Topic
Modular exponentiation
About: Modular exponentiation is a research topic. Over the lifetime, 1354 publications have been published within this topic receiving 24376 citations.
Papers published on a yearly basis
Papers
Journal Article•
TL;DR: In this paper, an Atmel ATmega128 at 8 MHz was used to implement ECC point multiplication over fields using pseudo-Mersenne primes as standardized by NIST and SECG.
Abstract: Strong public-key cryptography is often considered to be too computationally expensive for small devices if not accelerated by cryptographic hardware. We revisited this statement and implemented elliptic curve point multiplication for 160-bit, 192-bit, and 224-bit NIST/SECG curves over GF(p) and RSA-1024 and RSA-2048 on two 8-bit microcontrollers. To accelerate multiple-precision multiplication, we propose a new algorithm to reduce the number of memory accesses. Implementation and analysis led to three observations: 1. Public-key cryptography is viable on small devices without hardware acceleration. On an Atmel ATmega128 at 8 MHz we measured 0.81s for 160-bit ECC point multiplication and 0.43s for a RSA-1024 operation with exponent e = 2 16 +1. 2. The relative performance advantage of ECC point multiplication over RSA modular exponentiation increases with the decrease in processor word size and the increase in key size. 3. Elliptic curves over fields using pseudo-Mersenne primes as standardized by NIST and SECG allow for high performance implementations and show no performance disadvantage over optimal extension fields or prime fields selected specifically for a particular processor architecture.
1,113 citations
1 Apr 1990
TL;DR: This work presents general techniques for constructing simple to program self-testing/correcting pairs for a variety of numerical functions, including integer multiplication, modular multiplication, matrix multiplication, inverting matrices, computing the determinant of a matrix, Computing the rank of a Matrix, integer division, modular exponentiation, and polynomial multiplication.
Abstract: Suppose someone gives us an extremely cast program P that we can call as a black box to compute a function f. Should we trust that P works correctly? A self-testing/correcting pair for f allows us to: (1) estimate the probability that P(x) ¬= f(x) when x is randomly chosen; (2) on any input x, compute f(x) correctly as long as P is not too faulty on average. Furthermore, both (1) and (2) take time only slightly more than the original running time of P. We present general techniques for constructing simple to program self-testing/correcting pairs for a variety of numerical functions, including integer multiplication, modular multiplication, matrix multiplication, inverting matrices, computing the determinant of a matrix, computing the rank of a matrix, integer division, modular exponentiation, and polynomial multiplication
1,032 citations
TL;DR: This work provides an explicit construction of quantum networks effecting basic arithmetic operations: from addition to modular exponentiation, and shows that the auxiliary memory required to perform this operation in a reversible way grows linearly with the size of the number to be factorized.
Abstract: Quantum computers require quantum arithmetic We provide an explicit construction of quantum networks effecting basic arithmetic operations: from addition to modular exponentiation Quantum modular exponentiation seems to be the most difficult (time and space consuming) part of Shor's quantum factorizing algorithm We show that the auxiliary memory required to perform this operation in a reversible way grows linearly with the size of the number to be factorized \textcopyright{} 1996 The American Physical Society
911 citations
TL;DR: The operations involved in computing the Montgomery product are studied, several high-speed, space-efficient algorithms for computing MonPro(a, b), and their time and space requirements are described.
Abstract: Montgomery multiplication methods constitute the core of modular exponentiation, the most popular operation for encrypting and signing digital data in public-key cryptography. In this article, we study the operations involved in computing the Montgomery product, describe several high-speed, space-efficient algorithms for computing MonPro(a, b), and analyze their time and space requirements. Our focus is to collect several alternatives for Montgomery multiplication, three of which are new. However, we do not compare the Montgomery techniques to other modular multiplication approaches.
656 citations
TL;DR: A new type of signature scheme, which consists of two phases, which is supposed to be very fast, is proposed, which uses one-time signature schemes, which are very fast for the on-line signing and an ordinary signature scheme is used for the off-line stage.
Abstract: A new type of signature scheme is proposed. It consists of two phases. The first phase is performed off-line, before the message to be signed is even known. The second phase is performed on-line, once the message to be signed is known, and is supposed to be very fast. A method for constructing such on-line/off-line signature schemes is presented. The method uses one-time signature schemes, which are very fast, for the on-line signing. An ordinary signature scheme is used for the off-line stage.
In a practical implementation of our scheme, we use a variant of Rabin's signature scheme (based on factoring) and DES. In the on-line phase all we use is a moderate amount of DES computation and a single modular multiplication. We stress that the costly modular exponentiation operation is performed off-line. This implementation is ideally suited for electronic wallets or smart cards.
507 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 7 |
| 2024 | 17 |
| 2023 | 20 |
| 2022 | 40 |
| 2021 | 33 |
| 2020 | 38 |