Abstract:
An apparatus for executing cryptographic calculation on the basis of an elliptic point on an elliptic curve includes: a memory for storing a first value including a plurality of digits; and a processor for executing a process including: obtaining a second value representing a point on the elliptic curve; calculating output values by using a predetermined equation, each digit of the first value, and the second value; determining whether at least one of the second value and the output values indicates a point of infinity; terminating the calculation when at least one of the second value and the output values indicates the point at infinity; and completing calculation when both the second value and the output values do not indicate the point at infinity, so as to obtain a result of the cryptographic calculation.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
       [0001]    This application is based upon and claims the benefit of priority of the prior Japanese Patent Application No. 2009-009091, filed on Jan. 19, 2009 the entire contents of which are incorporated herein by reference. 
       FIELD 
       [0002]    The embodiments discussed herein are related to an information security technique. 
       BACKGROUND 
       [0003]    Along with advancement of the information society, it is to be anticipated that services using an information network such as an electronic payment or resident registry network become widely available. An information security technique is indispensable to safely deliver such services. In addition, a public key cryptosystem is used as the basic technique for the information security. RSA and elliptic curve cryptosystem (hereinafter referred to as ECC: Elliptic Curve Cryptosystem) are known as major public key cryptosystems. The use of these cryptosystems makes it possible to realize information encryption, digital signature, and an authentication function, and to prevent the unauthorized third person from accessing personal information. 
         [0004]    Further, a smart card is known as a device of an end user of the above services. The smart card is a card with a built-in IC chip. Confidential information about a user is stored in a memory area in the IC chip of the smart card. Further, the IC chip of the smart card is furnished with an encryption function, a digital signature function, and an authentication function. The confidential information about a user is used as a key to the processing of these functions. Since the confidential information is stored in the internal memory of the card, the card ensures higher level of security against unauthorized access by the third party, i.e., tamper-proofing than a magnetic recording card. 
         [0005]    To describe analyses of such a smart card, however, power analysis attack (Power Analysis; hereinafter referred to as PA) is known. For example, non-patent document of P. Kocher, 3, Jaffe and B. Jun “Differential Power Analysis”, Crypto&#39; 99, LNCS 1666, pp. 388-397, Springer-Verlag, 1999 discloses power analysis attack. The outline of the PA will be described below with reference to the drawings.  FIG. 15  illustrates the outline of the PA. 
         [0006]    As illustrated in  FIG. 15 , the PA is a method for measuring how much a smart card consumes power during processing executed using an encryption function with user&#39;s confidential information K used as a key (hereinafter cryptographic processing) to estimate and analyze the confidential information K using the measured data. The PA is an attack aimed at analysis based on observation of a power change, so a target of attack is not limited to the smart card. For example, there is known another PA that measures an amount of electromagnetic waves generated during power consumption in a PDA device. For example, non-patent document of Catherine H. Gebotys, Simon ho, and C. C. Tiu, “EM Analysis of Rijndael and ECC on a Wireless Java-Based PDA”, Cryptographic Hardware and Embedded System, CHES 2005, pp. 250-264, LNCS 3659 discloses the another PA. In other words, the PA targets all built-in devices that would consume a power. 
         [0007]    Next, PA against built-in devices using RSA and ECC is described in detail. Since the PA is an attack utilizing the mechanism of arithmetic operations of RSA and ECC, the arithmetic calculation thereof is described first.  FIG. 16  illustrates a correspondence relationship between the arithmetic calculation of the RSA and the arithmetic calculation of the ECC. 
         [0008]    The RSA and the ECC have a correspondence relationship as illustrated in  FIG. 16 . Based on the correspondence relationship, the RSA calculation and the ECC calculation are described below. 
         [0009]    The arithmetic calculation of the RSA is described first. According to the RSA, processing is performed based on modular exponentiation. The modular exponentiation is calculation for finding z=a x (mod n) based on radix a, exponent x, and modulo n. In the RSA, processing is performed with x used as confidential information through the modular exponentiation. For example, processing for decoding a text coded by RSA is to find m satisfying the condition of m=c d (mod n) where c represents the code text and d represents a private key. As for electronic signature based on RSA, the above calculation is performed on target data c, a private key d, and modulo n to obtain electronic signature m. In either processing, the third person who does not know the private key d may not obtain correct decoding result and electronic signature. 
         [0010]    Next, the arithmetic calculation of the ECC is described. In the ECC, the following relation between x and y is referred to as elliptic curve. The elliptic curve is composed of two elements, prime field (prime field) and binary field (binary field). In addition, parameters a and b for uniquely determining the elliptic curve are referred to as elliptic curve parameters. The elliptic curve (prime field) is expressed by y 2 =x 3 +ax+b(mod p) where p represents the prime number. In this expression, p is the prime number, 0≦a, and b&lt;p. Further, the elliptic curve (binary field) is expressed by y 2 +xy=x 3 +ax 2 +b(mod f(x)). In this expression, f is a polynomial in GF(2 m ), and a, b ⊂ GF(2 m ). Further, (x, y) satisfying the relation representing the elliptic curve (prime field) and the elliptic curve (binary field) is referred to as an elliptic point (elliptic point). 
         [0011]    In the ECC, processing is performed based on elliptic scalar multiplication (Elliptic Scalar Multiplication). The elliptic scalar multiplication is arithmetic calculation to find a point V on the elliptic curve satisfying V=sA with an integer s called a scalar value. For example, ECDH key exchange in the ECC finds a point V on the elliptic curve satisfying V=dA where A represents a point on the elliptic curve as a public key of the other end, and d represents a private key. In this way, key sharing is safely realized. The third person who does not know a value of the private key d may not obtain a correct value of a shared key. 
         [0012]    In the above RSA encryption, electronic signature using RSA, and ECC encryption, a value of a private key d does must not be leaked to the third person who tries to attack encrypted data (hereinafter referred to as an attacker). In short, it is a tamper-proof function to protect a value of d in the RSA and the ECC. From the mathematical point of view, as is known, even if values other than d leak to an attacker, a calculation amount to derive a value of d from these values and thus, the value of d may not be easily obtained within a realistic time range. For example, it is known that upon decoding RSA-encrypted data, if n is 1024 bit or more, an attacker may not easily obtain a value of d even if the attacker knows values of c, n, and m. In addition, it is known that upon decoding data encrypted by ECC, if an elliptic curve parameter is 160 bit or more, an attacker may not easily obtain a value of d even if the attacker knows values of A and V. 
         [0013]    As described above, it is known that, although a value of a private key d may not be easily obtained from the mathematical point of view in the RSA and the ECC, the value can be easily revealed when using PA. The fundamental mechanism of the PA is considerably related to the modular exponentiation and scalar multiplication in the ECC. Accordingly, procedures for arithmetic calculation thereof are described prior to explanation about the PA against the RSA and the PA against the ECC. 
