Prime Field Research Articles

Efficient Algorithms for Large Prime Characteristic Fields and Their Application to Bilinear Pairings

We propose a novel approach that generalizes interleaved modular multiplication algorithms for the computation of sums of products over large prime fields. This operation has widespread use and is at the core of many cryptographic applications. The method reformulates the widely used lazy reduction technique, crucially avoiding the need for storage and computation of “double-precision” operations. Moreover, it can be easily adapted to the different methods that exist to compute modular multiplication, producing algorithms that are significantly more efficient and memory-friendly. We showcase the performance of the proposed approach in the computation of multiplication over an extension field Fpk , and demonstrate its impact with record-breaking implementations of bilinear pairings. Specifically, we accomplish a full optimal ate pairing computation over the popular BLS12-381 curve, designed for the 128-bit security level, in under half a millisecond on a 3.2GHz Intel Coffee Lake processor, which is about 1.40× faster than the state-of-the-art. Similarly, we perform the same computation over the BLS24-509 curve, targeting the 192-bit security level, in ~ 2.6 milliseconds, achieving a speedup of more than 1.30x. We also report a significant impact on other applications, including protocols based on supersingular isogenies.

Modular Matrix Multiplication on GPU for Polynomial System Solving

The bottleneck of the SPARSE-FGLM algorithm for Gröbner bases change of order is an iterative matrix - tall and skinny matrix product over a finite prime field. Our contribution is twofold. First, we port existing CPU-only algorithms for matrix products over prime fields to GPU architectures, and carry out a performance analysis of our implementation that shows that we can nearly achieve the maximum theoretical throughput of the hardware. Second, existing CPU-only algorithms could not handle primes with more than 26 bits, other than the GMP-based implementation in FLINT; we overcome this limitation by proposing an efficient multiword matrix product algorithm that can deal with primes with at most 35 bits; we benchmarked it on GPU.

Reducibility of quartic polynomials and its relation to Galois groups

This paper deals with special classes of quartic polynomials and properties pertaining to their Galois groups and reducibility over certain fields. The existence of quartic polynomials irreducible over Q but reducible over every prime field is first proven, after which criteria are established for the Galois group of polynomials with this property. By constructing classes of V4-generic polynomials and comparing them with criteria put forth in previous studies for determining polynomials with this property, it can be shown that a polynomial of the biquadratic form x4 + ax2 + b has this property if and only if it can be written as x^4 - 2(u + v)x^2 + (u -v)^2 with u, v Q such that none of u, v, or uv can be expressed as ratio of two squares, and 2(u+v),(uv)2 Z . The general form for biquadratic polynomials irreducible over Q and reducible modulo every integer n is found to have a general form similar to this one.

FPGA Implementation for Elliptic Curve Cryptography Algorithm and Circuit with High Efficiency and Low Delay for IoT Applications.

The Internet of Things requires greater attention to the security and privacy of the network. Compared to other public-key cryptosystems, elliptic curve cryptography can provide better security and lower latency with shorter keys, rendering it more suitable for IoT security. This paper presents a high-efficiency and low-delay elliptic curve cryptographic architecture based on the NIST-p256 prime field for IoT security applications. A modular square unit utilizes a fast partial Montgomery reduction algorithm, demanding just a mere four clock cycles to complete a modular square operation. The modular square unit can be computed simultaneously with the modular multiplication unit, consequently improving the speed of point multiplication operations. Synthesized on the Xilinx Virtex-7 FPGA platform, the proposed architecture completes one PM operation in 0.08 ms using 23.1 k LUTs at 105.3 MHz. These results show significantly better performance compared to that in previous works.

Bounds on bilinear forms with Kloosterman sums

AbstractWe prove new bounds on bilinear forms with Kloosterman sums, complementing and improving a series of results by É. Fouvry, E. Kowalski and Ph. Michel (2014), V. Blomer, É. Fouvry, E. Kowalski, Ph. Michel and D. Milićević (2017), E. Kowalski, Ph. Michel and W. Sawin (2019, 2020) and I. E. Shparlinski (2019). These improvements rely on new estimates for Type II bilinear forms with incomplete Kloosterman sums. We also establish new estimates for bilinear forms with one variable from an arbitrary set by introducing techniques from additive combinatorics over prime fields. Some of these bounds have found a crucial application in the recent work of Wu (2020) on asymptotic formulas for the fourth moments of Dirichlet ‐functions. As new applications, an estimate for higher moments of averages of Kloosterman sums and the distribution of divisor function in a family of arithmetic progressions are also given.

