Long Modular Multiplication for Cryptographic Applications

Hars, Laszlo

doi:10.1007/978-3-540-28632-5_4

Cited by 18 publications

(12 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In [12], Hars proposes a long modular multiplication method that also simplifies an intermediate quotient evaluation. The method is based on Quisquater's algorithm and requires a preprocessing of the modulus by increasing its length.…”

Section: Related Workmentioning

confidence: 99%

Faster Interleaved Modular Multiplication Based on Barrett and Montgomery Reduction Methods

Knežević

Vercauteren

Verbauwhede

2010

IEEE Trans. Comput.

View full text Add to dashboard Cite

This paper proposes two improved interleaved modular multiplication algorithms based on Barrett and Montgomery modular reduction. The algorithms are simple and especially suitable for hardware implementations. Four large sets of moduli for which the proposed methods apply are given and analyzed from a security point of view. By considering state-of-the-art attacks on public-key cryptosystems, we show that the proposed sets are safe to use, in practice, for both elliptic curve cryptography and RSA cryptosystems. We propose a hardware architecture for the modular multiplier that is based on our methods. The results show that concerning the speed, our proposed architecture outperforms the modular multiplier based on standard modular multiplication by more than 50 percent. Additionally, our design consumes less area compared to the standard solutions.

show abstract

Section: Related Workmentioning

confidence: 99%

Faster Interleaved Modular Multiplication Based on Barrett and Montgomery Reduction Methods

Knežević

Vercauteren

Verbauwhede

2010

IEEE Trans. Comput.

View full text Add to dashboard Cite

show abstract

“…In contrast to the classical modular multiplication algorithm [1], contemporary algorithms [3], [6], do not use the operation of division, which is ineffectively realized on the general purpose processors. Based on this, as a criterion of the productivity evaluation of the software implementation of modular multiplication algorithms, the total operation time of multiplication and addition, which are the basic operations of the contemporary algorithms of modular multiplication, is usually examined [5].…”

Section: Principal Notations and Effectiveness Estimation Model Of Thmentioning

confidence: 99%

“…Since the division of the n-bits numbers on the k-bits processor (n>>k) is carried out very ineffectively, the calculation of the reduction in the classical algorithm requires s(s+2 5) operations of multiplication and s operations of the integer division [3].…”

Section: Principal Notations and Effectiveness Estimation Model Of Thmentioning

confidence: 99%

“…At present, various algorithms are proposed [2], [3], [5], which increase the performance of the software implementation of the modular multiplication operation. The largest part of the aforementioned algorithms realize the increase in the performance of modular multiplication due to the acceleration of modular reduction by the exception of the operation of integer division, which is used in the classical algorithm [1].…”

Section: Principal Notations and Effectiveness Estimation Model Of Thmentioning

confidence: 99%

“…Table 1 gives the quantity of the multiplication operations and the word divisions, which are utilized in the most known modular multiplication algorithms for the calculation of the product A·B and the modular reduction implementation [3].…”

Section: Analysis Of the Possibilities Of Accelerating The Modular Mumentioning

confidence: 99%

See 2 more Smart Citations

Accelerated Modular Multiplication Algorithm of Large Word Length Numbers with a Fixed Module

Bardis

Drigas

Markovskyy

et al. 2010

Knowledge Management, Information Systems, E-Learning, and Sustainability Research

View full text Add to dashboard Cite

Abstract.A new algorithm is proposed for the software implementation of modular multiplication, which uses pre-computations with a constant module. The developed modular multiplication algorithm provides high performance in comparison with the already known algorithms, and is oriented at the variable value of the module, especially with the software implementation on micro controllers and smart cards with a small number of bits.

show abstract

Fast Truncated Multiplication for Cryptographic Applications

Hars

2005

Cryptographic Hardware and Embedded Systems – CHES 2005

Self Cite

View full text Add to dashboard Cite

Abstract. The Truncated Multiplication computes a truncated product, a contiguous subsequence of the digits of the product of 2 integers. A few truncated polynomial multiplication algorithms are presented and adapted to integers. They are based on the most often used n-digit full multiplication algorithms of time complexity O(n α ), with 1< α ≤ 2, but a constant times faster. For example, the least significant half products with Karatsuba multiplication need only 80% of the full multiplication time. The faster the multiplication, the less relative time saving can be achieved. [x] denotes the integer part of x, and 0 ≤ {x} < 1 is the fractional part, such that x = [x] + {x} lg n = log 2 n = log n / log 2 LS stands for Least Significant, the low order bit/s or digit/s of a number MS stands for Most Significant, the high order bit/s or digit/s of a number (Grammar) School multiplication, division: the digit-by-digit multiplication and division algorithms, as taught in elementary schools A B, A B denote the MS or LS half of the digit-sequence of A×B (or A·B), respectively A⊗B denotes the middle third of the digit-sequence of A×B M α (n) the time complexity of the Toom-Cook type full multiplication, O(n α ), with 1< α ≤ 2 γ α = the speedup factor of the half multiplication, relative to M α (n) δ α = the speedup factor of the middle-third product, relative to M α (n)

show abstract

Long Modular Multiplication for Cryptographic Applications

Cited by 18 publications

References 10 publications

Faster Interleaved Modular Multiplication Based on Barrett and Montgomery Reduction Methods

Faster Interleaved Modular Multiplication Based on Barrett and Montgomery Reduction Methods

Accelerated Modular Multiplication Algorithm of Large Word Length Numbers with a Fixed Module

Fast Truncated Multiplication for Cryptographic Applications

Contact Info

Product

Resources

About