Intern. J. Computer Math., 2002, Vol. 79(7), pp. 797–806 APPLICATION OF PARALLEL VIRTUAL MACHINE FRAMEWORK TO THE STRONG PRIME PROBLEM DER-CHUYAN LOU,* CHIA-LONG WU and RONG-YI OU Department of Electrical Engineering, Chung Cheng Institute of Technology, National Defense University, Tahsi, Taoyuan 33509, Taiwan (Received 9 April 2001) This paper use the well-discussed PVM (Parallel Virtual Machine) software with several personal computers, and adopt the widespread Microsoft Windows ‘98 operating system as our operation platform to construct a heterogeneous PCs cluster. By engaging the related researches of PC cluster system and cluster computing theory, we apply our heterogeneous PC cluster computing system to generate more secure parameters for some public key cryptosystems such as RSA. Copes with each parameter’s related mathematic theory’s restriction, enormous computation power is needed to get better computation performance in generating these parameters. In this paper, we contribute heterogeneous PCs combined with the PVM software to cryptosystem parameters, which is conformed to today’s safety specification and requirement. We practically generate these data to prove that computer cluster can effectively accumulate enormous computation power, and then demonstrate the cluster computation application in finding strong primes which are needed in some public key cryptosystems. Keywords: Keywords:#Parallel virtual machine; Cluster computing; Cryptography; Primality test; Strong prime C.R. Categories: C.R. Categories: E.3, F.1.2, I.1.1.2, K.6.5 1. INTRODUCTION In this section, the Parallel Virtual Machine (PVM) system that is based on the message- passing model will be introduced. Message-passing parallel programming can be considered and designed among those different machines for our integrated system based on their unique information and data format, and allow different machines make communication. Based on this property, we can have PVM [1, 2] connect through different working platforms to each other, combine them as one virtual machine with strong operation power, even each machine might has its different specification, this also specifies how the name ‘‘PVM’’ comes from. In 1989, a parallel computation program called PVM is proceeded in Oak Ridge National Lab [3]. This project was expected to offer a parallel computing environment with heteroge- neous and general properties, which not only can support multi-party protocols effectively but also can be adapted to the distributed computation algorithm. Although the PVM was motioned as the most popular distributed computation operation system in 1992, and has most of the user population, it doesn’t necessarily means PVM can finish all jobs automati- *Corresponding Author. Fax: 886-3-3801407; E-mail: [email protected] ISSN 0020-7160 print; ISSN 1029-0265 online # 2002 Taylor & Francis Ltd DOI: 10.1080=00207160290029228 798 D.-C. LOU et al. cally. PVM [4] can only provides an environment that makes the parallel program executable. Program designers must depend on their manual processes and clearly specify those program instructions where the parallel computation task is needed. PVM does not have the ability to distribute the instruction and data automatically. That means, it does not offer the automatic parallel mechanism. PVM provides for a software environment for message passing between homogeneous computers. In PVM main design program, users must define all the parallel procedures and they must understand the fact that even though PVM is a parallel computation interface, but all the controlling main programs are still controlled by sequential pattern. Its proceeding control can let PVM process be interrupted and become an Unix or a Window 32 procedure (which doesn’t have the parallel capability), or become a PVM procedure in general process. In general speaking, PVM is still a sequential control procedure. In this paper, we utilize the well-discussed PVM software that uses message-passing model as interface, accompanied with our personal computers and windows operating system Window’98 to build an experimental personal computer cluster. The PVM software can constructs a framework through different computer platforms. Different computers are used in this paper to construct a powerful computation virtual machine to satisfy the computer cryptosystem requirement that is urging the computation power. In this paper, we use three different rank’s PCs to demonstrate the heterogeneous property and to show homely personal-computers can also accumulate adequate computation power in solving the strong prime problem. Here are these computers’ specifications shown as Table I. The rest of the paper is organized as follows. Section 2 has focus on the strong prime problem and the bottleneck of the RSA public-key cryptosystem as well as the popular ‘‘cluster computing’’ topic. In Section 3, we then introduce and discuss several different theo- rems for primality test. Section 4 and Section 5 we here have demonstrated our experimental design and experimental performance results using primality test algorithms for RSA public- key. Finally, we put our research contribution and future work aspect in Section 6 as our conclusion. 