         [0014]    First, the arithmetic calculation procedure for the modular exponentiation in the RSA and the PA against the RSA are described.  FIG. 17  illustrates an algorithm of modular exponentiation based on a binary method.  FIG. 18  schematically illustrates modular exponentiation based on a binary method.  FIG. 19  schematically illustrates PA against modular exponentiation based on a binary method. 
         [0015]    As for the modular exponentiation in the RSA (RSA encryption and electronic signature based on RSA), if lengths of all of n, c, and d are 1024 bit or more, in the case of performing modular exponentiation following mathematical expression, multiplication may be performed with (mod n) d times. This arithmetic calculation requires a calculation amount of 2 1024  or more and thus is unrealistic. To that end, a binary method is known as a calculation method for reducing the calculation amount to log 2 d=1024. Hereinbelow, the binary method applied to modular exponentiation is described. According to the binary method, as illustrated in  FIG. 17 , if a u-bit private key d is represented by d=d u-1 ∥ . . . ∥d 1 ∥d 0  (where d i  corresponds to a 1-bit value), a bit value of d i  is scanned in the order from the higher-order bit to the lower-order bit (i.e., in the order from i=u−1 to i=0). Then, arithmetic calculation is performed in accordance with a bit valve of d i . Considering that d i ==1, multiplication (v: =v×a(mod n) in  FIG. 17 ) follows squaring (v: =v×v(mod n) in  FIG. 17 ). On the other hand, considering that d i ==0, only squaring is performed. To elaborate, an arithmetic calculation sequence of squaring and multiplication illustrated in  FIG. 18  directly correlates to a bit value of d i , and The PA against the RSA decodes d utilizing this characteristic. 
         [0016]    As illustrated in  FIG. 19 , the PA against the RSA reveals a private key d by measuring power consumption of a device that performs processing based on the binary method and distinguishing between power waveforms in multiplication and squaring. In the case where multiplication follows squaring, a bit value of d is revealed as 1. In the case of performing squaring alone, a bit value of d is revealed as 0. All bits of d are revealed in this way to thereby successfully perform PA against RSA. 
         [0017]    The arithmetic calculation procedure for point scalar multiplication in the ECC and PA against the ECC are described.  FIG. 20  illustrates an algorithm of the point scalar multiplication based on a binary method.  FIG. 21  schematically illustrates the point scalar multiplication based on a binary method.  FIG. 22  schematically illustrates PA against the point scalar multiplication based on a binary method. 
         [0018]    Similar to the modular exponentiation in the RSA, a binary method is also known as a calculation method for reducing a calculation amount thereof. Hereinbelow, the binary method for the scalar multiplication is described. According to the binary method, as illustrated in  FIG. 20 , if a u-bit private key d is represented by d=d u-1 ∥ . . . ∥d 1 ∥d 0  (where d i  corresponds to a 1-bit value), a bit value of d i  is scanned in the order from the higher-order bit to the lower-order bit (i.e., in the order from i=u−1 to i=0). Then, arithmetic calculation is performed in accordance with a bit valve of d i . Considering that d i ==1, addition of the point (V: =V+A in  FIG. 20 ; hereinafter referred to as ECADD: Elliptic Curve Addition)  17 ) follows doubling of the point (V=2V in  FIG. 20 ; hereinafter referred to as ECDBL: Elliptic Curve Doubling). On the other hand, considering that d i ==0, only ECDBL is performed. To elaborate, an arithmetic calculation sequence of ECDBL and ECADD illustrated in  FIG. 20  directly correlates to a bit value of d i , and The PA against the ECC reveals d utilizing this characteristic. 
         [0019]    As illustrated in  FIG. 22 , the PA against the ECC reveals a private key d by measuring power consumption of a device that performs processing based on the binary method and distinguishing between power waveforms in ECDBL and ECADD. In the case where ECDBL follows ECADD, a bit value of d is revealed as 1. In the case of performing ECDBL alone, a bit value of d is revealed as 0. All bits of d are revealed in this way to thereby successfully perform PA against ECC. 
         [0020]    The PA against the RSA and the PA against the ECC determine the type of processing for determining a bit value of d based on a power waveform of a consumed power in a device for performing the processing to thereby reveal a private key d. Here, as a measure against the above PA, Add-and-double-always method (hereinafter referred to as A&amp;D method) is known. For example, non-patent document of Jean-Sebastien Coron, “Resistance against Differential Power Analysis for Elliptic Curve Cryptosystems”, Cryptographic Hardware and Embedded System, CHES 1999, pp. 2920392, LNCS 1717 discloses the A&amp;D method. Hereinbelow, a measure against the PA against the RSA based on the A&amp;D method and a measure against the PA against the RSA based on the A&amp;D method are described. 
         [0021]    First, the measure against the PA to the RSA based on the A&amp;D method is described.  FIG. 23  illustrates an algorithm of modulo exponentiation based on the binary method in accordance with the A&amp;D method.  FIG. 24  schematically illustrates the PA against modulo exponentiation based on the binary method in accordance with the A&amp;D method. 
         [0022]    As illustrated in  FIG. 23 , modular exponentiation using the A&amp;D method is performed while a calculation result is stored constantly in v[0]. Further, in the processing in  1206 , even if a bit value d i  of d is 0, multiplication is performed. The multiplication result in v[1] is stored in v[0] upon the processing in  1208  if d i =1 and used in the next loop. On the other hand, if d i =0, the multiplication result v[1] in the processing in  1206  is not stored in v[0]. Instead, the squaring result v[0] in the processing of  1204  is stored in v[0] instead and used in the next loop if d 1 =0. In other words, although the multiplication in the processing in  1206  is performed regardless of a value of d i , the calculation result thereof is used for calculation in the next loop only when d i =1. According to such processing, as illustrated in  FIG. 24 , a power waveform of consumed power in the device for performing modular exponentiation based on the binary method is kept constant regardless of a value of d i . This makes it possible to prevent the PA utilizing a difference between power waveforms in multiplication and squaring. Further, since the multiplication result is used in the next loop only when d i =1, consistency of modular exponentiation is kept. 
         [0023]    Next, the measure against the PA to the ECC based on the A&amp;D method is described.  FIG. 25  illustrates an algorithm of the point scalar multiplication based on the binary method in accordance with the A&amp;D method.  FIG. 26  schematically illustrates PA against the point scalar multiplication based on the binary method in accordance with the A&amp;D method. 
         [0024]    As illustrated in  FIG. 25 , the point scalar multiplication using the A&amp;D method based on the binary method is performed while a calculation result is stored constantly in V[0]. Further, in the processing of  1406 , even if a bit value d i  of d is 0, ECADD is performed. In addition, in copying to V[0] according to a value in  1408 , if d i =0, the ECDBL result V[0] in the processing in  1404  is stored in V[0]. Further, if d i =1, the ECADD result V[1] in the processing in  1406  is stored in V[0]. In this way, V[0] storing calculation results is used for calculation in the next loop. According to such processing, as illustrated in  FIG. 26 , a power waveform of consumed power in the device for performing point scalar multiplication based on the binary method is kept constant regardless of a value of d i . This makes it possible to prevent the PA utilizing a difference between power waveforms in point doubling and ECADD. 
