Solutions of xqk+⋯+xq+x=ax^{q^k}+\cdots+x^{q}+x=a in GF2nGF{2^n}

Though it is well known that the roots of any affine polynomial over a finite field can be computed by a system of linear equations by using a normal base of the field, such solving approach appears to be difficult to apply when the field is fairly large. Thus, it may be of great interest to find an explicit representation of the solutions independently of the field base. This was previously done only for quadratic equations over a binary finite field. This paper gives an explicit representation of solutions for a much wider class of affine polynomials over a binary prime field

A Proof of the Beierle-Kranz-Leander’s Conjecture related to Lightweight Multiplication in F2nF_{2^n}

Lightweight cryptography is an important tool for building strong security solutions for pervasive devices with limited resources. Due to the stringent cost constraints inherent in extremely large applications, the efficient implementation of cryptographic hardware and software algorithms is of utmost importance to realize the vision of generalized computing. In CRYPTO 2016, Beierle, Kranz and Leander have considered lightweight multiplication in ${F}_{2^n}$. Specifically, they have considered the fundamental question of optimizing finite field multiplications with one fixed element and investigated which field representation, that is which choice of basis, allows for an optimal implementation. They have left open a conjecture related to two XOR-count. Using the theory of linear algebra, we prove in the present paper that their conjecture is correct. Consequently, this proved conjecture can be used as a reference for further developing and implementing cryptography algorithms in lightweight devices

A Case of Primary Adrenal Diffuse Large B-cell Lymphoma Achieving Complete Remission with Rituximab-CHOP Chemotherapy

Primary adrenal lymphoma is a very rare extranodal lymphoma; its clinical features consist of a high incidence of bilateral adrenal involvement and diffuse large B-cell lymphoma. We report a patient with primary bilateral adrenal diffuse large B-cell lyphoma who achieved complete remission with R-CHOP (rituximab-cyclophosphamide, doxorubicin, vincristine, and prednisone) chemotherapy. A 52-yr-old man presented with fever and progressive fatigue for 3 months. Computed tomography (CT) scans of the abdomen and pelvis demonstrated large bilateral adrenal masses, and a needle biopsy of the left adrenal mass revealed diffuse large B-cell lymphoma. After 6 cycles of R-CHOP chemotherapy, CT scans showed no residual disease. To our knowledge, this is the second report to date of a patient with primary bilateral adrenal diffuse large B-cell lymphoma who achieved complete remission using R-CHOP chemotherapy

Comparison of first-line treatment with CHOP versus ICED in patients with peripheral T-cell lymphoma eligible for upfront autologous stem cell transplantation

IntroductionUpfront autologous stem cell transplantation (ASCT) has been recommended for patients who are newly diagnosed with peripheral T-cell lymphoma (PTCL), and CHOP (cyclophosphamide, doxorubicin, vincristine, and prednisone), an anthracycline-based chemotherapy has been the frontline chemotherapy for PTCL. However, it is not clear whether anthracycline-based chemotherapies such as CHOP could be standard induction therapy for PTCL.MethodsWe conducted a randomized phase II study to compare CHOP with fractionated ifosfamide, carboplatin, etoposide, and dexamethasone (ICED) for patients eligible for ASCT. The primary endpoint was progression-free survival (PFS) and secondary endpoints included objective response rate, overall survival (OS), and safety profiles.ResultsPatients were randomized into either CHOP (n = 69) or ICED (n = 66), and the characteristics of both arms were not different. PTCL-not otherwise specified (NOS, n = 60) and angioimmunoblastic T-cell lymphoma (AITL, n = 53) were dominant. The objective response rate was not different between CHOP (59.4%) and ICED (56.1%), and the 3-year PFS was not different between CHOP (36.7%) and ICED (33.1%). In AITL patients, CHOP was favored over ICED whereas ICED was associated with more cytopenia and reduced dose intensity. Patients who received upfront ASCT after achieving complete response to CHOP or ICED showed 80% of 3-year OS.DiscussionIn summary, our study showed no therapeutic difference between CHOP and ICED in terms of response and PFS. Thus, CHOP might remain the reference regimen especially for AITL based on its better outcome in AITL, and upfront ASCT could be recommended as a consolidation of complete response in patients with PTCL

Preimages of p −Linearized Polynomials over Fp{\mathbb {F}}_{p}

International audienc

Solving some affine equations over finite fields

Let $l$ and $k$ be two integers such that $l|k$. Define $T_l^k(X):=X+X^{p^l}+\cdots+X^{p^{l(k/l-2)}}+X^{p^{l(k/l-1)}}$ and $S_l^k(X):=X-X^{p^l}+\cdots+(-1)^{(k/l-1)}X^{p^{l(k/l-1)}}$, where $p$ is any prime. This paper gives explicit representations of all solutions in \GF{p^n} to the affine equations $T_l^{k}(X)=a$ and $S_l^{k}(X)=a$, a\in \GF{p^n}. For the case $p=2$ that was solved very recently in \cite{MKCL2019}, the result of this paper reveals another solution

A proof of the Beierle–Kranz–Leander conjecture related to lightweight multiplication in F2n\mathbb {F}_{2^n}

International audienc