JP2010148036A - Device, method and program for arithmetically operating elliptic curve - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To properly detect a failure-use attack while considering a detection rate to the failure-use attack and the quantity of an operation for the detection rate. <P>SOLUTION: A method for detecting the failure-use attack includes a scalar-multiplication arithmetic step of: computing a scalar-multiplication point [d]C<SB>1</SB>on the elliptic curve E on the basis of the elliptic curve E on a finite field, a point C<SB>1</SB>on the elliptic curve E and a scalar (d); and storing the scalar-multiplication point in a storage device. The method for detecting the failure-use attack further includes a failure-use attack detecting step of: computing pairing on the elliptic curve E by using the point C<SB>1</SB>and the scalar-multiplication point [d]C<SB>1</SB>; and deciding the presence of the failure-use attack when the result of computation of the pairing is not 1. <P>COPYRIGHT: (C)2010,JPO&INPIT

Description

  The present technology relates to failure detection attack detection technology for an apparatus that performs elliptic curve calculation processing.

  Conventionally, public key cryptosystems and common key cryptosystems are known as cryptosystems. The public key cryptosystem uses a pair of different keys for encryption and decryption. In a typical public key cryptosystem, a public key used for encryption is disclosed in advance, and a secret key used for decryption is held as secret information. For example, the sender encrypts the plaintext using the recipient's public key and sends the ciphertext. The recipient decrypts the ciphertext using the recipient's private key to obtain plaintext.

One of the encryption methods known as public key cryptosystems is elliptic curve cryptography. Elliptic curve cryptography is a general term for cryptosystems that use computation rules between points on an elliptic curve. prime number p, when the a natural number m, the finite GF (q) on the elliptic curve E number of elements q = ^{p m,}
E: y ² + a ₁ xy + a ₃ y = x ³ + a ₂ x ² + a ₄ x + a ₆
This is a set obtained by adding a point ∞ called a point at infinity to a set of points (x, y) satisfying. The infinity point ∞ may be expressed as 0. Here, a ₁ , a ₂ , a ₃ , a ₄ , a ₆ , x, and y are elements of the finite field GF (q). A point on an elliptic curve can be expressed in a coordinate format such as (x, y) (referred to as affine coordinates), but an infinite point ∞ is the only point that cannot be expressed in a coordinate format such as (x, y). Thus, in affine coordinates, points other than the point at infinity are represented as a set of two elements of the finite field GF (q).

Between the points on the elliptic curve E, addition of points can be defined as follows. First, let point P be a point on elliptic curve E on finite field GF (q). Then, the inverse element -P of the point P is defined as follows.
(A) When P = ∞, −P = ∞
(B) In cases other than (a) above, if P = (x, y), -P = (x, -y)
Further, the _{P 1} or _{P 2} and the point on the elliptic curve E over GF (q). Then, defined as follows sum _{P 3 P 1} and _{P 2.}
(A) When P ₁ = ∞, P ₃ = P ₂
(B) When P ₂ = ∞, P ₃ = P ₁
(C) When P ₁ = −P ₂ , P ₃ = ∞
(D) When P ₁ = (x ₁ , y ₁ ), P ₂ = (x ₂ , y ₂ ), and P ₃ = (x ₃ , y ₃ ) in cases other than (a) to (c) above,
x ₃ = λ ² + a ₁ λ-a ₂ -x ₁ -x ₂
y ₃ = (x ₁ −x ₂ ) λ−y ₁ + a ₁ x ₃ −a ₃
However,
When P ₁ ≠ P ₂ λ = (y ₂ −y ₁ ) / (x ₂ −x ₁ )
When P ₁ = P ₂ , λ = (3 × ₁ ² + 2a ₂ x ₁ + a ₄ −a ₁ y ₁ ) / (2y ₁ + a ₁ x ₁ + a ₃ )
P ₁ ≠ elliptic curve addition calculating the _P 1 + _{P 2} when _{P 2,} P 1 = the time of the _{P 2 P 1 + P 2 =} [2] to compute the _{P 1} of elliptic curve doubling.

  Given an elliptic curve E, a point P above it, and a scalar (here an integer scalar) d, calculate the point [d] P = P + P +... + P (sum of d Ps) Is called scalar multiplication. At this time, the point P is called a base point. A problem of obtaining a scalar d when an elliptic curve E, a base point P thereon, and a scalar multiple [d] P thereof are given is called an elliptic curve discrete logarithm problem. The security of elliptic curve cryptography depends on the fact that scalar multiplication can be easily calculated, but it is difficult to solve the elliptic curve discrete logarithm problem. For this reason, the elliptic cryptosystem often uses the scalar d as the secret key and the scalar multiple [d] P as part of the public key.