         [0025]    However, chosen message PA is known as a higher-level attack to the measure against the PA based on the A&amp;D method. For example, non-patent document of Sung-Ming Yen, Wei-Chih Lien, SangJae Moon, and JaeCheol Ha, “ 
         [0026]    Power Analysis by Exploiting Chosen Message and Internal Collisions-Vulnerability of Checking Mechanism for RSA-Decryption”, Mycrypt 2005, pp. 183-195, LNCS 3715 discloses a chosen message PA. Hereinbelow, the chosen message PA against the RSA and the chosen message PA against the ECC are described. 
         [0027]    First, the chosen message PA against the RSA is described.  FIG. 27  schematically illustrates the chosen message PA against the RSA. 
         [0028]    The aforementioned PA against the RSA is an attack to input a random value a upon calculating a remainder a d (mod n) of modulo exponentiation to measure a power waveform of consumed power in the processing and reveal a private key d. The chosen message PA against the RSA differs therefrom in that a special value is selected and input as a. According to this method, even the RSA using the A&amp;D method is attacked successfully. More specifically, as illustrated in  FIG. 27 , a difference is produced in power waveform according to the value of d i  by inputting a =−1(mod n). By determining the value of d i  based on the difference in power waveform, the private key d can be decoded. Here, a correspondence relationship between the type of doubling (processing in  1204  in  FIG. 23 ) with a loop variable i and a value of d i  is d 1 +1=0 upon doubling of 1×1, and is d i+1 =1 upon doubling of (−1)×(−1). Here, the correspondence relationship follows the algorithm illustrated in  FIG. 23 . Provided that d i =0, the squaring result v[0](=(−1)×(−1)=1) in the processing in  1204  is copied to v[0] through the processing in  1208 . Thus, squaring of 1×1 is performed in the next loop. On the other hand, if d i =1, the squaring result v[1](=1×(−1)=−1) in the processing in  1204  is copied to v[0] through the processing in  1208 . Thus, squaring of (−1)×(−1) is performed in the next loop. 
         [0029]    Subsequently, the chosen message PA against the ECC is described. Prior to a description about an attack to the aforementioned chosen message PA applied to the ECC, the point at infinity in the calculation procedure for point addition and point doubling based on the ECC is discussed. In the following description, the ECC is one utilizing the A&amp;D method.  FIG. 28  illustrates an algorithm of point addition of a prime field elliptic curve parameter.  FIG. 29  illustrates an algorithm of point doubling of a prime field elliptic curve parameter.  FIG. 30  illustrates an algorithm of point addition of a square elliptic curve parameter.  FIG. 31  illustrates an algorithm of point doubling of a square elliptic curve parameter. 
         [0030]    In the algorithm of the point addition or point doubling as illustrated in  FIGS. 28 to 31 , coordinates of the point are represented in three dimensions like (X, Y, Z). Further, the point at infinity is represented with the Z coordinate set to 0, i.e., Z=0. The point is as follows. That is, provided that the point at infinity is 0, the point at infinity satisfies a relation of A+O=0+A=A with any point A. All the algorithms illustrated in  FIGS. 28 to 31  involve processing related to the point at infinity. This processing includes exception processing applied to the case where an input or output value is the point at infinity. Examples of the exception processing include processing in  800  and  817  in  FIG. 28 , processing in  904  in  FIG. 29 , processing in  1000  and  1018  in  FIG. 30 , and processing in  1105  in  FIG. 31 . Hereinbelow, branching applied to the case of performing exception processing is explained as special branching. 
         [0031]    Upon calculation of A+B (B=ECDBL(V)) in the processing in  800  in  FIG. 28  and the processing in  1000  in  FIG. 30 , a point B is output as a calculation result if a point A is the point at infinity (Az==0). If the point B is the point at infinity (Bz==0), i.e., the point A or point B is the point at infinity (special branch  1 ), a point not regarded as the point at infinity is output as a calculation result. 
         [0032]    Further, if A==B upon the calculation of A+B (special branch  2 ), calculation of 2A (ECDBL(A)) is performed based on the calculation of A+B. Conceivable examples of the special branch  2  include the case where T 4 ==0 and T 5 ==0 in the processing in  817  in  FIG. 28  or the case where T 1 ==0 and T 2 ==0 in the processing in  1018  in  FIG. 30 . 
         [0033]    Further, if the calculation result is the point at infinity upon the calculation of A+B (special branch  3 ), coordinates (1, 1, 0) of the point at infinity are output as a calculation result. Conceivable examples of the special branch  3  include the case where T 4 ==0 and T 5 ≠0 in the processing in  817  in  FIG. 28  or the case where T i ==0 and T 2 ≠0 in the processing in  1018  in  FIG. 30 . 
         [0034]    Further, upon the calculation of 2A, if the point A as input data is the point at infinity, or 2A as output data is the point at infinity (special branch  4 ), coordinates (1, 1, 0) of the point at infinity are output as a calculation result. At the special branch  4 , the case where the point A as input data is the point at infinity corresponds to the case where T 3 ==0 in the processing in  904  in  FIG. 29  or in  1105  in  FIG. 31 . Further, the case where 2A as output data is the point at infinity corresponds to the case where T 2 ==0 in the processing in  904  in  FIG. 29  or the case where T 1 ==0 in the processing in  1105  in  FIG. 31 . 
         [0035]    Based on the above, the chosen message PA against the ECC is described. It is known that a point on the elliptic curve corresponding to a special value a=−1 in the above chosen message PA is a point A satisfying a relation of 2A=0 and A (0. This is because a satisfying a relation of a =−1 is a value satisfying a relation of a 2 =1 and a≠1. Applying the value to arithmetic calculation of the elliptic curve gives 2A=0 and A≠0. Further, in the chosen message PA against the ECC, A=P is used as input of point scalar multiplication. Here, the P is a point different from the point A. The point is such that a relation of 2P=0 and P≠0 is satisfied, a Y coordinate is 0 if an elliptic curve parameter is a prime field, and an X coordinate is 0 if an elliptic curve parameter is square. 
         [0036]    Considering that A=P is used as an input of scalar multiplication, 2P=0 as a result of ECDBL calculation in the processing in  1404  illustrated in  FIG. 25 , and an even multiple point is always the point at infinity  0 . Hence, a value stored in V[0] is always the point at infinity  0 . In the case where the ECDBL calculation in the processing in  1404  is performed based on the algorithm illustrated in  FIG. 29  (in the case where the arithmetic curve parameter is a prime field) or the algorithm illustrated in  FIG. 31  (in the case where the elliptic curve parameter is square), coordinates (1, 1, 0) as the point at infinity is output as a calculation result by the special branch  4  to thereby complete the ECDBL calculation. To elaborate, if A=P is used as an input of point scalar multiplication, special branch appears and thus, the ECDBL calculation is terminated, and no main calculation is performed. Hence, a power waveform in the ECDBL calculation involving the special branch differs from a power waveform in the ECDBL calculation involving no special branch. 
