Abstract:
In encryption techniques using an elliptic curve, in order to use a homogeneous coordinate system [X, Y, Z], a high speed [X 1 , Y 1 , Z 1 , (Z 1 ) 2 , (Z 1 ) 3 ] for the addition and a high speed [X 1 , Y 1 , Z 1 ] for the doubling the following schemes are provided: (1) Addition is executed by [X 3 , Y 3 , Z 3 ]=[X 1 , Y 1 , Z 1 , (Z 1 ) 2 , (Z 1 ) 3 ]+[X 2 , Y 2 , Z 2 ]. (2) Doubling is executed by a conventional [X 3 , Y 3 , Z 3 ]=2[X 1 , Y 1 , Z 1 ] and an addition operation is executed by [X 3 , Y 3 , Z 3 ]=[X 1 , Y 1 , Z 1 , (Z 1 ) 2 , (Z 1 ) 3 ]+[X 2 , Y 2 , Z 2 ]. It is also required to speed up the multiplication modulo operation. The Montgomery multiplication modulo operation is speeded up by using the following forms of the definition order (prime): (3) the multiplication modulo operation is executed at high speed by using a prime having a form of p=Ab n +B where 0&lt;A&lt;2 w , 0&lt;B&lt;2 w , b=2 w ; and w, A, b, n and B are positive integers.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to techniques of retaining security of a computer network, and more particularly to encryption techniques using an elliptic curve. 
     2. Description of the Related Art 
     Elliptic curve encryption is public key cryptography independently invented by V. Miller and N. Koblitz. 
     The public key cryptosystem has been developed in order to eliminate disadvantages of a common key cryptosystem, the disadvantages being the security which may possibly be lowered at the stage when a private key which is kept secret from third parties is shared by two partners exchanging enciphered information. In this public key cryptosystem, a pair of a private and a public key is used. The private key secret from third parties belongs to a particular individual, and the public key is obtained through arithmetic operation of the private key and made public to third parties. 
     One feature of this public key cryptosystem resides in that a text enciphered by the public key of a particular person cannot be deciphered unless the private key (paired with the public key) of the person is used. This feature can be utilized when a text is transmitted to a partner while the text is kept secret from third parties. For example, when Mr. A transmits a text to Mr. B, the text enciphered by using the public key of Mr. B is transmitted. The enciphered text can be deciphered only with the private key of Mr. B paired with the public key so that only Mr. B can recover the original plain text. 
     A text enciphered with the private key of a particular person can be verified, by using the public key paired with the private key of the person, as to whether or not the text was enciphered by the secret key. This feature can be applied to digital signature. The digital signature is data obtained through arithmetic operation of a text to be signed and through encipher with the private key of the signer. For example, verification of the digital signature of Mr. A can be made depending upon whether or not the data obtained through decipher of the digital signature with the public key of Mr. A is coincident with the data obtained through arithmetic operation of a text to be signed and through encipher with the private key of the signer. If coincident, it can be verified that the digital signature is a correct signature made by Mr. A and that the text with the digital signature was not illegally altered. This feature can therefore be applied to identification of a particular person and prevention of illegal alteration on a network such as the Internet. Verification of a correct signature can be applied to prevent a hostile pretender from purchasing some goods. Verification of no alteration of a text can be applied to prevent alteration of a price entered in a contract note or a receipt. 