Next, an elliptic Elgamal cipher that is one of elliptic curve ciphers will be described as an example. Here, it is assumed that the finite field GF (p ^m ), the elliptic curve E and the base point P thereon are set in advance. Further, it is assumed that the receiver's private key is d and the public key is Y = [d] P.

When the message M is sent to the recipient, the following encryption processing is performed using the recipient's public key Y. Here, it is assumed that the message M is represented as a point on the elliptic curve E.
(1) generates a random number r (2) points to calculate the [r] P, held as _{C 1} (3) to calculate the point M + [r] Y, holds as _{C 2} (4) _{C 1} and C ₂ is sent to the recipient as ciphertext C (C ₁ , C ₂ )

On the other hand, the recipient who has received the ciphertext C performs the following decryption process using the recipient's private key d.
(1) Calculate the point [d] C ₁ and hold it as K (2) Calculate the point C ₂ -K to obtain the decrypted message M ′

On the other hand, it is assumed that the attacker who intercepted the communication tried to decrypt the ciphertext. First, the attacker can obtain the public key Y = [d] P and the base point P. However, calculating the secret key d from the public key Y means solving the elliptic curve discrete logarithm problem and is difficult. In addition, the attacker can obtain (that is, intercept) the ciphertext C (C ₁ , C ₂ ) = ([r] P, M + [r] Y). However, calculating the random number r from C ₁ = [r] P and the base point P means solving the elliptic curve discrete logarithm problem, which is difficult. Therefore, it can be said that d and r cannot be obtained, and it is difficult to calculate the scalar multiple [r] Y = [r] [d] P or to obtain the message M from C ₂ = M + [r] Y. is there.

  Elliptic curve cryptography has a feature that a similar security level can be secured with a shorter key length as compared with the RSA cryptosystem which is also one of public key cryptosystems. For example, it is considered that the security level of elliptic curve cryptography with a key length of 160 bits is almost the same as that of RSA cryptography with a key length of 1024 bits. For this reason, elliptic curve cryptography is often used in small devices such as a memory, an IC card having a small processing capability, and a mobile phone.

  On the other hand, there are various decryption techniques for identifying secret information such as a secret key from available information such as ciphertext. As a decryption technique for an encryption device, one that identifies secret information by physically observing the operation state of the encryption device or a change thereof is known. A failure use attack, which is one of analysis attacks, intentionally misleads the processing result by changing the environment significantly during the processing of the cryptographic device, such as by rapidly changing the supply voltage. Then, the attacker tries to specify confidential information based on the error.

Next, an example of a failure use attack that targets an authentication system using the elliptic El Gamal encryption will be described. For example, the authentication system includes an IC card and an authentication device, and the following authentication processing is performed between the IC card and the authentication device. Here, it is assumed that the finite field GF (p ^m ), the elliptic curve E and the base point P thereon are set in advance. Further, it is assumed that the IC card private key is d and the IC card public key is Y = [d] P. Further, it is assumed that the secret key d is held as secret information in the IC card.

(1) The authentication device randomly generates a message M and a random number r. Here, it is assumed that the message M is represented as a point on the elliptic curve E.
(2) The authentication device calculates C ₁ = [r] P and C ₂ = M + [r] Y, and transmits the ciphertext C = (C ₁ , C ₂ ) to the IC card.
(3) The IC card calculates K = [d] C ₁ and M ′ = C ₂ −K using the ciphertext C = (C ₁ , C ₂ ) and the secret key d, and authenticates the message M ′. Send to device.
(4) The authentication device compares the message M generated in (1) with the received message M ′, and if they match, the authentication is successful.

For the above authentication system, the attacker specifies the secret key d as follows. First, in the above (3), the attacker gives a physical attack such as a high voltage to the IC card. As a result, the IC card miscalculates K = [d] C ₁ and, as a result, sends an incorrect M ′ = C ₂ −K to the authentication device. The attacker acquires and analyzes this erroneous output in order to specify the secret key d of the IC card.

  Thus, there is a possibility that the secret key d is specified by the attacker by the failure use attack. Therefore, the IC card needs to detect a failure use attack and prevent the calculation result from being transmitted to the authentication device.

