Bounds on Factors in Z[x]

We gather together several bounds on the sizes of coefficients which can appear in factors of polynomials in Z[x]; we include a new bound which was latent in a paper by Mignotte, and a few minor improvements to some existing bounds. We compare these bounds and show that none is universally better than the others. In the second part of the paper we give several concrete examples of factorizations where the factors have "unexpectedly" large coefficients. These examples help us understand why the bounds must be larger than you might expect, and greatly extend the collection published by Collins.Comment: 35 pages, no figure

Faktorisasi Polinomial Square-Free dan bukan Square-Free atas Lapangan Hingga Zp

Abstrak: Faktorisasi polinomial merupakan suatu proses penguraian suatu  polinomial berderajat n menjadi polinomial-polinomial lain yang berderajat lebih kecil dari n.  Faktorisasi polinomial atas lapangan hingga  merupakan suatu proses pengerjaan yang relative tidak mudah. Oleh karena itu, diperlukan suatu metode yang berupa algoritma untuk memproses faktorisasi polinomial. Algoritma Faktorisasi Berlekamp merupakan salah satu metode terbaik dalam memfaktorisasi polinomial atas lapangan hingga . Polinomial atas lapangan terbagi dua kategori berdasarkan faktorisasinya, yaitu polinomial square-free dan bukan square-free. Polinomial square-free adalah polinomial dimana setiap faktorisasi tak tereduksi tunggal. Sedangkan bukan square-free adalah sebaliknya. Penelitian ini bertujuan untuk membuat suatu algoritma untuk menfaktorkan polinomial square-free dan bukan square-free atas lapangan hingga. Adapun (Divasὀn, Joosten, Thiemann, & Yamada, 2017) yang menjadi referensi utama dalam penelitian ini adalah berdasarkan. Namun, dibatasi hanya untuk polinomial square-free saja. Untuk itulah dengan menggunakan konsep polinomial faktorisasi ganda. Pada bagian akhir penelitian akan mengimplementasikan algoritma baru yang telah disusun. Abstract:  Polynomial factorization is a decomposition of a polynomial of degree n into other polynomials whose degree is less than n. Polynomial factorization over finite field  is a relatively easy in process. Therefore, it’s needed a method in the form of an algorithm to process polynomial factorization. Algorithm Factorization Berlekamp is one of the best methods in factoring polynomials over a finite field  . Polynomials over field are divided into two category based on its factorization, namely square-free and not square-free polynomials. Square-free polynomials are polynomials in which each irreducible factorization is single. When non square-free is the opposite. This research aims to set an algorithm for factoring square-free polynomials and non square-free polynomials over a finite field   . The main reference in this research is based on (Divasὀn, Joosten, Thiemann, & Yamada, 2017) (Saropah, 2012). However, it is restricted only  to square-free polynomials. For this reason, this research will use the concept of repeated factorization polynomials. At the end of the research will implement a new algorithm that has been set

Curves and Their Applications to Factoring Polynomials

We present new methods for computing square roots and factorization of polynomials over finite fields. We also describe a method for computing in the Jacobian of a singular hyperelliptic curve. There is a compact representation of an element in the Jacobian of a smooth hyperelliptic curve over any field. This compact representation leads an efficient method for computing in Jacobians which is called Cantor's Algorithm. In one part of the dissertation, we show that an extension of this compact representation and Cantor's Algorithm is possible for singular hyperelliptic curves. This extension lead to the use of singular hyperelliptic curves for factorization of polynomials and computing square roots in finite fields. Our study shows that computing the square root of a number mod p is equivalent to finding any of the particular group elements in the Jacobian of a certain singular hyperelliptic curve. This is also true in the case of polynomial factorizations. Therefore the efficiency of our algorithms depends on only the efficiency of the algorithms for computing in the Jacobian of a singular hyperelliptic curve. The algorithms for computing in Jacobians of hyperelliptic curves are very fast especially for small genus and this makes our algorithms especially computing square roots algorithms competitive with the other well-known algorithms. In this work we also investigate superelliptic curves for factorization of polynomials

On The Applications of Lifting Techniques

Lifting techniques are some of the main tools in solving a variety of different computational problems related to the field of computer algebra. In this thesis, we will consider two fundamental problems in the fields of computational algebraic geometry and number theory, trying to find more efficient algorithms to solve such problems. The first problem, solving systems of polynomial equations, is one of the most fundamental problems in the field of computational algebraic geometry. In this thesis, We discuss how to solve bivariate polynomial systems over either k(T ) or Q using a combination of lifting and modular composition techniques. We will show that one can find an equiprojectable decomposition of a bivariate polynomial system in a better time complexity than the best known algorithms in the field, both in theory and practice. The second problem, polynomial factorization over number fields, is one of the oldest problems in number theory. It has lots of applications in many other related problems and there have been lots of attempts to solve the problem efficiently, at least, in practice. Finding p-adic factors of a univariate polynomial over a number field uses lifting techniques. Improving this step can reduce the total running time of the factorization in practice. We first introduce a multivariate version of the Belabas factorization algorithm over number fields. Then we will compare the running time complexity of the factorization problem using two different representations of a number field, univariate vs multivariate, and at the end as an application, we will show the improvement gained in computing the splitting fields of a univariate polynomial over rational field

On the factorization of polynomials over algebraic fields

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A Key-Recovery Side-Channel Attack on Classic McEliece

In this paper, we propose the first key-recovery side-channel attack on Classic McEliece, a KEM finalist in the NIST Post-quantum Cryptography Standardization Project. Our novel idea is to design an attack algorithm where we submit special ciphertexts to the decryption oracle that correspond to cases of single errors. Decoding of such cipher-texts involves only a single entry in a large secret permutation, which is part of the secret key. Through an identified leakage in the additive FFT step used to evaluate the error locator polynomial, a single entry of the secret permutation can be determined. Reiterating this for other entries leads to full secret key recovery. The attack is described using power analysis both on the FPGA reference implementation and a software implementation running on an ARM Cortex-M4. We use a machine-learning-based classification algorithm to determine the error locator polynomial from a single trace. The attack is fully implemented and evaluated in the Chipwhisperer framework and is successful in practice. For the smallest parameter set, it is using about 300 traces for partial key recovery and less than 800 traces for full key recovery, in the FPGA case. A similar number of traces are required for a successful attack on the ARM software implementation

A Key-Recovery Side-Channel Attack on Classic McEliece Implementations

In this paper, we propose the first key-recovery side-channel attack on Classic McEliece, a KEM finalist in the NIST Post-quantum Cryptography Standardization Project. Our novel idea is to design an attack algorithm where we submit special ciphertexts to the decryption oracle that correspond to cases of single errors. Decoding of such ciphertexts involves only a single entry in a large secret permutation, which is part of the secret key. Through an identified leakage in the additive FFT step used to evaluate the error locator polynomial, a single entry of the secret permutation can be determined. Iterating this for other entries leads to full secret key recovery. The attack is described using power analysis both on the FPGA reference implementation and a software implementation running on an ARM Cortex-M4. We use a machine-learning-based classification algorithm to determine the error locator polynomial from a single trace. The attack is fully implemented and evaluated in the Chipwhisperer framework and is successful in practice. For the smallest parameter set, it is using about 300 traces for partial key recovery and less than 800 traces for full key recovery, in the FPGA case. A similar number of traces are required for a successful attack on the ARM software implementation