Scheme for arithmetic operations in finite field and group operations over elliptic curves realizing improved computational speed

[0002] This is a divisional of U.S. application No. 09/014,891 filed Jan. 28, 1998 and now U.S. Pat. No. 6,038,581.

[0003] 1. Field of the Invention

[0004] The present invention relates to a scheme for arithmetic operations in finite field and group operations over elliptic curves, and more particularly, to a computational scheme for arithmetic operations in finite fields such as GF(2^m) which is to be utilized in realizing error correction coding (such as algebraic geometric coding) and information security technique (such as elliptic curve cryptosystem) including key distribution and authentication using group operations over elliptic curves.

[0005] 2. Description of the Background Art

[0006] As a fast implementation of multiplicative inverse calculation in GF(2^m), a scheme based on multiplication using a normal basis has been proposed by J. L. Massey and J. K. Omura (see U.S. Pat. No. 4,587,627). This scheme is based on the principle that, when the Fermat's little theorem over finite fields holds in a form of:

[0007] x^2m−1=1 for an element x(≠0) of GF(2^m)

[0008] it is possible to calculate multiplicative inverse in a form of:

[0009] x¹=x²ⁿ⁻²

[0010] Other schemes based on this same principle are also disclosed, for example, in Agnew et al.: “Arithmetic operations in GF(2^m)”, Journal of Cryptology, Vol. 6, pp. 3-13, 1993, and P. C. van Oorschot, S. A. Vanstone: “A Geometric Approach to Root Finding in GF(q^m), IEEE Transactions of Information Theory, Vol. 35, No. 2, pp. 444-453, March 1989.

[0011] Either scheme utilizes the fact that multiplication in GF(2^m) can be efficiently realized by hardware by using a normal basis, and realizes multiplicative inverse calculation in GF(2^m) as a combination of multiplication and shift (including rotate) operations in GF(2^m). In the presently known algorithm, it is known that multiplications in GF(2^m) are required for [log₂m]+{the number of 1 in the binary representation of (m−1)}−1 times, bit shift operations are required for (m−1) times, and when GF(2^m) is a quadratic extension of GF(2^m/2), by the use of subfield in multiplicative inverse calculation in GF(2^m), two multiplications in GF(2^m) and one shift operation in GF(2^m/2) constitute one multiplicative inverse calculation in GF(2^m/2).

[0012] However, when this multiplication algorithm is straightforwardly implemented by software, there arises a problem of lowering of efficiency because of tedious bit unit handling.

[0013] For this reason, there is a known scheme for calculating multiplication in GF(2^m) by using subfield (see A. Pincin: “A New Algorithm for Multiplication in Finite Fields”, IEEE Transactions on Computers, Vol. 38, No. 7, pp. 1045-1049, July 1989, for example).

[0014] In the case of realizing finite field arithmetic by software, because of the looser constraint on memory size compared with the case of hardware implementation, the fast implementation becomes possible by providing a table of calculation results obtained by preliminary calculations and reading out necessary information from the table subsequently. A very fast algorithm utilizing this fact is disclosed in E. De Win et al.: “A Fast Software Implementation for Arithmetic Operations in GF(2ⁿ)”, Advances in Cryptology—ASIACRYPT'96, Lecture Notes in Computer Science 1163, pp. 65-76, Springer-Verlag, 1996, for example.

[0015] Now, many secret key cryptosystems improve their security by iterating F functions several times. It is known that the security can be guaranteed by utilizing exponential calculations in F function (see K. Nyberg: “Differentially Uniform Mappings for Cryptography”, Advances in Cryptology—EUROCRYPT'93, Lecture Notes in Computer Science 765, pp. 55-64, Springer-Verlag, 1994, and K. Nyberg, L. R. Knudsen: “Provable Security Against a Differential Attack”, Journal of Cryptology, Vol. 8, pp. 27-37, 1995). In these references, it is recommended to construct F function by using cube calculations or multiplicative inverse calculations.

[0016] However, when conventionally used input data are represented by using a normal basis on prime field GF(2) and multiplicative inverse calculation in GF(2²ⁿ) is straightforwardly implemented by software using the algorithm of van Oorschot et al., there arises a problem of lowering of efficiency because of tedious bit unit handling.

[0017] Now, elements of a group E(K) of elliptic curves over a field K can be expressed in terms of either homogeneous coordinates formed by a set of three elements of K or affine coordinates formed by a set of two elements of K. Addition of E(K) can be calculated by arithmetic operations over field K in ether expression using homogeneous coordinates or affine coordinates.

[0018] In constructing a device for realizing group operations over elliptic curves, a field K can be chosen to be a finite field GF(q), and in particular, a finite field GF(2ⁿ) with characteristic 2 is often employed because it is possible to realize a fast implementation.

[0019] Among arithmetic operations over finite field, the very fast implementation is possible for addition and additive inverse by the conventional implementation scheme, but considerable time is required for calculating multiplication and multiplicative inverse (hereafter inverse refers to multiplicative inverse unless otherwise indicated). Consequently, a time required for addition of groups over elliptic curves can be evaluated by the required number of multiplication and inverse calculations over field K.

[0020] On the other hand, conventionally, inverse calculation over finite field with characteristic 2 requires an enormous amount of calculations compared with multiplication. For this reason, the conventional schemes for implementing group operations over elliptic curves are mainly the implementation using homogeneous coordinates which does not require inverse calculations, even though the required number of multiplication calculations becomes rather large (see A. J. Menczes, S. A. Vanstone: “Elliptic Curve Cryptosystems and Their Implementation”, Journal of Cryptology, Vol. 6, pp. 209-289, 1993, for example).

[0021] However, in recent years, a scheme for implementing inverse calculation in finite field with characteristic 2 has been developed, and schemes using affine coordinates for expressing elements of group over elliptic curves have been proposed, for example, in E. De Win et al.: “A Fast Software Implementation for Arithmetic Operations in GF(2ⁿ)”, Advances in Cryptology—ASIACRYPT'96, Lecture Notes in Computer Science 1163, pp. 65-76, Springer-Verlag, 1996. In the following, this scheme will be referred to as De Win's scheme.

[0022] Outline of the implementation of finite field according to the De Win's scheme is as follows. When a number of bits for basic operations of a processor is w (8 or 16, for example), all the operations over ground field are calculated in advance by using GF(2^w) as ground field. Also, using an odd degree three term irreducible polynomial over GF(2) in a form of:

[0023] x^d+x^t+1(d>t),

[0024] operations in GF(2^wd) are represented as:

[0025] GF(2^wd)≅GF(2^w)[x]/(x^d+x^t+1)

[0026] where a symbol ≅ denotes isomorphism of fields (see S. MacLane, G. Birkhoff: “Algebra”, Chelsea Publishing, 1967, for detail), and then using this representation, E(GF(2^wd) is implemented. In the De Win's scheme, inverse calculation in finite field utilizes the extended Euclidean algorithm over GF(2^w) which is the general inverse calculation method, and many multiplications and divisions are required in executing the extended Euclidean algorithm.

[0027] Note that finite fields with characteristic 2 are important because they have data structures suitable for computers, and they can be utilized in error correction coding and cryptography. Individual element of a finite field GF(2ⁿ) can be represented by using n-th degree irreducible polynomial f(X) over GF(2) as:

[0028] GF(2ⁿ)≅GF(2)[x]/(f(x))

[0029] so that it can be represented by polynomial of (n−1)-th degree or less. In other words, by regarding coefficients GF(2) of polynomial as bits, GF(2ⁿ) can be represented In terms of n bits.

[0030] When such a representation is used, addition can be represented by exclusive OR of n bits (note that subtraction is the same as addition in the case of field with characteristic 2) so that it can be implemented easily and efficiently. As for the implementation of multiplication and division, there are known schemes which are more efficient than the straightforward scheme for calculating a product of n−1)-th degree polynomials and then calculating a residue of f(X).

[0031] It is therefore an object of the present invention to provide a scheme for calculating multiplicative inverse and multiplication which realizes efficient multiplicative inverse calculation in GF(2²ⁿ), by storing preliminary calculation results in a table in view of the fact that more memory capacities are available for software implementation compared with hardware implementation.

[0032] It is another object of the present invention to provide a scheme for arithmetic operations in finite field and group operations over elliptic curves capable of realizing a very fast implementation by using an optimal normal basis.

[0033] According to one aspect of the present invention there is provided a method for calculating a multiplicative inverse in finite field GF(2²ⁿ), comprising the steps of: expressing an element mεGF(2²ⁿ) as

[0034] m=xα+y(α+1)(x, yεGF(2ⁿ))

[0035] where αεGF(2²ⁿ)\(2ⁿ), α²α+a=0, and aεGF(2²) so that a multiplicative inverse m⁻¹of the element m in the finite field GF(2²ⁿ) is expressed as a combination of multiplications, additions and a multiplicative inverse calculation in subfield GF(2ⁿ) given by

[0036] m⁻¹=(a(x+y)²+xy)⁻¹yα+(a(x+y)²+xy)⁻¹x(α+1)

[0037] by combining a normal basis [α α+1] with extended Euclidean algorithm; and calculating the multiplicative inverse m⁻¹of the element m in the finite field GF(2²ⁿ) by executing said combination of multiplications, additions and a multiplicative inverse calculation in the subfield GF(2ⁿ).

[0038] According to another aspect of the present invention there is provided a method for calculating a multiplication in finite field GF(2²ⁿ), comprising the steps of: reducing a multiplication of two elements m₁and m₂in GF(2²ⁿ) into multiplications and additions in subfield GF(2ⁿ) by expressing m₁, m₂εGF(2²ⁿ) as

[0039] m₁=x₁α+y₁(α+1), m₂=x₂α+y₂(α+1)

[0040] where x_i, y_iεGF(2ⁿ), i=1, 2, αεGF(2²ⁿ)\GF(2ⁿ), α²+α+a =0, and aεGF(2ⁿ) so that a multiplication m₀of m₁, m₂εGF(2²ⁿ) is given by

[0041] m₀=m₁m₂=(x₁x₂+a(x₁+y₂)(x₂+y₂))α+(y₁y₂+a(x₁+y₁)(x₂+y₂))(α+1);

[0042] and calculating the multiplication me by executing said multiplications and additions in the subfield GF(2ⁿ).

