1. Elliptic Curve Cryptography (ECC) for a public-key

1.    
Preliminaries
and Literature Review1.1.                    
Introduction
N. Koblitz 1 and V. Miller 2 have introduced Elliptic Curve
Cryptography (ECC) for a public-key cryptography. ECC provides same level of
security with smaller key sizes which lead to a better performance in
performing encryption and decryption algorithm. In other word, the interest in
the ECC is growing because the implementations of ECC have produced better
performance in term of calculation time, processing time, power consumption and
memory usage. There are many schemes work based on elliptic curves such as keys
exchange, encryption/decryption and digital signature.  The security of ECC schemes is based on the
resolution of an underlying mathematical problem called the Elliptic Curve
Discrete Logarithm Problem (ECDLP) which is very hard to solve. In ECC, public
and private keys allows encryption and decryption the cryptographic system.
Locking and unlocking algorithms are based on point multiplications. This
operation is considered as the basis of any ECC structure. Point multiplication
has been implemented based on the basic operations of point addition and
doubling. These two operations are implemented based on finite-fields
arithmetic. The figure 1 illustrates these dependencies. Fig. 1. Layers
of Point multiplication implementationThere are several forms of elliptic curves used in ECC such as the
Weierstrass, Hessian and Edwards and kolbitz curves. Each curves has different
characteristics. Koblitz curves 27 are a family of curves on which point
multiplication is noticeably faster than on other curves and also it can be   computed very efficiently in hardware.  Points on elliptic curves can be represented
by different coordinate systems. There are many coordinate systems like Affine,
Projective and mixed coordinates are in ECC for number representation. The
choice of the curve and point coordinate system are very important and have a
significant effect on the performance of the elliptic curve arithmetic
operations. In fact, point multiplication operations can be accelerated and
secured when efficient representation of elliptic curve points is used. In the
following sections, we review mathematical background needed for the elliptic
curve cryptography which are categorized below:1.     
Finite
fields. 2.     
Implementation
of the field operations 3.     
Elliptic
Curves over 4.     
Implementation
of the point multiplication algorithm.   1.2.                    
Finite
FieldsA Finite Field contains of a Finite set of objects called field
elements together with the description of two operations, Finite Field
arithmetic plays an important role in ECC and all the low-level operations are
carried out in these Fields. Finite fields regularly used in cryptography are
either prime fields or binary filed. 1.3.                    
Binary
Fields The binary Field of characteristic two is a Finite Field 3 that contains  different elements. The elements of  is are represented as a vector space over  is which contains 0 and 1 with respect to a
basis. As the two elements of   can be represented with a bit,  bits are required to represent elements of. Addition of
two elements in  is simply performed but the multiplication
depends on the Field basis and dependencies between the Field elements. Finite
fields regularly used in cryptography are either prime fields or binary filed.
These fields has been used in conventional hardware and software applications
and recommended by the international standards such as IEEE and NIST. 
Normal basis is the most efficient Field as it is offering free squaring in hardware
architecture using just cycle shift. However, the multiplication is so complex.
But there is a special subset of NB which is classed Gaussian Normal Basis
(GNB) which present more efficient multiplication.  We have used GNB for
our implementation.1.4.                    
Normal
Basis It is shown that there exists a normal basis for the binary
extension field for all
positive integers . The normal
basis is constructed by finding a normal element, where ? is a
root of an irreducible polynomial of degree m Then set N=  is a basis for  and its elements are linearly independent. In
this case, can be represented as, where. Gaussian normal basis (GNB) 5 is a special class of normal basis  which is included in the IEEE 1363 and NIST
41 standards for the Elliptic Curve Digital Signature Algorithm (ECDSA) and
exists whenever  is not divisible by 8 31. For such a given
m, there always exists a type , , GNB. Type T
GNB provides low complexity multiplication as compared with other normal bases
over.1.5.                    
Implementation
of the Binary field operationsHere, we review the basic field operations in a binary field.
Specifically, the operations of addition, multiplication, squaring and
inversion are reviewed with their Implementation in over GNB.1.5.1. 
Field
AdditionAddition of two field elements, say, where   are in  can be obtained by pair-wise addition of the
coordinates of  and  over .Thus, this is
a simple addition of each linearly independent digit. Each digit is represented
as a single bit and there are no carries. The identity element of addition, i.e., 0, is (0, 0, · · ·, 0, 0).1.5.2. 
Field
SquaringFinite-field squaring performs, where where   are in. Squaring
operation is performed by right cyclic shift of the coordinates of:It is free in hardware if all coordinates are available in
parallel.1.5.3. 
Field
Multiplication Let A and B be elements in in, and assume
their product is , .Then, we can
obtain  can be obtained as 1,
13: where  , , , and  denotes the th element of  matrix. Then, the other coordinates of  could be obtained by shifting the input
operands A and B. Bit-level multipliers provide the lowest possible area complexity.
Massey and Omura invented the first bit-level normal basis multiplier 12. Digit-level multipliers are alternatives for bit-level
multipliers in which the digit size can be chosen depending on the amount of
the resources available. There are some Low-complexity GNB multipliers have
been proposed in 13, 9,
4, 11. The results of such schemes are available after clock cycles,
where d is the digit-size in digit-level architectures, , and  is the field size. 1.5.4. 
Field
InversionInversion for a given element, finding an
element such that , is considered
an expensive operation. It is commonly required in cryptographic applications
of finite fields and its efficient implementation is important. Based on Fermat
Little Theorem, an inversion can be calculated by   the inversion
based on Fermat’s Little Theorem uses consecutive squaring and multiplication
and is more suitable while field elements are represented by normal basis. The
complexity of it can be further reduced by using the Itoh-Tsujii method. In
Itoh and Tsujii algorithm (ITA) 4, the number of multiplications is reduced
based on decomposing Itoh-Tsujii
reduces the complexity of the exponentiation to  squarings and the Hamming weight of (m-1)
multiplications, 1.6.                    
Binary
Koblitz curvesThe most standard binary elliptic
curves is called Binary Weierstrass curves (BWCs) and the curve is defined by
following equationIn this equation:   and . This equation
is suitable for cryptographic applications. For this family of curves, NIST
recommended standard elliptic curves over  fields consist of {B-163, B-233, B-283, B-409
and B-571}. In following point addition and point doubling on BWCs in affine
coordinate are presented.  Let  and  be two points on the BWCs with  Then the addition of points  is the point  denoted by
 ,   Where 
And also for the point doubling we have  ,   Where 
In this case, point addition and
point doubling are computed by , where ,  and  are cost of computation field inversion, field
multiplication and field squaring respectivelyIn the binary Weierstrass curves if  and , it is called
Koblitz curves 11. Therefore, the Koblitz curves are defined over   by following equation:

Koblitz curves offer considerable computational advantages compared
to the binary Weierstrass curves. They have special attractiveness among
elliptic curves, because point doublings can be replaced by efficiently
computable Frobenius endomorphism 27. Mapping in Frobenius endomorphism is performed
by raising every element to the power of two. Therefore in normal basis,
Frobenius endomorphism can be computed very efficiently 11.  Frobenius map ? is an endomorphism that raises
every element to its power of two, therefore, Frobenius maps cost or depending
on the coordinate system. Notice that squaring is cheap, as squaring in with NB
is only a cyclic shift of the bit vector. 

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