     From the viewpoint of security, requirements for the public key cryptosystem are that it is practically impossible to find a private key from a paired public key made public to third parties. Other requirements for the public key cryptosystem which fundamentally takes a longer encipher and decipher time than a private key cryptosystem are a shorter encipher and decipher time. As the techniques of the public key cryptosystem satisfying these to contradictory requirements of security and speed, an elliptic curve encryption has been paid much attention, which is better than conventional RSA and ElGamal cryptosystems. 
     An elliptic curve is represented by a standard formula y 2 =x 3 +ax+b (4a+27b≠0) of an elliptic curve in a finite field having a characteristic of 5 or higher. If a point of infinity is added to this curve, the Abelian group is established. The Abelian operation is represented by a symbol “+”. 
     A typical elliptic curve used for encryption is represented by the following standard forms of Weierstrass. 
     0: Unit element (a point of infinity on a two-dimensional projective plane of an elliptic curve). 
     0+0=0 
     2) (x, y)+0=(x, y) 
     3) (x, y)+(x, −y)=0 
     4) Commutativity (x 1 , y 1 )+(x 2 , y 2 )=(x 2 , y 2 )+(x 1 , y 1 ) 
     5) Addition (x 3 , y 3 )=(x 1 , y 1 )+(x 2 , y 2 ) x 3 =λ 2 −x 1 −x 2 ; y 3 =λ(x 1 −x 3 )−y 1 ; λ=(y 2 −y 1 )/(x 2 −x 1 ) 
     6) Doubling (x 3 , y 3 )=(x 1 , y 1 )+(x 1 , y 1 )=2(x 1 , y 1 ) x 3 =λ 2 −2*x 1 ; y 3 =λ(x 1 −y 3 )−y 1 ; λ=(3*x 1   2 =a)/(2*y 1 ) 
     An elliptic curve cryptograph uses an elliptic curve in a finite field and a set of points constituting the finite filed. 
     As the finite field, a set Fp of remainders of integers congruent modulo of a prime p is used. 
     F p ={0, 1, . . . , p−1} 
     The order of a finite field is the number of elements of the finite field. The order of an elliptic curve is the number of points on an elliptic curve. 
     A result of s-time addition of P (p+ . . . +P) is called an s-multiple point of P, and an operation of obtaining the s-multiple point of P is represented by sP. 
     The order of a point P on an elliptic curve is n which satisfies nP=0, 1&lt;=m&lt;n, and mp≠0. 
     Keys of the elliptic curve cryptograph include the following elliptic curve, base point, public key, and private key. 
     Coefficients of an elliptic curve are a and b. 
     Base point: a point P having a prime as the order. 
     Private key: finite field element d. 
     Public key: a point of private-key-multiplication of the base point Q (Q=dP). 
     An elliptic curve, base point, and public key are public information. The public key and private key are different for each user, whereas the elliptic curve and base point are common to each user. 
     Data encipher, data decipher, digital signature generation, and digital signature verification respectively of the elliptic curve encryption uses a sR operation of an arbitrary point R. This operation can be executed by a combination of the above-described addition and doubling. The above-described addition arithmetic and doubling each require to perform a division once. However, it takes a very long time to perform a division in a finite field. A method of avoiding a division in the finite field has been desired. 
     According to the document of D. V. Chudnovsky, G. V. Chudnovsky “Sequences of Numbers Generated by Addition in Formal Groups and New and New Primality and Factorization Tests”, Advances in Applied Mathematics, 7, 385-434, 1986, formulas of the addition and doubling are derived in a projective space. These formulas will be described in the following. 
     Chudnovsky Formulas 1 
     Addition: 
     