Systematic KMTNet Planetary Anomaly Search. VIII. Complete Sample of 2019 Subprime Field Planets

We complete the publication of all microlensing planets (and “possible planets”) identified by the uniform approach of the KMT AnomalyFinder system in the 21 KMT subprime fields during the 2019 observing season, namely, KMT-2019-BLG-0298, KMT-2019-BLG-1216, KMT-2019-BLG-2783, OGLE-2019-BLG-0249, and OGLE-2019-BLG-0679 (planets), as well as OGLE-2019-BLG-0344 and KMT-2019-BLG-0304 (possible planets). The five planets have mean log mass ratio measurements of (−2.6, −3.6, −2.5, −2.2, −2.3), median mass estimates of (1.81, 0.094, 1.16, 7.12, 3.34) M Jup, and median distance estimates of (6.7, 2.7, 5.9, 6.4, 5.6) kpc, respectively. The main scientific interest of these planets is that they complete the AnomalyFinder sample for 2019, which has a total of 25 planets that are likely to enter the statistical sample. We find statistical consistency with the previously published 33 planets from the 2018 AnomalyFinder analysis according to an ensemble of five tests. Of the 58 planets from 2018–2019, 23 were newly discovered by AnomalyFinder. Within statistical precision, half of the planets have caustic crossings, while half do not; an equal number of detected planets result from major- and minor-image light-curve perturbations; and an equal number come from KMT prime fields versus subprime fields.

English

is the prime modal field with elements as presented by Ayissi Eteme in [5] in order to define on a notion of Hamming code which respects its structure of set. We show a relation between protocol and Hamming code. By using this relation, we give a method to construct good steganographic protocols.

Generalized Lucas polynomials over finite fields

In this paper, we give necessary and sufficient conditions on n for the unsigned and signed generalized Lucas polynomials fn(x) and gn(x) being permutations over the prime field. Moreover, the permutation behaviours of these polynomials over non-prime fields are also discussed.

Lee weight distributions of trace codes over F_q+uF_q from irreducible cyclic codes

The construction of two or few-weight codes from trace codes over the ring Fq + uFq, where u2 = 0, was recently presented in [4]. For such construction, the defining sets for the trace codes are given in terms of cyclotomic classes, and for some of these classes, it is shown that it is possible to obtain the Lee weight distributions of the corresponding trace codes. Motivated by this construction, and by the p-ary semiprimitive irreducible cyclic codes over a prime field Fp, the Lee weight distributions of an infinite family of p-ary three-weight codes from trace codes over the ring Fq + uFq, was recentely found in [11]. In this work, we prove that the Lee weight distribution problem for the trace codes constructed in accordance with either [4] or [11], is equivalent to the weight distribution problem for the irreducible cyclic codes. With this equivalence in mind, and by using the already known weight distributions of an infinite family irreducible cyclic codes (semiprimitive and not semiprimitive), we follow the open problem suggested in the Conclusion of [11] to determine the Lee weight distribution of an infinite family of trace codes over the ring Fq + uFq , that includes the infinite family found in [11].

Unimodular Perfect and Nearly Perfect Sequences: a Variation of Björck’s Scheme

Constant Amplitude (CA), Zero Auto Correlation (ZAC) sequences (or CAZAC sequences, aka perfect sequences) have numerous applications. We generalize the CAZAC notion to what we term as CASAC by permitting small autocorrelations (SAC). We extend Björck’s classification result of two-valued CAZAC sequences by providing a complete classification of all almost 2-valued (i.e., two-valued except for the first position which uses a third value) CASAC sequences. While Björck’s original work dealt only with primes p, we extend his ideas to any abelian group of order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v\equiv 1\pmod {4}$ </tex-math></inline-formula> , as opposed to restricting just to the prime fields GF(p). Björck sequences have better ambiguity function than Zadoff-Chu sequences, making them suitable for radar and communications applications in the presence of high Doppler shifts. In fact, the discrete narrow band ambiguity function has an optimal bound in case of Björck sequences (as opposed to Gauss sequences). A one-parameter infinite family of CASAC we construct would have applications in Multiple-Input Multiple-Output (MIMO) areas. Toward MIMO applications, we introduce a performance measure we term as cross merit factor to study cross correlation behavior, generalizing the well-known notion of Golay Merit Factor (GMF).

