68 research outputs found
Obtaining More Karatsuba-Like Formulae over The Binary Field
The aim of this paper is to find more Karatsuba-like formulae for a fixed set of moduli polynomials in GF(2)[x]. To this end, a theoretical framework is established. We first generalize the division algorithm, and then present a generalized definition of the remainder of integer division. Finally, a previously generalized Chinese remainder theorem is used to achieve our initial goal. As a by-product of the generalized remainder of integer division, we rediscover Montgomery’s N-residue and present a systematic interpretation of definitions of Montgomery’s multiplication and addition operations
Efficient Bit-parallel Multiplication with Subquadratic Space Complexity in Binary Extension Field
Bit-parallel multiplication in GF(2^n) with subquadratic space complexity has been explored in recent years due to its lower area cost compared with traditional parallel multiplications. Based on \u27divide and conquer\u27 technique, several algorithms have been proposed to build subquadratic space complexity multipliers. Among them, Karatsuba algorithm and its generalizations are most often used to construct multiplication architectures with significantly improved efficiency. However, recursively using one type of Karatsuba formula may not result in an optimal structure for many finite fields. It has been shown that improvements on multiplier complexity can be achieved by using a combination of several methods. After completion of a detailed study of existing subquadratic multipliers, this thesis has proposed a new algorithm to find the best combination of selected methods through comprehensive search for constructing polynomial multiplication over GF(2^n). Using this algorithm, ameliorated architectures with shortened critical path or reduced gates cost will be obtained for the given value of n, where n is in the range of [126, 600] reflecting the key size for current cryptographic applications. With different input constraints the proposed algorithm can also yield subquadratic space multiplier architectures optimized for trade-offs between space and time. Optimized multiplication architectures over NIST recommended fields generated from the proposed algorithm are presented and analyzed in detail. Compared with existing works with subquadratic space complexity, the proposed architectures are highly modular and have improved efficiency on space or time complexity. Finally generalization of the proposed algorithm to be suitable for much larger size of fields discussed
Non-polynomial Worst-Case Analysis of Recursive Programs
We study the problem of developing efficient approaches for proving
worst-case bounds of non-deterministic recursive programs. Ranking functions
are sound and complete for proving termination and worst-case bounds of
nonrecursive programs. First, we apply ranking functions to recursion,
resulting in measure functions. We show that measure functions provide a sound
and complete approach to prove worst-case bounds of non-deterministic recursive
programs. Our second contribution is the synthesis of measure functions in
nonpolynomial forms. We show that non-polynomial measure functions with
logarithm and exponentiation can be synthesized through abstraction of
logarithmic or exponentiation terms, Farkas' Lemma, and Handelman's Theorem
using linear programming. While previous methods obtain worst-case polynomial
bounds, our approach can synthesize bounds of the form
as well as where is not an integer. We present
experimental results to demonstrate that our approach can obtain efficiently
worst-case bounds of classical recursive algorithms such as (i) Merge-Sort, the
divide-and-conquer algorithm for the Closest-Pair problem, where we obtain
worst-case bound, and (ii) Karatsuba's algorithm for
polynomial multiplication and Strassen's algorithm for matrix multiplication,
where we obtain bound such that is not an integer and
close to the best-known bounds for the respective algorithms.Comment: 54 Pages, Full Version to CAV 201
Generalised Mersenne Numbers Revisited
Generalised Mersenne Numbers (GMNs) were defined by Solinas in 1999 and
feature in the NIST (FIPS 186-2) and SECG standards for use in elliptic curve
cryptography. Their form is such that modular reduction is extremely efficient,
thus making them an attractive choice for modular multiplication
implementation. However, the issue of residue multiplication efficiency seems
to have been overlooked. Asymptotically, using a cyclic rather than a linear
convolution, residue multiplication modulo a Mersenne number is twice as fast
as integer multiplication; this property does not hold for prime GMNs, unless
they are of Mersenne's form. In this work we exploit an alternative
generalisation of Mersenne numbers for which an analogue of the above property
--- and hence the same efficiency ratio --- holds, even at bitlengths for which
schoolbook multiplication is optimal, while also maintaining very efficient
reduction. Moreover, our proposed primes are abundant at any bitlength, whereas
GMNs are extremely rare. Our multiplication and reduction algorithms can also
be easily parallelised, making our arithmetic particularly suitable for
hardware implementation. Furthermore, the field representation we propose also
naturally protects against side-channel attacks, including timing attacks,
simple power analysis and differential power analysis, which is essential in
many cryptographic scenarios, in constrast to GMNs.Comment: 32 pages. Accepted to Mathematics of Computatio
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 using a -term Karatsuba algorithm (KA), where is defined by an irreducible trinomial, . We show that such a type of trinomial combined with the -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 is further studied.
As the main contribution of this study, the lower bound of the space complexity of our proposal is about . 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
Some New Results on Binary Polynomial Multiplication
This paper presents several methods for reducing the number of bit operations for multiplication of polynomials over the binary field. First, a modified Bernstein’s 3-way algorithm is introduced, followed by a new 5-way algorithm. Next, a new 3-way algorithm that improves asymptotic arithmetic complexity compared to Bernstein’s 3-way algorithm is introduced. This new algorithm uses three multiplications of one-third size polynomials over the binary field and one multiplication of one-third size polynomials over the finite field with four elements. Unlike Bernstein’s algorithm, which has a linear delay complexity with respect to input size, the delay complexity of the new algorithm is logarithmic. The number of bit operations for the multiplication of polynomials over the finite field with four elements is also computed. Finally, all these new results are combined to obtain improved complexities
Elliptic Curve Arithmetic for Cryptography
The advantages of using public key cryptography over secret key
cryptography include the convenience of better key management and
increased security. However, due to the complexity of the
underlying number theoretic algorithms, public key cryptography
is slower than conventional secret key cryptography, thus
motivating the need to speed up public key cryptosystems.
A mathematical object called an elliptic curve can be used in the
construction of public key cryptosystems. This thesis focuses on
speeding up elliptic curve cryptography which is an attractive
alternative to traditional public key cryptosystems such as RSA.
Speeding up elliptic curve cryptography can be done by speeding
up point arithmetic algorithms and by improving scalar
multiplication algorithms. This thesis provides a speed up of
some point arithmetic algorithms. The study of addition chains
has been shown to be useful in improving scalar multiplication
algorithms, when the scalar is fixed. A special form of an
addition chain called a Lucas chain or a differential addition
chain is useful to compute scalar multiplication on some elliptic
curves, such as Montgomery curves for which differential addition
formulae are available. While single scalar multiplication may
suffice in some systems, there are others where a double or a
triple scalar multiplication algorithm may be desired. This
thesis provides triple scalar multiplication algorithms in the
context of differential addition chains. Precomputations are
useful in speeding up scalar multiplication algorithms, when the
elliptic curve point is fixed. This thesis focuses on both
speeding up point arithmetic and improving scalar multiplication
in the context of precomputations toward double scalar
multiplication. Further, this thesis revisits pairing
computations which use elliptic curve groups to compute pairings
such as the Tate pairing. More specifically, the thesis looks at
Stange's algorithm to compute pairings and also pairings on
Selmer curves. The thesis also looks at some aspects of the
underlying finite field arithmetic
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