What do recent announcements about solving the DLP in $GF(2^{6120})$ mean for RSA
Göloğlu, Granger, McGuire, and Zumbrägel broke the DLP in the Galois finite field with $2^n$ elements for $n=6120$, and Antoine Joux did it for $n=6168$, with modest computing power. These recent announcements directly imply that cryptosystems (if any) based on the DLP in a field with $2^n$ elements are no longer secure, including for $n$ big enough that this finite field was previously thought to be of cryptographic interest (e.g. for pairing schemes), at least when $n$ is of the forms explained here enabling the new attacks.

One way to see the Galois finite field with $2^n$ elements, $GF(2^n)$ or $\mathbb{F}_{2^n}$, is as the set of $n$-bit vectors representing the coefficients of a binary polynomial of degree $n-1$, with as additive law the addition of binary polynomials, and as multiplicative law (of interest in the DLP) the multiplication of binary polynomial followed by reduction modulo some suitable binary polynomial of degree $n$.

RSA and DSA work on different algebraic structures:


RSA works on the finite unit ring of integers modulo $N$, with $N$ a public composite of secret factorization, which factors are huge primes. Notice that the number of elements in this ring is $N$, and thus not of the form $2^n$.
DSA works on the finite field of integers modulo $P$ for a public prime $P$ (such that $P-1$ is divisible by some suitably large prime, which is the order of a multiplicative subgroup; thanks to Samuel Neves for pointing my earlier serious mistake on that). Notice that the number of elements in this field is $P$, and thus not of the form $2^n$.


Because of these different algebraic structures, I can confidently tell that the algorithms used in these recent announcements are not directly usable against RSA and DSA. In my not so informed opinion, for the same reason, they are not directly usable against schemes based on the DLP on elliptic curve groups, including when the base field of the elliptic curve is some of the $GF(2^n)$ that are common practice for ECDSA. In fact I can not spot any cryptographic scheme in wide use that is directly impacted.

As of future advances against RSA, DSA, and Elliptic Curve cryptographic schemes, that may be enabled by these recent developments, my opinion is near worthless, for I'm not a competent number theorist. But I can still commit my feeling: I'm not too worried for the security of RSA, DSA, DH; and ECDSA or ECDH over elliptic curve groups with base field $GF(P)$, or even $GF(2^n)$.

Additions: The $n$ in the new attacks is large (compared to $2048$ or $4096$, previously thought safe and very safe levels) because the attackers can, and bigger values better demonstrate the power of their techniques.