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This section describes the Intel® Integrated Performance Primitives
Cryptography (Intel® IPP Cryptography) functions that implement arithmetic
operations with elements of the following finite fields [ANT]:
GF(
p
)
A finite field of
p
elements.
GF(
q
)
If
q
is an odd prime number, then the finite
field is represented by integers modulo
q
. This field is also known as the
prime finite field
.
GF(
p
d
)
If
p
=
q
,
q
is an odd prime number and
d
> 1, the finite field is represented by
polynomials modulo
g
(
x
),
GF(
p
)[
x
]/
g
(
x
),
where
g
(
x
) is an irreducible
polynomial over GF(
p
). This field is also known as
a degree
d
extension of the GF(
p
)
field
.
GF(((
q
n1
)
n2
)
n3
)
A very complex extension of the prime finite field
GF(
q
). The initial prime field GF(
q
) used
at the lowest level of the construct is frequently called the
basic finite field
with respect to the extension.
The finite field arithmetic functions use context structures of the
IppsGFpState
and
IppsGFpElement
types to store data of the finite field
and the field elements, respectively.
The
IppsGFpElement
type structure is used for
internal
representation of field elements. In application (
or
external
) representation of field element is straightforward. Each element
E
of the prime field GF(
q
) is an
unsigned number in the range [0,
q
- 1], which is represented by a data array
Ipp32u qe[len32]
, so that
where
is the length
of the prime
q
, expressed in
dwords
(32-bit chunks).
Each element
E
of
GF(
p
d
) is represented by a
polynomial of degree less than
d
. This polynomial is represented by an array of
coefficients
is an external (application-side) representation of an element
that belongs to GF(
q
2
) - a degree 2 extension of
some prime field GF(
q
), bitsize(
q
)=128.
You can use Intel IPP Cryptography finite field functions to convert
between the internal and the external representations of a finite field
element.
Prime finite fields are the basic mathematical objects of Elliptic Curve
(EC) cryptography. Intel IPP Cryptography supports different kinds of EC over
finite fields and, in particular, the
standard
elliptic curves - elliptic curves with pre-defined
parameters, including the underlying finite field. The performance of EC
functionality directly depends on the efficiently of the implementation of
operations with finite field elements such as addition, multiplication, and
squaring.
Intel IPP Cryptography contains several different optimized
implementations of finite field arithmetic functions. These implementations,
referred to in this document as "methods", are grouped together in structures.
Intel IPP Cryptography does not reveal the content of these structures. The
implementations, including those optimized for a particular prime
q
, are accessed by special Intel IPP Cryptography
functions. For example,
ippsGFpMethod_p192r1()
returns a pointer to the structure
containing optimized arithmetic over prime
),
additional knowledge concerning the predefined field polynomial
g
(
x
) allows Intel IPP Cryptography to
provide a more efficient implementation of finite field arithmetic than in the
case of an arbitrary field polynomial
g
(
x
). Intel IPP Cryptography contains
methods
dedicated to certain predefined
g
(
x
). For example, the functions
ippsGFpxMethod_binom2()
returns a pointer to the structure
containing optimized arithmetic over GF(
p
2
).
The comparison function
GFpCmpElement
returns the result of comparison:
#define IPP_IS_EQ (0) // elements are equal
#define IPP_IS_GT (1) // the first element is greater than the second one
#define IPP_IS_LT (2) // the first element is less than the second one
#define IPP_IS_NE (3) // elements are not equal
#define IPP_IS_NA (4) // elements are not comparable
is the length
of the prime
