N-term Karatsuba Algorithm and its Application to Multiplier designs for Special Trinomials

In this paper, we propose a new type of non-recursive Mastrovito multiplier for $GF(2^m)$ using a $n$-term Karatsuba algorithm (KA), where $GF(2^m)$ is defined by an irreducible trinomial, $x^m+x^k+1, m=nk$. We show that such a type of trinomial combined with the $n$-term KA can fully exploit the spatial correlation of entries in related Mastrovito product matrices and lead to a low complexity architecture. The optimal parameter $n$ is further studied. As the main contribution of this study, the lower bound of the space complexity of our proposal is about $O(\frac{m^2}{2}+m^{3/2})$. Meanwhile, the time complexity matches the best Karatsuba multiplier known to date. To the best of our knowledge, it is the first time that Karatsuba-based multiplier has reached such a space complexity bound while maintaining relatively low time delay

Mastrovito Form of Non-recursive Karatsuba Multiplier for All Trinomials

We present a new type of bit-parallel non-recursive Karatsuba multiplier over $GF(2^m)$ generated by an arbitrary irreducible trinomial. This design effectively exploits Mastrovito approach and shifted polynomial basis (SPB) to reduce the time complexity and Karatsuba algorithm to reduce its space complexity. We show that this type of multiplier is only one $T_X$ slower than the fastest bit-parallel multiplier for all trinomials, where $T_X$ is the delay of one 2-input XOR gate. Meanwhile, its space complexity is roughly 3/4 of those multipliers. To the best of our knowledge, it is the first time that our scheme has reached such a time delay bound. This result outperforms previously proposed non-recursive Karatsuba multipliers

Efficient Square-based Montgomery Multiplier for All Type C.1 Pentanomials

In this paper, we present a low complexity bit-parallel Montgomery multiplier for $GF(2^m)$ generated with a special class of irreducible pentanomials $x^m+x^{m-1}+x^k+x+1$. Based on a combination of generalized polynomial basis (GPB) squarer and a newly proposed square-based divide and conquer approach, we can partition field multiplications into a composition of sub-polynomial multiplications and Montgomery/GPB squarings, which have simpler architecture and thus can be implemented efficiently. Consequently, the proposed multiplier roughly saves 1/4 logic gates compared with the fastest multipliers, while the time complexity matches previous multipliers using divide and conquer algorithms

On the Complexity of non-recursive nn-term Karatsuba Multiplier for Trinomials

The $n$-term Karatsuba algorithm (KA) is an extension of 2-term KA, which can obtain even fewer multiplications than the original one. In this contribution, we proposed a novel hybrid $GF(2^m)$ Karatsuba multiplier using $n$-term KA for irreducible trinomials of arbitrary degree, i.e., $x^m+x^k+1$ where $m\geq2k$. We multiply two $m$-term polynomials using $n$-term KA, by decomposing $m$ into $n\ell+r$, such that $r<n, \ell$. Combined with shifted polynomial basis (SPB), a new approach other than Mastrovito approach is introduced to exploit the spatial correlation between different subexpressions. Then, exact complexity formulations for proposed multipliers are determined. Based on these formulae, we discuss the optimal choice of parameters $n, \ell$ and the effect of $k$. Some upper and lower bounds with respect to these complexities are evaluated as well. As a main contribution, the space complexity of our proposal can achieve to ${m^2}/{2}+O({\sqrt{11}m^{3/2}}/{4})$, which roughly matches the best result of current hybrid multipliers for special trinomials. Meanwhile, its time complexity is slightly higher than the counterparts, but can be improved for some special trinomials. In particular, we demonstrate that the hybrid multiplier for $x^m+x^{k}+1$, where $k$ is approaching $\frac{m}{2}$, can achieve a better space and time trade-off than any other trinomials

A Comparative Study on Two Typical Schemes for Securing Spatial-Temporal Top-k Queries in Two-Tiered Mobile Wireless Sensor Networks

A novel network paradigm of mobile edge computing, namely TMWSNs (two-tiered mobile wireless sensor networks), has just been proposed by researchers in recent years for its high scalability and robustness. However, only a few works have considered the security of TMWSNs. In fact, the storage nodes, which are located at the upper layer of TMWSNs, are prone to being attacked by the adversaries because they play a key role in bridging both the sensor nodes and the sink, which may lead to the disclosure of all data stored on them as well as some other potentially devastating results. In this paper, we make a comparative study on two typical schemes, EVTopk and VTMSN, which have been proposed recently for securing Top-k queries in TMWSNs, through both theoretical analysis and extensive simulations, aiming at finding out their disadvantages and advancements. We find that both schemes unsatisfactorily raise communication costs. Specifically, the extra communication cost brought about by transmitting the proof information uses up more than 40% of the total communication cost between the sensor nodes and the storage nodes, and 80% of that between the storage nodes and the sink. We discuss the corresponding reasons and present our suggestions, hoping that it will inspire the researchers researching this subject