Electrical computers: arithmetic processing and calculating – Electrical digital calculating computer – Particular function performed
Reexamination Certificate
1999-01-28
2002-11-19
Ngo, Chuong Dinh (Department: 2124)
Electrical computers: arithmetic processing and calculating
Electrical digital calculating computer
Particular function performed
Reexamination Certificate
active
06484192
ABSTRACT:
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to a root finding method and a root finding circuit of a one-element quadratic polynomial over a finite field used for elliptic cryptosystem and the like.
2. Description of Related Art
Prior to describing a conventional technique, computation of a finite field will first be described.
(Computation of Finite Field),
A finite field GF(2
m
) is a set formed of 2
m
elements, and each element is represented using vector representation. An element of the degree 2
m
−1 is referred to as primitive root. According to the vector representation, GF(2
m
) is regarded as an m dimensional vector space of GF(2), and an arbitrary element “a” is represented by an m dimensional numerical vector (a
0
, a
1
. . . , a
m−1
), where each component a
i
of the vector is an element over GF(2), i.e., 0 or 1. In vector representation, a vector space is not limited to one type of basis, and representation of an element can vary depending on the basis used.
As for the basis, there are a normal basis and a polynomial basis. The normal basis uses a basis as shown below.
(&agr;, &agr;
2
, &agr;
2
2
, . . . , &agr;
2
m−1
)
where the primitive root &agr; is used so that the followings are linearly independent.
&agr;, &agr;
2
, &agr;
2
2
, . . . , &agr;
2
m−1
The polynomial basis is a basis (1, z, z
2
, . . . z
m−1
) which is generated from a monic irreducible polynomial of degree m “f” over GF(2) serving as a generation polynomial using an element “z” which is the root of “f”. Here, a=(a
0
, a
1
, . . . , a
m−1
) is regarded as an element over GF(2) [x] where x is a variable, and “a” is represented by a=a
m−1
x
m−1
+. . . +a
1
x +a
0
. This representation is referred to as polynomial representation.
Addition of two elements “a” and “b” over GF(2
m
) is represented by a+b=(a
0
+b
0
, a
1
+b
1
, . . . , a
m−1
+b
m−1
). That is, the two elements may be added over GF(2) for each component. The addition over GF(2) is carried out as Exclusive OR. As for the multiplication of two elements “a” and “b” over GF(2m), methods each employing a normal basis are described in U.S. Pat. No. 4,587,627, “Computational Method and Apparatus for Finite Field Arithmetic” and U.S. Pat. No. 4,745, 568, “Computational Method and Apparatus for Finite Field Multiplication.” Each of these methods has a drawback that a circuit for implementation is complicated and the circuit scale becomes very large when m is large. The multiplication using a normal basis is described in detail in A. J. Menezes, Ed, “Applications of Finite Fields”, Kluwer Academic Pub. On the other hand, the method using a polynomial basis is described in detail in J. Weldolon, Jr., “Error-Correcting Codes”, MIT Press. As compared with the multiplication using a normal basis, the method using the polynomial basis has an advantage that the circuit is simple, the circuit scale does not become so large even when m is large, and high speed operation with a high rate clock is possible.
(Elliptic Cryptosystem)
The elliptic cryptosystem is a cryptosystem using addition of GF(2
m
) rational points of an elliptic curve
E:y
2
+xy=x
3
+c
1
x
2
+c
2
, c
1
&egr;GF(
2
m
)
over GF(2
m
). In this cryptosystem, a message is mapped onto a rational point over an elliptic curve E to form an encrypted sentence at the time of encryption. At the time of decryption, the encrypted sentence is mapped onto a rational point over an elliptic curve E to restore the original message.
In this elliptic cryptosystem, the encrypted sentence is formed of a rational point (x
c
, y
c
) on the elliptic curve E. A message having m bits becomes 2m bits when encrypted. This results in a drawback that the size of the encrypted sentence becomes twice as compared with other cryptosystems using an information group over a finite field. For eliminating this drawback, there is such a method as to make the encrypted sentence have a size which is equal to one bit plus the size obtained when other cryptosystems using an information group over a finite field are employed. For implementing this, it is necessary to find roots of a quadratic polynomial over GF(2
m
).
(Mapping from Message onto Rational Point)
For mapping a message onto a rational point, typically the message is subjected to binary expansion, and blocking every m′ bits, where m′<m. This message is used as components of m′ high-order bits of an element M of GF(2
m
) represented as a vector, and components of m-m′ low-order bits are filled up with a random number. This element M is associated with an x coordinate of the elliptic curve E. A rational point having an x coordinate equivalent to the element M is calculated. In other words, y satisfying the relation
y
2
+My=M
3
+c
1
M
2
+c
2
is found. If y does not exist, then the m-m′ low-order bits are filled up with a different random number, and y is found again. If y exists and y is found to be Y, the map of the message onto a rational point is defined to be (M, Y).
If at this time the elliptic curve E is converted in variable by z=y/x, then z satisfying the relation
z
2
+z=a
where
a
=
M
+
c
1
+
c
2
M
2
is found. From this z, Y=Mz is found. Mapping from the message to the point has thus been conducted.
(Reduction of Encrypted Sentence)
Since the elliptic curve E can be represented by a quadratic polynomial as described above, there are only two rational points on the elliptic curve E each of which has an element X over GF(2m) as the value of its x coordinate. Therefore, a cryptosystem sentence (X, Y) can be represented by X and 1-bit information. If z=Y/X is found and its lowest order bit z
0
is used as the cryptograph sentence together with X, therefore, then the cryptograph sentence is reduced by m−1 bits. In the case where this method is used, Y corresponding to X can be found by letting the least significant bit be z
0
and solving z
2
+z=a and ,
a
=
X
+
c
1
+
c
2
X
2
and letting Y=Xz at the time of decryption. The cryptograph sentence (X, Y) can be thus reconstructed. The configuration of the elliptic cryptosystem is described in detail in A. J. Menezes, “Elliptic Curve Public Key Cryptosystems”, Kluwer Academic Pub.
As for the root finding method of the quadratic polynomial, a method of the case where a normal basis is used is generally known. As described above, however, multiplication using a normal basis involves a complicated circuit. In addition, a root finding method, and root finding apparatus of the quadratic polynomial has not been known to the person in the art.
SUMMARY OF THE INVENTION
The present invention has been conceived to solve the above described problem. An object of the present invention is to provide a root finding method, and root finding apparatus, of a quadratic polynomial over a finite field using a polynomial basis capable of operating at high speed and making the circuit scale in implementation small.
The above described object is achieved by the following aspects of the present invention.
In accordance with a first aspect of the present invention, assuming that an expansion degree m is selected so that a polynomial f=x
m
+x
m−1
+. . . x+1 over GF(2) is irreducible and the polynomial f is used as a generation polynomial of GF (2
m
), a root finding circuit of a quadratic polynomial includes m−3 cascade-connected exclusive OR gates X(1, 0) to X(1, m−4) each supplied with a corresponding bit of the element “a” at a first input thereof and each supplied with output of an exclusive OR gate of an immediately preceding stage at a second input, a second input of only X(1, 0) being supplied with a
m−1
instead of the output of an exclusive OR gate of an immediately preceding stage, and m/2−1 exclusive OR gates X(2, 0) to X(2, m/2−2) respectively supplied with a
m−1
and outputs of X
Koda & Androlia
Ngo Chuong Dinh
Toyo Communication Equipment Co., Ltd.
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