         [0037]      FIG. 32A  and  FIG. 32B  illustrate a power waveform in the ECDBL calculation involving the special branch and a power waveform in the ECDBL calculation involving no special branch. As described above, the special branch interrupts the ECDBL calculation, as illustrated in  FIG. 32A  and  FIG. 32B , a power wavelength thereof becomes shorter than that in the ECDBL calculation involving no special branch. 
         [0038]    Further, if A=P is used as an input of point scalar multiplication, the point at infinity is stored in V[0] as a result of ECDBL calculation in the processing in  1404  illustrated in  FIG. 25 . Thus, in the processing in  1406  illustrated in  FIG. 25 , arithmetic calculation of ECADD(0, P) is continuously performed. In the case where one input of the ECADD calculation is the point at infinity, main calculation is not performed due to the special branch  1 , and the ECADD calculation is terminated. To be specific, the ECADD calculation is terminated due to the special branch in the processing in  1000  illustrated in  FIG. 30  and the special branch in the processing in  800  illustrated in  FIG. 28 . Hence, a power waveform in the ECADD calculation involving the special branch differs from a power waveform in the ECADD calculation involving no special branch. 
         [0039]      FIG. 33A  and  FIG. 33B  illustrate the power waveform in the ECADD calculation involving the special branch and the power waveform in the ECADD calculation involving no special branch. As described above, since the special branch interrupts the ECDBL calculation, as illustrated in  FIG. 33A  and  FIG. 33B , the power wavelength thereof becomes shorter than that in the ECDBL calculation involving no special branch. 
         [0040]    As described above, A=P is used as an input of point scalar multiplication, so a special branch is caused by the ECDBL calculation and the ECADD calculation. Hereinbelow, a power waveform of the entire point scalar multiplication with a special branch is explained.  FIG. 34  illustrates a power waveform of the entire point scalar multiplication with the special branch. 
         [0041]    As illustrated in  FIG. 34 , if A=P is used as an input of point scalar multiplication, a power waveform of the entire point scalar multiplication is such that a power waveform in the ECADD calculation with the special branch and a power waveform in the ECADD calculation with the special branch appear alternately. In the chosen message PA against the ECC based on the A&amp;D method, processing corresponding to 1×1 and (−1)×(−1) in the chosen message PA against the RSA based on the A&amp;D method is not performed. Hence, a pattern for identifying a bit value of a scalar value d does not appear in the power waveform, with the result that a private key may not be revealed. 
         [0042]    However, a public key encoded in the ECC based on the A&amp;D method can be revealed by chosen message PA different from the above chosen message PA (hereinafter referred to as special branch PA). The special branch PA is based on a unique analysis of the inventor of the subject application. Hereinbelow, the special branch PA is described. 
         [0043]    The special branch PA uses A=Q as an input of scalar multiplication where Q represents a point satisfying the relation of 4Q=0 and 2Q≠0. If A=Q is input in the point scalar multiplication based on the A&amp;D method illustrated in  FIG. 25 , the processing of the ECDBL calculation with the loop variable i is performed in accordance with a value of d i+1 . If d i+1 ==0, ECDBL((2k)Q) is calculated with a predetermined integer k. In other words, if the ECDBL calculation is performed on the point of even multiple of Q, a calculation result is (4k)Q=0. In addition, the special branch occurs to terminate the ECDBL calculation, with the result that V[0]=0. On the other hand, if d i+1 ==1, ECDBL((2k+1)Q) is calculated with a predetermined integer k. In other words, if ECDBL is performed on the point of odd multiple of Q, a calculation result is (4k+2)Q (0, no special branch occurs, and all ECDBL calculations are performed and completed. As a result, V[0]=(4k+2)Q=2Q. 
         [0044]    Similar to the above ECDBL calculation, the ECADD calculation with the loop variable i is performed in accordance with a value of d i+1 . If d i+1 ==0, a result of ECDBL calculation is V[0]=0. Thus, ECADD(0, Q) is calculated. In this calculation, since one input of ECDBL is the point at infinity, the special branch occurs, and the processing is terminated. On the other hand, if d i+1 ==1, the ECDBL calculation result is V[0]=2Q. Thus, ECADD(2Q, Q) is calculated. In this calculation, both inputs of ECADD calculation are not the point at infinity, no special branch is involved, and all ECADD calculations are performed to terminate the processing. 
         [0045]    As is apparent from the above, if a correspondence relationship between steps of ECDBL calculation and a bit value of d i+1  is correct, a correspondence relationship between steps of ECADD calculation based on the ECDBL calculation result and a bit value of d i+1  is correct. Hereinbelow, the correspondence relationship between the steps of ECDBL calculation and a bit value of d i+1  is described. 
         [0046]    As for the loop variable i, a value copied to V[0] through copying processing in  1408  illustrated in  FIG. 25  (V[0]=V[d i ]) varies depending on a value of d i . If d i ==0, the ECDBL calculation result for V[0] in  1404  is a point of even multiple of Q, and 2kΩ is copied to V[0]. On the other hand, if d i ==1, the ECADD calculation result of V[1] in  1406  is a point +Q that is even multiple of Q, and (2k+1)! Is copied to V[0]. As described above, a value of V[0] with d i ==0 or d i ==1 is used as an input of ECDBL(V[0]) calculation in the next loop, i.e., at the time when the loop variable is i−1. Hence, a correspondence relationship between steps of the ECDBL calculation and a bit value of d i+1  is determined. 
         [0047]    As understood from the above correspondence relationship, the following correspondence holds: if A=Q is used as an input of scalar multiplication, processing is terminated due to the special branch in both of the ECDBL calculation and the ECADD calculation in the case where d i+1 ==0, while all calculations are performed to terminate the processing in both of the ECDBL calculation and the ECADD calculation in the case where d i+1 ==1. In short, a value of d i  can be estimated from a power waveform based on the correspondence relationship.  FIG. 35  illustrates the special branch PA against the ECC based on the A&amp;D method. 
         [0048]    As illustrated in  FIG. 35 , since a correspondence relationship holds between a value of d i  and a power waveform of the ECDBL calculation and the ECADD calculation in the next loop, a private key d can be decoded based on a power waveform of a device for performing point scalar multiplication with A=Q used as an input. 
         [0049]    However, as a technique for preventing a private key from being revealed by the above special branch PA, public key validation (hereinafter referred to as PKV: public Key Validation) is known. For example, non-patent document of STANDARDS FOR EFFICIENT CRYPTOGRAPHY, SEC 1: Elliptic Curve Cryptography, http://www.secg.org/download/aid-385/sec1_final.pdf discloses the PKV. According to the ECC using the PKV and the A&amp;D method (hereinafter referred to as PKV method), a point Q satisfying a relation of 4Q=0 and 2Q (0 is prevented from being used as an input of point scalar multiplication.  FIG. 36  illustrates an algorithm of PKV processing. 