       
         [ X   3   , Y   3   , Z   3   ]=[X   1   , Y   1   , Z   1   ]+[X   2   , Y   2   , Z   2 ] 
       
     
     
       
           X   3 =−( U   1   +U   2 ) P   2   +R   2   
       
     
      2 Y   3   =R (−2 R   2 +3 P   2 ( U   1   +U   2 ))− P   3 ( S   1   +S   2 ) 
     
       
         
           Z 
           3 
           =Z 
           1 
           Z 
           2 
           P 
         
       
     
     
       
           U   1   =X   1 ( Z   2 ) 2   ; U   2   =X   2 ( Z   1 ) 2   ; S   1   =Y   1 ( Z   2 ) 3   
       
     
     
       
           S   2   =Y   2 ( Z   1 ) 3   
       
     
     
       
         
           P=U 
           2 
           −U 
           1 
           ; R=S 
           2 
           −S 
           1 
         
       
     
     Doubling: 
     
       
         [ X   3   , Y   3   , Z   3 ]=2 [X   1   , Y   1   , Z   1 ] 
       
     
     
       
         
           X=T 
         
       
     
     
       
           Y =−8(( Y   1 ) 2 ) 2   +M ( S−T ) 
       
     
     
       
           Z =2 Y   1   Z   1   
       
     
     
       
           S =4 X   1 ( Y   1 ) 2   ; M =3( X   1 ) 2   +a (( Z   1 ) 2 ) 2   ; T =−2 S+M   2   
       
     
     
       
           M =3( X   1 −( Z   1 ) 2 ( X   1 +( Z   1 ) 2  if  a =−3 
       
     
     
       
           M =3( X   1 ) 2  if  a =0 
       
     
     The X 1 , Y 1 , and Z 1  are finite field elements whose data can be expressed by a multiple-precision integer (larger than 2 160 ) 
     Multiple-precision multiplication modulo arithmetic generally takes a longer time than multiple-precision addition subtraction. Therefore, a calculation time can be evaluated from the multiplication modulo arithmetic. The above-cited document describes that the addition arithmetic requires to perform a multiplication modulo operation 16 times and the doubling requires to perform it 10 times. It also describes that if the coefficient a of an elliptic curve is a=−3, it is required to perform the multiplication modulo operation 8 times, and if a=0, it is required to perform it 8 times. 
     The document further describes a method using an expression of [X 1 , Y 1 , Z 1 , (Z 1 ) 2 , (Z 1 ) 3 ] which is described in the following. 
     Chudnovsky Formulas 2 
     Addition arithmetic: 
     
       
         [ X   1   , Y   1   , Z   1 , ( Z   1 ) 2 , ( Z   1 ) 3   ]=[X   1   , Y   1   , Z   1 , ( Z   1 ) 2 , ( Z   1 ) 3   ]+[X   2   , Y   2   , Z   2 , ( Z   2 ) 2 , ( Z   2 ) 3 ] 
       
     
     Doubling: 
     
       
         [ X   3   , Y   3   , Z   3 , ( Z   3 ) 2 , ( Z   3 ) 3 ]=2 [X   1   , Y   1   , Z   1 , ( Z   1 ) 2 , ( Z   1 ) 3 ] 
       
