 Hi, I'm Maxime and I'm a PhD student at Ecole Polytechnique and in RIA. And today, I'm really thrilled to introduce you to our joint work with Alain Couveur and Thomas de Brissal-Azard, entitled On Codes and Learning with Errors Over Function Fiat. In this work, we focus on search-to-decision reductions for structured variants of the decoding problem. search reductions are useful for cryptography because many cryptographic primitives or different applications rely on the hardness of the decision version instead of the usual syndrome decoding problem. But first, let me give you the intuition behind search-to-decision reductions. If I give you a picture of many people and I ask you, is there water on this picture? How would you solve this problem? Well, you will probably look for water on this picture for, let's say, one minute. And if you are able to find it, you would say, yeah, there is water on this picture. On the other hand, if you can't find it in this minute, then you would say, well, with high probability, there is no water on the picture. In other words, in order to tackle the decision problem, is there water on the picture? You try to solve the computational version finding water. And this is a very natural strategy, because more often computational problems are easier to analyze and to understand than their decision counterpart. However, the latter may be strictly easier and therefore one needs to be very careful when instantiating cryptosystems based on the hardness of the decision problem. For example, it is well known that decision-deficient man is easy in some instantiations, in some groups, while the computational version is still believed to be hard. For instance, for elliptic curves and dodway sparrings. Therefore, such decision reductions are here to bridge search and decision problems in order to understand more precisely the hardness of decision problems. In postpontum cryptography, and especially for codes and lattices, it has been proven that both versions are equivalent in the unstructured case. More precisely, for decoding problems, Fisher and Stern proved in 1996 that if I give you binary vector and I ask you whether it is completely random or it is a noisy codeword of some random code, mg plus e, if you're able to tell me that, then you can recover the message. In lattice-based cryptography, especially for LWE, it has been proved by Regaff in his all-fine famous paper that both LWE problems are equivalent. Nevertheless, modern cryptosystems rely also on structural variants such as ring LWE or module LWE, for which there exist reductions using tools from algebraic number theory. On the code-by-setting, though, it is a long-standing open problem to find such reductions for our structure codes, for instance for quasi-cyclic codes. In this work, we build upon an old analogy between number fields and function fields to provide a general framework to give such reductions. This yields the first reductions for our structure codes and also proves assumptions made to design efficient pseudo-random correlations generators or PCGs in multi-party computation. If you are interested in learning more about this topic, come attend the talk on Tuesday afternoon during the coding theory session in Santa Barbara. I will be excited to discuss it further with you in person. You can also check out the paper on e-print. Thank you for watching.