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With the fast development of cloud computing, clients without enough computational power can widely outsource their heavy computations to cloud service providers. One of the most widely used and costly operations in cryptographic protocols is modular exponentiation, which can be computed at a lower cost by enjoying advantages of cloud computing, however, at the same time we need to address new challenges such as data privacy and verification of results. In this paper, first, we propose a secure outsourcing of single modular exponentiation protocol with verifiability one. Although the proposed single exponentiation scheme has the same verifiability as Ren’2018, but our scheme requires one less modular multiplication. However, the main contribution of this paper is proposing a scheme for outsourcing of multiplications of several modular exponentiations, hereafter called as composite exponentiation, which to the best of our knowledge, and is the first outsourcing scheme with full verification for composite exponentiation. As the evaluation results show, the advantages of this scheme, in comparison with state of the art schemes, are evident in terms of performance and verifiability criteria. [ABSTRACT FROM AUTHOR]

Side-Channel Attacks target the recovery of key material in cryptographic implementations by measuring physical quantities such as power consumption during the execution of a program. Simple Power Attacks consist in deducing secret information from a trace using a single or a few samples, as opposed to differential attacks which require many traces. Software cryptographic implementations usually contain a data-independent execution path, but often do not consider variations in power consumption associated to data. In this work, we show that a technique commonly used to select a value from different possible values in a control-independant way leads to significant power differences depending on the value selected. This difference is actually so important that a single sample can be considered for attacking one condition, and no training on other traces is required. We exploit this finding to propose a single-trace attack without any knowledge gained on previous executions, using trace folding. We target the two modular exponentiation implementations in Libgcrypt, getting respectively 100% and 99.998% correct bits in average on 30 executions using 2,048-bit exponents. We also use this technique to attack the scalar multiplication in ECDSA, successfully recovering all secret nonces on 1,000 executions. Finally, the insights we gained from this work allow us to show that a proposed countermeasure from the literature for performing the safe loading of precomputed operands in the context of windowed implementations can be attacked as well.

Let $ N = \left\{{n}_{1}, {n}_{2}, \;\dots , {n}_{k}\right\} $ be a finite set of positive numbers. The problem of finding the minimal number of additions required to compute all elements of N starting from 1 (called the addition sequence problem) is NP-complete. It is equivalent to finding the minimum number of multiplications needed to compute a group exponentiation $ {g}^{{n}_{1}}, {g}^{{n}_{2}}, \;\dots , {g}^{{n}_{k}} $, where g is an element in a group. This paper aims to propose a new metaheuristic algorithm using a simulated annealing strategy to generate a short addition sequence. The performance of the proposed algorithm is measured by considering two parameters: The size of N and the domain of $ {n}_{\mathrm{i}}, 1\le i\le k $. The proposed algorithm is a new trade-off between the length of the generated addition sequence and the average running time of generating addition sequences. It sometimes produces longer addition sequences than exact algorithms that are slower, and it is slower than suboptimal algorithms that produce longer addition sequences.

Let N = {n1, n2, ..., nk} be a finite set of positive numbers. The problem of finding the minimal number of additions required to compute all elements of N starting from 1 (called the addition sequence problem) is NP-complete. It is equivalent to finding the minimum number of multiplications needed to compute a group exponentiation gn1, gn2, ..., gnk, where g is an element in a group. This paper aims to propose a new metaheuristic algorithm using a simulated annealing strategy to generate a short addition sequence. The performance of the proposed algorithm is measured by considering two parameters: The size of N and the domain of ni,1 ≤ i ≤ k. The proposed algorithm is a new trade-off between the length of the generated addition sequence and the average running time of generating addition sequences. It sometimes produces longer addition sequences than exact algorithms that are slower, and it is slower than suboptimal algorithms that produce longer addition sequences. [ABSTRACT FROM AUTHOR]

Modular exponentiation is crucial for secure data exchange in cryptography, especially for resource-constrained Internet of Things (IoT) devices. These devices often rely on third-party servers to handle computationally intensive tasks like modular exponentiation. However, existing outsourcing solutions for the RSA algorithm may have security vulnerabilities. This work identifies a critical flaw in a recent outsourcing protocol for RSA proposed by Hu et al. We demonstrate how this flaw compromises the security of the entire RSA system. Subsequently, we propose a robust solution that strengthens the RSA algorithm and mitigates the identified vulnerability. Furthermore, our solution remains resilient against existing lattice-based attacks. The proposed fix offers a more secure and efficient way for IoT devices to leverage the power of third-party servers while maintaining data integrity and confidentiality. An extensive performance evaluation confirms that our solution offers comparable efficiency while significantly enhancing security compared to existing approaches. [ABSTRACT FROM AUTHOR]