Next, a method for detecting a failure use attack by the IC card will be described. In general, it is determined whether the IC card has been subjected to a failure use attack based on C ₁ , K, and M ′. And when it determines with having received the failure use attack, IC card does not transmit message M 'to an authentication apparatus. As a specific example, there is a method using the property that the calculation result of K may not be a point on the elliptic curve E when a failure use attack is received. That is, the IC card determines whether the point is on the elliptic curve E by substituting K into the equation of the elliptic curve E. If it is not a point on the elliptic curve E, the IC card determines that a failure use attack has occurred. More specifically, it can be expressed by the following procedure.

(1) The elliptic curve E is E: y ² + a ₁ xy + a ₃ y = x ³ + a ₂ x ² + a ₄ x + a ₆
When expressed by an equation, function f (x, y) _{^{f (x, y) = x}} 3 + a 2 x 2 + a 4 x + a 6 - (y 2 + a 1 xy + a 3 y)
It is determined. Then, when K is expressed as (a, b), f (a, b) is calculated.
(2) When f (a, b) ≠ 0 (zero), it is determined that the IC card has been subjected to a failure use attack. On the other hand, when f (a, b) = 0 (zero), it is determined that the IC card is not subjected to a failure use attack.

However, in this method, even when a failure use attack is received, if K becomes a point on the elliptic curve E, the failure use attack cannot be detected. Actually, it is known that even when a failure use attack is received, it becomes a point on the elliptic curve E at a rate of about once every two times. Will occur.
JP 2004-252433 A

  As described above, when the elliptic curve calculation process is performed with an IC card or the like, it is necessary to take measures against a failure use attack. However, the conventional method has a problem in that there are many detection failures of failure use attacks. In addition, the calculation processing capability of the IC card is generally low, and there is a problem that the calculation time becomes longer as the calculation amount increases.

  Accordingly, an object of the present technology is to provide a new technique for appropriately detecting a failure use attack while taking into consideration the detection rate for the failure use attack and the amount of calculation for the detection rate.

The elliptic curve calculation device calculates a scalar multiple [d] C ₁ on the elliptic curve E based on the elliptic curve E on the finite field, the point C ₁ on the elliptic curve E, and the scalar d. The pairing on the elliptic curve E is calculated using the arithmetic means, the point C ₁ and the scalar multiple [d] C _1, and when the calculation result of the pairing is not 1, it is determined that a failure use attack has been received. First failure utilization attack detection means.

  It is possible to appropriately detect a failure use attack while taking into consideration the detection rate for the failure use attack and the amount of computation for that purpose.

[Description of pairing]
In this technology, a function called pairing on an elliptic curve is used to detect a failure use attack. Here, the premise pairing will be described. In the description of this pairing and the embodiment described later, the same symbol or character may be used in a different meaning.

Pairing is a function having the following properties when two points on an elliptic curve defined on a finite field (here, P and Q) are calculated as input values.
(A) (P ₁ + P ₂ , Q) = (P ₁ , Q) · (P ₂ , Q)
Here, P = P ₁ + P ₂ (b) (P, Q) = (Q, P)
(C) (P, P) = 1

Due to the above properties, when C ₁ and [d] C ₁ are input to the pairing function, they can be modified as follows.
(C ₁ , [d] C ₁ )
= (C ₁ , [d−1] C ₁ ) · (C ₁ , C ₁ )
= (C ₁ , [d−1] C ₁ ) · 1
If this is repeated until d becomes 1, the output value becomes 1. Therefore, it can be seen that the output value becomes 1 when the two points as the input value have a scalar multiplication relationship.

Further, in the present embodiment, it is assumed that Weil Pairing is employed. Baye pairing is defined as a function _er as follows.
_{e r (S, T) =} f S (Q + S) / f S (Q) · f T (P) / f T (P + T) ······ (1)
Here, r is a natural number, and E [r] is a set of points P on the elliptic curve E that satisfies [r] P = ∞. The point S, T is an input value of _{e r} is an element of the set E [r]. _Further, f S, _{f T} are each point S, a function derived from the T, Determination is described later in detail. Points P and Q are arbitrary points on the elliptic curve E in which P + T and Q + S do not become the infinity point ∞, respectively. Further, f _S and f _T are functions having a zero in the r-position only at points S and T, respectively.
Note that the output value of pairing is a value identified with a group of 1's roots of the original r, and is a value obtained by raising the above e _r (S, T) to the power of {(q−1) / r}. .