     
     The document describes that the addition arithmetic requires to perform the multiplication modulo operation 14 times and the doubling requires to perform it 11 times. 
     The addition arithmetic can be performed at high speed by &lt;Chudnovsky Formulas 2&gt;, whereas the doubling can be performed at high speed by &lt;Chudnovsky Formulas 1&gt;. 
     As an example of the multiple-precision multiplication modulo arithmetic, Montgomery modulo arithmetic is known which is described in the document of A. Menezes, P. Oorschot, S. Vanstone, “Handbook of Applied Cryptography”, CRC Press, p. 600 (1996), Section 14.3 Multiple-precision modular arithmetic. 
     The Montgomery modulo arithmetic described in this document will be described. 
     Input: 
     p=(p n−1 , . . . , p 2 , p 1 ); gcd(p, b)=1; R=b n ; 
     p′=−1/p mod b; T=(t 2n−1 , . . . , t 1 , t 0 )&lt;pR; 
     b=2 w    
     p is a modulo by which an integer is divided to obtain a remainder. 0&lt;=p i &lt;b. 0&lt;=t i &lt;b. w is a positive integer. T is a multiplication result of integers x and y which is smaller than p. 
     An output is assumed to be T/R (mod p). 
     Step  1 : A←T 
     Step  2 : The following Steps  2 . 1  and  2 . 2  are executed from i=0 to i=(n−1). 
     Step  2 . 1 : u i ←a i p′ mod b 
     Step  2 . 2 : A←A+u i pb i    
     Step  3 : A←A/b n    
     Step  4 : If A&gt;=p, then A←A−p 
     Step  5 : A is an output. 
     An elliptic curve used for the elliptic curve encryption is expressed by an elliptic curve y 2 =x 2 +ax+b which uses as the definition field a prime field Fp having a prime p as its order. In order to form a perfect elliptic curve, it is necessary to set the parameters a and b which have prime factors r having a large order #E(Fp) of an elliptic curve, where 
     #E(Fp)=kr, k is a small integer, and r is a large prime. 
     A method of setting parameters of an elliptic curve having primes with a large order is described in the document of Henri Cohen, “A Course in Computational Algebraic Number Theory”, GTM138, Springer (1993), p.464, Atkin&#39;s Test. 
     As a primality test used when the prime p is generated, a Miller-Rabin primality test is widely used which is described in the document of A. Menezes, P. Oorschot, S. Vanstone, “Handbook of Applied Cryptography”, CRC Press, p. 139 (1996), Section 4.1.3. 
     Elliptic curve cryptograph using specific primes are described in U.S. Pat. No. 5,271,061 and U.S. Pat. No. 5,463,690. These patents disclose techniques of using the prime p in the form of “p=2 e −a; e is a positive integer; and a&lt;232 or a=1” in the elliptic curve encryption having as its definition field a finite field Fq with q=p k , i.e., a finite field of characteristic p. An operation of obtaining an s-multiple point of a point P is similar to an exponentiation modulo operation of an integer of a raised to a power of e. As a high speed exponent operation, a sliding-window method is known which is described in the document of A. Menezes, P. Oorschot, S. Vanstone, “Handbook of Applied Cryptography”, CRC Press, p. 616 (1996), Section 14.6.1 (ii), “Sliding-window exponentiation”. 
     Both &lt;Chudnovsky Formulas 1&gt; and &lt;Chudnovsky Formulas 2&gt; are not satisfactory for high speed operations. The inventors consider that an operation method is desired which provides an efficiency of &lt;Chudnovsky Formulas 2&gt; for the addition and an efficiency of &lt;Chudnovsky Formulas 1&gt; for the doubling. 
     In order to further speed up a calculation time, it is necessary to speed up the multiplication modulo operation used in &lt;Chudnovsky Formulas 1&gt; and &lt;Chudnovsky Formulas 2&gt;. 
     SUMMARY OF THE INVENTION 
     In order to solve the above-described problems, the following means are provided in order to use a high speed [X 1 , Y 1 , Z 1 , (Z 1 ) 2 , (Z 1 ) 3 ] for the addition and a high speed [X 1 , Y 1 , Z 1 ] for the doubling. 
     (1) Addition: execute [X 3 , Y 3 , Z 3 ]=[X 1 , Y 1 , Z 1 , (Z 1 ) 2 , (Z 1 ) 3 ]+[X 2 , Y 2 , Z 2 ]. 