Evolutionary computation techniques have always provided fascinating results in all the fields of science and engineering. However in the area of computer security, their contribution has been comparatively very less. More specifically if we consider the side-channel attacks, use of these nature based techniques have been very nominal. Therefore, we proposed a secure protocol in this paper to combat against the Higher Order Differential Power Analysis attacks on modular exponentiation based cryptosystems using one of the popular evolutionary computation techniques. The proposed work first uses Genetic Algorithm for splitting the huge exponent within the modular exponentiation into multiple non-uniform shares. Then, this shares are randomly chosen for computing individual modular exponentiation with the help of nearest neighbor algorithm. Using Genetic Algorithm, our proposed protocol can generate reasonable number of shares which exposes secret exponent at the least. As a result, it provides significant resistance to Higher Order Differential Power Analysis attacks. Moreover, randomization in computing individual modular exponentiation secures the cryptosystem from generic power analysis attacks like SPA and DPA.

With the development of the Internet of Things, smart grids have become indispensable in our daily life and can provide people with reliable electricity generation, transmission, distribution and control. Therefore, how to design a privacy-preserving data aggregation protocol has been a research hot-spot in smart grid technology. However, these proposed protocols often contain some complex cryptographic operations, which are not suitable for resource-constrained smart meter devices. In this paper, we combine data aggregation and the outsourcing of computations to design two privacy-preserving outsourcing algorithms for the modular exponentiation operations involved in the multi-dimensional data aggregation, which can allow these smart meter devices to delegate complex computation tasks to nearby servers for computing. By utilizing our proposed outsourcing algorithms, the computational overhead of resource-constrained smart meter devices can be greatly reduced in the process of data encryption and aggregation. In addition, the proposed algorithms can protect the input's privacy of smart meter devices and ensure that the smart meter devices can verify the correctness of results from the server with a very small computational cost. From three aspects, including security, verifiability and efficiency, we give a detailed analysis about our proposed algorithms. Finally, through carrying out some experiments, we prove that our algorithms can improve the efficiency of performing the data encryption and aggregation on the smart meter device side. [ABSTRACT FROM AUTHOR]

With the advancements of communication technology, security threats are also increasing. RSA is a robust cryptosystem to protect classified information. The efficiency of RSA cryptosystem depends on the efficient execution of Montgomery modular multiplication. A low latency and throughput efficient Montgomery modular multiplier is proposed in this paper. The input and final output of this multiplier are binary, but intermediate input and output are carry-saved i.e. sum and carry bits are stored in separate registers. Sum and carry bits are obtained from the addition of operands using carry-save adder (CSA). Montgomery multipliers require quotient calculation for subsequent iteration. The proposed multiplier computes two subsequent quotients in parallel with carry-save addition. Output of carry-save addition is right shifted by two bits to obtain the next-to-next intermediate output. This calculation also requires integer multiples of the inputs. Computation of next output is skipped. Format conversion of output is performed using a carry look-ahead unit (CLU). CLU along with one cycle of carry save addition is also utilized for pre-computation of integer multiples of inputs. The proposed multiplier is implemented on NEXYS4DDR and VIRTEX VII FPGA and the following result is obtained. NEXYS4DDR: Area (LUT + REG): 38903, Delay: 5.84 ns, Cycles: 1158, Latency: 6.76 µs, Throughput: 302.96 Mbps. VIRTEX VII: Area (LUT): 16447, Delay: 1.62 ns, Cycles: 1158, Latency: 1.88 µs, Throughput: 1089.4 Mbps. Results verify the improved area, latency and throughput performance of the proposed work.

Copyright of Radio Electronics, Computer Science, Control is the property of Zaporizhzhia National Technical University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

With the growing popularity of cloud computing in recent years, data owners (DOs) now prefer to outsource their data to cloud servers and allow the specific data users (DUs) to retrieve the data. Searchable encryption is an important tool to provide secure search over the encrypted cloud data without infringing data confidentiality and data privacy. In this work, we consider a secure search service providing fine-grained and search functionality, called attribute-based multiple keyword search (ABMKS), which can be seen as an extension of searchable encryption. In the existing ABMKS schemes, the computation operations in the encrypted keyword index generation are time-consuming modular exponentiation, and the number of which is linearly growing with the factor $m$ . Here $m$ is the number of keywords embedded in a file. To reduce the computation overhead, in this paper, we propose an ABMKS with only multiplication operations in encrypted keyword index generation. As a result, the computation cost of the encrypted keyword index generation is more efficient than the existing schemes. In addition, the encrypted keyword indexes are aggregated into one item, which is regardless of the number of underlying keywords in a file data. Finally, the security and the performance analysis demonstrate that our scheme is both efficient and secure.

