Efficient evaluation of polynomials over finite fields

A method is described which allows to evaluate efficiently a polynomial in a (possibly trivial) extension of the finite field of its coefficients. Its complexity is shown to be lower than that of standard techniques when the degree of the polynomial is large with respect to the base field. Applications to the syndrome computation in the decoding of cyclic codes, Reed-Solomon codes in particular, are highlighted.Comment: presented at AusCTW 201

On the Decoding Complexity of Cyclic Codes Up to the BCH Bound

The standard algebraic decoding algorithm of cyclic codes $[n,k,d]$ up to the BCH bound $t$ is very efficient and practical for relatively small $n$ while it becomes unpractical for large $n$ as its computational complexity is $O(nt)$. Aim of this paper is to show how to make this algebraic decoding computationally more efficient: in the case of binary codes, for example, the complexity of the syndrome computation drops from $O(nt)$ to $O(t\sqrt n)$, and that of the error location from $O(nt)$ to at most $\max \{O(t\sqrt n), O(t^2\log(t)\log(n))\}$.Comment: accepted for publication in Proceedings ISIT 2011. IEEE copyrigh

The Rabin cryptosystem revisited

The Rabin public-key cryptosystem is revisited with a focus on the problem of identifying the encrypted message unambiguously for any pair of primes. In particular, a deterministic scheme using quartic reciprocity is described that works for primes congruent 5 modulo 8, a case that was still open. Both theoretical and practical solutions are presented. The Rabin signature is also reconsidered and a deterministic padding mechanism is proposed.Comment: minor review + introduction of a deterministic scheme using quartic reciprocity that works for primes congruent 5 modulo

Improvements on Cantor-Zassenhaus Factorization Algorithm

After revisiting Cantor-Zassenhaus polynomial factorization algorithm, we describe a new simplified version of it, which requires less computational cost. Moreover we show that it is able to find a factor of a fully splitting polynomial of degree $t$ over $\mathbb F_{2^m}$ with $O(\frac{2^m}{3^{t}})$ attempts and over $\mathbb F_{p^m}$ for odd $p$ with $O(\frac{p^m}{2^{t}})$ attempts.Comment: extended and revised version; case s>1 adde

Dual plane breast implant reconstruction in large sized breasts: how to maximise the result following first stage total submuscular expansion

Introduction: Women who were good candidates for a skin reducing mastectomy, but were instead treated with a skin-sparing mastectomy and reconstruction with expanders, show discrepancy of volume and form between the healthy breast (voluminous and ptotic) and the expanded mastectomy envelope and muscle, which has a smaller size as well as excessive amount of skin at the lower pole. Methods: From January 2014 to March 2015, we recruited 18 women with breasts of medium to large volume and with moderate to severe ptosis, already treated at a different centre with a one-side mastectomy and reconstruction by means of an expander. These women were treated at our unit for the second reconstructive step with a dual plane technique and a contralateral reduction/mastopexy. Results: The minimum duration of follow-up was 2 years (range 24–30 months). The average volume of the implants was 613 g. The reconstructive outcome at the final follow-up (at least 24 months) was judged by the specialist as excellent in 5 cases, very good in 10 cases and good in 3 cases. Breast Q average score was 87.08. Discussion: The disinsertion of the expanded muscle dome and the use of a dual plane technique for the placement of the definitive implant provide a solution to the skin-volume mismatch problem. The subcutaneous placement of the implant at the level of the lower pole extends the excessive amount of skin and gives the reconstructed breast fullness and natural ptosis. Further validation of our results is neede