[0043] According to another aspect of the present invention there is provided a method for calculating a rational expression in finite field GF(2²ⁿ) in a form of: $\frac{p\left(X_1,X_2,\cdots ,X_r\right)}{q\left(X_1,X_2,\cdots ,X_r\right)}$

[0044] where p, qεGF(2²ⁿ)[X₁, X₂, . . . , X_r], comprising the steps of: calculating a multiplication in GF(2²ⁿ) that arises in calculations using polynomials p, qεGF(2²ⁿ)[X₁, X₂, . . . , X_r] according to a multiplication formula: $\left(x_1\alpha +y_1\left(\alpha +1\right)\right)\times \left(x_2\alpha +y_2\left(\alpha +1\right)\right)=\left(x_1{x}_2+a\left(x_1+y_1\right)\left(x_2+y_2\right)\right)\alpha +\left(y_1{y}_2+a\left(x_1+y_1\right)\left(x_2+y_2\right)\right)\left(\alpha +1\right)$

[0045] where x₁, x₂, y₁, y₂aεGF(2ⁿ), αεGF(2ⁿ), α²+α+a=0 and GF(2²ⁿ)=GF(2ⁿ)(α)); calculating an inverse q⁻¹of q in GF(2²ⁿ) according to an inverse calculation formula:

[0046] (xα+y(α+1)⁻¹=(a(x+y)²+xy)⁻¹(yα+x(α+1))

[0047] where x, y, aεGF(2ⁿ), α ∉GF(2ⁿ), α²+α+a=0 and GF(2²ⁿ)=GF(2ⁿ)(α)); and calculating p×q⁻¹by using said multiplication formula.

[0048] According to another aspect of the present invention there is provided a method for calculating a multiplication in finite field GF(2²ⁿ), comprising the steps of: expressing elements m₁, m₂εGF(2²ⁿ)≅GF(2ⁿ)[x]/(x²+x+a) as

[0049] m₁=x₁+y₁α

[0050] m₂=x₂+y₂α(x₁, x₂, y₁, y₂εGF(2ⁿ))

[0051] where α∉GF(2ⁿ), α²+α+a=0, and aεGF(2ⁿ) so that a multiplication m₁m₂of the elements m₁and m₂in the finite field GF(2²ⁿ) is expressed as a combination of multiplications and additions in subfield GF(2ⁿ) given by

[0052] m₁m₂=(x₁x₂+ay₁y₂)+((x+y₁)(x₁+y₂)+x₁x₂)α

[0053] by using a standard basis [1 α]; and calculating the multiplication m₁m₂of the elements m₁and m₂in the finite field GF(2²ⁿ) by executing said combination of multiplications and additions in the subfield GF(2ⁿ).

[0054] According to another aspect of the present invention there is provided a method for calculating a square in finite field GF(2²ⁿ), comprising the steps of: expressing an element mε GF(2²ⁿ)≅GF(2ⁿ)[x]/(x²+x+a) as

[0055] m=x+yα(x, yεGF(2ⁿ))

[0056] where α∉GF(2ⁿ), α²+α+a=0, and aεGF(2ⁿ) so that a square m²of the element m in the finite field GF(2²ⁿ) is expressed as a combination of multiplications and additions in subfield GF(2ⁿ) given by:

[0057] m²=(x²+ay²)+y²α

[0058] by using a standard basis [1 α]; and calculating the square m²of the element m in the finite field GF(2²ⁿ) by executing said combination of multiplications and additions in the subfield GF(2ⁿ).

[0059] According to another aspect of the present invention there is provided a method for calculating a multiplicative inverse in finite field GF(2²ⁿ), comprising the steps of: expressing an element mεGF(2²ⁿ)≅GF(2ⁿ)[x]/(x²+x+a) as

[0060] m=x+yα(x, yεGF(2ⁿ))

[0061] where α∉GF(2ⁿ), a²+α+a=0, and aεGF(2ⁿ) so that a multiplicative inverse m⁻¹of the element m in the finite field GF(2²ⁿ) is expressed as a combination of multiplications, additions and a multiplicative inverse calculation in subfield GF(2ⁿ) given by

[0062] m⁻¹=(x(x+y)+ay²)⁻¹((x+y)+yα)

[0063] by using a standard basis [1 α]; and calculating the multiplicative inverse m⁻¹of the element m in the finite field GF(2²ⁿ) by executing said combination of multiplications, additions and a multiplicative inverse calculation in the subfield GF(2ⁿ).

[0064] According to another aspect of the present invention there is provided a method for calculating a rational expression in finite field GF(2²ⁿ) in a form of: $\frac{p\left(X_1,X_2,\cdots ,X_r\right)}{q\left(X_1,X_2,\cdots ,X_r\right)}$

[0065] where p, qεGF(2²ⁿ)[X₁, X₂, . . . X_r], comprising the steps of: calculating a multiplication in GF(2²ⁿ) that arises in calculations using polynomials p, qεGF(2²ⁿ)[X₁, X₂, . . . , X_r] according to a multiplication formula:

[0066] (x₁+y₁α)×(x₂+y₂α)==(x₁x₂+ay₁y₂)+((x₁+y₁)(x₂+y₂)+x₁x₂)α

[0067] and a square calculation formula

[0068] (x₁+y₁α)²=(x₁²+ay₁²)+y₁²α

[0069] where x₁, x₂, y₁, y₂, aεGF(2ⁿ), α∉GF(2n ), α²+α+a=0 and GF(2²ⁿ)=GF(2ⁿ)(α); calculating an inverse q⁻¹of q in GF(2²ⁿ) according to an inverse calculation formula:

[0070] (x₁+y₁α)⁻¹=(x₁(x+y₁)+ay₁²)⁻¹((x₁+y₁)+y₁α)

[0071] where x₁, y₁, aεGF(2ⁿ), a∉GF(2ⁿ), α²+α+a=0 and GF(2²ⁿ)=GF(2ⁿ)(α); and calculating p×q⁻¹by using said multiplication formula.

[0072] These methods can be implemented in forms of corresponding devices or articles of manufacture.

[0073] Other features and advantages of the present invention will become apparent from the following description taken in conjunction with the accompanying drawings.

[0074] FIG. 1 is a block diagram of a 2n bit inverse calculation device according to the first embodiment of the present invention.

[0075] FIG. 2 is a block diagram of an n bit multiplication unit that can be used in the 2n bit inverse calculation device of FIG. 1.

[0076] FIG. 3 is a block diagram of an n bit inverse calculation unit that can be used in the 2n bit inverse calculation device of FIG. 1.

[0077] FIG. 4 is a block diagram of another n bit multiplication unit that can be used in the 2n bit inverse calculation device of FIG. 1.

[0078] FIG. 5 is a block diagram of another n bit multiplication unit that can be used in the 2n bit inverse calculation device of FIG. 1.

[0079] FIG. 6 is a block diagram of a 2n bit multiplication device according to the first embodiment of the present invention.

[0080] FIG. 7 is a block diagram of a GF(2^e) addition device according to the second embodiment of the present invention.

[0081] FIG. 8 is a block diagram of a GF(2^e) multiplication device according to the second embodiment of the present invention.

[0082] FIG. 9 is a block diagram of a GF(2^e) inverse calculation device according to the second embodiment of the present invention.

[0083] FIG. 10 is a block diagram of a GF(2²ⁿ) addition device according to the second embodiment of the present invention.

[0084] FIG. 11 is a block diagram of a GF(2²ⁿ) multiplication device according to the second embodiment of the present invention.

[0085] FIG. 12 is a block diagram of a GF(2²ⁿ) square calculation device according to the second embodiment of the present invention.

[0086] FIG. 13 is a block diagram of a GF(2²ⁿ) inverse calculation device according to the second embodiment of the present invention.

[0087] FIG. 14 is a table showing comparison of performances by the first and second embodiments according to the present invention.

[0088] FIG. 15 is a block diagram of an elliptic curve group inverse calculation device according to the third embodiment of the present invention.

[0089] FIG. 16 is a block diagram of an elliptic curve group addition device according to the third embodiment of the present invention.

[0090] FIG. 17 is a block diagram of an elliptic curve group comparison device according to the third embodiment of the present invention.

[0091] FIG. 18 is a block diagram of an elliptic curve group special addition device according to the third embodiment of the present invention.

[0092] FIG. 19 is a block diagram of an elliptic curve group double calculation device according to the third embodiment of the present invention.

[0093] FIG. 20 is a block diagram of an elliptic curve group special double calculation device according to the third embodiment of the present invention.

[0094] FIG. 21 is a block diagram of an elliptic curve group natural number multiple calculation device according to the third embodiment of the present invention.

[0095] Referring now to FIG. 1 to FIG. 6, the first embodiment of a scheme for calculating multiplicative inverse and multiplication in finite field according to the present invention will be described in detail. In the following description, inverse refers to multiplicative inverse unless otherwise indicated.