         [0050]    The PKV is an algorithm for determining whether a target point A for scalar multiplication is a correct value applicable to cryptographic calculation based on a mathematical relationship. More specifically, as illustrated in  FIG. 36 , determination processing using the PKV is performed as pre-processing, and only a point A validated (regarded as valid) through this determination processing is input to the point scalar multiplication. Consider that the point scalar multiplication is performed on the point A validated by the PKV through n-fold multiplication. Provided that an order is r, it is known that, as long as a relation of d&lt;r is satisfied, all calculations are always performed in both of the ECADD calculation and the ECDBL calculation. In other words, since the scalar value d in the encryption satisfies the relation of d&lt;r all the time, no special branch is involved in the point scalar multiplication. As a result, the special branch PA against the PKV method becomes impossible. 
         [0051]    However, there is a problem that a private key can be revealed using an attack called Fault attack against the PKV method. Next, the Fault attack is described.  FIG. 37  schematically illustrates the Fault attack. 
         [0052]    As illustrated in  FIG. 37 , in this Fault attack, an encryption circuit on a built-in device such as a smart card is applied with various types of stress (abnormal clock, overvoltage, and high temperature). The stress leads to an abnormal value of internal data of the encryption circuit. Confidential information in the encryption circuit is read based on the abnormal value. The Fault attack against the PKV method is described below.  FIG. 38  schematically illustrates the Fault attack against the PKV method.  FIG. 39  illustrates an example of a point A selected upon the Fault attack against the PKV method.  FIG. 40  illustrates a real-time operation against the attack. 
         [0053]    The Fault attack against the PKV method is to bring about abnormality in the encryption circuit to falsify a point A regarded as valid upon the above determination processing to a point A′ that is different from the point A and satisfies the relation of A′=Q (4Q=0, 2Q≠0). The above special branch PA becomes possible due to the falsification. Here, a failure rate of the Fault attack is proportional to the number of bits falsified. By selecting an input value, the number of bits falsified can be reduced. A specific example of the Fault attack against the PKV method is described below. First, in the case of inputting a value with affine coordinates to a target encryption circuit for attack by an attacker, an attacker inputs a point A=(A x , A y ) closest to Q=(Q x , Q y ) in the affine coordinate system illustrated in  FIG. 39  is input. If the input point A is regarded as valid upon determination processing, the attacker applies stress to the encryption circuit to falsify coordinate data (A x , A y ) of the point A to (Q x , Q y ). These points are close to each other in the affine coordinate system, so tampering can be performed only by changing several bits. In addition, similar tampering could be performed in the other coordinate system such as Jacobian. As described above, the PKV method is not safe against falsification of an input value after the determination processing. To elaborate, as illustrated in  FIG. 40 , such a method as does not detect improper data in real time may not be protected against any attack after detection and thus has the lower security level against any attack than a method having a real-time detection function. 
       SUMMARY 
       [0054]    According to an aspect of the invention, an apparatus for executing cryptographic calculation on the basis of an elliptic point on an elliptic curve includes: a memory for storing a first value including a plurality of digits; and a processor for executing a process including: obtaining a second value representing a point on the elliptic curve; calculating output values by using a predetermined equation, each digit of the first value, and the second value; determining whether at least one of the second value and the output values indicates a point of infinity; terminating the calculation when at least one of the second value and the output values indicates the point at infinity; and completing calculation when both the second value and the output values do not indicate the point at infinity, so as to obtain a result of the cryptographic calculation. 
         [0055]    The object and advantages of the invention will be realized and attained by means of the elements and combinations particularly pointed out in the claims. 
         [0056]    It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory and are not restrictive of the invention, as claimed. 
     
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         [0057]      FIG. 1  illustrates the hardware configuration of an elliptic curve arithmetic processing unit according to a first embodiment of the present invention. 
           [0058]      FIG. 2  is a block diagram of the functional configuration of an elliptic curve arithmetic processing unit according to the first embodiment. 
           [0059]      FIG. 3  is a flowchart of point scalar multiplication according to the first embodiment of the present invention. 
           [0060]      FIG. 4  illustrates an algorithm of point scalar multiplication with a prime field elliptic curve parameter. 
           [0061]      FIG. 5  is a flowchart illustrating an operation of ECDBL calculation processing. 
           [0062]      FIG. 6  illustrates an algorithm of ECDBL calculation processing using a prime field elliptic curve parameter. 
           [0063]      FIG. 7  is a flowchart illustrating an operation of ECADD calculation processing. 
           [0064]      FIG. 8  illustrates an algorithm of ECADD calculation processing with a prime field elliptic curve parameter. 
           [0065]      FIG. 9  illustrates an algorithm of ECDBL calculation with a square elliptic curve parameter. 
           [0066]      FIG. 10  illustrates an algorithm of ECADD calculation with a square elliptic curve parameter. 
           [0067]      FIG. 11  illustrates a result of comparison between advantages of conventional methods and advantages of the present invention. 
           [0068]      FIG. 12  illustrates an algorithm of point scalar multiplication according to a second embodiment of the present invention. 
           [0069]      FIG. 13  illustrates an algorithm of ECADDDBL calculation according to the second embodiment. 
           [0070]      FIG. 14  illustrates an algorithm of point scalar multiplication according to a third embodiment of the present invention. 
           [0071]      FIG. 15  schematically illustrates PA. 
           [0072]      FIG. 16  illustrates a correspondence relationship between arithmetic calculation of RSA and arithmetic calculation of ECC. 
           [0073]      FIG. 17  illustrates an algorithm of modulo exponentiation based on a binary method. 
           [0074]      FIG. 18  schematically illustrates modulo exponentiation based on a binary method. 
           [0075]      FIG. 19  schematically illustrates PA against modulo exponentiation based on a binary method. 
           [0076]      FIG. 20  illustrates an algorithm of point scalar multiplication based on a binary method. 
           [0077]      FIG. 21  schematically illustrates point scalar multiplication based on a binary method. 
           [0078]      FIG. 22  schematically illustrates PA against point scalar multiplication based on a binary method. 
           [0079]      FIG. 23  illustrates an algorithm of modulo exponentiation based on a binary method in accordance with an A&amp;D method. 
           [0080]      FIG. 24  schematically illustrates PA against modulo exponentiation based on a binary method in accordance with an A&amp;D method. 
           [0081]      FIG. 25  illustrates an algorithm of point scalar multiplication based on a binary method in accordance with an A&amp;D method. 
           [0082]      FIG. 26  schematically illustrates PA against point scalar multiplication based on a binary method in accordance with an A&amp;D method. 
           [0083]      FIG. 27  schematically illustrates chosen message PA against RSA. 
           [0084]      FIG. 28  illustrates an algorithm of point addition of a prime field elliptic curve parameter. 
           [0085]      FIG. 29  illustrates an algorithm of point doubling of a prime field elliptic curve parameter. 
           [0086]      FIG. 30  illustrates an algorithm of point addition of a square elliptic curve parameter. 