     (2) A doubling point calculation is executed by a conventional [X 3 , Y 3 , Z 3 ]=2[X 1 , Y 1 , Z 1 ] and an addition operation is executed by [X 3 , Y 3 , Z 3 ]=[X 1 , Y 1 , Z 1 , (Z 1 ) 2 , (Z 1 ) 3 ]+[X 2 , Y 2 , Z 2 ]. 
     It is also required to speed up the multiplication modulo operation. The Montgomery multiplication modulo operation is speeded up by using the following forms of the definition order (prime). 
     (3) The multiplication modulo operation is executed at high speed by using a prime having a form of p=Ab n +B (0&lt;A&lt;2 w ; 0&lt;B&lt;2 w ; b=2 w ; and w, A, b, n and B are positive integers). 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a block diagram illustrating an elliptic curve encryption method according to a first embodiment of the invention. 
     FIG. 2 is a flow chart illustrating an elliptic curve operation of the elliptic curve encryption method of the first embodiment. 
     FIG. 3 is a flow chart illustrating a prime generation operation capable of performing an elliptic curve encryption method at high speed according to a second embodiment of the invention. 
     FIG. 4 is a flow chart illustrating a prime generation operation capable of performing an elliptic curve encryption method at high speed according to a third embodiment of the invention. 
     FIG. 5 is a diagram showing a bit train of primes used by the second embodiment. 
     FIG. 6 is a diagram showing a bit train of primes used by the third embodiment. 
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS 
     FIG. 1 illustrates a first embodiment. An elliptic curve encryption system of this invention illustrated in FIG. 1 executes enciphering and deciphering operations. 
     An elliptic curve generating unit  101  generates an elliptic curve to be used for elliptic curve encryption. A public/private key generating unit  102  generates a public key  116  and a private key  117  in accordance with an input elliptic curve generated by the elliptic curve generating unit  101 . The public key  116  is distributed over a network (not shown) or the like similar to a conventional public key cryptosystem. 
     An enciphering unit  103  receives a plain text  113  and a public key  116  and outputs an enciphered text  114 . The enciphering unit  103  may be formed on an electronic apparatus other than a computer if the electronic apparatus has an arithmetic unit capable of operating the enciphering unit  103  in accordance with the elliptic curve encryption of this invention. 
     A deciphering unit  104  receives the enciphered text  114  and private key  117  and outputs a plain text  115  which is the same as the original plain text  103 . Similar to the enciphering unit  103 , the deciphering unit  104  may be formed on an electronic apparatus other than a computer if the electronic apparatus has an arithmetic unit capable of operating the deciphering unit  104  in accordance with the elliptic curve encryption of this invention. 
     The prime p is generated by a prime generating unit  105  of the elliptic curve generating unit  101  by the following sequence. 
     1: A random number p is generated to follow the next step. 
     2: The prime of p is judged by the Miller-Rabin primality test. If the random number is judged as the prime, the sequence is terminated, whereas if it is judged as a composite number, the sequence returns to step  1 . 
     The invention may be practiced by using another primality test. 
     An elliptic curve parameter setting unit  106  sets the parameters a and b of an elliptic curve y 2 =x 3 +ax+b which uses as the definition field a prime field Fp having the prime p generated by the prime generating unit  105  as its order. In order to form a perfect elliptic curve, it is necessary to set the parameters a and b which have prime factors r having a large order #E(Fp) of the elliptic curve, where 
     #E(Fp)=kr, k is a small integer, and r is a large prime. 
     By using the method described in the document of Henri Cohen, “A Course in Computational Algebraic Number Theory”, GTM138, Springer (1993), p.464, Atkin&#39;s Test, an elliptic curve is generated which has large prime factors r as the order. The invention may be practiced by using another elliptic curve parameter setting method capable of setting an elliptic curve having large prime factors r as the order of the elliptic curve. 
