Elliptic Curve Point Research Articles

Parallelization of scalable elliptic curve cryptosystem processors in GF(2m)

The parallelization of scalable elliptic curve cryptography (ECC) processors (ECPs) is investigated in this paper. The proposed scalable ECPs support all 5 pseudo-random curves or all 5 Koblitz curves recommended by the National Institute of Standards and Technology (NIST) without the need to reconfigure the hardware. The proposed ECPs parallelize the finite field arithmetic unit and the elliptic curve point multiplication (ECPM) algorithm to gain performance improvement. The finite field multiplication is separated such that the reduction step is executed in parallel with the next polynomial multiplication. Subsequently, the finite field arithmetic of the ECPs are further parallelized and the performance can be further improved by over 50%. Since the multiplier blocks consume a low number of hardware resources, the latency reduction outweighs the cost of the extra multiplier resulting in more efficient ECP designs. The technique is applied for both pseudo-random curve and Koblitz curve algorithms. A novel ECPM algorithm is also proposed for Koblitz curves that take advantage of the proposed finite field arithmetic architecture. The implementation results show that the proposed parallelized scalable ECPs have better performance compared to state-of-the-art scalable ECPs that support the same set of elliptic curves.

Hardware realization of fast elliptic curve point multiplication using balanced ternary representation and pre-computation over GF(p)

We present a new hardware realization of fast elliptic curve Point Multiplication (PM) using the balanced ternary representation of the scalar multiplier. This hardware implementation uses multiple Elliptic Curve Point Tripler's (ECPT) and Elliptic Curve Point Adders (ECPA). Here, we pre-compute 3P, 9P, 27P, etc. using fast tripling and use them in calculating the final product over GF(p). However, to the best of our knowledge there exist no previous implementations of Balanced Ternary representation and pre-computation over GF(p) on FPGA platform. The FPGA synthesis results show that the proposed TPM6 and TPM24 methods can reach a speed up to 48.147 MHz and 50.814MHz; It needs 11.05µs and 49.45µs for the computation, consumes a power of 1.083W and 1.104W, occupies 2,657 and 5,449 slices respectively. The modular hardware approach used here provides high speed and scalability.

Scalable Elliptic Curve Cryptosystem FPGA Processor for NIST Prime Curves

The architecture and the implementation of a high-performance scalable elliptic curve cryptography processor (ECP) are presented. The proposed ECP is able to support all five prime field elliptic curves recommended by the National Institute of Standards and Technology (NIST). The design takes advantage of the high-performance capabilities of the DSP48E slices available in Xilinx field-programmable gate arrays (FPGAs) to achieve high speed and low hardware resource utilization. The proposed design parallelizes the underlying prime field operations to reduce the latency of the elliptic curve point multiplication (ECPM) operation. Prime field inversion is performed efficiently using the same arithmetic blocks as the ones used for prime field multiplication and addition/subtraction. To the best of the authors' knowledge, the proposed scalable ECP is the fastest and smallest ECP that can support all five NIST recommended prime curves without the need to reconfigure the hardware. It can compute the ECPM between 1.709 and 28.04 ms using a Xilinx Virtex-5 FPGA.

Ramification of the Kummer extension generated from torsion points of elliptic curves

For a prime p, let ζp denote a fixed primitive pth root of unity. Let E be an elliptic curve over a number field k with a p-torsion point. Then the p-torsion subgroup of E gives a Kummer extension over k(ζp). In this paper, for p = 5 and 7, we study the ramification of such Kummer extensions using explicit Kummer generators directly computed by Verdure in 2006.

Performance Analysis of Reversible Finite Field Arithmetic Architectures Over GF(p) and GF(2m) in Elliptic Curve Cryptography

Elliptic curve cryptosystems (ECC) are becoming more and more popular and are included in many standards, as they offer high security strength when compared with other conventional public-key cryptosystems, for the same key length. But the security strength of hardware implementations of ECC is challenged by side channel attacks (SCA) such as power analysis. Reversible logic circuits ideally consume zero energy, which serves as the motivation to implement cryptographic algorithms against power analysis attacks. This paper proposes two new hardware architectures for performing montgomery multiplication in GF(p) and GF(2m), as they are the power consuming operations in ECC. The two architectures are optimized to reduce the hardware cost and they are then implemented in reversible logic with reduced number of quantum cost. In this work, the reversible logic synthesis is performed with Toffoli family of reversible gates. The performance metrics of all the multipliers are analyzed and properly tabulated. Scalar multiplication on elliptic curve points, which is the core operation used in every elliptic curve cryptosystem, has been implemented in reversible logic by using the proposed reversible montgomery multipliers.