2. THE STRONG PRIME PROBLEM As we know number theory has play an important role in the public-key cryptographic system [5]. Prime number is an essential issue in number theory. It has been well discussed to construct the strong prime as the mainly secure parameter in some the public-key crypto- systems. Here we will discuss the RSA public-key cryptosystem and its bottleneck as well as the strong prime number problem, next we concentrate on the cluster computing and PVM system concepts. 2.1. Bottleneck of the RSA Public-key Cryptosystem In 1978, three MIT professors: Rivest, Shamir, and Adleman brought the public-key crypto- system using security-based modular exponential function with complex factoring large prime numbers difficulties, is what people known the RSA public-key cryptosystem [6]. TABLE I System specifications Name Specification D-Celeron CPU: Celeron-450  2, Memory 128 MB Celeron CPU: Celeron-300, Memory 64 MB Pentium CPU: Pentium-75, Memory 48 MB CLUSTER COMPUTING PRIME NUMBERS 799 The RSA algorithm is widely used in public-key cryptosystems [7]. Public-key crypto- system, though to some extent advantages, still its disadvantages does exist. Especially in encryption=decryption operations respect, these operation processes are quite complex, enor- mous operation capability is needed. Comparing the RSA public-key cryptosystem with the DES (Data Encryption Standard) secret-key cryptosystem. The DES hardware chip can reach the speed with approximately 45 Mega bits per second, while the RSA cryptosystem only has 50 Kilo bits per second, there is approximate 1000 times difference, enough to specify the bottleneck of the RSA public-key cryptosystem. Nowadays, the DES cryptosystem is no longer secure and its major safety concern is coming from the Wiener’s [8] assumption (based on a known plaintext attack). Because these systems are vulnerable to a shortcut attack, they must use key sizes substan- tially greater than those required for comparable levels of security with traditional single-key methods. The AES [9] now has its secret-key length extended to 128 192 bits, the RSA cryptosystem is also being recommended to extend its public key from 512 bits to 1024 bits to keep its safety, therefore the computation capability we need to have is then enor- mously increased. 2.2. Strong Prime Number The RSA cryptosystem is a block cipher that will process the input one block of elements at a time and produce an output block for each input block. Plaintext is encrypted in blocks, and every binary value in each block is no greater than some number N. Assume we have two given prime numbers p and q, such that N can be calculated as N ¼ pq. By using the Euler’s theorem, we can then have fð pqÞ¼ðp À 1Þðq À 1Þ and d eÀ1 mod fðnÞ: That is ed is of the form ed ¼ kfðnÞþ1; therefore ed 1 mod fðnÞ: According to the statement shown above, we can understand the RSA cryptosystem is build its security-based property on the complexity of the factorization problem. It is obliv- ious that for in the public key (e, N) of the RSA cryptosystem, if N can be successfully factor- ized by factor p or q, then the trapdoor T ¼ fðNÞ¼ðp À 1Þðq À 1Þ and decryption key d which are the decryption process depending on is no place to hide. Therefore, the decryption key d can no longer keeps itself as a ‘‘secret’’ key, that means, ‘‘there exist no security’’ what- soever. Although it is not yet ‘‘identify’’ or ‘‘prove’’ the difficulty of how to break the RSA public key cryptosystem is as same as the effort of how we factorize the number N, but in general it is ‘‘believed’’ that the difficulty of breakdown the RSA cryptosystem is equal to factorize the number N. Therefore, for the RSA cryptosystem, how to choose its parameters should be considered most prudently and carefully. Since the RSA cryptosystem build its security-based property on the complexity of breaking down number N, the prime factors of N should satisfy the property of strong prime to assure that: it is computationally infeasible. The strong prime property is introduced as follows. r1, s1, r2, s2 be four extreme large prime numbers, we call them as ‘‘simple primes’’. Let xjy demote y is divisible by x.Ifwe have r1jp1 À 1; s1jp1 þ 1; r2jp2 À 1; s2jp2 þ 1; such p1, p2, we call them as ‘‘complex primes’’. To process these assemble steps furthermore we can have p1jp À 1; p2jp þ 1; then we can get p as so called ‘‘strong prime’’ [10]. The structure of a strong prime is shown as Figure 1. It is truly oblivious that any general prime number can also be called as simple prime.