Polynomial Interpolation and Identity Testing from High Powers Over Finite Fields

We consider the problem of recovering (that is, interpolating) and identity testing of a “hidden” monic polynomial f, given an oracle access to (Formula presented.) for (Formula presented.), where (Formula presented.) is finite field of q elements (extension fields access is not permitted). The naive interpolation algorithm needs (Formula presented.) queries and thus requires (Formula presented.). We design algorithms that are asymptotically better in certain cases; requiring only (Formula presented.) queries to the oracle. In the randomized (and quantum) setting, we give a substantially better interpolation algorithm, that requires only (Formula presented.) queries. Such results have been known before only for the special case of a linear f, called the hidden shifted power problem. We use techniques from algebra, such as effective versions of Hilbert’s Nullstellensatz, and analytic number theory, such as results on the distribution of rational functions in subgroups and character sum estimates. © 2017 Springer Science+Business Media New Yor

Security considerations for Galois non-dual RLWE families

We explore further the hardness of the non-dual discrete variant of the Ring-LWE problem for various number rings, give improved attacks for certain rings satisfying some additional assumptions, construct a new family of vulnerable Galois number fields, and apply some number theoretic results on Gauss sums to deduce the likely failure of these attacks for 2-power cyclotomic rings and unramified moduli

KAJIAN KEDUDUKAN DAN NILAI PEMBUKTIAN SAKSI MAHKOTA SEBAGAI ALAT BUKTI DALAM PEMBUKTIAN TINDAK PIDANA KORUPSI (STUDI KASUS NO.REG.PERK : PDS-01/SKRTA/Ft.1/03/2010 BERKAIT KORUPSI DI RUMAH SAKIT JIWA DAERAH SURAKARTA)

Penulisan penelitian hukum ini bertujuan untuk mengetahui dasar hukum menurut jaksa penuntut umum digunakannya saksi mahkota serta kedudukan dan nilai pembuktian saksi mahkota dalam pandangan hakim sebagai alat bukti dalam kasus perkara No. Reg. Perk : PDS-01/SKRTA/Ft.1/03/2010. Pengertian saksi mahkota dalam putusan Mahkamah Agung RI No.1986 K/Pid/1989 adalah teman terdakwa yang dilakukan secara bersama-sama yang diajukan sebagai saksi untuk membuktikan dakwaan penuntut umum dalam hal ini perkaranya dipisah dikarenakan kurangnya alat bukti. Tetapi dalam perkembangannya di dalam Putusan Mahkamah Agung RI No. 1174/K/Pid/1994 tanggal 3 Mei 1995, Putusan Mahkamah Agung RI No. 1590/K/Pid/1995 tanggal 3 Mei 1995 dan Putusan Mahkamah Agung RI No. 1592/K/Pid/1995 tanggal 3 Mei 1995 tidak membenarkan adanya penggunaan saksi mahkota. Menurut putusan ini saksi mahkota juga pelaku yang diajukan sebagai terdakwa dalam dakwaan yang terpisah sehingga hal ini dianggap sebagai pelanggaran hak asasi terdakwa. Pada kenyataannya dalam praktek peradilan di Indonesia masih sering digunakannya saksi mahkota dalam mengatasi masalah kurangnya alat bukti saksi. P e n u lisa n H u k u m ini term asu k dala m je nis p en elitia n h u k u m e m p iris ata u non doctrinal y a itu pe n elitia n ya n g d ilak u ka n se ca ra la n gsu n g de n ga n m e m b a n d in gk a n h u k u m da la m ha l te o ritis de n ga n m e n ga m ati pe rila k u ya n g te rjad i d idala m m a sya rak at. Penulisan hukum ini bersifat deskriptif dengan pendekatan kualitatif. Hasil yang diperoleh dari penelitian ini yaitu bahwa selain dari Putusan Mahkamah Agung RI tidak ada dasar hukum mengenai saksi mahkota dan penggunaan saksi mahkota dalam perkara No.Reg.Perk : PDS- 01/SKRTA/Ft.1/03/2010 berkait korupsi di Rumah Sakit Jiwa Daerah Surakarta dibenarkan didasarkan pada prinsip-prinsip tertentu yaitu terdapat kekurangan alat bukti, dalam perkara delik penyertaan (Deelneming), diperiksa dengan mekanisme pemisahan (Splitsing). S aksi mahkota dalam kasus ini berkedudukan murni sebagai saksi karena memenuhi syarat sebagai saksi sesuai Pasal 1 angka 26 KUHAP maka sah untuk dapat diperiksa sebagai saksi, sehingga majelis hakim akan menerima dan mengakui kesaksian dari saksi mahkota ini dan akan digunakan sebagai pertimbangkan dalam menyusun putusan. Kata kunci : saksi mahkota