Edge computing provides users with nearby services, promotes the development of the Internet of Things and transforms the mode of traditional electronic medical system that improves the speed and accuracy of medical diagnosis. However, edge computing has introduced new security challenges, such as the difficulty in applying standard encryption algorithms because of the limited computing capability of devices on the edge and concerns of privacy leakage. In this article, we propose a data access scheme by employing a lightweight attribute-based encryption (ABE) to address the security weaknesses of edge computing for e-health. Our novel modular exponential outsourcing algorithm outsources the shared data to the untrusted edge server and verifies the correctness of the returned results while achieving a lightweight computational load. The proposed Mask Algorithm along with blind pairs conceals the bases and exponents of the encrypted data, thus preserving the privacy of sensitive data in the process of outsourced computing. Security analysis and performance evaluation confirmed the security and efficiency of the proposed method.

The aim of this paper is to propose the alternative algorithm to finish the process in public key cryptography. In general, the proposed method can be selected to finish both of modular exponentiation and point multiplication. Although this method is not the best method in all cases, it may be the most efficient method when the condition responds well to this approach. Assuming that the binary system of the exponent or the multiplier is considered and it is divided into groups, the binary system is in excellent condition when the number of groups is small. Each group is generated from a number of 0 that is adjacent to each other. The main idea behind the proposed method is to convert the exponent or the multiplier as the subtraction between two integers. For these integers, it is impossible that the bit which is equal to 1 will be assigned in the same position. The experiment is split into two sections. The first section is an experiment to examine the modular exponentiation. The results demonstrate that the cost of completing the modular multiplication is decreased if the number of groups is very small. In tables 7 – 9, four modular multiplications are required when there is one group, although number of bits which are equal to 0 in each table is different. The second component is the experiment to examine the point multiplication process in Elliptic Curves Cryptography. The findings demonstrate that if the number of groups is small, the costs to compute point additions are low. In tables 10 – 12, assigning one group is appeared, number of point addition is one when the multiplier of a point is an even number. However, three-point additions are required when the multiplier is an odd number. As a result, the proposed method is an alternative way that should be used when the number of groups is minimal in order to save the costs. [ABSTRACT FROM AUTHOR]

Copyright of Radio Electronics, Computer Science, Control is the property of Zaporizhzhia National Technical University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper presents a secured computation of modular exponentiation to resist higher-order differential power analysis (HODPA) attacks in asymmetric cryptosystems like RSA. HODPA attacks can be resisted by segmenting secret sensitive data and its intermediate values into multiple shares. In modular exponentiation-based cryptosystems, the exponent plays a significant part in the secret key. We have used inner product with differential evolution algorithm to segment the exponent into multiple shares. Using entropy-based nearest neighbor algorithm, we have randomly computed independent modular exponentiation to resist SPA and DPA attacks. Analysis was done on 1024, 1536 and 2048 bit RSA. With a pre-computation complexity, the proposed approach can provide significant resistance to SPA, DPA and HODPA attacks against modular exponentiation-based cryptosystems. [ABSTRACT FROM AUTHOR]

Attribute-based encryption (ABE) is a promising cryptographic tool for data owner (DO) to realize fine-grained date sharing in the cloud computing. In the encryption of most existing ABE schemes, a substantial number of modular exponentiations are often required; the computational cost of it is growing linearly with the complexity of the access policy. Besides, in the most existing ABE with outsourced decryption, the computation cost of generating transformation key is growing linearly with the number of attributes associated with user private key; these computations are prohibitively high for mobile device users, which becomes a bottleneck limiting its application. To address the above issues, we propose a secure outsourcing algorithm for modular exponentiation in one single untrusted server model and a new method to generate the transformation key. Based on these techniques and Brent Waters's ciphertext-policy ABE scheme, we propose an ABE scheme with verifiable outsourced both encryption and decryption, which can securely outsource encryption and decryption to untrusted encryption service provider (ESP) and decryption service provider (DSP), respectively, leaving only a constant number of simple operations for the DO and eligible users to perform locally. In addition, both DO and the eligible users can check the correctness of results returned from the ESP and the DSP with a probability, respectively. Finally, we provide the experimental evaluation and security analysis of our scheme, which indicates that our construction is suitable for the mobile environment.