     A base point generating unit  107  obtains a generator of a cyclic group having the prime factor r as the order in an Abelian group on the elliptic curve. If #E(Fp)=kr, the following sequence is executed. 
     1: An arbitrary point (x 1 , y 1 ) on E(Fp) is obtained. 
     2: If r(x 1 , y 1 )=0 and k(x 1 , y 1 )≠0, the base point is G=(x 1 , y 1 ). If not, the sequence returns to step  1 . 
     An operation r(x 1 , y 1 ) is a scalar multiplication (r-multiplication) operation of (x 1 , y 1 ), which will be later described with an elliptic curve calculating unit  109 . 
     As described above, in the elliptic curve generating unit  101 , the definition field order p, the parameters a and b of the elliptic curve y 2 =x 3 +ax+b, base point G, and base point order r are generated which are information to be made public. 
     The public/private key generating unit  102  generates the public key and private key by the following sequence. 
     Input: 
     Definition field order p, parameters a and b of the elliptic curve y 2 =x 3 +ax+b, and base point G 
     Output: 
     Public key Q and private key d 
     1: A random number 2&lt;d&lt;p−1 is generated. 
     2: Q=dG, a d-multiple of G is obtained, which will be later described with the elliptic curve calculating unit  109 . 
     The public key is information to be made public, and the private key is information to be kept in secret. A problem of obtaining d from Q and G is called a discrete logarithm problem. A calculation amount of calculating d from the elliptic curve is in the order of an exponent of a bit length of the order r of the base point. Therefore, if r is a large prime, e.g, r&gt;2 159 , it is practically impossible to obtain d from Q and G. This is a principle of an elliptic curve encryption. 
     The enciphering unit  103  converts the plain text  113  into the enciphered text by the following sequence. 
     Input: 
     Plain text M, public key Q, definition field order p, parameters a and b of the elliptic curve y 2 =x 3 +ax+b, and base point G 
     Output: 
     Enciphered Text C 
     Step  1 : A random number k is generated (random number generating unit  108 ). 
     Step  2 : (x 1 , y 1 )=kG (elliptic curve calculating unit  109 ). 
     Step  3 : (x 2 , y 2 )=kQ (elliptic curve calculating unit  109 ). 
     Step  4 : M′=M XOR x 2  (data enciphering unit  110 ). 
     Step  5 : Enciphered text C=x 1 | |y 1 | |M′ Y 1  (data enciphering unit  110 ). 
     The elliptic curve calculating unit  109  executes a scalar multiplication operation kR of an arbitrary point R. 
     In this operation, an addition operation is executed by [X 3 , Y 3 , Z 3 ]=[X 1 , Y 1 , Z 1 , (Z 1)   2 , (Z 1 ) 3 ]+[X 2 , Y 2 , Z 2 ] and a doubling point calculation is executed by a conventional [X 3 , Y 3 , Z 3 ]=2[X 1 , Y 1 , Z 1 ]. This method will be described with reference to the flow chart of FIG.  2 . Steps  2  and  3  are executed as illustrated in the flow chart of FIG.  2 . 
     Input: 
     k, R=[X 1 , Y 1 , Z 1 ] 
     Output: 
     kR 
     At Step  201 , the operation starts. 
     At Step  202 , k and R=[X 1 , Y 1 , Z 1 ] are input. 
     At Step  203 , k is expressed by binary notation as H m , . . . , H 0 . 
     At Step  204 , it is set that [X k , Y k , Z k ]←[X 1 , Y 1 , Z 1 ] and j←m, and (Z 1 ) 2  and (Z 1 ) 3  are calculated. 
     At Step  205 , it is set that j←j−1. 
     At Step  206 , it is checked whether j=0, in order to repeat Steps  206  to  208  from m to 1 of the variable j in the descending order. If j=0, the flow skips to Step  210 , whereas if not the flow advances to Step  207 . 
     At Step  207 , it is set that [X k , Y k , Z k ]←2[X k , Y k , Z k ]. 
     At Step  208 , it is judged whether H j =0. If 0, the flow advances to Step  209 , whereas if 1, the flow returns to Step  205 . 
     At Step  209 , it is set that [X k , Y k , Z k ]←[X k , Y k , Z k ]+[X 1 , Y 1 , Z 1 , (Z 1 ) 2 , (Z 1 ) 3 ]. 