Comparing design approaches for elliptic curve point multiplication over [formula omitted] with polynomial basis representation

Point Multiplication (PM) is considered the most computationally complex and resource hungry Elliptic Curve Cryptography (ECC) mathematical operation. PM hardware accelerator design can follow several approaches that lead to a fast, small or flexible implementation, meeting related application specifications. However, each PM design decision has certain outcomes in utilized hardware resources and computation speed. Such a key design decision is related to the structure of the GF(2k) multipliers to be employed in the PM accelerator. In this paper, we highlight the GF(2k) multiplication role in the overall PM performance and investigate what are the trade-offs on a PM accelerator when using bit serial or bit parallel multiplication approach in terms of speed, chip covered area and flexibility. To achieve this goal, we estimate these tradeoffs for a single point operation and specify realistic design cases for bit serial and bit parallel multiplier based PM design approaches. To evaluate the theoretical modeling, a point operation design methodology based on the parallelism and rescheduling of GF(2k) operations is proposed. This design approach is adapted to two characteristic PM algorithm realizations, the traditional double & add algorithm and the side channel attack resistant Montgomery power ladder algorithm. Our goal is to assess the resulting PM accelerator overall performance so as to achieve high speed with an acceptable cost on chip covered area (hardware resources). Using this methodology, PM is performed in series of GF(2k) parallelism stages. To test the proposed methodology, 8 PM accelerator use cases are identified that can offer high speed, flexibility, side channel attack resistance or small chip covered area. To provide fair comparisons and results, a common PM architecture is devised and the use case PM accelerators are implemented in FPGA technology. Depending on the designers goal, the proposed architectures and 8 implementations can offer the benefit of either high speed (the proposed work is currently one of the fastest known GF(2k) bit parallel multiplier based PM realization) or flexibility with reasonable compromises in chip covered area.

The mean-value of a product of shifted multiplicative functions and the average number of points of elliptic curves

In this paper, we consider the mean value of the product of two real valued multiplicative functions with shifted arguments. The functions F and G under consideration are close to two nicely behaved functions A and B, such that the average value of A(n−h)B(n) over any arithmetic progression is only dependent on the common difference of the progression. We use this method on the problem of finding mean value of NK(N)/ϕ(N), where NK(N)/ϕ(N)log⁡N is the expected number of primes such that a random elliptic curve over rationals has N points when reduced over those primes.

Mathematical analysis of the computational complexity of integer sub-decomposition algorithm

In this paper, the computational complexity to compute a scalar multiplication on classes of elliptic curve over a prime field that have efficiently-computable endomorphismsis analyzed mathematically. This scalar multiplication is called integer sub-decomposition (ISD) method, which is based on the GLV method of Gallant, Lambert and Vanstone that was initially proposed in the year 2001. In this work, the mathematical proof of the computational cost of ISD implementation is presented. The computational cost of ISD algorithm has been proven based on the operations counting. Two types of the operations are used, namely, elliptic curve operations represented by elliptic curve point addition A and elliptic curve point doubling D and finite field operations represented by field inversion I, field multiplication M and field squaring S that design the computation of the running time of ISD scalar multiplication.

An Encryption Scheme Using Chaotic Map and Genetic Operations for Wireless Sensor Networks

Over the past decade, the application domain of wireless sensor networks has expanded steadily, ranging from environmental management to industry control, and from structural health monitoring to strategic surveillance. With the proliferation of sensor networks at home, work place, and beyond, securing data in the network has become a challenge. A number of security mechanisms have been proposed for sensor networks to provide data confidentiality: 1) advanced encryption system; 2) KATAN; 3) LED; and 4) TWINE. However, these schemes have drawbacks, including security vulnerabilities, need for hardware-based implementation, and higher computational complexity. To address these limitations, we propose a lightweight block cipher based on chaotic map and genetic operations. The proposed cryptographic scheme employs elliptic curve points to verify the communicating nodes and as one of the chaotic map parameters to generate the pseudorandom bit sequence. This sequence is used in XOR, mutation, and crossover operations in order to encrypt the data blocks. The experimental results based on Mica2 sensor mote show that the proposed encryption scheme is nine times faster than the LED protocol and two times faster than the TWINE protocol. We have also performed a number of statistical tests and cryptanalytic attacks to evaluate the security strength of the algorithm and found the cipher provably secure.

Algebraic functional equations and completely faithful Selmer groups

Let E be an elliptic curve — defined over a number field K — without complex multiplication and with good ordinary reduction at all the primes above a rational prime p ≥ 5. We construct a pairing on the dual p∞-Selmer group of E over any strongly admissible p-adic Lie extension K∞/K under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group G = Gal(K∞/K). Under some mild additional hypotheses, this gives an algebraic functional equation of the conjectured p-adic L-function. As an application, we construct completely faithful Selmer groups in case the p-adic Lie extension is obtained by adjoining the p-power division points of another non-CM elliptic curve A.