           [0087]      FIG. 31  illustrates an algorithm of point doubling of a square elliptic curve parameter. 
           [0088]      FIG. 32A  and  FIG. 32B  illustrate a power waveform in ECDBL calculation involving special branch and a power waveform in ECDBL calculation involving no special branch. 
           [0089]      FIG. 33A  and  FIG. 33B  illustrates a power waveform in ECADD calculation involving special branch and a power waveform in ECADD calculation involving no special branch. 
           [0090]      FIG. 34  illustrates a power waveform of the entire point scalar multiplication with special branch. 
           [0091]      FIG. 35  illustrates special branch PA against ECC based on an A&amp;D method. 
           [0092]      FIG. 36  illustrates an algorithm of PKV processing. 
           [0093]      FIG. 37  schematically illustrates a Fault attack. 
           [0094]      FIG. 38  schematically illustrates a Fault attack against a PKV method. 
           [0095]      FIG. 39  illustrates an example of a point A selected upon a Fault attack against a PKV method. 
           [0096]      FIG. 40  illustrates a real-time operation against any attack. 
       
    
    
     DESCRIPTION OF EMBODIMENTS 
       [0097]    Hereinafter, embodiments of the present invention will be described with reference to the drawings. 
       First Embodiment 
       [0098]    To begin with, the hardware configuration of an elliptic curve arithmetic processing unit according to an embodiment of the present invention is described.  FIG. 1  illustrates the hardware configuration of an elliptic curve arithmetic processing unit according to the embodiment of the present invention. 
         [0099]    As illustrated in  FIG. 1 , an elliptic curve arithmetic processing unit  10  (apparatus) of this embodiment includes an ECC (Elliptic Curve Cryptosystem) processor  101 , a CPU (Central Processing Unit)  102 , a ROM (Read-Only Memory)  103 , an I/F  104 , an EEROM (Electrically Erasable ROM)  105 , a RAM (Random Access Memory)  106 , and a data bus  107  connecting these components with one another. Further, the elliptic curve arithmetic processing unit  10  includes an oscilloscope  20  for measuring power consumption, which is connected to Vcc and GND. The ECC processor  101  performs elliptic curve arithmetic processing related to encryption and electronic signature. Further, the CPU  102  controls the elliptic curve arithmetic processing unit  10 . Further, the ROM  103  stores programs executed by the ECC processor  101  and the CPU  102 . Further, the I/F  104  mediates data input/output to/from the elliptic curve arithmetic processing unit  10 . Further, the EEROM  105  is a ROM that can electrically erase data, and stores a private key d. In addition, the RAM  106  temporarily stores programs executed by the ECC processor  101  and the CPU  102 . 
         [0100]    Next, the functional configuration of the elliptic curve arithmetic processing unit of this embodiment is described.  FIG. 2  is a block diagram of the functional configuration of the elliptic curve arithmetic processing unit of this embodiment. 
         [0101]    As illustrated in  FIG. 2 , the elliptic curve arithmetic processing unit  10  of this embodiment includes a determination unit  301 , a computation unit  302  (calculation unit), and a termination unit  303  as functional units. The computation unit  302  performs calculation related to ECC. The determination unit  301  performs determination as to calculation of the calculation unit  302 . Further, the termination unit  303  terminates the calculation of the calculation unit  302  based on the determination of the determination unit  301 . Here, functions of each unit are realized by the ECC processor  101  and the CPU  102 . 
         [0102]    If an input value or output result is the point at infinity upon scalar multiplication of a point on the elliptic curve, the elliptic curve arithmetic processing unit  10  of this embodiment regards the input or output as attack and terminates the calculation processing. Hereinbelow, an operation of the elliptic curve arithmetic processing unit  10  of this embodiment is described. First, an operation of the entire point elliptic curve arithmetic processing unit  10  of this embodiment.  FIG. 3  is a flowchart of point scalar multiplication of this embodiment.  FIG. 4  illustrates an algorithm of point scalar multiplication with a prime field elliptic curve parameter. In the following description, A represents an input to the elliptic curve arithmetic processing unit  10  (predetermined point on the elliptic curve), d represents a scalar value (private key), i represents a bit position of d (the default is the highest-order bit position; considering u bits, i=u−2), d i  represents a value of ith bit, and d u-1 =1, 0 represents the point at infinity. The value d has a plurality of digits. Further, V[0] represents an array variable storing an input value and calculation result (output value), and V[1] represents an array variable storing a calculation result of the ECADD calculation. Further, a point on the elliptic curve is expressed by jacobian coordinates. 
         [0103]    As illustrated in  FIG. 3 , the computation unit  302  substitutes A to V[0] as an initial value of V[0]. This is because, as described above, the calculation processing is terminated if an input value is the point at infinity upon point scalar multiplication. In other words, not the point at infinity but A is set as an initial value of V[0], making it possible to prevent the calculation processing from stopping due to an initial value 0. 
         [0104]    Next, the computation unit  302  selects an ith bit (d i ) of d (S 102 ), performs ECDBL calculation with an input value, V[0], and then substitutes a calculation result to V[0] (S 103 , calculation step). Next, the determination unit  301  determines whether a return value in the ECDBL calculation is ERROR (S 104 , determination step). 
         [0105]    If the return value in the ECDBL calculation is not ERROR (S 104 , NO), the computation unit  302  performs ECADD calculation with A and v[0] as input values and then substitutes a calculation result to V[1] (S 105 , calculation step). Next, the determination unit  301  determines whether a return value in the ECADD calculation is ERROR (S 106 , determination step). The ECDBL calculation and the ECADD calculation are described later as ECDBL calculation processing and ECADD calculation processing. 
         [0106]    If the return value in the ECADD calculation is not ERROR (S 106 , NO), the computation unit  302  substitutes V[d i ] into V[0] (S 107 , calculation step). In other words, a value to be substituted to V[0] is determined according to a bit value. Next, the computation unit  302  subtracts 1 from i (S 108 , calculation step) to determine whether i is 0 or more (S 109 , calculation step). 
         [0107]    If i is smaller than 0 (S 109 , NO), the computation unit  302  outputs V[0] (S 110 , calculation step). 
         [0108]    On the other hand, if i is 0 or more (S 109 , YES), the calculation unit  302  selects the ith bit (d 1 ) of d again (S 102 ). 
         [0109]    Further, if the return value in the ECADD calculation is ERROR in step S 106  (S 106 , YES), the termination unit  303  terminates the scalar multiplication (S 111 , termination step). 
         [0110]    Further, if the return value in the ECADD calculation is ERROR in step S 104  (S 104 , YES), the termination unit  303  terminates the scalar multiplication (S 111 , termination step). 