     At Step  210 , Ei is output. 
     At Step  211 , the operation is terminated. 
     The deciphering unit  104  converts the enciphered text  114  into the original plain text  115  same as the plain text  113  by the following sequence. 
     Input: 
     Enciphered text C=x 1 | |y 1 | |M′ Y 1 , private key d, definition field order p, parameters a and b of the elliptic curve y 2 =x 3 +ax+b, and base point G 
     Output: 
     Plain text M 
     Step  1 : (x 2 , y 2 )=k(x 1 , y 1 ) (elliptic curve calculating unit  111 ). 
     Step  2 : Plain text M=M′ XOR x 2    
     Step  1  is executed as illustrated in the flow chart of FIG.  2 . 
     Next, a second embodiment will be described. In the second embodiment, the prime generating unit  105  shown in FIG. 1 generates a specific prime to allow a high speed operation. By using a prime having the form of p=Ab n +B (0&lt;A&lt;2 w ; 0&lt;B&lt;2 w ; b=2 w ; and w, b, n and B are positive integers), the multi-precision modulo operation can be performed at high speed. The first and second embodiments may be practiced independently or a combination thereof may be practiced at higher speed. 
     By using the definition field order (prime) having the form of p=Ab n +B, the Montgomery multiplication modulo operation used by the multi-precision integer multiplication modulo operation can be performed at high speed. This method will be described with reference to the flow chart shown in FIG.  3 . 
     At Step  301 , the operation starts. 
     At step  302 , b and n are input. 
     At Step  303 , a random number p=Ab n +B is generated. 
     At Step  304 , the prime of p is judged. In this example, the Miller-Rabin primality test is used. If it is judged that p is a prime, the flow advances to Step  305 , whereas if it is judged that p is a composite number, the flow returns to Step  303 . 
     At Step  305 , p is output. 
     At Step  306 , the operation is terminated. 
     Since the prime having the above-described form is used, in the Montgomery modulo operation described with the conventional techniques, a multiplication of p in Step  2 . 2 : A←A+u i pb i  can be performed at high speed. 
     In a conventional operation, a multiplication is performed for all p i  and u i . With the prime of this embodiment, a multiplication is performed only for the highest p n =A and the lowest p 0 =B. With the prime of this embodiment, p i  excepting the highest and lowest is 0. For example, as shown in FIG. 5, if w=32, and n=5, the bits  502 ,  503 , and  504  excepting the highest 32 bits  501  and lowest 32 bits  505  are 0 so that a multiplication work between 0 and u i  can be omitted. 
     Next, a third embodiment will be described. In the third embodiment, the prime generating unit  105  shown in FIG. 1 generates a prime having a form of p=Ab n +1 to allow a high speed operation. This prime is a specific example of the prime described with FIG.  3 . As compared to the prime described with FIG. 3, since a multiplication of the lowest p 0  is 1×p 0  and is not necessary, a multiplication is performed only for the highest p n . For example, as shown in FIG. 6, if w=32 and n=5, a multiplication is performed only for the highest 32 bits  601 . The bits  602 ,  603 , and  604  are 0 so that a multiplication work between 0 and u i  can be omitted. Since the lowest  605  is 1, the multiplication by u i  can be omitted. This method will be described with reference to the flow chart shown in FIG.  4 . 
     At Step  401 , the operation starts. 
     At Step  402 , b and n are input. 
     At Step  403 , a random number p=Ab n +1 is generated. 
     At Step  404 , the prime of p is judged. In this example, the Miller-Rabin primality test is used. If it is judged that p is a prime, the flow advances to Step  405 , whereas if it is judged that p is a composite number, the flow returns to Step  403 . 
     At Step  405 , p is output. 
     At Step  406 , the operation is terminated. 
