## Section: New Results

### The Number Field Sieve – High-Level Results

#### A New Ranking Function for Polynomial Selection in the Number Field Sieve

Participant : Paul Zimmermann.

With Nicolas David (ÉNS Paris-Saclay, France), we designed a new
ranking function for polynomial selection in the Number Field Sieve.
The previous ranking function was only considering the *mean* of the
so-called $\alpha $-value, which measures how small primes divide the norm
of the polynomial. The new function also takes into account the
*variance* of the corresponding distribution. This partially explains
why the previous function did sometimes fail to correctly identify the best
polynomials. The new ranking function is implemented in Cado-NFS (branch
`dist-alpha` ) and is detailed in [3].

#### On the Alpha Value of Polynomials in the Tower Number Field Sieve Algorithm

Participant : Aurore Guillevic.

With Shashank Singh from IISER Bhopal (former post-doc at CARAMBA in 2017), we generalized the ranking function $\alpha $ for the Tower setting of the Number Field Sieve in [22]. In the relation collection of the NFS algorithm, one tests the smoothness of algebraic norms (computed with resultants). The $\alpha $ function measures the bias of the average valuation at small primes of algebraic norms, compared to the average valuation at random integers of the same size. A negative $\alpha $ means more small divisors than average. We then estimate the total number of relations with a Monte-Carlo simulation, as a generalized Murphy's $E$ function, and finally give a rough estimate of the total cost of TNFS for finite fields ${\mathbb{F}}_{{p}^{k}}$ of popular pairing-friendly curves.

#### Faster Individual Discrete Logarithms in Finite Fields of Composite Extension Degree

Participant : Aurore Guillevic.

We improved the previous work [30] on speeding-up the first phase of the individual discrete logarithm computation, the initial splitting, a.k.a. the smoothing phase. We extended the algorithm to any non-prime finite field ${\mathbb{F}}_{{p}^{n}}$ where $n$ is composite. We also applied it to the new variant Tower-NFS. The paper was finally published in 2019 [4].