[0096] This first embodiment is directed to a scheme for calculating an inverse in finite field GF(2²ⁿ) by using arithmetic operations in GF(2ⁿ). This first embodiment is abased on the fact that GF(2²ⁿ) is a two-dimensional vector space in GF(2ⁿ) so that by using:

[0097] αεGF(2²)\GF(2ⁿ), α²+α+aε0, aεGF(2ⁿ)

[0098] where a symbol \ denotes a difference set, an element mεGF(2²ⁿ) can be expressed as:

[0099] m=xα+y(α+1)(x, yεGF(2ⁿ))

[0100] Then, when an element mε GF(2²ⁿ) is expressed by:

[0101] m=xα+y(α+1)(x, yεGF(2ⁿ))

[0102] an inverse of m in GF(2²ⁿ) can be calculated by utilizing the extended Euclidean algorithm as:

[0103] (a(x+y)²+xy)⁻¹yα+(a(x+y)²+xy)⁻¹x(α+1)

[0104] Therefore, the inverse calculation requires two addition calculations in GF(2ⁿ) (namely t₇=x+y=t₃+t₅and t₁₅=a(x+y)²+xy=t₁₁+t₁₄in a configuration of FIG. 1)), five multiplication calculations (namely t₁₀=(x+y)²=t₈×t₉, t₁₁=a(x+y)²=at₁₀, t₁₄=xy=t₁×t₁₂, t₁₉=t₁₇×t₄, and t₂₀=t₁₃×t₁₈in a configuration of FIG. 1), and one inverse calculation in GF(2ⁿ) (namely t₁₆=t₁₅⁻¹in a configuration of FIG. 1).

[0105] FIG. 1 shows a configuration of a 2n bit inverse calculation device according to this first embodiment. This 2n bit inverse calculation device of FIG. 1 comprises two n bit exclusive OR units (add) 106 and 112, five n bit multiplication units (mul) 103, 107, 110, 111 and 113, and one n bit inverse calculation unit (inv) 114, which are configured to carry out the following processing.

[0106] STEP 1: A 2n bit input m is split into two n bit parts x and y.

[0107] STEP 2: x is duplicated to yield t₁and t₂, and t₁is set as an input to the n bit multiplication unit 103.

[0108] STEP 3: t₂is duplicated to yield t₃and t₄, and t₃is set as an input to the n bit exclusive OR unit 106 while t₄is set as an input to the n bit multiplication unit 107.

[0109] STEP 4: y is duplicated to yield t₅and t₆, and t₅is set as an input to the n bit exclusive OR unit 106.

[0110] STEP 5: An exclusive OR t₇of the inputs t₃and t₅is calculated by the n bit exclusive OR unit 106.

[0111] STEP 6: t₇is duplicated to yield t₈and t₉, and t₈and t₉are set as inputs to the n bit multiplication unit 110.

[0112] STEP 7: A product t₁₀of the inputs t₈and t₉is calculated by the n bit multiplication unit 110, and set as an input to the n bit multiplication unit 111.

[0113] STEP 8: A product t₁₁of a constant a and the input to is calculated by the n bit multiplication unit 111, and set as an input to the n bit exclusive OR unit 112.

[0114] STEP 9: t₆is duplicated to yield t₁₂and t₁₃, and t₁₂is set as an input to the n bit multiplication unit 103 while t₁₃is set as an input to the n bit multiplication unit 113.

[0115] STEP 10: A product t₁₄of the inputs t₁and t₁₂is calculated by the n bit multiplication unit 103, and set as an input to the n bit exclusive OR unit 112.

[0116] STEP 11: An exclusive OR t₁₅of the inputs t₁and t₁₄is calculated by the n bit exclusive OR unit 112, and set as an input to the n bit inverse calculation unit 114.

[0117] STEP 12: An inverse t₁₆of the input t₁₅is calculated by the n bit inverse calculation unit 114.

[0118] STEP 13: t₁₆is duplicated to yield t₁₇and t₁₈, and t₁₇is set as an input to the n bit multiplication unit 107 while tie is set as an input to the n bit multiplication unit 113.

[0119] STEP 14: A product t₁₉of the inputs t₄and t₁₇is calculated by the n bit multiplication unit 107.

[0120] STEP 15: A product t₂₀of the inputs t₁₃and t₁₈is calculated by the n bit multiplication unit 113.

[0121] STEP 16: t₁₉and t₂₀are joined and outputted as an inverse m⁻¹of the input m (where the output result is in 2n bits).

[0122] Note that the n bit inverse calculation unit 114 can be realized by recursively regarding t₁₅as m in this configuration of FIG. 1, if it cannot be implemented in a form of FIG. 3 described below because of the limitation on cache memory.

[0123] Note also that, in this configuration of FIG. 1, t₁and t₄are the same value (x), so that there is no need to look up logarithmic conversion table to be described below in the n bit multiplication units 103 and 107 twice. By using the value obtained by the n bit multiplication unit 103 directly in the n bit multiplication unit 107, it is possible to reduce a number of times for looking up the logarithmic conversion table. The similar remarks also apply to t₈and t₉pair, t₁₂and t₁₃pair, and t₁₇and t₁₈pair.

[0124] FIG. 2 shows one exemplary configuration of the n bit multiplication unit that can be used in the configuration of FIG. 1.

[0125] In this n bit multiplication unit 200, in carrying out multiplication in GF(2ⁿ)^x, where GF(2ⁿ)^xdenotes the multiplicative group of GF(2ⁿ), when a logarithmic conversion table and an exponential conversion table can be stored in a memory, a logarithmic conversion table 202 and an exponential conversion table 205 are provided as shown in FIG. 2.

[0126] Then, for two elements x₁and x₂in GF(2ⁿ)^xentered by a query unit 201, corresponding two logarithms e₁=log_gx₁and e₂=log_gx₂are obtained by looking up the logarithmic conversion table 202, and e=e₁+e₂mod(2ⁿ−1) is calculated by an addition unit 203. Then, for this e entered by a query unit 204, an exponential g^eis obtained by looking up the exponential conversion table 205, so as to obtain a product x=x₁×x₂. Here, an element g is set to be a primitive root GF(2ⁿ)^x=GF(2ⁿ)\{0} in GF(2ⁿ)^x.

[0127] More specifically, this n bit multiplication unit 200 of FIG. 2 carries out the following processing.

[0128] STEP 1: The query unit 201 enters two inputs x₁and x₂into the logarithmic conversion table 202.

[0129] STEP 2: Two logarithms e₁=log_gx₁and e₂=log_gx₂are obtained by using the logarithmic conversion table 202, and returned to the query unit 201.

[0130] STEP 3: The query unit 201 enters e₁and e₂into the addition unit 203.

[0131] STEP 4: The addition unit 203 calculates e=e₁+e₂mod(2ⁿ−1), and enters it into the query unit 204.

[0132] STEP 5: The query unit 204 enters e into the exponential conversion table 205.

[0133] STEP 6: The exponential x=g^eis obtained by using the exponential conversion table 205, and returned to the query unit 204.

[0134] STEP 7: The query unit 204 outputs x as the product of x₁and x₂.

[0135] FIG. 3 shows an exemplary configuration of the n bit inverse calculation unit that can be used in the configuration of FIG. 1.

[0136] In this n bit inverse calculation unit 300, the inverse calculation in GF(2ⁿ)^xis carried out as follows. Namely, for an element x in GF(2n)^xentered by a query unit 301, a logarithm e=log_gx is obtained by looking up a logarithmic conversion table 302, and f=−e mod(2ⁿ−1) is calculated by a complement calculation unit 303. Then, for this f entered by a query unit 304, an exponential g^fis obtained by looking up an exponential conversion table 305, so as to obtain an inverse x′=x⁻¹. Note that the logarithmic conversion table 202 and the exponential conversion table 205 of FIG. 2 can be directly utilized as the logarithmic conversion table 302 and the exponential conversion table 305 of FIG. 3.

[0137] More specifically, this n bit inverse calculation unit 300 of FIG. 3 carries out the following processing.

[0138] STEP 1: The query unit 301 enters an input x into the logarithmic conversion table 302.

[0139] STEP 2: The logarithm e=log_gx is obtained by using the logarithmic conversion table 302, and returned to the query unit 301.

[0140] STEP 3: The query unit 301 enters e into the complement calculation unit 303.

[0141] STEP 4: The complement calculation unit 303 calculates f=−e mod (2ⁿ−1), and enters it into the query unit 304.

[0142] STEP 5: The query unit 304 enters f into the exponential conversion table 305.

[0143] STEP 6: The exponential x′=g^fis obtained by using the exponential conversion table 305, and returned to the query unit 304.

[0144] STEP 7: The query unit 304 outputs x′ as the inverse of x.

[0145] FIG. 4 shows another exemplary configuration of the n bit multiplication unit that can be used in the configuration of FIG. 1.

[0146] In this n bit multiplication unit 400, in carrying out multiplication in GF(2ⁿ), when a multiplication table can be stored in a memory, a multiplication table 402 is provided as shown in FIG. 4.

[0147] Then, for two elements x₁and x₂in GF(2ⁿ) entered by a query unit 401, a corresponding multiplication result x=x₁×x₂is obtained by looking up the multiplication table 402.

[0148] More specifically, this n bit multiplication unit 400 of FIG. 4 carries out the following processing.

[0149] STEP 1: The query unit 401 enters two inputs x₁and x₂into the multiplication table 402.

[0150] STEP 2: The multiplication result x=x₁×x₂is obtained by using the multiplication table 402, and returned to the query unit 401.

[0151] STEP 3: The query unit 401 outputs x as the product of x₁and x₂.

[0152] FIG. 5 shows another exemplary configuration of the n bit multiplication unit that can be used in the configuration of FIG. 1.

[0153] This n bit multiplication unit 500 is for a case where one of multiplying number (x₁=a) is constant, as in the n bit multiplication unit 111 of FIG. 1. Consequently, a multiplication table 502 for x=a×x₂(or a×x₂×x₃) is provided as shown in FIG. 5.

[0154] Then, for another multiplying number x₂entered by a query unit 501, a corresponding multiplication result x=a×x₂is obtained by looking up the multiplication table 502. Note that this n bit multiplication unit 500 can be similarly used for a case where a number of multiplying numbers is increased, as in a case of obtaining ax₁x₂.