This paper presents an approach to resist modular exponentiation-based cryptosystems like RSA from higher-order DPA attacks. The best way to prevent higher-order DPA attacks is by splitting the sensitive data into independent shares and execute them individually. We have incorporated the concept of inner product to split the large exponent or secret key in modular exponentiation into multiple shares. With these independent shares, individual modular exponentiation has been computed using multiply-always binary method. These shares have also been arithmetically blinded with a 32-bit blinding factor prior to computation. The entire analysis has been executed in RSA of sizes: 1024 bits, 1536 bits and 2048 bits. The results show that the proposed work can provide significant resistance to SPA, DPA and higher-order DPA attacks on modular exponentiation-based cryptosystems. [ABSTRACT FROM AUTHOR]

At present,in discrete-log based cryptographic protocols,most of the computational models of modular exponentiation are for primes,while less work has been done for composite.What’s more,most schemes are based on two servers,in which it requires at least one server to be honest.However,there may not be a fully trusted cloud server in the actual environment.Then a new secure method for outsourcing exponentiation modular a composite which based on a single server was proposed.The scheme used a new mathematical division method,it could ensure the privacy of the base and exponentiation.Compared with the existing schemes,the checkability of our scheme can be greatly improved.The user can detect the error result with the probability of 119/120.

Outsourcing computing allows users with resource-constrained devices to outsource their complex computation workloads to cloud servers that may not be honest. In this paper, we propose a new algorithm for securing the outsourcing of composite modular exponentiation, which is one of the most complex computing tasks in discrete-log based cryptographic protocols. Unlike algorithms based on two untrusted servers, we outsource modular exponentiation operation to only a single server, which eliminates the potential for a collusion attack when using two servers. Moreover, our proposed algorithm can hide the base and exponent of the outsourced data, which prevents the exposure of sensitive information to cloud servers. In addition, compared with the state-of-the-art algorithms, our scheme has remarkably better checkability. The user could detect any misbehavior with a probability of one if the server returns a fault result.

Several ways to perform data encryption have been found, and one of the functions involved in standard algorithms such as RSA is the modular exponentiation. Basically, the RSA algorithm uses some properties of modular arithmetic to cipher and decipher plain text, with a certain performance dependence on text lengths. The growth in computing capacity has created the need to use robust systems that can perform calculations with significantly large numbers and the formulation of procedures focused on improving the speed to achieve it. One of these is the M-ary algorithm for the execution of the modular exponential function. This paper describes an implementation of this algorithm in reprogrammable hardware (FPGA) to evaluate its performance. The first section of this work introduces the M-ary algorithm. The second section uses block description for implementation understanding. The third section shows the results in time diagrams, and finally, the last section conclusions.

This paper presents an FPGA implementation of the most critical operations of Public Key Cryptography (PKC), namely the Modular Exponentiation (ME) and the Modular Multiplication (MM). Both operations are integrated as Programmable System on Chip (PSoC) where the processor Microblaze of Xilinx is used for flexibility. Our objective is to achieve a best trade-off between time execution, occupied area and flexibility. The implementation of these operations on such environment requires taking into account several criteria. Indeed, the Hardware (HW) architectures data bus should be smaller than the input data length. The design must be scalable to support different security levels. The implementation achieves optimums execution time and HW resources number. In order to satisfy these constraints, Montgomery Power Ladder (MPL) and Montgomery Modular Multiplication (MMM) algorithms are utilized for the ME and the MM implementations as HW accelerators, respectively. Our implementation approach is based on the digit-serial method for performing the basic arithmetic operations. Efficient parallel and pipeline strategies are developed at the digit level for the optimization of the execution times. The application for 1024-bits data length shows that the MMM run in 6.24 μ s and requires 647 slices. The ME is executed in 6.75 ms using 2881 slices. [ABSTRACT FROM AUTHOR]

