galois group of t^3-12*t^2-9*t+12 [JoLe00].

Precisely, we found in less than 6 hours, using our very crude
implementation, two algebraic integers
      num = 136919628471533453465*t^2
          - 109185518772042207040*t
          - 218010383119442982304

and

      den = 90752177247861263294*t^2
          + 5976502381138861785*t
          + 161979899979266169279

such that

                         19^2*y = num/den modulo p

and such that,

        in GP-PARI notation, the principal ideal (num) is equal to 

[[53, [7, 2, 0]~, 1, 1, [-18, 19, 12]~] 1] *
[[431, [-20, 2, 0]~, 1, 1, [168, 20, 12]~] 1] *
[[2179, [140, 2, 0]~, 1, 1, [-502, -300, 12]~] 1] *
[[16831, [-3928, 2, 0]~, 1, 1, [-1478, 7836, 12]~] 1] *
[[156781, [-45467, 2, 0]~, 1, 1, [15351, -65867, 12]~] 1] *
[[7691507, [-3118090, 2, 0]~, 1, 1,
 [-1592322, -1455347, 12]~] 1] *
[[5847120361, [-944554227, 2, 0]~, 1, 1,
 [-2613536996, 1889108434, 12]~] 1] *
[[7689099923, [-1979641955, 2, 0]~, 1, 1,
 [-1763503409, -3729816033, 12]~] 1] *
[[14023824873312563677, [2910448673841826685, 2, 0]~, 1, 1,
 [-4872413772263364932, -5820897347683653390, 12]~] 1]

        and the principal ideal (den) is equal to

[[2, [2, 0, 0]~, 1, 3, [1, 0, 0]~] 1] *
[[3, [1, -1, 1]~, 3, 1, [2, 2, 0]~] 3] *
[[19, [8, 2, 0]~, 1, 1, [-3, 2, -7]~] 1] *
[[1873, [-237, 2, 0]~, 1, 1, [889, 454, 12]~] 1] *
[[110359, [36889, 2, 0]~, 1, 1, [-37268, 36561, 12]~] 1] *
[[2672473789, [-932319595, 2, 0]~, 1, 1,
 [1125968488, -807834619, 12]~] 1] *
[[626844366559, [-255501920253, 2, 0]~, 1, 1,
 [-211702090482, -115840526073, 12]~] 1] *
[[685495972547, [-197652881054, 2, 0]~, 1, 1,
 [-276404069769, -290190210459, 12]~] 1] *
[[641614040507139551, [33835176915624305, 2, 0]~, 1, 1,
 [159262188369356649, -67670353831248630, 12]~] 1]

Antoine JOUX    (DCSSI, Issy les Moulineaux, France, Antoine.Joux@ens.fr),
Reynald LERCIER (CELAR, Rennes, France, lercier@celar.fr).

References:
===========

[JoLe01] A. Joux and R. Lercier, ``Discrete logarithms in GF(p)'',
         January 19 2001. Announce on the NMBRTHRY Mailing List.
         Available at http://www.medicis.polytechnique.fr/~lercier.

[JoLe00] A. Joux and R. Lercier, Improvements to the general Number
         Field Sieve for discrete logarithms in prime fields, acceped
         for publication at Math. of Comp., 2000. Preprint
         available at http://www.medicis.polytechnique.fr/~lercier.

[Schi93] O. Schirokauer, Discrete Logarithms and local units. Phil.
         Trans. R. Soc Lond. A 345, pages 409-423, 1993.

[LaOd91] B.A. Lamacchia and A.M. Odlyzko, Computation of discrete
         logarithm in prime fields, Designs, Codes and Cryptography,
         1991, volume 1, pages 47-62.

[GoLo89] G.H. Golub and C.F. van Loan, Matrix computations, chapter 9,
         The John Hopkins University Press, 1989, Mathematical
         Sciences.