[0155] More specifically, this n bit multiplication unit 500 of FIG. 5 carries out the following processing.

[0156] STEP 1: The query unit 501 enters an input x into the multiplication table 502.

[0157] STEP 2: The multiplication result x′=a×x is obtained by using the multiplication table 502, and returned to the query unit 501.

[0158] STEP 3: The query unit 501 outputs x′ as the product of a and x.

[0159] Now, when the n bit multiplication unit cannot be implemented in any of the configurations of FIG. 2, FIG. 4 and FIG. 5 described above because of the limitation on cache memory, it is possible to implement the n bit multiplication unit from n/2 bit multiplication units as follows. Note that this implementation can be used recursively.

[0160] Namely, the multiplication in arbitrary 2n bits can be reduced to multiplications in n bits and additions in n bits. When n is sufficiently small, the n bit multiplication can be realized by the fast implementation of any of FIG. 2, FIG. 4 and FIG. 5 described above, so that the following description is given in terms of a parameter n.

[0161] For two elements m₁, m₂εGF(2²ⁿ) expressed by:

[0162] m₁=x₁α+y₁(α+1),

[0163] m₂=x₂α+y₂(α+1), (x_i, y_iεGF(2ⁿ), i=1, 2),

[0164] a product m₀of m₁and m₂can be expressed as:

[0165] m₀=m_{1 m2}=(x₁x₂+a(x₁+y₁)(x₂+y₂))α+(y₁y₂+a(x₁+y₁)(x₂+y₂))(α+1)

[0166] Thus the multiplication in GF(2ⁿ) can be reduced to the arithmetic operations in its subfield, and the multiplication table or the logarithmic conversion table and the exponential conversion table required for the arithmetic operations in the subfield can be realized in smaller size.

[0167] FIG. 6 shows an exemplary configuration of a 2n bit multiplication device according to this first embodiment. This 2n bit multiplication device of FIG. 6 comprises four n bit exclusive OR units (add) 605, 609, 614 and 615, and three n bit multiplication units (mul) 604, 607 and 611, which are configured to carry out the following processing.

[0168] STEP 1: A 2n bit input m₁is split into two n bit parts x₁and y₁.

[0169] STEP 2: A 2n bit input m₂is split into two n bit parts x₂and y₂.

[0170] STEP 3: x₁is duplicated to yield t₁and t₂, and t₁is set as an input to the n bit multiplication unit 604 while t₂is set as an input to the n bit exclusive OR unit 605.

[0171] STEP 4: y₁is duplicated to yield t₃and t₄, and t₃is set as an input to the n bit exclusive OR unit 605 while t₄is set as an input to the n bit multiplication unit 607.

[0172] STEP 5: y₂is duplicated to yield t₅and t₆, and t₅is set as an input to the n bit exclusive OR unit 609 while t₆is set as an input to the n bit multiplication unit 607.

[0173] STEP 6: x₂is duplicated to yield t₇and t₈, and t₇is set as an input to the n bit exclusive OR unit 609 while t₈is set as an input to the n bit multiplication unit 604.

[0174] STEP 7: An exclusive OR t₉of the inputs t₅and t₇is calculated by the n bit exclusive OR unit 609, and set as an input to the n bit multiplication unit 611.

[0175] STEP 8: An exclusive OR tie of the inputs t₂and t₃is calculated by the n bit exclusive OR unit 605, and set as an input to the n bit multiplication unit 611.

[0176] STEP 9: A product t₁₂of two inputs t₅and t₁₀and a constant a is calculated by the n bit multiplication unit 611. Here, the n bit multiplication unit 611 can be in a configuration of FIG. 5 described above.

[0177] STEP 10: A product t₁₃of the inputs t₁and t₈is calculated by the n bit multiplication unit 604, and set as an input to the n bit exclusive OR unit 614.

[0178] STEP 11: A product t₁₄of the inputs t₄and t₈is calculated by the n bit multiplication unit 607, and set as an input to the n bit exclusive OR unit 615.

[0179] STEP 12: t₁₂is duplicated to yield t₁₅and t₁₆, and t₁₅is set as an input to the n bit exclusive OR unit 614 while t₁₆is set as an input to the n bit exclusive OR unit 615.

[0180] STEP 13: An exclusive OR t₁₇of the inputs t₁₃and t₁₅is calculated by the n bit exclusive OR unit 614.

[0181] STEP 14: An exclusive OR t₁₈of the inputs t₁₄and t₁₆is calculated by the n bit exclusive OR unit 615.

[0182] STEP 15: t₁₇and tie are joined and outputted as a product me of the inputs m₁and m₂.

[0183] Thus it can be seen that the product of elements In GF(2²ⁿ) can be calculated by four multiplications and four additions in GF(2ⁿ).

[0184] Note that this reduction of 2n bit multiplication to n bit multiplications and additions can be applied to each 2n bit multiplication appearing in the 2n bit inverse calculation of FIG. 1 described above.

[0185] Note also that the above noted fact that, when an element mεGF(2²ⁿ) is expressed by:

[0186] m=xα+y(α+1)(x, yεGF(2ⁿ))

[0187] an inverse m⁻¹of m in GF(2²ⁿ) can be expressed by:

[0188] m⁻¹=(a(x+y)²+xy)⁻¹yα+(a(x+y)²+xy)⁻¹x(α+1)

[0189] can be demonstrated as follows (assuming that m≠0).

[0190] Namely, for two elements m₁, m₂GF(2²ⁿ) expressed by:

[0191] m₁=x₁α+y₁(α+1),

[0192] m₂=x₂α+y₂(α+1), (x_i, y_iεGF(2ⁿ), i=1, 2),

[0193] a product m₀of m₁and m₂can be expressed as:

[0194] m₀=m₁m₂=(x₁x₂+a(x₁+y₁)(x₂+y₂))α+(y₁y₂+a(x₁+y)(x₂+y₂))(α+1)

[0195] and therefore it follows that: $\begin{array}{c}m\times m^{-1}=\left\{\frac{\mathrm{xy}}{{a\left(x+y\right)}^2+\mathrm{xy}}+a\left(x+y\right)\frac{x+y}{{a\left(x+y\right)}^2+\mathrm{xy}}\right\}\alpha +\\ \left\{\frac{\mathrm{xy}}{{a\left(x+y\right)}^2+\mathrm{xy}}+a\left(x+y\right)\frac{x+y}{{a\left(x+y\right)}^2+\mathrm{xy}}\right\}\left(\alpha +1\right)\\ =\alpha +\left(\alpha +1\right)=1\end{array}$

[0196] Now, the algorithm of this first embodiment is compared with a combination of conventional algorithms. Namely, by using the algorithm of Agnew et al. mentioned above, the inverse calculation in GF(2²ⁿ) can be reduced to multiplications in GF(2²ⁿ) and inverse calculation and shift operation in GF(2ⁿ). That is,

[0197] Agnew:

[0198] one inverse calculation in GF(2ⁿ)

[0199] two multiplications in GF(2²ⁿ)

[0200] one shift operation in GF(2ⁿ)

[0201] Then, when the multiplication in GF(2²ⁿ) is realized by arithmetic operations in subfield GF(2ⁿ) by using the scheme of Pincin mentioned above, it requires four multiplications and four additions in GF(2²). Consequently,

[0202] Agnew+Pincin:

[0203] one inverse calculation in GF(2ⁿ)

[0204] eight multiplications in GF(2ⁿ)

[0205] eight additions in GF(2ⁿ)

[0206] one shift operation in GF(2ⁿ)

[0207] In contrast, the required arithmetic operations in the first embodiment are as follows.

[0208] First Embodiment:

[0209] one inverse calculation in GF(2ⁿ)

[0210] five multiplications in GF(2ⁿ)

[0211] two additions in GF(2ⁿ)

[0212] one shift operation in GF(2ⁿ)

[0213] Thus it can be seen that the first embodiment can save three multiplications in GF(2ⁿ) and six additions in GF(2ⁿ).

[0214] Next, an amount of calculations in an application to cryptographic processing will be described.

[0215] For F function used in the encryption processing of 64 bit block cipher, the use of cube calculation and inverse calculation is recommended. For an exemplary case of 64 bit cipher, a case of realizing the cube calculation by the scheme of Pincin and a case of realizing the inverse calculation by the first embodiment can be compared as follows.

[0216] Cube calculation:

[0217] seven multiplications in GF(2¹⁶)

[0218] (Pincin) three additions in GF(2¹⁶)

[0219] Inverse calculation:

[0220] one inverse calculation in GF(2¹⁶)

[0221] (First Embodiment) five multiplications in GF(2¹⁶)

[0222] two additions in GF(2¹⁶)

[0223] Consider a case of implementation on workstation. Most of the present-day CPUs have a cache memory in size of 256 KB or more, so that the logarithmic conversion table and the exponential conversion table for subfield GF(2 ¹⁶) can be realized as fast accessible tables.

[0224] When the required number of times for looking up table and calculation contents for e and f in the multiplication and the inverse calculation described above, it can be seen that the inverse calculation can be realized faster than the multiplication because the inverse calculation is a monomial operation. Thus it can be seen that, in comparison with the cube calculation, the first embodiment can save one multiplication or more in GF(2¹⁶) and one addition in GF(2¹⁶).

[0225] As described, according to this first embodiment, the normal basis and the extended Euclidean algorithm are combined to reduce the inverse calculation in GF(2²ⁿ) to multiplications, additions, and an inverse calculation in GF(2ⁿ), so as to reduce a required number of multiplications and additions in subfield GF(2ⁿ) compared with the conventional scheme.