Efficient hardware implementations of public-key cryptosystems have been gaining interest in the past few decades. To achieve the goal, a high frequency as well as low latency Rivest-Shamir-Adleman (RSA) cryptosystem is reported in this paper. To configure such cryptosystem shift-add multiplier have been re-constructed and binary digit based modular exponentiation circuitry is proposed. Such exponentiation circuitry has been implemented through binary bit distribution technique, where, most significant bit (MSB) has been discarded for the implementation, owing to increase the operating frequency. The functionality of the reported algorithms were justified and compared in Hardware Description Language (HDL), simulated in Modelsim and synthesized in Xilinx ISE 14.2 platform. The proposed hardware implementation of RSA algorithm has a maximum frequency of operation of 545 MHz and 298 MHz for the bit sizes of 8 and 64 respectively. The proposed method shows improvements in terms of speed as well as in number of Look-up-tables (LUTs). Moreover, application-specific integrated circuit (ASIC) implementation of such cryptosystem of RSA was carried out through Encounter® RTL Compiler v11.10-p005_1 of Cadence® tool. [ABSTRACT FROM AUTHOR]

Public cloud storage auditing allows a file owner or a public verifier to conduct integrity checking without downloading the whole file from cloud server. Plenty of unaffordable modular exponentiations for resource-constraint devices are required on the client side in the process of signature generation for public cloud storage auditing. In this paper, we introduce a secure outsourcing algorithm to make cloud storage public auditing more feasible and practical for the computationally limited clients. Compared with the existing algorithms for secure outsourcing of modular exponentiations, the proposed algorithm designed for this scenario takes advantage of that the bases of modular exponentiations do not need to be protected. Therefore, our algorithm is efficient. Furthermore, the evaluation shows that it has high efficiency and checkability as well. Specifically, the client-side cost is reduced to 5.6% of the original computation cost, and the probability that the client detects the misbehavior of server is close to 1. [ABSTRACT FROM AUTHOR]

Modular multiplication forms the basis of many cryptographic functions such as RSA, Diffie-Hellman key exchange, and ElGamal encryption. For large RSA moduli, combining the fast Fourier transform (FFT) with McLaughlin's Montgomery modular multiplication (MLM) has been validated to offer cost-effective implementation results. However, the conditional selections in McLaughlin's algorithm are considered to be inefficient and vulnerable to timing attacks, since extra long additions or subtractions may take place and the running time of MLM varies. In this work, we restrict the parameters of MLM by a set of new bounds and present a modified MLM algorithm involving no conditional selection. Compared to the original MLM algorithm, we inhibit extra operations caused by the conditional selections and accomplish constant running time for modular multiplications with different inputs. As a result, we improve both area-time efficiency and security against timing attacks. Based on the proposed algorithm, efficient FFT-based modular multiplication and exponentiation are derived. Exponentiation architectures with dual FFT-based multipliers are designed obtaining area-latency efficient solutions. The results show that our work offers a better efficiency compared to the state-of-the-art works from and above 2048-bit operand sizes. For single FFT-based modular multiplication, we have achieved constant running time and obtained area-latency efficiency improvements up to 24.3 percent for 1,024-bit and 35.5 percent for 4,096-bit operands, respectively. [ABSTRACT FROM AUTHOR]

Constant-time technique is of crucial importance to prevent secrets of cryptographic algorithms from leakage by cache attacks. In this paper, we propose Permute-Scatter-Gather, a novel constant-time method for the modular exponentiation that is used in the RSA cryptosystem. On the basis of the scatter-gather design, our method utilizes pseudo-random permutation to obfuscate memory access patterns. Based on this strategy, the resistance against fine-grained cache attacks is ensured, i.e., providing the higher level of security than the existing scatter-gather implementations. Evaluation shows that our method outperforms the OpenSSL library at most 11% in the mainstream Intel processors. [ABSTRACT FROM AUTHOR]

Modular multiplication (MM) is the main operation in cryptography algorithms such as elliptic‐curve cryptography (ECC) and Rivest–Shamir–Adleman, where repeated MM is used to perform elliptic curve point multiplication and modular exponentiation, respectively. The algorithm for the proposed architecture is derived from the Chinese remainder theorem and performs MM completely within a residue number system (RNS). Moreover, a 40‐channel RNS moduli‐set is proposed for this architecture to benefit from the short‐channel width of the RNS moduli‐set. The throughput of the architecture is enhanced by pipelining and pre‐computations. The proposed architecture is fabricated as an ASIC using 65‐nm CMOS technology. The measurement results are obtained for energy dissipation at different voltage levels from 0.43 to 1.25 V. The maximum throughput of the proposed design is 1037 Mbps while operating at a frequency of 162 MHz with an energy dissipation of 48 nJ. The proposed architecture enables the construction of low‐voltage and energy‐efficient ECCs. [ABSTRACT FROM AUTHOR]

