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## 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.

#### 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 ${𝔽}_{{p}^{k}}$ 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 ${𝔽}_{{p}^{n}}$ where $n$ is composite. We also applied it to the new variant Tower-NFS. The paper was finally published in 2019 [4].