         [0111]    More specifically, the above processing corresponds to an algorithm illustrated in  FIG. 4 . According to the algorithm, initialization with V[0]: =0 is not performed similar to the point scalar multiplication. Instead, in the processing in  3401 , initialization with V[0]: =0 is performed. Further, processing for maintaining calculation consistency is performed in  3402  to  3404  in parallel with the initialization with V[0]: =0. This processing finds the maximum value of i satisfying d i ==1, and i is decremented by 1. As a result, loop processing is started with the second-highest-order bit after the highest order bit with a bit value of 1. In addition, upon the loop processing in  3402  to  3404 , if the return values in the ECDBL calculation and the ECADD calculation are ERROR, point scalar multiplication is terminated. 
         [0112]    Next, the ECDBL calculation processing is explained. This ECDBL calculation processing corresponds to processing in step S 103  in  FIG. 3 .  FIG. 5  is a flowchart illustrating an operation of the ECDBL calculation processing.  FIG. 6  illustrates an algorithm of the ECDBL calculation processing using a prime field elliptic curve parameter. 
         [0113]    As illustrated in  FIG. 5 , first, the computation unit  302  determines whether V[0]=0 or 2V[0]==0 (S 201 ). 
         [0114]    If the relation of neither V[0]=0 nor 2V[0]==0 is satisfied (S 201 , NO), i.e., if an input value in the ECDBL calculation processing is not the point at infinity, the computation unit  302  performs point doubling with V[0] as an input value (S 202 ), and then substitutes a calculation result into V[0] (S 203 ). 
         [0115]    On the other hand, if V[0]=0 nor 2V[0]==0 (S 201 , YES), the computation unit  302  sends back ERROR as a return value (S 204 ). 
         [0116]    In other words, the computation unit  302  outputs ERROR as a return value if an input value or output value in the ECDBL calculation processing is the point at infinity. 
         [0117]    More specifically, the above processing corresponds to an algorithm illustrated in  FIG. 6 . In this algorithm, processing in  3004  corresponds to processing in steps S 201  and S 204 . Further, in the processing in  3004 , if T 2 ==0 or T 3 ==0, the processing corresponds to the aforementioned special branch  4 , and then return Error; corresponds to exception processing accompanying the special branch  4 . 
         [0118]    Next, the ECADD calculation processing is described. The ECADD calculation processing corresponds to processing in step S 104  illustrated in  FIG. 3 .  FIG. 7  is a flowchart illustrating an operation of the ECADD calculation processing.  FIG. 8  illustrates an algorithm of the ECADD calculation processing with a prime field elliptic curve parameter. 
         [0119]    As illustrated in  FIG. 7 , the computation unit  302  first determines whether A==0 (S 301 ). 
         [0120]    If a relation of A==0 is not satisfied (S 301 , NO), the computation unit  302  determines whether V[0]==0 (S 302 ). 
         [0121]    If a relation of V[0]==0 is not satisfied (S 302 , NO), the computation unit  302  performs point addition  1  (corresponding to processing in  2901  to  2916  in  FIG. 8  (S 303 ) to determine whether A+V[0]==0 (S 304 ). 
         [0122]    If a relation of A+V[0]==0 is not satisfied (S 304 , NO), the computation unit  302  performs point addition  2  (corresponding to processing in  2918  to  2935  in  FIG. 8  (S 305 ) and then substitutes a calculation result to V[1] (S 306 ). 
         [0123]    On the other hand, if A+V[0]==0 (S 304 , YES), the computation unit  302  outputs ERROR as a return value (S 307 ). As a result, if an input value or output value in the ECADD calculation processing is the point at infinity, point scalar multiplication is terminated. 
         [0124]    Further, if V[0]==0 in step S 302  (S 302 , YES), the computation unit  302  outputs ERROR as a return value (S 307 ). 
         [0125]    Further, if A==0 in step S 301  (S 301 , YES), the computation unit  302  outputs ERROR as a return value (S 307 ). 
         [0126]    In other words, the computation unit  302  outputs ERROR as a return value if an input value or output value in the ECADD calculation processing is the point at infinity. 
         [0127]    More specifically, the above processing corresponds to an algorithm illustrated in  FIG. 8 . In this algorithm, processing in  2900  corresponds to processing in steps S 301 , S 302 , and S 307 . Further, in the processing in  2900 , if A z ==0 or B z ==0, the processing corresponds to the aforementioned special branch  1 , and then return Error; corresponds to exception processing accompanying the special branch  1 . Moreover, in the processing in  2917 , if T 1 ==0 or T 2 ==0, the processing corresponds to the aforementioned special branch  3 , and then return Error; corresponds to exception processing accompanying the special branch  3 . 
         [0128]    The algorithms illustrated in  FIGS. 6 and 8  are related to point scalar multiplication with a prime field elliptic curve parameter. However, the above processing is applicable to point scalar multiplication with a square elliptic curve parameter.  FIG. 9  illustrates an algorithm of ECDBL calculation with a square elliptic curve parameter.  FIG. 10  illustrates an algorithm of ECADD calculation with a square elliptic curve parameter. 
         [0129]    The processing illustrated in  FIG. 5  corresponds to processing using the square elliptic curve parameter, more specifically, an algorithm illustrated in  FIG. 9 . According to the algorithm, the processing in  3205  corresponds to processing in steps S 201  and S 204 . Further, in the processing in  3205 , if T 1 ==0 or T 3 ==0, the processing corresponds to the aforementioned special branch  4 , and then return Error; corresponds to exception processing accompanying the special branch  4 . 
         [0130]    In addition, the processing illustrated in  FIG. 7  corresponds to processing using the square elliptic curve parameter, more specifically, an algorithm illustrated in  FIG. 10 . According to the algorithm, the processing in  3100  corresponds to processing in steps S 301 , S 302 , and S 307 . Further, in the processing in  3100 , if A z ==0 or B z ==0, the processing corresponds to the aforementioned special branch  1 , and then return Error; corresponds to exception processing accompanying the special branch  1 . Moreover, in the processing in  3118 , if T 1 ==0 or T 3 ==0, the processing corresponds to the aforementioned special branch  3 , and else return Error; corresponds to exception processing accompanying the special branch  3 . 
         [0131]    As described above, if the special branch (special branches  1 ,  3 , and  4 ) occurs, i.e., an input value or output value is the point at infinity in either the ECDBL calculation or the ECADD calculation, ERROR is output. By outputting ERROR, not only the EDBDL calculation but point scalar multiplication as higher-level processing is totally terminated. As a result, a short power waveform and a long power waveform do not appear together upon measuring power consumption of the elliptic curve arithmetic processing unit  10 , and only long power waveform is measured. This makes it possible to prevent an attack based on special branch PA utilizing a different in power waveform therebetween. In addition, since determination as to whether an attack is detected is similar to the determination in the ECADD calculation, an overhead of processing time to detect an attack can be eliminated. Moreover, in the point scalar multiplication, each time the ECDBL calculation and ECADD calculation, which are repeated with high frequency, are performed, an attack is detected. This enhances a real-time detection function. Moreover, if the real-time detection function is enhanced, it is possible to prevent an attack to falsify data at a predetermined timing like a Fault attack. 