On lattice profile of the elliptic curve linear congruential generators

Lattice tests are quality measures for assessing the intrinsic structure of pseudorandom number generators. Recently a new lattice test has been introduced by Niederreiter and Winterhof. In this paper, we present a general inequality that is satisfied by any periodic sequence. Then, we analyze the behavior of the linear congruential generators on elliptic curves (EC-LCG) under this new lattice test and prove that the EC-LCG passes it up to very high dimensions. We also use a result of Brandstätter and Winterhof on the linear complexity profile related to the correlation measure of order k to present lower bounds on the linear complexity profile of some binary sequences derived from the EC-LCG

An iterative algorithm for parametrization of shortest length shift registers over finite rings

The construction of shortest feedback shift registers for a finite sequence S_1,...,S_N is considered over the finite ring Z_{p^r}. A novel algorithm is presented that yields a parametrization of all shortest feedback shift registers for the sequence of numbers S_1,...,S_N, thus solving an open problem in the literature. The algorithm iteratively processes each number, starting with S_1, and constructs at each step a particular type of minimal Gr\"obner basis. The construction involves a simple update rule at each step which leads to computational efficiency. It is shown that the algorithm simultaneously computes a similar parametrization for the reciprocal sequence S_N,...,S_1.Comment: Submitte

Hardness of Computing Individual Bits for One-way Functions on Elliptic Curves

We prove that if one can predict any of the bits of the input to an elliptic curve based one-way function over a finite field, then we can invert the function. In particular, our result implies that if one can predict any of the bits of the input to a classical pairing-based one-way function with non-negligible advantage over a random guess then one can efficiently invert this function and thus, solve the Fixed Argument Pairing Inversion problem (FAPI-1/FAPI-2). The latter has implications on the security of various pairing-based schemes such as the identity-based encryption scheme of Boneh–Franklin, Hess’ identity-based signature scheme, as well as Joux’s three-party one-round key agreement protocol. Moreover, if one can solve FAPI-1 and FAPI-2 in polynomial time then one can solve the Computational Diffie--Hellman problem (CDH) in polynomial time. Our result implies that all the bits of the functions defined above are hard-to-compute assuming these functions are one-way. The argument is based on a list-decoding technique via discrete Fourier transforms due to Akavia--Goldwasser–Safra as well as an idea due to Boneh–Shparlinski

Pseudorandom Sequences from Elliptic Curves

In this article we will generalize some known constructions to produce pseudorandom sequences with the aid of elliptic curves. We will make use of both additive and multiplicative characters on elliptic curves

On the complexity of arithmetic secret sharing

Since the mid 2000s, asymptotically-good strongly-multiplicative linear (ramp) secret sharing schemes over a fixed finite field have turned out as a central theoretical primitive in numerous constant-communication-rate results in multi-party cryptographic scenarios, and, surprisingly, in two-party cryptography as well. Known constructions of this most powerful class of arithmetic secret sharing schemes all rely heavily on algebraic geometry (AG), i.e., on dedicated AG codes based on asymptotically good towers of algebraic function fields defined over finite fields. It is a well-known open question since the first (explicit) constructions of such schemes appeared in CRYPTO 2006 whether the use of “heavy machinery” can be avoided here. i.e., the question is whether the mere existence of such schemes can also be proved by “elementary” techniques only (say, from classical algebraic coding theory), even disregarding effective construction. So far, there is no progress. In this paper we show the theoretical result that, (1) no matter whether this open question has an affirmative answer or not, these schemes can be constructed explicitly by elementary algorithms defined in terms of basic algebraic coding theory. This pertains to all relevant operations associated to such schemes, including, notably, the generation of an instance for a given number of players n, as well as error correction in the presence of corrupt shares. We further show that (2) the algorithms are quasi-linear time (in n); this is (asymptotically) significantly more efficient than the known constructions. That said, the analysis of the mere termination of these algorithms does still rely on algebraic geometry, in the sense that it requires “blackbox application” of suitable existence results for these schemes. Our method employs a nontrivial, novel adaptation of a classical (and ubiquitous) paradigm from coding theory that enables transformation of existence results on asymptotically good codes into explicit construction of such codes via concatenation, at some constant loss in parameters achieved. In a nutshell, our generating idea is to combine a cascade of explicit but “asymptotically-bad-yet-good-enough schemes” with an asymptotically good one in such a judicious way that the latter can be selected with exponentially small number of players in that of the compound scheme. This opens the door t