     Next, a fourth embodiment will be described. In the fourth embodiment, a higher speed operation is realized by the elliptic curve calculating unit  109  shown in FIG.  1 . In the fourth embodiment, the sliding-window method is used for an s-multiple point operation of a point P, an addition operation is executed by [X 3 , Y 3 , Z 3 ]=[X 1 , Y 1 , Z 1 , (Z 1 ) 2 , (Z 1 ) 3 ]+[X 2 , Y 2 , Z 2 ], and a doubling point calculation is executed by a conventional [X 3 , Y 3 , Z 3 ]=2[X 1 , Y 1 , Z 1 ]. 
     An algorithm described in the document of A. Menezes, P. Oorschot, S. Vanstone, “Handbook of Applied Cryptography”, CRC Press, p. 616 (1996), Section 14.6.1 (ii) “Sliding-window exponentiation” will be described. 
     Sliding-window Exponentiation 
     Input: 
     g, e=(e t , e t−1 , . . . , e 1 , e 0 )e t =1, k (integer) 
     Output: 
     g e    
     Preliminary Calculations: 
     g 1 ←1; g 2 ←g 2 ; g 2i+1 ←g 2i−1 *g 2  from 1 to (2 k−1 −1) of i 
     Step  1 : A←1; i←t 
     Step  2 : the following Steps are executed if i&gt;=0 
     Step  21 : if e t =0, A←A 2  and i←i−1 
     Step  22 : if not, the longest bit string e t , e t−1 , . . . e 1  where i−1+1&lt;=k and e 1 =1 is searched, and it is set that A←A 2 *g(e t , e t−1 , . . . , e 1 ) and i←1−1. 
     This algorithm is utilized for an algorithm of obtaining eP by using the addition operation of the invention. The algorithm of obtaining eP will be described. 
     Calculation of an e-multiple Point of a Point P using Sliding-window Method 
     Input: 
     g, e=(e t , e t−1 , . . . , e 1 , e 0 )e t =1, k (integer) 
     Output: 
     eP 
     Preliminary Calculations: 
     P 1 ←P; P 2 ←2P; P 2i+1 ←P 2i−1 +P 2  from 1 to (2 k−1 −1) of i 
     Step  1 : A←P; i←t 
     Step  2 : the following Steps are executed if i&gt;=0 
     Step  21 : if e t =0, A←2A; i←i−1 
     Step  22 : if not, the longest bit string e t , e t−1 , . . . e 1  where i−1+1&lt;=k and e 1 =1 is searched, and it is set that A←2A+P(e t , e t−1 , . . . , e 1 ) and i←1−1. 
     In the preliminary calculations of this algorithm, each P 2i+1  is expressed by [X 1 , Y 1 , Z 1 , (Z 1 ) 2 , (Z 1 ) 3 ]. The calculation of A←2A at Steps  21  and  22  is performed by [X 3 , Y 3 , Z 3 ]=2[X 1 , Y 1 , Z 1 ]. An addition calculation “+” in 2A+P(e t , e t−1 , . . . , e 1 ) at Step  22  is performed by [X 3 , Y 3 , Z 3 ]=[X 2 , Y 2 , Z 2 ]+[X 1 , Y 1 , Z 1 , (Z 1 ) 2 , (Z 1 ) 3 ]. 
     The above embodiments are practiced by software which executes the operations illustrated in FIGS. 3 to  5 . Obviously, the embodiments can be practiced by using electronic circuits. 
     With each embodiment of the invention, data enciphering and deciphering processes using the elliptic curve encryption can be executed faster than conventional processes. Also with each embodiment of the invention, it is obvious that a person skilled in the art can execute a digital signature generation process and a digital signature verification process at high speed by using the elliptic curve encryption. 
     As described in each embodiment of the invention, by using a high speed [X 1 , Y 1 , Z 1 , (Z 1 ) 2 , (Z 1 ) 3 ] for the addition arithmetic and a high speed [X 1 , Y 1 , Z 1 ] for the doubling in the elliptic curve encryption, the following advantages can be obtained. 
     (1) Addition arithmetic: By using [X 3 , Y 3 , Z 3 ]=[X 1 , Y 1 , Z 1 , (Z 1 ) 2 , (Z 1 ) 3 ]+[X 2 , Y 2 , Z 2 ], the addition calculation can be executed by performing a multi-precision integer multiplication modulo operation 14 times. 
     (2) In the scalar multiplication operation of an arbitrary point, a so-called window method is used to express a segment as [X 1 , Y 1 , Z 1 , (Z 1 ) 2 , (Z 1 ) 3 ] so that the above-described 14 operations can be used for the addition calculation. 
     (3) The doubling calculation can be performed by the operations of 10 times (by 8 times if a=−3) by executing a conventional [X 3 , Y 3 , Z 3 ]=2[X 1 , Y 1 , Z 1 ]. 
     According to the present invention, data enciphering and deciphering processes using the elliptic curve encryption can be executed faster than conventional processes, and a digital signature generation process and a digital signature verification process can be executed at high speed.