Efficient Subquadratic Space Complexity Architectures for Parallel MPB Single- and Double-Multiplications for All Trinomials Using Toeplitz Matrix-Vector Product Decomposition

Subquadratic multiplication algorithm has received significant attention of cryptographic hardware researchers for efficient implementation public-key cryptosystems. In this paper, we derive a new shifted MPB (SMPB) representation based on modified polynomial basis (MPB). We have shown that by using MPB and SMPB, the proposed double basis multiplication can be transformed into Toeplitz matrix-vector product (TMVP) structure. Furthermore, by employing this formulation of double basis multiplication, we show that three-operand multiplication over GF(2 m ) for all trinomials can be realized efficiently by the recursive TMVP (RTMVP) formulation. To perform the three-operand multiplication with the RTMVP formulation, we have derived a new RTMVP decomposition scheme. The proposed single- and double-multiplications can, respectively, use TMVP and RTMVP decompositions to achieve subquadratic space complexity architectures. By theoretical analysis, it is shown that the proposed subquadratic multipliers involve significantly less space complexity and less computation time compared to the existing subquadratic multipliers using TMVP and Karatsuba algorithms. Moreover, our proposed double-multiplication design can be used in several applications involving successive multiplications, such as exponentiation, inversion, and elliptic curve point multiplication.

The Geometry of Representations of 3-Dimensional Sklyanin Algebras

The representation scheme ${\tt rep}_n A$ of the 3-dimensional Sklyanin algebra $A$ associated to a plane elliptic curve and n-torsion point contains singularities over the augmentation ideal $\mathfrak{m}$. We investigate the semi-stable representations of the noncommutative blow-up algebra $B=A \oplus \mathfrak{m}t \oplus\mathfrak{m}^2 t^2 \oplus ...$ to obtain a partial resolution of the central singularity ${\tt proj} Z(B) \rightarrow {\tt spec} Z(A)$ such that the remaining singularities in the exceptional fiber determine an elliptic curve and are all of type $\mathbb{C} \times \mathbb{C}^2/\mathbb{Z}_n$.

The arithmetic of curves defined by iteration

We show how the size of the Galois groups of iterates of a quadratic polynomial $f(x)$ can be parametrized by certain rational points on the curves $C_n:y^2=f^n(x)$ and their quadratic twists. To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem for the Galois groups of the fourth iterate of quadratic polynomials $x^2+c$, using techniques in the theory of rational points on curves. Moreover, we show that the Hall-Lang conjecture on integral points of elliptic curves implies a Serre-type finite index result for these dynamical Galois groups, and we use conjectural bounds for the Mordell curves to predict the index in the still unknown case when $f(x)=x^2+3$. Finally, we provide evidence that these curves defined by iteration have geometrical significance, as we construct a family of curves whose rational points we completely determine and whose geometrically simple Jacobians have complex multiplication and positive rank.

Area Efficient Implementation of Elliptic Curve Point Multiplication Algorithm

Elliptic Curve Cryptography (ECC) has established itself as the most preferred and secured cryptography algorithm for the secure data transfer and secure data storage in embedded system environment. Efficient implementation of point multiplication algorithm is crucial activity for designing area efficient, low footprint ECC cryptoprocessors. In this paper, an area efficient implementation of double point multiplication algorithm over binary elliptic curve is presented. Area analysis of double point multiplication algorithm based on differential addition chains method is carried out and area report is generated. Area optimization is achieved by using pipelined structure and by reutilizing idle resources from previous stages in processing unit. The proposed architecture for double point multiplication is implemented on Xilinx Virtex-4 FPGA device. Architecture is modeled in verilog-HDL and synthesized using Xilinx ISE 14.1 design software and is found to be more efficient in terms of area than the existing such architectures.

Hardware Realization of Fast Multi-Scalar Elliptic Curve Point Multiplication by Reducing the Hamming Weights Over GF(p)

We present a new hardware realization of fast elliptic curve Multi-Scalar Point Multiplication (MSPM) using the sum of products expansion of the scalars. In Elliptic curve point Multiplication latency depends on the number of one's (Hamming Weight) in the binary representation of the scalar multiplier. By reducing the effective number of one's in the multiplier, the multiplication speed is automatically increased. Therefore we describe a new method of effectively reducing the Hamming weight of the scalar multipliers thereby reduces the number of Point Adders when multi scalar multiplication is needed. The increase in speed achieved outweighs the hardware cost and complexity. Index Terms—Sum of products expansion, Hamming Weight, multi-scalar point multiplication, triple-scalar point multiplication, elliptic curve point multiplication, elliptic curve point addition.