         [0132]    As understood from the above, the present invention is superior to conventional methods in security, processing time, a real-time attack detection function.  FIG. 11  illustrates a result of comparison between advantages of the conventional methods and advantages of the present invention. 
         [0133]    As illustrated in  FIG. 11 , although the ECC using the A&amp;D method realizes a higher processing speed, the ECC is not resistant to an attack with an input of Q and may not detect an attack in real time. In addition, the PKV method is inferior in processing speed and may not detect an attack in real time albeit resistant to an attack with an input of Q. In contrast thereto, the present invention ensures security against an attack with an input of Q, high processing speed, and real-time attack detection. In short, according to the present invention, problems inherent to conventional ECC such as low resistance to an attack with an input of Q, low processing speed, and the lack of real-time detection function can be all solved. 
       Second Embodiment 
       [0134]    The elliptic curve arithmetic processing unit  10  according to another embodiment of the present invention has the same hardware configuration as that of the first embodiment but differs from the first embodiment in that the ECDBL calculation and the ECADD calculation are performed at the same time (ECADDDBL calculation) upon point scalar multiplication. The ECADDDBL calculation uses a prime field elliptic curve parameter and jacobian coordinates. The following description is focused on differences from the first embodiment.  FIG. 12  illustrates an algorithm of point scalar multiplication of this embodiment.  FIG. 13  illustrates an algorithm of ECADDDBL calculation of this embodiment. 
         [0135]    As illustrated in  FIG. 12 , the scalar multiplication of this embodiment differs from the first embodiment in that V[0]: =A; in  3501  and V[1]: =A; in  3502  are calculated for initialization. This is because the ECDBL calculation and the ECADD calculation are concurrently performed. Further, the multiplication differs from that in the first embodiment in that the whole point scalar multiplication is terminated in the case where the ECADDDBL calculation is performed in  3507 , and ERROR is sent back as a return value as a result of the ECADDDBL calculation. 
         [0136]    Further, the ECADDDBL calculation performs ECDBL calculation and ECADD calculation without calculating y coordinates as illustrated in  FIG. 13 . Thus, calculation is not applied to y coordinate values R y , S y  as an output result. Further, the point scalar multiplication algorithm is a so-called Montgomery-Ladder method. Further, an x coordinate I x  of a target point for scalar multiplication is used as an input for the ECADDDBL calculation. 
         [0137]    Further, in the processing in  3305  according to the algorithm illustrated in  FIG. 13 , if T 2 ==0 or T 4 ==0, the processing corresponds to the aforementioned special branch  1 , and then return Error; corresponds to exception processing accompanying the special branch  1 . Moreover, in the processing in  3311 , if T 3 ==0, the processing corresponds to the aforementioned special branch  3 , and then return Error; corresponds to exception processing accompanying the special branch  3 . Furthermore, in the processing in  3334 , if T 1 ==0, the processing corresponds to the aforementioned special branch  4 , and then return Error; corresponds to exception processing accompanying the special branch  4 . 
         [0138]    As described above, in the elliptic curve arithmetic processing unit  10  of this embodiment, similar to the first embodiment, if ERROR is sent back as a return value as a result of the ECADDDBL calculation, and the entire processing is terminated upon point scalar multiplication as a higher-level processing. Upon the point scalar multiplication, the ECADDDBL calculation processing is performed in place of the ECDBL calculation processing and the ECADD calculation processing to thereby reduce a table memory area and a calculation amount. 
       Third Embodiment 
       [0139]    In the scalar multiplication of the first embodiment, the ECADD calculation is performed once each time the ECDBL calculation is performed. However, point scalar multiplication according to another embodiment of the present invention uses a window method to perform ECADD calculation once every k ECDBL calculations. Here, the number of ECDBL calculations in this embodiment and the number of ECDBL calculations in the first embodiment are the same. In other words, by applying the window method, a frequency of ECADD calculation is reduced. The following description is focused on different operations from the first embodiment.  FIG. 14  illustrates an algorithm of point scalar multiplication of this embodiment. 
         [0140]    As illustrated in  FIG. 14 , the point scalar multiplication using the window method generates pre-calculation table data in order to reduce the ECADD calculation frequency. The pre-calculation table data is generated upon processing in  3601  to  3602 , and applied as W[x]=xA(0&lt;x&lt;2 k ). In addition, upon the point scalar multiplication using the window method, the maximum value of i satisfying a relation of (d ik+k-1 , d ik ) i  (0 is obtained, and initialization processing with V: =W[d ik+k-1 , . . . , d ik )] (point other than the point at infinity) is performed in the processing in  3603  to  3605 . Further, the loop processing in  3606  to  3611  is basically similar to a conventional window method but is different therefrom in that scalar multiplication is stopped if a return value in the ECADD calculation and the ECDBL calculation is ERROR. 
         [0141]    As described above, by applying the window method to the point scalar multiplication of the first embodiment, the number of ECADD calculations is reduced, with the result that a calculation amount for the point scalar multiplication can be reduced. 
         [0142]    Here, the ECADD calculation and the ECDBL calculation with jacobian coordinates are described above, but the processing for terminating the point scalar multiplication accompanying the special branch in each calculation is also applicable to the ECADD calculation and the ECDBL calculation with projection coordinates or affine coordinates. Further, in the above embodiments, if a return value in the ECADD calculation and the ECDBL calculation is ERROR, the point scalar multiplication is terminated. However, hardware resetting may be executed instead of terminating the processing. In addition, a flag indicating a detected attack may be set in a nonvolatile memory of the elliptic curve arithmetic processing unit  10  (for example, EEROM  105 ). By setting this flag, if a return value in the ECADD calculation and the ECDBL calculation is ERROR, the flag is set ON. If the flag is ON upon reboot following the hardware resetting, processing is performed to disable the device itself. The above algorithm of the point scalar multiplication is discussed for illustrative purposes, and the present invention is applicable to any algorithm of point scalar multiplication that does not perform initialization with the point at infinity. In other words, the initialization could be performed with any point other than the point at infinity. 
         [0143]    Further, the elliptic curve arithmetic processing unit of this embodiment can be provided in the form of computer. Further, a program that prompts a computer that implements the elliptic curve arithmetic processing unit to perform the above steps may be provided as an elliptic curve arithmetic processing program. The above program can be stored in a computer-readable recording medium. Here, examples of the computer-readable recording medium include an internal storage device incorporated into a computer such as a ROM or a RAM, a portable storage medium such as a CD-ROM, a flexible disk, a DVD disk, a magneto-optical disk, and an IC card, a database storing computer programs, or the other computers or database thereof or a transmission medium on a line. 
         [0144]    All examples and conditional language recited herein are intended for pedagogical purposes to aid the reader in understanding the invention and the concepts contributed by the inventor to furthering the art, and are to be construed as being without limitation to such specifically recited examples and conditions, nor does the organization of such examples in the specification relate to a showing of the superiority and inferiority of the invention. Although the embodiments of the present inventions have been described in detail, it should be understood that the various changes, substitutions, and alterations could be made hereto without departing from the spirit and scope of the invention.