[0226] Also, according to this first embodiment, binomial operation (multiplication) in subfield GF(2ⁿ)^xis converted into binomial operation (addition) in Z/(2ⁿ−1)Z (additive cyclic group of order (2ⁿ−1)) where calculation using monomial operation (logarithmic conversion) is easier, and this calculation result is re-converted into monomial operation (exponential conversion) in subfield GF(2ⁿ). Here, in order to provide a multiplication table in GF(2ⁿ) for use in binomial operations, a memory of (2ⁿ)²×n bits will be required, but the logarithmic conversion table and the exponential conversion table storing preliminary calculation results for monomial operations of the logarithmic conversion and the exponential conversion will be required to have a size of about 2n×n bits each.

[0227] Also, according to this first embodiment, when a sufficient amount of fast read accessible memory is available, a multiplication table storing calculation results for binomial operations (multiplications) in GF(2ⁿ) can be provided, so as to reduce a processing load required for table look up and calculation in Z/(2ⁿ−1)Z.

[0228] Also, according to this first embodiment, when one of the numbers to be multiplied together in multiplication is fixed (x₁=a), a multiplication table storing calculation results for multiplications (x=a×x₂or x=a×x₂×x₃) in GF(2ⁿ) can be provided.

[0229] Also, according to this first embodiment, multiplication in GF(2ⁿ) is reduced to arithmetic operations in subfield GF(2^/2). A table for storing multiplication result in GF(2ⁿ) requires about (2ⁿ)²×n bits, but a table for storing multiplication result in GF(2^n/2) requires about (2^n/2)²×(n/2)=(2ⁿ)×(n/2) bits. Consequently, by iterating the reduction of multiplication into multiplication in subfield, it becomes possible to utilize a subfield that has a parameter n for which a table size can be reduced to that of a fast read accessible memory (cache memory).

[0230] Also, according to this first embodiment, when the logarithmic conversion table and the exponential conversion table cannot be stored because of a limited size of cache memory, multiplication in GF(2ⁿ) is reduced to multiplication in GF(2^n/2) and a calculation algorithm in that subfield is applied. Similarly, when an inverse calculation in GF(2ⁿ) cannot be executed, it is reduced to multiplications and an inverse calculation in GF(2^n/2) and a calculation algorithm in that subfield is applied.

[0231] Also, according to this first embodiment, the number of table accesses is reduced by storing values read out from tables for one calculation operation and utilizing them in other calculation operations.

[0232] It is to be noted that, according to the above described scheme of the first embodiment, when a finite field GF(2^{2 n}) is represented by:

[0233] GF(2^{2 n})≈GF(2ⁿ)[x]/(x²+x+a) (aεGF(2ⁿ))

[0234] and a normal basis [a α+l] is taken as a basis when GF(2²ⁿ) is regarded as a two-dimensional vector space in GF(2ⁿ) where α is a root of x^2+x+a=0, multiplication in GF(2²ⁿ) can be calculated by arithmetic operations in GF(2ⁿ) because of the following equation. $\left(x_1\alpha +y_1\left(\alpha +1\right)\right)\times \left(x_2\alpha +y_2\left(\alpha +1\right)\right)=\left(x_1{x}_2+a\left(x_1+y_1\right)\left(x_2+y_2\right)\right)\alpha +\left(y_1{y}_2+a\left(x_1+y_1\right)\left(x_2+y_2\right)\right)\left(\alpha +1\right)$

[0235] Similarly, the square can be calculated by the following equation:

[0236] (xα+y(α+1))²=(x²+a(x²+y²))α+(y²+a(x²+y²))(α+1)

[0237] while the inverse can be calculated by the following equation:

[0238] (xα+y(α+1)⁻¹=(a(x+y)²+xy)⁻¹(yα+x(α+1))

[0239] According to this scheme of the first embodiment, when n is divisible by a large number (such as 16 or more), it becomes possible to obtain an inverse in finite field GF(2ⁿ) more efficiently than the De Win's scheme mentioned above. As a consequence, it becomes possible to realize a faster implementation of group operations over elliptic curves than the De Win's scheme by reducing a required number of multiplications and divisions.

[0240] Also, according to this first embodiment, when elements other than a point at infinity 0 of a group over elliptic curves E(GF(2^2N)) in finite field GF(2²ⁿ) are expressed in terms of affine coordinates (x, y), group operations over elliptic curves can be expressed as rational expressions in x and y, and it is possible to calculate a rational expression in a form of: $\frac{p\left(X_1,X_2,\cdots ,X_r\right)}{q\left(X_1,X_2,\cdots ,X_r\right)}$

[0241] where p, qεGF(2²ⁿ)[X₁, X₂, . . . , X₅] by calculating a multiplication in GF(2²ⁿ) that arises in calculations using polynomials p, qεGF(2²ⁿ)[X₁, X₂, . . . , X_r] according to: $\left(x_1\alpha +y_1\left(\alpha +1\right)\right)\times \left(x_2\alpha +y_2\left(\alpha +1\right)\right)=\left(x_1{x}_2+a\left(x_1+y_1\right)\left(x_2+y_2\right)\right)\alpha +\left(y_1{y}_2+a\left(x_1+y_1\right)\left(x_2+y_2\right)\right)\left(\alpha +1\right)$

[0242] where x₁, x₂, y₁, y₂, aεGF(2ⁿ), α∉GF(2ⁿ), α²+α+a=0 and GF(2²ⁿ)=GF(2ⁿ)(α);

[0243] and calculating an inverse q⁻¹of q in GF(2²ⁿ) according to:

[0244] (xα+y(α+1))⁻¹=(a(x+y)²+xy)⁻¹(yα+x(α+1))

[0245] where x, y, aεGF(2ⁿ), α∉GF(2ⁿ), αC²+α+a=0 and GF(2²ⁿ)=GF(2ⁿ)(α);

[0246] and then calculating p×q⁻¹by repeating the same multiplication as described above for calculations using polynomials p and q.

[0247] Referring now to FIG. 7 to FIG. 14, the second embodiment of a scheme for arithmetic operations in finite field according to the present invention will be described in detail.

[0248] The first embodiment described above is based on the fact that, when a finite field GF(2²ⁿ) is represented by:

[0249] GF(2²ⁿ)≅GF(2ⁿ)[x]/(x²+x+a)(aεGF(2ⁿ))

[0250] and a normal basis [α α+1 ] is taken as a basis when GF(22n) is regarded as a two-dimensional vector space in GF(2ⁿ) where α is a root of x²+x+a=0, multiplication in GF(2²ⁿ) can be calculated by arithmetic operations in GF(2ⁿ) because of the following equation. $\left(x_1\alpha +y_1\left(\alpha +1\right)\right)\times \left(x_2\alpha +y_2\left(\alpha +1\right)\right)=\left(x_1{x}_2+a\left(x_1+y_1\right)\left(x_2+y_2\right)\right)\alpha +\left(y_1{y}_2+a\left(x_1+y_1\right)\left(x_2+y_2\right)\right)\left(\alpha +1\right)$

[0251] Similarly, the square can be calculated by the following equation:

[0252] (xα+y(α+1))²=(x²+a(x²+y²))α+(y²a(x²+y²))(α+1)

[0253] while the inverse can be calculated by the following equation:

[0254] (xα+y(α+1))⁻¹=(a(x+y)²+xy)⁻¹(yα+x(α+1))

[0255] so that operations in GF(2²ⁿ) can be calculated by using arithmetic operations in GF(2ⁿ).

[0256] This first embodiment realizes the fast implementation by utilizing successive quadratic extensions of finite field. However, there is no evidence that a basis taken in realizing quadratic extensions is an optimal one. In this second embodiment, the faster implementation is realized by taking a basis different from the normal basis used in the first embodiment. That is, this second embodiment realizes the faster implementation by using a standard basis instead of the normal basis of the first embodiment.

[0257] More specifically, in this second embodiment, when a finite field GF(2²ⁿ) is represented by:

[0258] GF(2²ⁿ)≅GF(2ⁿ)[x]/(x²+x+a)(aεGF(2ⁿ))

[0259] and a standard basis [1 α] is taken as a basis when GF(2²ⁿ) is regarded as a two-dimensional vector space in GF(2ⁿ) where αm is a root of x²+x+a=0, arithmetic operations can be calculated as follows.

[0260] Addition:

[0261] (x₁+Y₁α)+(x₂+y₂α)=(x₁+x₂)+(y₁+y₂)α  (1)

[0262] Multiplication:

[0263] (x₁+y₁α)×(x₂+y₂α)=(x₁x₂+ay₁y₂)+((x₁+y₁)(x₂+y₂)+x₁x₂)α  (2)

[0264] Square:

[0265] (x₁+y₁α)²=(x₁²+ay₁²)α+y²α  (3)

[0266] Inverse:

[0267] (x₁+y₁α)⁻¹=(x₁(x₁+y₁)+ay₁²)⁻¹((x₁+y₁)+y₁α)  (4)

[0268] Now this second embodiment will be described in further detail with references to the drawings.

[0269] A finite field GF(22^e2) has a subfield GF(2^e2), and can be regarded as a vector space in GF(2^e2). From this fact it follows that operations in GF(2^e2) can be reduced to operations in GF(2^e2), and similarly, they can be reduced to operations in GF(2^e2), . . . , GF(2^e) as explicitly noted above.

[0270] In further detail, in this second embodiment, operations in GF(2^e2) are reduced to operations in GF(2^e2), operations in GF(2^e2) are reduced to operations in GF(2^e2), and so on so forth sequentially, until operations are finally reduced to those in GF(2^e). Then, arithmetic operations in GF(2^e) are realized as follows.

[0271] FIG. 7 shows a configuration of a GF(2^e) addition device to be used for arithmetic operations in finite field according to this second embodiment. This GF(2^e) addition device comprises an e bit exclusive OR (XOR) unit 20a, and operates as follows.

[0272] STEP 101: An exclusive OR value x+(XOR) y of inputs x and y are calculated by the e bit exclusive OR unit 20a.

[0273] STEP 102: An output x+(XOR) y of the e bit exclusive OR unit 20ais outputted as x+y.

