Heres the righttoleft method with modular reductions at each step. Modular exponentiation is used in public key cryptography. This implementation is seamlessly integrated into openssl, by patching over openssl 1. Efficient implementation of modular multiplication using carry look ahead adder 1muhammad aqeel aslam. Abstract this paper presents the fpga implementation of modular exponentiation me, based on software hardware swhw approach. Modular exponentiation is composed of sequence of modular multiplications. Your method will only work if b equals 2, which is same as exponentiation by squaring but it will fail in cases with b 2. Since modular exponentiation is an additive function of the exponent similar to that of multiplier from scalar multiplication, both operations are adoptable to the idea of ac. Index terms cryptography, rsa, modular multiplication, montgomery algorithm. Efficient software implementations of modular exponentiation shay gueron abstract. If the underlying group or semigroup is commutative, then it is often possible to reduce the number of multiplications by computing the product simultaneously. So these are the two simple ways we have seen before how we can implement a to the power e mod n.
However, it is also very computationally expensive in terms of hardware implementation. Ill come back to this issue when ill discuss modular exponentiation. Modular exponentiation is a basic operation in cryptosystems. Indeed, in rivest, shamir and adleman rsa cryptosystem, me which is computed by. Fpga implementation of modular exponentiation using single.
Fpga implementation of modular exponentiation using single modular multiplier. Rsa computations have a significant effect on the workloads of ssltls servers, and therefore their software implementations on general purpose processors are an important target for optimization. The first one is the one exponentiation and modular, where we first do the exponentiation and then we do the modular. Modular exponentiation an overview sciencedirect topics. Below is the fundamental modular property that is used for efficiently computing power under modular arithmetic. You dont need to wait until the end of the computation to compute the remainder, you can do that in each step of the exponentiation. Efficient modular exponentiation architectures for rsa. The advantage of that approach is that it runs in logn time. Both operations are integrated as programmable system on chip psoc where the processor microblaze of xilinx is. The most efficient way to implement an integer based power. Efficient modular exponential algorithms compatible with. Modular exponentiation you are encouraged to solve this task according to the task description. This paper presents an implementation of rivest, shamir and adleman rsa cryptosystem based on hardware software hwsw codesign. To this end, we study here efficient software implementations of modular exponentiation, which are also protected against software side channel analyses.
There are two wellknown methods to evaluate modular exponentiation, in binary form, namely, lefttoright binary exponential method and righttoleft binary exponential method. Few years later in 1978, one of the most used publickey cryptosystems, rsa, is also. It is useful in computer science, especially in the field of publickey cryptography. Efficient software implementations of modular exponentiation efficient software implementations of modular exponentiation gueron, shay 20120405 00. Fast modular exponentiation if youre seeing this message, it means were having trouble loading external resources on our website. Let m be some odd integer modulus, a, b be two integers such that 0. The operation of modular exponentiation calculates the remainder when an integer b the base raised to the e th power the exponent, b e, is divided by a positive integer m the.
Whats much more useful is modular exponentiation, raising integers to high powers. See how we can use the fast power algorithm to find modular multiplicative inverse of a number. Efficient software implementations of modular exponentiation 3 definition 1. At a glance, the sequence \3, 2, 6, 4, 5, 1\ seems to have no order or structure whatsoever. Tabulated modular exponentiation tme algorithm for. The paper proposes optimizations in several directions. Hardware implementation of the modular exponentiation using the slidingwindow method with constantlength partitioning ieee conference publication. There are many algorithms to compute the exponentiation itself, the simplest is squareandmultiply. In fact, although there are things we can say about this sequence for example, members three elements apart add up to 7, it turns out that so little is known about the behaviour of this sequence that the following problem is difficult to solve efficiently. In this paper, we explain efficient novel modular exponentiation. Modular exponentiation power in modular arithmetic. Using the big integer implementation from a cryptographic library.
Efficient software implementations of modular exponentiation. Exponentiation by squaring may also be used to calculate the product of 2 or more powers. Computing modular exponentiation efficiently is in. Luckily, we can reuse the efficient algorithms developed in the previous article, with very few modifications to perform modular exponentiation as well. This is possible because of some convenient properties of modular arithmetic. Read and learn for free about the following article. The performance of this operations has a tremendous impact on the efficiency of the whole application. Therefore, many researchers devoted special interest to providing smart methods and efficient implementation for modular exponentiation. Most of the publickey cryptosystems use modular exponentiation in their calculation. Modular exponentiation represents a significant workload for public key cryptosystems. Examples include not only the classical rsa, dsa, and dh algorithms, but also the partially homomorphic. This implementation is seamlessly integrated into openssl. Efficient integer exponentiation algorithms march 21, 2009 at 19. The first is a modified convolution algorithm for modular multiplication while the second is a tabulated modular exponentiation tme algorithm based on the.