Next, an algorithm for deriving the function f _S will be described. First, f _S = g _r is set. Here, when the point S = (x _S , y _S ), g ₁ = 1, and g ₂ = x−x _S , g _{i + j} can be obtained as follows.
_{g i + j = g i ·} g j · l (x, y) / v (x, y) ······ (2)
The equation l (x, y) = 0 is a straight line as follows: i ≠ j, and when the straight line i = j passing through two points [i] S and [j] S on the elliptic curve E, A straight line passing through the point [i] S on the elliptic curve E and tangent to the elliptic curve E. Also, the equation v (x, y) = 0 is a straight line passing through [i + j] S and the infinity point ∞, and the above g _{i + j} from, determine the _{g r.} That is, r is decomposed into i + j iterations by binary expansion or the like, and g _{i + j} is repeatedly calculated to obtain g _r . For example, when r = 13, since r = 1 + 2 ² +2 ³ , g ₁ , g ₂ , g ₄ (= g _{2 + 2} ) and g ₈ (= g _{4 + 4} ) are obtained, and g ₅ = g _{1 + 4} , g ₁₃ = g _{5 + 8} .

Further, a specific calculation example of Bayeux pairing will be described. First, an elliptic curve E on the finite field GF (13) is determined as follows.
E: y ² = x ³ + x
With r = 2, a Bayeux pairing e ₂ (S, T) is calculated using the points S = (0, 0) and T = (5, 0) on the elliptic curve included in E [2] as input values. When P = (0, 0) and Q = (5, 0),
X coordinate of Q + S = λ ² + a ₁ λ−a ₂ −x ₁ −x ₂ = 0 ² + 0 · 0−0−5−0 = −5 = 8 (mod 13)
Y coordinate of Q + S = (x ₁ −x ₂ ) · λ−y ₁ −a ₁ x ₃ −a ₃ = (5-0) · 0-0-0 · (−5) −0 = 0 (mod 13)
Therefore, it is possible to calculate Q + S = (8, 0), and similarly P + T = (8, 0). Also,
f _S = x-0 = x
f _T = x-5
E ₂ (S, T) is as follows.
e ₂ (S, T) = 8/5 · (−5) / 3 = 12 (mod 13) / 2 (mod 13) = 6 (mod 13)
If this is equated with the group of 1 primitive roots, the output value of pairing is as follows.
e ₂ (S, T) = 6 ^{(13-1) / 2} = −1

[Embodiment]
A functional block diagram of an IC card and an authentication device according to an embodiment of the present technology is shown in FIG. In the present embodiment, a plurality of IC cards 1 (only one is shown in FIG. 1) and the authentication device 3 are connected to a predetermined type (for example, if the IC card 1 is a contact IC card) A communication method for a card, or a communication method for a non-contact type IC card (a communication method for a non-contact type IC card), and authenticating a user who has the IC card. Note that, as an encryption method used in the authentication process, an elliptic El Gamal encryption is adopted. However, the present technology can be applied not only to the elliptical Elgamal encryption but also to other elliptic curve encryptions.

  The IC card 1 shown in FIG. 1 includes a storage device 101, a public information storage unit 103, a secret key storage unit 105, a failure use attack detection unit 107, a decryption processing unit 109, a control unit 111, a communication Part 113. In the present embodiment, the public information storage unit 103 already stores the equation of the elliptic curve E on the finite field GF (q), the base point P, and the user's public key Y = [d] P, and stores the secret key. The storage unit 105 will be described assuming that the user's private key d is already stored.

  The communication unit 113 cooperates with the control unit 111 to store the ciphertext received from the authentication device 3 in the storage device 101 and transmit the message stored in the storage device 101 to the authentication device 3. The decryption processing unit 109 cooperates with the control unit 111 to decrypt the ciphertext using data stored in the storage device 101, the public information storage unit 103, and the secret key storage unit 105, and stores the decrypted ciphertext in the storage device 101. The failure use attack detection unit 107 cooperates with the control unit 111 to determine whether there is a failure use attack using data stored in the storage device 101 and the public information storage unit 103.

  The authentication device 3 includes a storage device 301, a public information storage unit 303, a random number generation unit 305, an authentication processing unit 307, an encryption processing unit 309, a control unit 311, and a communication unit 313. In the present embodiment, the public information storage unit 303 already stores the equation of the elliptic curve E on the finite field GF (q), the base point P, and the user's public key Y = [d] P. Will be described.

