Ring-LWE Cryptography for the Number Theorist

In this paper, we survey the status of attacks on the ring and polynomial learning with errors problems (RLWE and PLWE). Recent work on the security of these problems [Eisentr\"ager-Hallgren-Lauter, Elias-Lauter-Ozman-Stange] gives rise to interesting questions about number fields. We extend these attacks and survey related open problems in number theory, including spectral distortion of an algebraic number and its relationship to Mahler measure, the monogenic property for the ring of integers of a number field, and the size of elements of small order modulo q.Comment: 20 Page

SS-unit equations and the asymptotic Fermat conjecture over number fields

Recent attempts at studying the Fermat equation over number fields have uncovered an unexpected and powerful connection with $S$-unit equations. In this expository paper we explain this connection and its implications for the asymptotic Fermat conjecture

The Hasse norm principle in global function fields

Let $L$ be a finite extension of $\mathbb{F}_q(t)$. We calculate the proportion of polynomials of degree $d$ in $\mathbb{F}_q[t]$ that are everywhere locally norms from $L/\mathbb{F}_q(t)$ which fail to be global norms from $L/\mathbb{F}_q(t)$

Provably weak instances of Ring-LWE

The ring and polynomial learning with errors problems (Ring-LWE and Poly-LWE) have been proposed as hard problems to form the basis for cryptosystems, and various security reductions to hard lattice problems have been presented. So far these problems have been stated for general (number) rings but have only been closely examined for cyclotomic number rings. In this paper, we state and examine the Ring-LWE problem for general number rings and demonstrate provably weak instances of Ring-LWE. We construct an explicit family of number fields for which we have an efficient attack. We demonstrate the attack in both theory and practice, providing code and running times for the attack. The attack runs in time linear in q, where q is the modulus. Our attack is based on the attack on Poly-LWE which was presented in [EHL]. We extend the EHL-attack to apply to a larger class of number fields, and show how it applies to attack Ring-LWE for a heuristically large class of fields. Certain Ring-LWE instances can be transformed into Poly-LWE instances without distorting the error too much, and thus provide the first weak instances of the Ring-LWE problem. We also provide additional examples of fields which are vulnerable to our attacks on Poly-LWE, including power-of-2 cyclotomic fields, presented using the minimal polynomial of $\zeta_{2^n} \pm 1$

IL-6 mediated JAK/STAT3 signaling pathway in cancer patients with cachexia

CONCLUSION: STAT3 may be considered as a therapeutic target for cachectic patients with gastric, lung and breast cancer. Furthermore, IL-6 mediates STAT3 activation in cachectic gastric and breast cancer patients (Tab. 5, Fig. 2, Ref. 62)

Ring-LWE cryptography for the number theorist

In this paper, we survey the status of attacks on the ring and polynomial learning with errors problems (RLWE and PLWE). Recent work on the security of these problems (Eisentraeger et al., Weak Instances of PLWE. In: Proceedings of the selected areas of cryptography 2014. Lecture notes in computer science. Springer, New York, 2014; Elias Y., Lauter K., Ozman E., Stange K., Provably weak instances of ring-LWE. In: Advances in Cryptology – CRYPTO 2015. Springer, 2015 gives rise to interesting questions about number fields. We extend these attacks and survey related open problems in number theory, including spectral distortion of an algebraic number and its relationship to Mahler measure, the monogenic property for the ring of integers of a number field, and the size of elements of small order modulo q

Extracorporeal shock-wave lithotripsy for retained common bile duct stones

Retained common bile duct stones (CBDS) become a challenging problem when percutaneous and endoscopic methods fail. Extracorporeal shock-wave lithotripsy (ESWL) is a noninvasive and effective treatment modality, and can be used as an alternative treatment of retained CBDS. We report our experience with 20 patients who had retained CBDS, using a second-generation electromagnetic lithotriptor. Thirteen patients who had cholecystectomy and common bile duct exploration, with stone extraction and T-tube drainage, were in the early postoperative period. Seven patients had undergone previous endoscopic sphincterotomy and nasobiliary drainage. Fourteen patients had only one ESWL session. Stone fragmentation rates were 100% and 57% in patients with T-tube and nasobiliary catheter, respectively. The overall stone fragmentation rate was 85% and complete stone clearance was achieved in all these patients (85%). Complications were mild and relatively infrequent (20%). There was no mortality. We conclude that ESWL for retained CBDS is a safe, effective and minimally-invasive treatment modality. ESWL should be considered as a significant alternative to surgery when endoscopic and percutaneous treatment modalities are not successful