## Section: New Results

### Second-order entropy accumulation theorem

Device-independent cryptography is a way to use quantum mechanics to perform cryptographic tasks using equipment from an untrusted manufacturer. To prove the security of device-independent protocols, the main challenge is to show that a step-by-step procedure involving the untrusted device produces a certain of randomness even from the point of view of the manufacturer. The entropy accumulation theorem [38] provides a generic way to obtain such statements. However, while the bounds provided by this theorem are optimal in the first order (meaning the term that is linear in the number of steps in the process), the second-order sublinear term is bounded more crudely, in such a way that the bounds deteriorate significantly when the theorem is applied directly to protocols where parameter estimation is done by sampling a small fraction of the positions, as is done in most QKD protocols. In [25], we improve this second-order sublinear term and remedy this problem. This paper has been submitted to IEEE Transactions on Information Theory.