  The communication unit 313 cooperates with the control unit 311 to transmit the ciphertext stored in the storage device 301 to the IC card 1 and store the message received from the IC card 1 in the storage device 301. The random number generation unit 305 generates random data in cooperation with the control unit 311 and stores it in the storage device 301. The encryption processing unit 309 generates a ciphertext using data stored in the storage device 301 and the public information storage unit 303 in cooperation with the control unit 311, and stores the ciphertext in the storage device 301. The authentication processing unit 307 performs user authentication processing using data stored in the storage device 301 in cooperation with the control unit 311.

Next, the contents of the authentication process will be described with reference to FIGS. When the authentication process is started between the IC card 1 and the authentication device 3, the control unit 311 of the authentication device 3 displays a message M represented as a random point on the random number r and the elliptic curve E in the random number generation unit 305. Is generated and stored in the storage device 301. Next, the control unit 311 causes the encryption processing unit 309 to use C ₁ = [r] P using the random number r and the message M stored in the storage device 301 and the base point P stored in the public information storage unit 303. and _C 2 = M + [r] is calculated P, it is stored in the storage device 301 as a ciphertext _{C (C 1, C 2)} ( step S1). Then, the control unit 311 causes the communication unit 313 to transmit the ciphertext C stored in the storage device 301 to the IC card 1 (step S3).

On the other hand, when the communication unit 113 of the IC card 1 receives the ciphertext C, the control unit 111 of the IC card 1 causes the communication unit 113 to store the ciphertext C in the storage device 101 (step S5). Then, the control unit 111 calculates K = [d] C ₁ in the decryption processing unit 109 using the C ₁ stored in the storage device 101 and the secret key d stored in the secret key storage unit 105. And stored in the storage device 101 (step S7).

Next, the control unit 111 of the IC card 1, the failure to use the attack detecting section 107, K = [d] to execute a determining attack existence determining process or underwent fault attack in the calculation of _{C 1} (step S9). The details of the attack presence / absence determination process will be described later. During the process, the attack determination flag is set to ON or OFF in the storage device 101. And the control part 111 acquires the attack determination flag stored in the memory | storage device 101 via the failure utilization attack detection part 107, and when the attack determination flag is ON (step S11: Yes route), failure utilization is acquired. It is determined that an attack has been received, and predetermined abnormal processing is performed as necessary (step S13). On the other hand, when the attack determination flag is OFF (step S11: No route), the control unit 111 determines that a failure use attack has not been received, and continues the decoding process. The control unit 111 causes the decryption processing unit 109 to calculate the message M ′ = C ₂ −K using C ₂ and K stored in the storage device 101 and store the message M ′ = C ₂ −K in the storage device 101 (step S15). Then, the control unit 111 causes the communication unit 113 to transmit the message M ′ stored in the storage device 101 to the authentication device 3 (step S17).

  When the communication unit 313 of the authentication device 3 receives the message M ′, the control unit 311 of the authentication device 3 causes the communication unit 313 to store the message M ′ in the storage device 301 (Step S <b> 19). Then, the control unit 311 causes the authentication processing unit 307 to compare whether the message M and the message M ′ stored in the storage device 301 are the same, and notifies the control unit 311 of the result. If M and M ′ are the same, the control unit 311 determines that the authentication is successful (step S21). Note that the predetermined normal processing performed when authentication is successful and the predetermined abnormal processing performed when authentication fails are the same as those in the related art, and will not be described further.

[Attack presence / absence judgment processing]
Next, an attack presence / absence determination process using Baye pairing will be described with reference to FIGS. In the present embodiment, pairing is performed using C ₁ and K (= [d] C ₁ ) as input values, and it is determined whether or not a failure-use attack has occurred by determining whether the relationship is a scalar multiple. In the present embodiment, description will be made assuming that the public information storage unit 103 stores the number of elements q (= p ^m ) of the finite field GF (q) in advance. Further, it is assumed that all values or variables substituted in the attack presence / absence determination process are stored in the storage device 101.

First, the failure use attack detection unit 107 of the IC card 1 substitutes q stored in the public information storage unit 103 for a variable n (step S31). Next, the failure use attack detection unit 107 executes a function f _S derivation process to obtain a function f _C1 and a function f _K. The function f _C1 and the function f _K are functions required in the pairing calculation, and are obtained from the point C ₁ and the point K (= [d] C ₁ ), respectively. The fault attack detection section 107, a point _{C 1} of the ciphertext C stored in the storage device 101, and assigned to the S point is variable (step S33). Further, the failure use attack detection unit 107 performs a function f _S derivation process (step S35).