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Elliptic curve cryptography (ECC) is an approach to public key cryptography (PKC) that is based on algebraic operations with elliptic curves defined over finite fields. Security of elliptic curve cryptography is based on the hardness of the elliptic curve discrete logarithm problem (ECDLP). Although there is no theoretical proof that ECDLP is intractable, no general-purpose sub-exponential running time algorithm has been found for solving the ECDLP if the elliptic curve parameters are chosen properly. In this study, we develop a new security attack based on the binary division of elliptic curve points over prime fields that may be used to solve the ECDLP when the order q of elliptic curve satisfies the congruence q = 2 (mod 4). To perform the binary division, we devise a novel algorithm of point halving on elliptic curves defined over prime fields that applies to the cases when q = 1 (mod 2) and q = 2 (mod 4). The binary division attack has exponential worst-case asymptotic time complexity but in certain practical cases can be used to solve the ECDLP in a relatively efficient way. We therefore make a recommendation to avoid the case of q = 2 (mod 4) in elliptic curve cryptosystems.

HARDWARE REALIZATION OF HIGH SPEED ELLIPTIC CURVE POINT MULTIPLICATION USING PRECOMPUTATION OVER GF(P)

Two new theoretical approaches for the hardware realization of high speed elliptic curve point multiplication over a prime field (GF(p)) are presented. These hardware implementations use multiple units of elliptic curve point doublers, point adders and multiplexers. The modular hardware approach used here provides high speed and scalability.

A Flexible Architecture for Modular Arithmetic Hardware Accelerators based on RNS

Modular arithmetic is a building block for a variety of applications potentially supported on embedded systems. An approach to turn modular arithmetic more efficient is to identify algorithmic modifications that would enhance the parallelization of the target arithmetic in order to exploit the properties of parallel devices and platforms. The Residue Number System (RNS) introduces data-level parallelism, enabling the parallelization even for algorithms based on modular arithmetic with several data dependencies. However, the mapping of generic algorithms to full RNS-based implementations can be complex and the utilization of suitable hardware architectures that are scalable and adaptable to different demands is required. This paper proposes and discusses an architecture with scalability features for the parallel implementation of algorithms relying on modular arithmetic fully supported by the Residue Number System (RNS). The systematic mapping of a generic modular arithmetic algorithm to the architecture is presented. It can be applied as a high level synthesis step for an Application Specific Integrated Circuit (ASIC) or Field Programmable Gate Array (FPGA) design flow targeting modular arithmetic algorithms. An implementation with the Xilinx Virtex 4 and Altera Stratix II Field Programmable Gate Array (FPGA) technologies of the modular exponentiation and Elliptic Curve (EC) point multiplication, used in the Rivest-Shamir-Adleman (RSA) and (EC) cryptographic algorithms, suggests latency results in the same order of magnitude of the fastest hardware implementations of these operations known to date.

Exponential sums over points of elliptic curves

We derive a new bound for some bilinear sums over points of an elliptic curve over a finite field. We use this bound to improve a series of previous results on various exponential sums and some arithmetic problems involving points on elliptic curves.

Stopping Sets of Algebraic Geometry Codes

Stopping sets and stopping set distribution of a linear code play an important role in the performance analysis of iterative decoding for this linear code. Let C be an [n, k] linear code over Fq with parity-check matrix H, where the rows of H may be dependent. Let [n] = {1, 2,...,n} denote the set of column indices of H. A stopping set S of C with parity-check matrix H is a subset of [n] such that the restriction of H to S does not contain a row of weight 1. The stopping set distribution {Ti(H)}i=0n enumerates the number of stopping sets with size i of C with parity-check matrix H. Denote H*, the parity-check matrix, consisting of all the nonzero codewords in the dual code C⊥. In this paper, we study stopping sets and stopping set distributions of some residue algebraic geometry (AG) codes with parity-check matrix H*. First, we give two descriptions of stopping sets of residue AG codes. For the simplest AG codes, i.e., the generalized Reed-Solomon codes, it is easy to determine all the stopping sets. Then, we consider the AG codes from elliptic curves. We use the group structure of rational points of elliptic curves to present a complete characterization of stopping sets. Then, the stopping sets, the stopping set distribution, and the stopping distance of the AG code from an elliptic curve are reduced to the search, counting, and decision versions of the subset sum problem in the group of rational points of the elliptic curve, respectively. Finally, for some special cases, we determine the stopping set distributions of the AG codes from elliptic curves.