A Karatsuba-Based Montgomery Multiplier

Abstract

Modular multiplication of long integers is an important building block for cryptographic algorithms. Although several FPGA accelerators have been proposed for large modular multiplication, previous systems have been based on O(N 2 ) algorithms. In this paper, we present a Montgomery multiplier that incorporates the more efficient Karatsuba algorithm which is O(N (log 3/ log 2) ). This system is parameterizable to different bitwidths and makes excellent use of both embedded multipliers and fine-grained logic. The design has significantly lower LUT-delay product and multiplier-delay product compared with previous designs. Initial testing on a Virtex-6 FPGA showed that it is 60-190 times faster than an optimized multi-threaded software implementation running on an Intel Xeon 2.5 GHz CPU. The proposed multiplier system is also estimated to be 95-189 times more energy efficient than the software-based implementation. This high performance and energy efficiency makes it suitable for server-side applications running in a datacenter environment.

Implementing homomorphic encryption based secure feedback control

“…Many hardware designs for computing Montgomery multiplications exist. A design involving the Karatsuba multiplication algorithm can be used to evaluate very large multiplications [11]. While this proved to be computationally effective in [11], such a method may not be suitable for some applications due to prohibitive hardware resource required for evaluating Montgomery multiplications even with relatively small operands.…”

Attacking RSA–CRT signatures with faults on montgomery multiplication

“…Theoretical minimum number ℓ of zero higher-order h-bit faulty signatures required to factor a balanced 1024-bit RSA modulus N using the general Cohn-Heninger attack or the simplified linear one. After a discussion about the tools needed to get the desired effects, we focus on several implementation proposals [29,18,16,21,28,19,6], chosen for their relevance.…”

Attacking RSA–CRT Signatures with Faults on Montgomery Multiplication

“…After a discussion about the tools needed to get the desired effects, we focus on several implementation proposals [29,18,16,21,28,19,6], chosen for their relevance.…”

“…To reduce the full complexity of the big multiplication, interpolation techniques are used. In [6], a classical nested Karatsuba multiplication is used, whereas [21] proposes RNS.…”