Here, the function f _S derivation processing will be described with reference to FIG. First, the failure use attack detection unit 107 substitutes an initial value 2 for i which is a counter for repeated processing (step S51). Then, fault attack detection section 107 substitutes the _{x-x S} to the function _{g 1} 1, the function _{g 2} (step S53). Here, _{x S} is the x-coordinate of the point S. The fault attack detection section 107, the variable n stored in the storage device 101 to expand binary, necessary for obtaining the functions _{g n,} all _{g 2} ^ z ₍₂ ^ z a is a power of 2 Is specified (step S55). For example, when n = 13, binary expansion is performed as 13 = 2 ⁰ +2 ² +2 ^3, and the necessary g _2z is specified as g ₁ , g _4, and g ₈ .

Here, _{if g 2 ^ z} required for calculating the _{g n} is not calculated all (step S57: No route), fault attack detection section 107 calculates the _{g 2i} accordance previously described (2) . First, the failure use attack detection unit 107 calculates a straight line l (x, y) = 0 that passes through [i] S and touches E (step S59). Next, the failure use attack detection unit 107 calculates [2i] S and a straight line v (x, y) = 0 passing through S and the infinity point ∞ (step S61). Then, the failure use attack detection unit 107 calculates g _2i = g _i · g _i · l (x, y) / v (x, y) (step S63). Further, the failure use attack detection unit 107 substitutes 2i for the variable i (step S65), and returns to the process of step S57.

On the other hand, _{if g 2 ^ z} required for calculating the _{g n} is calculated all (step S57: Yes route), fault attack detection section 107, the _{g 2 ^ z} identified by binary expansion, g _n is calculated (step S67). For example, when n = 13, g ₅ = g _{1 + 4} = g ₁ · g ₄ · l (x, y) / v (x, y), g ₁₃ = g _{5 + 8} = _{g 5 · g 8 · l (} x, y) / v is calculated as (x, y). Then, by substituting the calculated _{g n} to the function _{f S} (step S69), the processing returns to the original processing. In this way, the function f _S derivation process is performed. By this processing, f _C1 and f _K necessary for pairing calculation can be obtained.

Returning to the description of the processing in FIG. 3, the failure use attack detection unit 107 substitutes the derived function f _S for the function f _C1 (step S37). Next, fault attack detection section 107, point _{K (= [d] C 1} ) was substituted for the point S (step S39), and executes a function _{f S} deriving process (step S41). The function f _S derivation process has already been described with reference to FIG. Then, the failure use attack detection unit 107 substitutes the derived function f _S for the function f _K (step S43), and the processing shifts to the processing of FIG.

Then, fault attack detection section 107, a point U and V on the elliptic curve E is randomly selected (step S71), it calculates a V + _{C 1} and U + K (step S73). Here, when V + C ₁ = ∞ or U + K = ∞ (step S75: No route), the process returns to step S71, and the points U and V are selected again. On the other hand, when V + C ₁ ≠ ∞ and U + K ≠ ∞ (step S75: Yes route), the process proceeds to step S77.

The fault attack detection section 107 is stored in the storage device 101, the function _f C1, _{f K,} using a point U, V, V + _{C 1} and U + K, pairing _e n of _(C 1, K) Calculation is performed (step S77). The formula for the pairing, as described above in accordance with equation _{(1), e n (C 1, K} ) = f C1 (V + C1) / f C1 (V) · f K (U) / f K (U + K) The output value is obtained by multiplying the calculation result by {(q-1) / n}.

  Then, the failure use attack detection unit 107 determines whether the pairing output value is 1, and when the output value is 1 (step S79: Yes route), sets the attack determination flag of the storage device 101 to OFF. (Step S81), the process returns to the original process. On the other hand, when the output value of pairing is not 1 (step S79: No route), the failure use attack detection unit 107 sets the attack determination flag of the storage device 101 to ON (step S83), and returns to the original process.

  In this way, the attack presence / absence determination process is performed. The control unit 111 of the IC card 1 acquires the attack determination flag set in the process, and determines that it has been subjected to a failure use attack when ON, and determines that it has not been subjected to a failure use attack when OFF. be able to.

