Let q be a generator of a cyclic group image = {1, 2, …, p − 1} of order p − 1 and y = qx (modulo p).

Proposition 14.5 (The Index-Calculus Algorithm): Initialization: Select a factor base image = {p1, p2, …, ps} consisting of elements of image. image is chosen so that a significant proportion of the elements of image can be expressed in the form image with ni ≥ 0.

14.5a Select a random k with 0 ≤ k < n and compute qk (modulo p).

Try to write qk (modulo p) as a product image with ci, ≥ 0:

  • if unsuccessful, return to Step 14.5a and choose another value for k;
  • if successful, write k = [c1 logq p1 + [c2 logq p2 + … + [cs logq ps](modulo p − 1).
14.5c Repeat Steps 14.5a–b until a sufficient number of linear relations as above are found in order to solve the system of equations to determine ...

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