# How to choose the appropriate Smoothness Bound while using the Index Calculus method

While implementing the Quadratic Sieve, the textbooks give a rough formula for what Smoothness bound you should use in your Factor Base.

To factor a number N using the Quadratic Sieve, we can use the following:

$$L = e^{\sqrt {\ln(N)ln(ln(N))}}$$, $$B = L^{\frac {1}{\sqrt 2}}$$

For the Index Calculus method for solving the Discrete Log problem in $$\mathbb F_p$$, is there a similar formula? Many of the texts I checked, just say choose an appropriate Smoothness Bound B. But don't give any indication of how one chooses an appropriate B.

Coppersmith, Odlyzko, and Schroeppel originally set $$B = L[1/2, 1/2]$$ for both the linear and Gaussian sieves. Pomerance set $$B = L[1/2, 1/\sqrt{2}]$$ for a rigorous index calculus variant using the elliptic curve method as the smoothness testing method.
• I tried this formula with some solved examples in some books & the result seems quite off from what the examples have used. For e.g. in Silverman's mathematical cryptography book, he solves $37^x \equiv 211 \pmod 18443$. If I use the $B = L^{\frac {1}{\sqrt 2}}$ formula, I get 28.5. However, to solve the problem, Silverman uses B=5 which is quite far off from the calculated B. Silverman also has an exercise problem $17^x \equiv 19 \pmod 19079$, where B calculated would be 28.5, but I am able to solve it again using B=5. Aug 27, 2021 at 1:19
• What do you mean by  with early aborts, as the smoothness tester.? Aug 27, 2021 at 1:20
• Why are there 2 formulas i.e. why $B = L[1/2, 1/\sqrt{2}]$ - what is the $L^{1/2}$ Aug 27, 2021 at 1:25
• $L[1/2, 1/\sqrt{2}]$ is the more general L-notation. It means the same as your $L^{1/\sqrt{2}}$. Aug 27, 2021 at 3:34