[Variation 1 of attack existence determination processing]
FIG. 6 shows a second attack presence / absence determination process that is a modification of the attack presence / absence determination process. In this modification, a second attack presence / absence determination process is performed instead of the attack presence / absence determination process described with reference to FIGS. In the attack presence / absence determination processing of FIGS. 3 to 5, the number of elements q (= p ^m ) of the finite field GF (q) is substituted for the variable n. substituting natural number _{n 1} of. As a result, the amount of processing for obtaining a function necessary for pairing can be reduced to n ₁ / q. Along with this, it is necessary to convert the input value of the pairing function, and as will be described later, the points C ₁ and K (= [d] C ₁ ) are converted to C ₁ to and K to, respectively. In a second attack existence determining process, the public information memory unit 103 will be described assuming that the number of elements q (= p ^m) and pre arbitrary natural number set of a finite field GF (q) has been stored . In addition, it is assumed that all values or variables substituted in the second attack presence / absence determination process are stored in the storage device 101.

First, the failure use attack detection unit 107 substitutes an arbitrary natural number n ₁ stored in the public information storage unit 103 for a variable n (step S91), and performs a modified pairing process (step S93).

Here, the modified pairing process will be described with reference to FIGS. First, the failure use attack detection unit 107 of the IC card 1 substitutes 1 as an initial value for j which is a counter for repeated processing (step S101). Then, the failure use attack detection unit 107 uses q stored in the public information storage unit 103 to determine whether q ^j −1 is divisible by n, and when it is not divisible (step S103: No route), j is j + 1. Is substituted (step S105), and the process returns to step S103. On the other hand, when q ^j −1 is divisible by n (step S103: Yes route), q ^j is substituted into variable l (step S107).

Then, fault attack detection section 107 uses the _{C 1, K (= [d} ] C 1) and n stored in the storage device 101, the point _{C 1 '= [1 / n} ] C 1 and A point K ′ = [1 / n] K is calculated (step S109). Here, the division on the elliptic curve is an existing technique and will not be described. Then, assuming that C ₁ ′ = (x _{C1 ′} , y _{C1 ′} ) and K ′ = (x _{K ′} , y _{K ′} ), the failure use attack detection unit 107 determines the x coordinates and y of C ₁ ′ and K ′. Points σ (C ₁ ) = (x _{C1 ′} ^l , y _{C1 ′} ^l ) and σ (K ′) = (x _{K ′} ^l , y _{K ′} ^l ) obtained by raising the coordinates to the power of ₁ are calculated (step S111). Here, each coordinate of a point q raised to the power points are mapped called Frobenius map, the coordinates l (= q ^j) raised to the power point is the point of application j times the Frobenius map. Further, the failure use attack detection unit 107 calculates C ₁ ~ = σ (C ₁ ′) −C ₁ ′ and K˜ = σ (K ′) − K ′ (step S113). In the modified pairing process, pairing can be calculated using an arbitrary natural number n ₁ set in advance by using C ₁ ~ and K ~ thus calculated as input values. In the modified pairing process, if n ₁ is reduced, the number of repetitions of steps S59 to S65 (FIG. 4) in the function f _S derivation process is reduced, and the amount of calculation can be reduced. However, when n ₁ is used, the output value of pairing is reduced to n ₁ and the failure detection attack detection rate is reduced.

Next, the failure use attack detection unit 107 executes the function f _S derivation processing using the points C ₁ to K and the points K to stored in the storage device 101, and obtains the functions f _{C1 to} and the functions f _{K to} , respectively. Ask. First, fault attack detection section 107, _{C 1} ~ point stored in the storage device 101 are substituted into S point is variable (step S115), it performs the function _{f S} deriving process (step S117). Since the function f _S derivation processing has already been described with reference to FIG. Then, the process proceeds to the process in FIG. 8 via the terminal B, fault attack detection section 107, the derived function _{f S, substituted-function f C1} (step S121).

Next, the failure use attack detection unit 107 substitutes the point K˜ for the point S (step S123), and executes a function f _S derivation process (step S125). Since the function f _S derivation processing has already been described with reference to FIG. The fault attack detection section 107, the derived function _{f S,} the function f _K Substituting _~ (step S127).

Next, the failure use attack detection unit 107 randomly selects points U and V on the elliptic curve E (step S129), and calculates V + C ₁ to and U + K to (step S131). Here, if V + C ₁ ~ = ∞ or U + K ~ = ∞ (step S133: No route), the process returns to step S129, and the points U and V are selected again. On the other hand, if V + C ₁ ~ ≠ ∞ and U + K˜ ≠ ∞ (step S133: Yes route), the process proceeds to step S135.