[0274] FIG. 8 shows a configuration of a GF(2^e) multiplication device 21 to be used for arithmetic operations in finite field according to this second embodiment. This GF(2^e) multiplication device 21 comprises a query unit 21aand a multiplication table 21b, which are configured to operate as follows.

[0275] STEP 201: The query unit 21alooks up the multiplication table 21bby using inputs x and y as retrieval key so as to retrieve a product x×y of the inputs x and y.

[0276] STEP 202: The query unit 21aoutputs a value x×y obtained from the multiplication table 21b.

[0277] As for a GF(2^e) square calculation to be used for arithmetic operations in finite field according to this second embodiment, it can be realized by entering identical inputs to the GF(2^e) multiplication device 21 of FIG. 8.

[0278] FIG. 9 shows a configuration of a GF(2^e) inverse calculation device 22 to be used for arithmetic operations in finite field according to this second embodiment. This GF(2^e) inverse calculation device 22 comprises a query unit 22aand a query unit 22aand an inverse table 22b, which are configured to operate as follows.

[0279] STEP 301: The query unit 22alooks up the inverse table 22bby using an input x as retrieval key so as to retrieve an inverse x⁻¹of the input x.

[0280] STEP 302: The query unit 22aoutputs a value x⁻¹obtained from the inverse table 22b.

[0281] FIG. 10 shows a configuration of a GF(2²ⁿ) addition device 23 according to this second embodiment which realizes calculation of the above described equation (1). This GF(2²ⁿ) addition device 23 comprises two GF(2ⁿ) addition devices 23aand 23b, which are configured to operate as follows.

[0282] STEP 401: The GF(2ⁿ) addition device 23acalculates a sum x₃of x₁in one input m₁=x₁+y₁α and y₂in another input m₂=x₂+y₂α.

[0283] STEP 402: The GF(2ⁿ) addition device 23bcalculates a sum y₃of y₁in the input m₁=x₁+y₁α and y₂in the input m₂=x₂+y₂α.

[0284] STEP 403: A set of x₃and y₃that gives a sum m_3=m1+m₂=x_3+y3α is outputted.

[0285] FIG. 11 shows a configuration of a GF(2²ⁿ) multiplication device 24 according to this second embodiment which realizes calculation of the above described equation (2). This GF(2²ⁿ) multiplication device 24 comprises four GF(2ⁿ) addition devices (add) 24a, 24c, 24fand 24hand four GF(2ⁿ) multiplication devices (mul) 24b, 24d, 24eand 24g, which are configured to operate as follows.

[0286] STEP 501: The add device 24acalculates a sum t₁of inputs x₁and y₁, and outputs this t₁to the mul device 24e.

[0287] STEP 502: The mul device 24bcalculates a product t₂of inputs x₁and y₁, and outputs this t₂to the add device 24f.

[0288] STEP 503: The add device 24ccalculates a sum t₃of inputs x₂and y₂, and outputs this t₃to the mul device 24e.

[0289] STEP 504: The mul device 24dcalculates a product t₄of inputs x₂and y₂, and outputs this t₄to the add device 24h.

[0290] STEP 505: The mul device 24ecalculates a product t₅of inputs t₁and t₃, and outputs this t₅to the add device 24fand the mul device 24g.

[0291] STEP 506: The add device 24fcalculates a sum y₃of inputs t₂and t₅, and outputs this y₃as part of the output of the GF(2²ⁿ) multiplication device 24.

[0292] STEP 507: The mul device 24gcalculates a product to of an input t₅and a constant a, and outputs this t₇to the add device 24h.

[0293] STEP 508: The add device 24hcalculates a sum x₃of inputs t₄and t₇, and outputs this x₃as part of the output of the GF(2²ⁿ) multiplication device 24.

[0294] Thus the GF(2²ⁿ) multiplication device 24 outputs a set x₃and y₃that gives a product m₃=m₁×m₂=x₃+y₂α.

[0295] FIG. 12 shows a configuration of a GF(2²ⁿ) square calculation device according to this second embodiment which realizes calculation of the above described equation (3). This GF(2²ⁿ) square calculation device comprises one GF(2ⁿ) addition device (add) 25d, one GF(2ⁿ) multiplication device (mul) 25c, and two GF(2ⁿ) square calculation devices (sqr) 25aand 25b, which are configured to operate as follows.

[0296] STEP 601: The sqr device 25acalculates a square t₁of an input x₁, and outputs this t₁to the add device 25d.

[0297] STEP 602: The sqr device 25bcalculates a square t₂of an input y₁, and outputs this t₂to the mul device 25c.

[0298] STEP 603: The mul device 25ccalculates a product y₃of an input t₂and a constant a, and outputs this y₃to the add device 25dwhile also outputting this y₃as part of the output of to the square calculation device 25.

[0299] STEP 604: The add device 25dcalculates a sum x₃of the inputs t₁and y₃, and outputs this x₃as part of the output of the square calculation device 25.

[0300] Thus the GF(2²ⁿ) square calculation device outputs a set x₃and y₃that gives a square m₁²=x₃α.

[0301] FIG. 13 shows a configuration of a GF(2²ⁿ) inverse calculation device 26 according to this second embodiment which realizes calculation of the above described equation (4). This GF(2²ⁿ) inverse calculation device 26 comprises two GF(2ⁿ) addition devices (add) 26aand 26e, four GF(2ⁿ) multiplication devices (mul) 26b, 26d, 26gand 26h, oneGF(2ⁿ) square calculation device (sqr) 26c, and one GF(2ⁿ) inverse calculation device (Inv) 26f, which are configured to operate as follows.

[0302] STEP 701: The add device 26acalculates a sum t₁of inputs x₁and y₁, and outputs this t₁to the mul device 26band the mul device 26g.

[0303] STEP 702: The mul device 26bcalculates a product t₂of inputs x₁and t₁, and outputs this t₂to the add device 26e.

[0304] STEP 703: The sqr device 26ccalculates a square t₃of an input y₁, and outputs this t₃to the mul device 26d.

[0305] STEP 704: The mul device 26dcalculates a product t₄of an input t₃and a constant a, and outputs this t₄to the add device 26e.

[0306] STEP 705: The add device 26ecalculates a sum t₅of inputs t₂and t₄, and outputs this t₅to the inv device 26f.

[0307] STEP 706: The inv device 26fcalculates an inverse t₆of an input t₅, and outputs this t₆to the mul device 26gand the mul device 26h.

[0308] STEP 707: The mul device 26gcalculates a product x₃of inputs t₁and t₆, and outputs this x₃as part of the output of the GF(2²ⁿ) inverse calculation device 26.

[0309] STEP 708: The mul device 26hcalculates a product y₃of inputs y₁and t₆, and outputs this y₃as part of the output of the GF(2²ⁿ) multiplication device 24.

[0310] Thus the GF(2²ⁿ) inverse calculation device 26 outputs a set x₃and y₃that gives an inverse m₁⁻¹=x_3+y3α.

[0311] FIG. 14 shows a comparison of finite field arithmetic operation performances by this second embodiment and the first embodiment described above.

[0312] As can be seen in FIG. 14, this second embodiment is superior to the first embodiment in terms of depth, for all the cases except for the addition. In addition, this second embodiment is superior to the first embodiment in terms of a required number of additions for the square calculation.

[0313] Since the inverse calculation internally uses the square calculation, this second embodiment can be implemented to be faster than the first embodiment for the inverse calculation as well.

[0314] It is to be noted that, when operations in finite field are to be reduced to operations in its subfield successively in a sequence of:

[0315] GF(2^e2)→GF(2^e2)→ . . . →GF(2^e)

[0316] it is possible to use the scheme of the first embodiment in one reduction stage and the scheme of the second embodiment in another reduction stage. For instance, it is possible to use the scheme of the first embodiment for the reduction of operations in GF(2^e2) to operations in GF(2^e2), while using the scheme of the second embodiment for the reduction of operations in GF(2^e2) to operations in GF(2^e2), and so on so forth.

[0317] Referring now to FIG. 15 to FIG. 21, the third embodiment of a scheme for group operations over elliptic curves according to the present invention will be described in detail. This third embodiment is an application of a scheme for arithmetic operations in finite field of the first or second embodiment described above to group operations over elliptic curves in finite field.

[0318] When appropriate field K is defined, group E(K) over elliptic curves can be expressed as:

[0319] E(K)={(x, y)εK²|f(x, y)=0}U{0}

[0320] where f(x, y)εK[x, y] (but f(x, y) cannot be chosen arbitrarily and is subjected to some constraints). For P_iεE(K) (i=1, 2, 3), assuming that P₃=P₁+P₂holds when P_i≠0(i=1, 2, 3), group operations over elliptic curves can be expressed in terms of appropriate polynomials p(x₁, x₂, y₁, y₂), q(x₁, x₂, y₁, y₂), r(x₁, x₂, y₁, y₂), s(x₁, x₂, y₁, y₂)εK(x₁, x₂, y₁, y₂) as follows. $\{\begin{array}{c}{x}_3=\frac{p\left(x_1,x_2,y_1,y_2\right)}{q\left(x_1,x_2,y_1,y_2\right)}\\ y_3=\frac{r\left(x_1,x_2,y_1,y_2\right)}{s\left(x_1,x_2,y_1,y_2\right)}\end{array}$

[0321] Note here that polynomials p, q, r, s are determined and not dependent on P_iwhen the elliptic curves E(K) are fixed, but polynomials p, q, r, s are different for a case of P_i=P₂and a case of P₁≠P₂. From the above it can be seen that group operations over elliptic curves can be constructed from arithmetic operations in field K.

[0322] Now this third embodiment will be described in further detail with references to the drawings.

[0323] Non-supersingular elliptic curves over GF(2^e2) can be defined in terms of parameters:

[0324] a², a⁶εGF(2^e2)(a₆≠0)

[0325] by using the affine coordinates as follows.