As most modular exponentiations do, return 1 if the exponent is 0. This section describes montgomery reduction scheme functions. Montgomery reduction is a technique for efficient implementation of modular multiplication without explicitly carrying out the classical modular reduction step. In this paper techniques for the software implementation of modular exponentiation are presented and analysed. Simple, unoptimised implementation of montgomery multiplication and modular exponentiation intended for rsa encryption and diffie hellman key exchange schemes. For this purpose, however, we need to dive into some mathematics. If youre behind a web filter, please make sure that the domains. It involves computing b to the power e mod m c efficient algorithms for this process in cryptography, the numbers involved are usually very large. It involves computing b to the power e mod m c efficient algorithms for this process to have any practical application. Also known as modular powers or modular high powers. There are other modular multiplications like barret or montgomery. For example, a typical problem related to encryption might involve solving one of the following two equations.
Compute the following exponentiations xe mod m applying the squareand multiply algorithm. Generally, the performance of this operation has a tremendous impact on the efficiency of the a hardware software codesign vs. Efficient hardware for modular exponentiation using the. Iterative sliding window method for shorter number of.
To this end, we applied our algorithm and generated an optimized avx2based software implementation of 1024bit modular exponentiation. Me implementation and vulnerability physical attacks and. In your code, you only check for digit 1, and in the case of b 7, there can be any digit in the range. In this lesson, we will see an efficient recursive algorithm to calculate xn%m x to power n modulo n prerequisite. Keywords rns multiplicative splitting digital signature fixedbase modular exponentiation scalar multiplication memory storage ef. Therefore, power is generally evaluated under modulo of a large number. Our results show that our implementation requires 51% less instructions than the current openssl 1.
This is done by fpga implementation using zedboard and comparing them with the previous methods. This section describes functions for montgomery modular reduction, montgomery modular multiplication, and montgomery modular exponentiation. When measured on the latest x8664 architecture, the 2 nd generation intel core processor, our implementation is 43 % faster than that of the current version of openssl 1. Modular exponentiation is a type of exponentiation performed over a modulus. Your string n can contain numbers in the range 0,b1 as it is the representation of number n in base b. Just as a follow up to comments on the efficiency of exponentiation by squaring. Computing modular exponentiation efficiently is inevitable for the practicability of rsa. It will be useful to develop an iterative implementation for the fast exponentiation algorithm. How to develop a defensive plan for your opensource software project.
The implementation of montgomery modular exponentiation is achieved on spartan3e, virtex4 and virtex6 series of fpgas for 4, 8, 16 and 32 bits respectively. If the modulus size is large, the algorithm is secure. The method is highly efficient in implementation due to its minimal bookkeeping in the process knuth, 1998. The significant cost of rsa computations affects the efficiency and responsiveness of ssltls servers, and therefore software implementations of rsa are an important target for optimization. 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. At the exponentiation level, we propose a method to reduce the cost of protecting the wary exponentiation algorithm against cachetiming side channel attacks. In order to reduce the computation time of algorithm, the modular exponentiation complexity is reduced. In both of the aforementioned algorithms, the frequency of modular multiplications to. Implementation of efficient modular exponentiation on reconfigurable platforms, august 2008, kashif latif, department of. We use exactly the same algorithm, but reduce every multiplication. Efficient psoc implementation of modular multiplication.
And we are going to sh, show that in such efficient implementation whe, whether therell be any security leak vulnerabilities. The main operation of rsa is the modular exponentiation me which is performed by repeated modular multiplications mms. Shay gueron 1, 2 1 department of mathematics, university of haifa, israel. Software implementation of modular exponentiation, using. So what well learn today is, were going to learn the definition of modular exponentiation. It is a very popular operation in modern cryptography. Together, these lead to an efficient software implementation of 512bit modular exponentiation, which outperforms the currently fastest publicly available alternative. Efficient implementation of modular multiplication using. Modular exponentiation me basics physical attacks and. Modular exponentiation is an essential operations for various applications, such as cryptography.
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