Welcome to the resource topic for
**2023/058**

**Title:**

SCALLOP: scaling the CSI-FiSh

**Authors:**
Luca De Feo, Tako Boris Fouotsa, Péter Kutas, Antonin Leroux, Simon-Philipp Merz, Lorenz Panny, Benjamin Wesolowski

**Abstract:**

We present SCALLOP: SCALable isogeny action based on

Oriented supersingular curves with Prime conductor, a new group action based on isogenies of supersingular curves. Similarly to CSIDH and

OSIDH, we use the group action of an imaginary quadratic order’s class

group on the set of oriented supersingular curves. Compared to CSIDH,

the main benefit of our construction is that it is easy to compute the

class-group structure; this data is required to uniquely represent— and

efficiently act by— arbitrary group elements, which is a requirement in,

e.g., the CSI-FiSh signature scheme by Beullens, Kleinjung and Vercauteren. The index-calculus algorithm used in CSI-FiSh to compute

the class-group structure has complexity L(1/2), ruling out class groups

much larger than CSIDH-512, a limitation that is particularly problematic in light of the ongoing debate regarding the quantum security of

cryptographic group actions.

Hoping to solve this issue, we consider the class group of a quadratic order of large prime conductor inside an imaginary quadratic field of small

discriminant. This family of quadratic orders lets us easily determine

the size of the class group, and, by carefully choosing the conductor,

even exercise significant control on it— in particular supporting highly

smooth choices. Although evaluating the resulting group action still has

subexponential asymptotic complexity, a careful choice of parameters

leads to a practical speedup that we demonstrate in practice for a security level equivalent to CSIDH-1024, a parameter currently firmly out of reach of index-calculus-based methods. However, our implementation

takes 35 seconds (resp. 12.5 minutes) for a single group-action evaluation at a CSIDH-512-equivalent (resp. CSIDH-1024-equivalent) security

level, showing that, while feasible, the SCALLOP group action does not

achieve realistically usable performance yet.

**ePrint:**
https://eprint.iacr.org/2023/058

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