[0326] E(GF(2^e2))={(x, y)εGF(2^e2)²|y²+xy=x³+a₂x²+a₆}U{1}

[0327] In this case, the addition over the elliptic curves is defined such that, when:

[0328] P_i(x_i, y_i)εE(GF(2^e2)) (i=1, 2)

[0329] assuming that −-P₁≠P₂, (x₃, y₃)=P₁+P₂can be given by: $\lambda =\frac{y_1+y_2}{x_1+x_2},$ $\begin{array}{cc}\{\begin{array}{c}{x}_3={\lambda }^2+\lambda +\left(x_1+x_2\right)+a_2\\ y_3=\lambda \left(x_1+x_3\right)+x_3+y_1\end{array}& \left(5\right)\end{array}$ $\lambda =x_1+\frac{y_1}{x_1}$ $\begin{array}{cc}\{\begin{array}{c}{x}_3={\lambda }^2+\lambda +a_2\\ y_3=\left(\lambda +1\right)x_3+x_1^2\end{array}& \left(6\right)\end{array}$

[0330] Also, the inverse over the elliptic curves can be expressed as follows.

[0331] −(x₁, y₁)=(x₁, x₁+y₁)  (7)

[0332] (For details of group operations over elliptic curves in field with characteristic 2, see A. J. Menezes: “Elliptic Curve Public Key Cryptosystems”, Kluwer Academic Publishers, pp. 21-23, 1993, for example.)

[0333] An element in finite field GF(2^e2) can be expressed by a bit sequence in e2^tdigits, and a point on an elliptic curve can be expressed by two elements in finite field, so that a point on an elliptic curve can be expressed by a bit sequence in 2e2^tbits. In the following, a point P_ion an elliptic curve is assumed to be expressed in this way. Note however that 0εE(GF(2^e2) can be expressed as 0=(0, 0) because:

[0334] (0, 0)∉E(GF(2^e2))

[0335] FIG. 15 shows a configuration of an elliptic curve group inverse calculation device 1 for an element of a group over elliptic curves according to this third embodiment which realizes calculation of the above equation (7). This elliptic curve group inverse calculation device 1 comprises a query unit 1aand a finite field addition device 1bsuch as that shown in FIG. 10 described above, which are configured to operate as follows.

[0336] STEP 801: The query unit 1agives input (x, y) to the finite field addition device 1b.

[0337] STEP 802: The finite field addition device 1bcalculates a sum x+y in GF(2^e2) of the input x and y, and returns it to the query unit 1a.

[0338] STEP 803: The query unit 1aoutputs an inverse (x, x+y) of an element of a group over elliptic curves by using x+y obtained from the finite field addition device 1b.

[0339] FIG. 16 shows an elliptic curve group addition device 3 for elements of a group over elliptic curves according to this third embodiment which realizes calculation of the above equation (5). This elliptic curve group addition device 3 comprises a query unit 3a, an elliptic curve group inverse calculation device 3bfor an element of a group over elliptic curves such as that shown in FIG. 15 described above, an elliptic curve group comparison device 3cfor elements of a group over elliptic curves such as that of FIG. 17 to be described below, and an elliptic curve group special addition device 3dfor elements of a group over elliptic curves such as that of FIG. 18 to be described below, which are configured to operate as follows.

[0340] STEP 901: The query unit 3achecks whether P₁=0 or not by using the elliptic curve group comparison device 3c, and if it is TRUE, the query unit 3asets Q=P₂and proceeds to the step 908.

[0341] STEP 902: The query unit 3achecks whether P₂=0 or not by using the elliptic curve group comparison device 3c, and if it is TRUE, the query unit 3asets Q=P_iand proceeds to the step 908.

[0342] STEP 903: The query unit 3agives an input P₁to the elliptic curve group inverse calculation device 3b.

[0343] STEP 904: The elliptic curve group inverse calculation device 3bcalculates an inverse −P₁of the input P₁, and returns it to the query unit 3a.

[0344] STEP 905: The query unit 3agives the output −P₁of the elliptic curve group inverse calculation device 3band its own input P₂to the elliptic curve group comparison device 3c.

[0345] STEP 906: The elliptic curve group comparison device 3ccompares inputs −P₁and P₂, and returns T (TRUE) when they coincide or F (FALSE) otherwise.

[0346] STEP 907: The query unit 3asets Q=0 when the output of the elliptic curve group comparison device 3cat the step 906 is T (TRUE), or obtains Q=P₁+P₂by giving P₁and P₂to the elliptic curve group special addition device 3dotherwise.

[0347] STEP 908: The query unit 3aoutputs Q (=P₁+P₂).

[0348] FIG. 17 shows an elliptic curve group comparison device 4 for elements of a group over elliptic curves according to this third embodiment which can be used in the elliptic curve group addition device of FIG. 16 described above. This elliptic curve group comparison device 4 comprises a query unit 4aand a 2e2^tbit sequence comparison unit 4b, which are configured to operate as follows.

[0349] STEP 1001: The query unit 4agives bit sequences X₁∥y₁and x₂∥y₂obtained by joining bit sequences x₁and y₁of an input P₁=(x₁, y₁) and bit sequences x₂and y₂of an input P₂=(x₂, y₂), respectively, to 2e2^tbit sequence comparison unit 4b.

[0350] STEP 1002: The 2e2^tbit sequence comparison unit 4bcompares x₁∥y₁and x₂∥y₂, and returns T (TRUE) when they coincide or F (FALSE) otherwise.

[0351] STEP 1003: The query unit 4aoutputs T/F value obtained by the 2e2^tbit sequence comparison unit 4b.

[0352] FIG. 18 shows an elliptic curve group special addition device 5 for elements of a group over elliptic curves according to this third embodiment which can be used in the elliptic curve group addition device of FIG. 16 described above. This elliptic curve group special addition device 5 comprises eight finite field addition devices (add) 5a, 5c. 5f, 5g, 5h, 5i, 5kand 5l, one finite field inverse calculation device 5b, two finite field multiplication devices (mul) 5dand 5j, and one finite field square calculation device (sqr) 5e, which are configured to operate as follows.

[0353] STEP 1101: The add device 5acalculates a sum t₁(=x₁+x₂) of inputs x₁and x₂, and outputs this t₁to the inv device 5band the add device 5g.

[0354] STEP 1102: The inv device 5bcalculates an inverse t₂(=(x₁+x₂)⁻¹) of an input t₁, and outputs this t₂to the mul device 5d.

[0355] STEP 1103: The add device 5ccalculates a sum t₃(=y₁+y₂) of inputs y₁and y₂, and outputs this t₃to the mul device 5d.

[0356] STEP 1104: The mul device 5dcalculates a product: $\lambda \left(=\frac{y_1+y_2}{x_1+x_2}\right)$

[0357] of inputs t₂and t₃, and outputs this λ to the sqr device 5e, the add device 5f, and the mul device 5j.

[0358] STEP 1105: The sqr device 5ecalculates a square to (=λ²) of an input λ, and outputs this t₅to the add device 5f.

[0359] STEP 1106: The add device 5fcalculates a sum t₅(=λ²+λ) of inputs λ and t₄, and outputs this t₅to the add device 5g.

[0360] STEP 1107: The add device 5gcalculates a sum t₆(=λ²+λ+x₁+x₂) of inputs t₁and t₅, and outputs this t₆to the add device 5h.

[0361] STEP 1108: The add device 5hcalculates a sum x₃(=λ²+λ+x₁+x₂+a²) of an input t₆and a constant a₂, and outputs this x₃to the add device 5iand the add device 5kwhile also outputting this x₃as part of the output of the elliptic curve group special addition device 5.

[0362] STEP 1109: The add device 5icalculates a sum t₇(=x₁+x₃) of inputs x₁and x₃, and outputs this t₇to the mu1 device 5j.

[0363] STEP 1110: The mul device 5Jcalculates a product t₈(=λ(x₁+x₃)) of inputs λ and t₇, and outputs this t₈to the add device 5k.

[0364] STEP 1111: The add device 5kcalculates a sum t₉(=π(x₁+x₃)+x₃) of inputs to and x₃, and outputs this t₉to the add device 5l.

[0365] STEP 1112: The add device 5lcalculates a sum y₃(=λ(x₁+x₃)+x₃+y₁) of inputs y₁and t₉, and outputs this y₃as part of the output of the elliptic curve group special addition device 5.

[0366] Note that each finite field multiplication device (mul) used in this elliptic curve group special addition device 5 can have a configuration as shown in FIG. 11 described above. The finite field square calculation device (sqr) used in this elliptic curve group special addition device 5 can be realized by a configuration of FIG. 11 described above, but can be realized in faster implementation by a special configuration for square calculation such as that of FIG. 12 described above. The finite field inverse calculation device (inv) used in this elliptic curve group special addition device 5 can have a configuration as shown in FIG. 13 described above.

[0367] FIG. 19 shows an elliptic curve group double calculation device for an element of a group over elliptic curves according to this third embodiment which realizes calculation of the above equation (6). This elliptic curve group double calculation device comprises a query unit 10aand an elliptic curve group special double calculation device 10bfor an element of a group over elliptic curves such as that of FIG. 20 to be described below, which are configured to operate as follows.

[0368] STEP 1201: The query unit 10asets Q=0 when x=0 in an input P=(x, y), or obtains Q=2P by giving the input P to the elliptic curve group special double calculation device 10botherwise.

[0369] STEP 1202: The query unit 10aoutputs Q (=2P).

[0370] FIG. 20 shows an elliptic curve group special double calculation device 11 for an element of a group over elliptic curves according to this third embodiment which can be used in the elliptic curve group double calculation device of FIG. 19 described above. This elliptic curve group special double calculation device 11 comprises five finite field addition devices (add) 11d, 11f, 11g, 11hand 11j, one finite field inverse calculation device 11b, two finite field multiplication devices (mul) 11cand 11i, and two finite field square calculation devices (sqr) 11aand 11e, which are configured to operate as follows.