Separating Rank Logic from Polynomial Time

In the search for a logic capturing polynomial time the most promising candidates are Choiceless Polynomial Time (CPT) and rank logic. Rank logic extends fixed-point logic with counting by a rank operator over prime fields. We show that the isomorphism problem for CFI graphs over ℤ 2 i cannot be defined in rank logic, even if the base graph is totally ordered. However, CPT can define this isomorphism problem. We thereby separate rank logic from CPT and in particular from polynomial time.

Hochschild cohomology of symmetric groups and generating functions

Abstract In this article, we compute the dimensions of the Hochschild cohomology of symmetric groups over prime fields in low degrees. This involves us in studying some partition identities and generating functions of the dimensions in any fixed degree of the Hochschild cohomology of symmetric groups. We show that the generating function of the dimensions of the Hochschild cohomology in any fixed degree of the symmetric groups differs from that in degree 0 by a rational function.

Prime-Field Masking in Hardware and its Soundness against Low-Noise SCA Attacks

A recent study suggests that arithmetic masking in prime fields leads to stronger security guarantees against passive physical adversaries than Boolean masking. Indeed, it is a common observation that the desired security amplification of Boolean masking collapses when the noise level in the measurements is too low. Arithmetic encodings in prime fields can help to maintain an exponential increase of the attack complexity in the number of shares even in such a challenging context. In this work, we contribute to this emerging topic in two main directions. First, we propose novel masked hardware gadgets for secure squaring in prime fields (since squaring is non-linear in non-binary fields) which prove to be significantly more resource-friendly than corresponding masked multiplications. We then formally show their local and compositional security for arbitrary orders. Second, we attempt to &gt;experimentally evaluate the performance vs. security tradeoff of prime-field masking. In order to enable a first comparative case study in this regard, we exemplarily consider masked implementations of the AES as well as the recently proposed AESprime. AES-prime is a block cipher partially resembling the standard AES, but based on arithmetic operations modulo a small Mersenne prime. We present cost and performance figures for masked AES and AES-prime implementations, and experimentally evaluate their susceptibility to low-noise side-channel attacks. We consider both the dynamic and the static power consumption for our low-noise analyses and emulate strong adversaries. Static power attacks are indeed known as a threat for side-channel countermeasures that require a certain noise level to be effective because of the adversary’s ability to reduce the noise through intra-trace averaging. Our results show consistently that for the noise levels in our practical experiments, the masked prime-field implementations provide much higher security for the same number of shares. This compensates for the overheads prime computations lead to and remains true even if / despite leaking each share with a similar Signal-to-Noise Ratio (SNR) as their binary equivalents. We hope our results open the way towards new cipher designs tailored to best exploit the advantages of prime-field masking.

The Legendre pseudorandom function as a multivariate quadratic cryptosystem: security and applications

AbstractSequences of consecutive Legendre and Jacobi symbols as pseudorandom bit generators were proposed for cryptographic use in 1988. Major interest has been shown towards pseudorandom functions (PRF) recently, based on the Legendre and power residue symbols, due to their efficiency in the multi-party setting. The security of these PRFs is not known to be reducible to standard cryptographic assumptions. In this work, we show that key-recovery attacks against the Legendre PRF are equivalent to solving a specific family of multivariate quadratic (MQ) equation system over a finite prime field. This new perspective sheds some light on the complexity of key-recovery attacks against the Legendre PRF. We conduct algebraic cryptanalysis on the resulting MQ instance. We show that the currently known techniques and attacks fall short in solving these sparse quadratic equation systems. Furthermore, we build novel cryptographic applications of the Legendre PRF, e.g., verifiable random function and (verifiable) oblivious (programmable) PRFs.