Copyright of Dyna is the property of Universidad Nacional de Colombia, Medellin, Facultad de Minas and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The Montgomery algorithm is widely used to reduce the computational complexity of large integer modular exponentiation. The SPA (simple power analysis) attacks against public-key cryptosystems based on Montgomery algo-rithm implementation were presented by exploitation of the inherent security vulnerability which that sensitive informa-tion leakage could be used by side-channel attack. The chosen-message SPA attacks were focused on, which enhance the differences of operating wave-forms between multiplication and squaring correlated to the secret key by using the input of particular messages. In particular, a SPA attack against RSA cryptosystem was showed based on large integer modular exponentiation. The results show that the attack algorithm is correct and effective.

Cryptography via public key cryptosystems (PKC) has been widely used for providing services such as confality, authentication, integrity and non-repudiation. Other than security, computational efficiency is another major issue of concern. And for PKC, it is largely controlled by either modular exponentiation or scalar multiplication operations such that found in RSA and elliptic curve cryptosystem (ECC), respectively. One approach to address this operational problem is via concept of addition chain (AC), in which the exhaustive single operation involving large integer is reduced into a sequence of operations consisting of simple multiplications or additions. Existing techniques manipulate the representation of integer into binary and m-ary prior performing the series of operations. This paper proposes an iterative variant of sliding window method (SWM) form of m-ary family, for shorter sequence of multiplications corresponding to the modular exponentiation. Thus, it is called an iterative SWM. Moreover, specific for ECC that imposes no extra resource for point negation, the paper proposes an iterative recoded SWM, operating on integers recoded using a modified non-adjacent form (NAF) for speeding up the scalar multiplication. The relative behaviour is also examined, of number of additions in scalar multiplications, with the integers hamming weight. The proposed iterative SWM methods reduce the number of operations by up to 6% than the standard SWM heuristic. They result to even shorter chains of operations than ones returned by many metaheuristic algorithms for the AC.

Constant-time technique is of crucial importance to prevent secrets of cryptographic algorithms from leakage by cache attacks. In this paper, we propose Permute-Scatter-Gather, a novel constant-time method for the modular exponentiation that is used in the RSA cryptosystem. On the basis of the scatter-gather design, our method utilizes pseudo-random permutation to obfuscate memory access patterns. Based on this strategy, the resistance against fine-grained cache attacks is ensured, i.e., providing the higher level of security than the existing scatter-gather implementations. Evaluation shows that our method outperforms the OpenSSL library at most 11% in the mainstream Intel processors.

In this work we re-examine a modular multiplication and a modular exponentiation method. The multiplication method, proposed by Hayashi in 1998, uses knowledge of the factorization of both N + 1 and N + 2 to compute a multiplication modulo N. If both N + 1 and N + 2 can be factored into k equally sized relatively prime factors then the computations are done modulo each of the factors and then combined using the Chinese Remainder Theorem. It was suggested that the (asymptotic) computational costs of the method is 1/k of simply multiplying and reducing modulo N. We show, however, that the computational costs of the method is (asymptotically) at least as costly as simply multiplying and reducing modulo N for both squarings and general multiplications when efficient arithmetic is used. The exponentiation method, proposed by Hwang, Su, Yeh and Chen in 2005, is based on Hayashi's method and uses knowledge of the factorization of P + 1 and P – 1 to compute an exponentiation modulo an odd prime P. We begin by showing that the method cannot be used as a general purpose exponentiation method and then modify the method so that it can work as a general purpose modular multiplication method. Like Hayashi's method, however, this method is at best (asymptotically) only as efficient as simply multiplying and reducing modulo P.

With the development of cloud computing, more and more mobile devices have the requirement to outsource expensive computations in an untrusted environment. Among all such computations, exponentiations modulo a large prime are basic and frequent operations in many discrete-logarithm-based cryptographic protocols. Currently the most efficient outsourcing algorithm of modular exponentiations is under a one-malicious version of a two-untrusted-program assumption. And the result of the algorithm is checkable with a probability at most 2/3. We here propose an efficient algorithm under a two-untrusted- program assumption and improve the probability of checkability to be about l-2/3s for a suitable s configured by a client. Further, we provide practical results to show the energy saving about outsourcing modular exponentiations. [ABSTRACT FROM AUTHOR]