[0371] STEP 1301: The sqr device 11acalculates a square t₁(=x²) of an input x, and outputs this t₁to the add device 11j.

[0372] STEP 1302: The inv device 11bcalculates an inverse t₂(=x⁻¹) of the input x, and outputs this t₂to the mul device 11c.

[0373] STEP 1303: The mul device 11ccalculates a product t₃(=x⁻¹×y) of inputs y and t₂, and outputs this t₃to the add device 11d.

[0374] STEP 1304: The add device 11dcalculates a sum: $\lambda \left(=x+\frac{y}{x}\right)$

[0375] of inputs x and y₃, and outputs this λ to the sqr device 11e, the add device 11fand the add device 11h.

[0376] STEP 1305: The sqr device 11ecalculates a square t₄(=λ²) of an input λ, and outputs this t₄to the add device 11f.

[0377] STEP 1306: The add device 11fcalculates a sum t₅(=λ²+λ) of inputs λ and t₄, and outputs this t₅to the add device 11g.

[0378] STEP 1307: The add device 11gcalculates a sum x₂(=λ²+λ+a²) of an input t₅and a constant a₂, and outputs this x₃as part of the output of the elliptic curve group special double calculation device 11.

[0379] STEP 1308: The add device 11hcalculates a sum t₆(=λ+1) of an input λ and a constant 1, and outputs this t₆to the mul device 11i.

[0380] STEP 1309: The mul device 11icalculates a product t₇(=(λ+1)x₃) of inputs x₃and t₆, and outputs this t₇to the add device 11j.

[0381] STEP 1310: The add device 11jcalculates a sum y₃(=(λ+1)x₃+x₁²) of inputs t₁and t₇, and outputs this y₃as part of the output of the elliptic curve group special double calculation device 11.

[0382] FIG. 21 shows an elliptic curve group natural number multiple calculation device 12 for an element of a group over elliptic curves according to this third embodiment. A natural number multiple of an element of a group over elliptic curves can be realized in various ways, and FIG. 21 shows an implementation using a binary calculation method (see B. Schneier, Applied Cryptography, 2nd Edition, pp. 242-244). This elliptic curve group natural number multiple calculation device 12 comprises a control unit 12a, an elliptic curve group comparison device 12bfor elements of a group over elliptic curves such as that shown in FIG. 17 described above, an elliptic curve group addition device 12cfor elements of a group over elliptic curves such as that shown in FIG. 16 described above, and an elliptic curve group double calculation device 12dfor an element of a group over elliptic curves such as that shown in FIG. 19 described above, which are configured to operate as follows.

[0383] STEP 1401: The control unit 12ainitializes internal variables Q and R as follows.

[0384] Q=O

[0385] R=P

[0386] STEP 1402: The control unit 12achecks a value of n, and outputs Q as the output of the elliptic curve group natural number multiple calculation device 12 and then stops operating when n coincides with 0.

[0387] STEP 1403: The control unit 12achecks a value of n, and sets:

[0388] n←n−1

[0389] when n is odd. Then, control unit 12acompares values of Q and R by using the elliptic curve group comparison device 12b, and calculates:

[0390] Q←Q+R

[0391] by using the elliptic curve group double calculation device 12dwhen Q and R coincide, or by using the elliptic curve group addition device 12cwhen Q and R do not coincide.

[0392] STEP 1404: The control unit 12acalculates:

[0393] n←n/2

[0394] STEP 1405: The control unit 12acalculates:

[0395] R←2R

[0396] by using the elliptic curve group double calculation device 12d.

[0397] STEP 1406: The control unit 12areturns the operation back to the step 1402 described above.

[0398] Now, this scheme for group operations over elliptic curves according to the third embodiment can be applied to various fields such as those of cipher communications and electronic money. Here, the key sharing, encryption, and digital signing in a case of applying this third embodiment to the fields of cipher communications and electronic money will be described.

[0399] First, the Diffie-Hellman key sharing scheme using elliptic curves will be described. Here, system parameters are assumed to be an elliptic curve E(GF(2ⁿ)) and an element PεE(GF(2ⁿ)) of large order.

[0400] In this case, at a time of key generation, a user U randomly generates a positive integer x_Uand calculates:

[0401] Y_U=x_UP

[0402] where x_Uis a secret key and Y_Uis a public key.

[0403] Next, the key sharing between users A and B can be realized as follows.

[0404] STEP 1501: The user A acquires the public key Y_Bof the user B somehow.

[0405] STEP 1502: The user A calculates:

[0406] K_A,B=x_AY_B

[0407] STEP 1503: The user B similarly calculates:

[0408] K_B=x_BY_A

[0409] As a result, the key K_A,B=K_B,Ais shared between the users A and B.

[0410] By applying the scheme for group operations over elliptic curves of the third embodiment to this procedure, it becomes possible to realize the faster processing speed

[0411] Next, the ElGamal encryption using elliptic curves will be described. Here, system parameters are assumed to be an elliptic curve E(GF(2ⁿ)) and an element PεE(GF(2ⁿ)) of large order.

[0412] In this case, at a time of key generation, a user U randomly generates a positive integer x_Uand calculates:

[0413] Y_U=x_UP

[0414] where x_Uis a secret key and Y_Uis a public key.

[0415] Next, a sender can encrypt a plaintext M and transmits a resulting ciphertext to a receiver user A as follows.

[0416] STEP 1601: The sender acquires the public key Y_Aof the user A somehow.

[0417] STEP 1602: The sender generates a positive integer random number r.

[0418] STEP 1603: The sender calculates the ciphertext (C₁, C₂) as follows.

[0419] C₁=rP

[0420] C₂=M+rY_A

[0421] STEP 1604: The user A can obtain the plaintext M by carrying out the decryption of the ciphertext (C₁, C₂) as follows.

[0422] M=C₂−x_AC₁

[0423] Next, the ElGamal digital signature using elliptic curves will be described. Here, system parameters are assumed to be an elliptic curve E(GF(2ⁿ)), an element PεE(GF(2ⁿ)) of large order, and an order ^#E(GF(2ⁿ)) of the elliptic curve, and a one-way hash function h is to be used.

[0424] In this case, at a time of key generation, a user U randomly generates a positive integer x_Uand calculates:

[0425] Y_U=x_UP

[0426] where x_Uis a secret key and Y_Uis a public key.

[0427] Next, a user A can digitally sign data m as follows.

[0428] STEP 1701: The user A randomly selects a positive integer k which is relatively prime with respect to ^#E(GF(2ⁿ)).

[0429] STEP 1702: The user A calculates the signature (R, s) as follows.

[0430] R=kP

[0431] s=(m−x_Ah(r))k⁻¹mod ¹⁹⁰E(GF(2ⁿ))

[0432] Then, the authenticity of this signature (R, s) can be verified as follows.

[0433] mP=h(R)Y_A+R

[0434] It is to be noted that, according to this scheme of the third embodiment, when elements other than a point at infinity 0 of a group over elliptic curves E(GF(2²ⁿ)) in finite field GF(2²ⁿ) are expressed in terms of affine coordinates (x, y), group operations over elliptic curves can be expressed as rational expressions in x and y, and it is possible to calculate a rational expression in a form of: $\frac{p\left(X_1,X_2,\cdots ,X_r\right)}{q\left(X_1,X_2,\cdots ,X_r\right)}$

[0435] where p, qεGF(2²ⁿ)[X₁, x₂, . . . , X_r] by calculating a multiplication in GF(2²ⁿ) that arises in calculations using polynomials p, qεGF(2²ⁿ)[X₁, X₂, X_r] according to:

[0436] (x₁+y₁α)×(x₂+y₂α)=(x₁x₂+ay₁y₂)+((x₁+y₁)(x₂+y₂)+x₁x₂)α

[0437] and

[0438] (x₁+y₁α)²=(x₁^2+ay₁²)+y₁²α

[0439] where x₁, x₂, y₁, y₂, aεGF(2ⁿ), α∉GF(2ⁿ), α²+α+a=0 and GF(2²)=GF(2ⁿ)(α); and calculating an inverse q⁻¹of q in GF(2²ⁿ) according to:

[0440] (x₁+y₁α)⁻¹=(x₁(x₁y₁)+ay₁²)⁻¹((x₁+y₁)+y₁α)

[0441] where x₁, y₁, aεGF(2ⁿ), α∉GF(2ⁿ), α²+α+a=0 and GF(2²ⁿ)=GF(2ⁿ)(α); and then calculating p×q⁻¹by repeating the multiplication as described above for calculations using polynomials p and q.

[0442] It is to be noted that the above described embodiments according to the present invention may be conveniently implemented using conventional general purpose digital computers programmed according to the teachings of the present specification, as will be apparent to those skilled in the computer art. Appropriate software coding can readily be prepared by skilled programmers based on the teachings of the present disclosure, as will be apparent to those skilled in the software art.

[0443] In particular, any of the devices shown in FIG. 1 to FIG. 13 and FIG. 15 to FIG. 21 described above can be conveniently implemented in forms of software package.

[0444] Such a software package can be a computer program product which employs a storage medium including stored computer code which is used to program a computer to perform the disclosed function and process of the present invention. The storage medium may include, but is not limited to, any type of conventional floppy disks, optical disks, CD-ROMs, magneto-optical disks, ROMs, RAMs, EPROMs, EEPROMs, magnetic or optical cards, or any other suitable media for storing electronic instructions.

[0445] It is also to be noted that, besides those already mentioned above, many modifications and variations of the above embodiments may be made without departing from the novel and advantageous features of the present invention. Accordingly, all such modifications and variations are intended to be included within the scope of the appended claims.