The geometrically m-step solvable Grothendieck conjecture for genus 0 curves over finitely generated fields

In this paper, we present some partial results for the geometrically m-step solvable Grothendieck conjecture in anabelian geometry. Among other things, we prove the geometrically 3-step solvable Grothendieck conjecture for genus 0 curves over fields finitely generated over the prime field of arbitrary characteristic.

Study of the open clusters in Kepler prime field

ABSTRACTWe present a detailed study of NGC 6791, NGC 6811, NGC 6819, and NGC 6866, the four open clusters that are located in the Kepler prime field. We use new CCD UBV(RI)KC photometry, which was combined with Gaia EDR3 photometric/astrometric data, to derive the astrophysical parameters with two-independent methods – one of them the fitCMD algorithm. Furthermore, we provide among others estimates of the mass and mass function, the cluster structure, derive the cluster orbits, and discuss the cluster dynamics. All objects belong to the older open cluster population (∼1–7 Gyr), are in an advanced dynamical stage with signs of mass segregation, and are located close to the solar circle, but show a large range in respect of radii, member stars or observed cluster mass ($\sim 100\!-\!2000\, \mathrm{M}_\odot$). For the three younger objects, we were also able to provide photometric metallicity estimates, which confirms their status as clusters with a roughly solar metallicity. The most outstanding object is clearly NGC 6791, a very old cluster with a high metallicity at a distance of about 4.5 kpc from the Sun. We estimate a probable radial migration by about 7 kpc, resulting in a birth position close to the Galactic centre.

Computation of Multiplicative Inverses of non-zero Polynomials of GF(73) by Gauss-Jordan method.

The present manuscript deals with the derivation of multiplicative inverses of all non-zero elemental polynomials of the Galois field with respect to the monic irreducible polynomials of degree 3 over the prime field by Gauss-Jordan method as the Extended Euclidean Algorithm (EEA) is not always successful over the prime field , when prime .

Cryptography based on algebraic perpendicularities

Cryptography, the mathematics of protecting secret or sensitive information, is a continuously evolving area of interest. A concrete example of this is the fact that worldwide spending on information security and risk management technology and services has been estimated to reach over $150 billion in 2021. Modern cryptography is actually not a single separate domain of mathematics but an advanced encryption scheme can be based on applications of results in number theory, such as the Euler–Fermat Theorem, or involve the discrete logarithm problem using either a primitive root of a large prime or an element of an elliptic curve over a prime field. In the background, probability theory, statistics, studies on computational models and finite geometries, etc. play a major role. Recent research has considered even DNA-based molecular cryptography systems.

Bounds for the Diameters of Orbital Graphs of Affine Groups

General bounds are presented for the diameters of orbital graphs of finite affine primitive permutation groups. For example, it is proved that the orbital diameter of a finite affine primitive permutation group with a nontrivial point stabilizer H ≤GL(V ), where the vector space V has dimension d over the prime field, can be bounded in terms of d and log |V|/log |H| only. Several infinite families of affine primitive permutation groups with large orbital diameter are constructed. The results are independent from the classification of finite simple groups.

An Innovative approach towards image encryption by using novel PRNs and S-boxes Modeling techniques

Efficient multiple pseudo-random number sequences (PRNS) and substitution boxes (S-boxes) are one of the most significant building blocks, which are jointly adopted normally for secure data encryption. Multiple aspects pave the way to handle large-scale multimedia data. However, the computational efforts on multiple constructions may certainly lead to limits the required ciphering through-put. Therefore, reducing the computational cost of multiple patterns such as PRNS and S-boxes is the core requirement for an efficient cryptosystem. For this achievement, we exploited the indexing technique over elliptic curves with small prime fields to introduce a computationally efficient mechanism for both multiple PRNS and multiple S-boxes. In the newly constructed PRNS and S-boxes, we used collectively EC group law and simple algebraic operations to get the security strength as well as low computational cost respectively. Based on statistical results, the proposed S-box mechanism is the most effective method that generates efficient multiple S-boxes on minimum prime fields. Likewise, the PRNS’s simulation results shows that the proposed mechanism is the highly effective model for generating multiple verified pseudo random patterns on small prime fields in a single round. These assessments indicates that the proposed mechanism offers desired key-space, better statistical features of encrypted data and less computational cost.