The fault attack detection section 107 is stored in the storage device 101, _~ function _{f C1,} the function f _{K ~,} the points U, V, using a ~ V + _{C 1} ~ and U + K, pairing _e n (C ₁ ~, K ~) is calculated (step S135), and the process returns to the original process. The formula for pairing according previously described _{(1), e n (C 1 ~,} K ~) = f C1 ~ (V + C 1 ~) / f C1 ~ (V) · f K ~ (U ) / F _{K ~} (U + K ~), and the result obtained by multiplying the calculation result by {(q-1) / n} is the output value. In this way, the modified pairing process is performed.

  Returning to the description of the processing in FIG. 6, the failure use attack detection unit 107 determines whether the output value of pairing is “1”. When the output value of pairing is 1 (step S95: Yes route), the failure use attack detection unit 107 sets the attack determination flag of the storage device 101 to OFF (step S97), and returns to the original process. On the other hand, when the output value of pairing is not 1 (step S95: No route), the failure use attack detection unit 107 sets the attack determination flag of the storage device 101 to ON (step S99), and returns to the original process.

In this way, the control unit 111 of the IC card 1 acquires the attack determination flag set in the processing, determines that it has been subjected to a failure use attack if it is ON, and receives a failure use attack if it is OFF. It can be determined that it is not. Further, as described above, if n ₁ set in advance is reduced, the amount of calculation can be reduced, but at the same time, the output value of pairing is degenerated, so the detection rate of failure use attacks is lowered. Therefore, in the second attack presence / absence determination process, it is necessary to set an appropriate n ₁ in consideration of the calculation amount and the detection rate. In the case where the normal Bayeux pairing is performed with the number of elements q = 2 ¹⁶⁰ of the finite field GF (q), when the second attack presence / absence determination processing is performed with n ₁ = 2 set, pairing is performed. The amount of calculation for obtaining the necessary function is 2/2 ¹⁶⁰ , and the failure detection attack detection rate is ½.

[Variation 2 of attack presence / absence determination processing]
FIG. 9 shows a third attack presence / absence determination process which is another modification of the attack presence / absence determination process. In the present modification, a third attack presence / absence determination process is performed instead of the second attack presence / absence determination process described with reference to FIG. In the third attack presence / absence determination process, a plurality of natural numbers are set in advance, and the second attack presence / absence determination process is performed on each of the natural numbers, thereby increasing the detection rate of the failure use attack. In the third attack presence / absence determination process, the public information storage unit 103 stores the number of elements q (= p ^m ) of the finite field GF (q) and a set N of arbitrary natural numbers set in advance (= {n ₁ , n ₂ ,..., N _t }) is already stored. In addition, it is assumed that all the values calculated in the third attack presence / absence determination process or the values assigned to the variables are stored in the storage device 101.

First, the failure use attack detection unit 107 assigns an initial value 1 to s, which is a counter for repeated processing, and an arbitrary set of natural numbers N = {n ₁ , n ₂ stored in the public information storage unit 103. , ..., a _{n s} of _{n t},} into the variable n (step S141), performs the deformation pairing process (step S143). Since the modified pairing process has already been described with reference to FIGS. 7 and 8, the description thereof will be omitted.

Next, the failure use attack detection unit 107 determines whether the output value of pairing is 1. When the output value of pairing is not 1 (step S145: No route), the failure use attack detection unit 107 sets the attack determination flag of the storage device 101 to ON (step S147), and returns to the original process. On the other hand, when the output value of pairing is 1 (step S145: Yes route), it is determined whether pairing calculation has been performed for all elements of the set N. If not n _s = _nt , that is, if pairing calculation has not been performed for all elements of N (step S149: No route), the failure use attack detection unit 107 substitutes s + 1 for s, n _s is substituted for n (step S151), and the process returns to step S143. On the other hand, when n _s = _nt, that is, when the pairing calculation is performed for all the elements of N (step S149: Yes route), the failure use attack detection unit 107 detects the attack of the storage device 101. The determination flag is set to OFF (step S153), and the process returns to the original process.

In this way, the control unit 111 of the IC card 1 acquires the attack determination flag set in the processing, determines that it has been subjected to a failure use attack if it is ON, and receives a failure use attack if it is OFF. It can be determined that it is not. Further, in the third attack presence / absence determination process, the amount of computation can be reduced by reducing the value of each natural number included in the set N of natural numbers set in advance, and each natural number included in N can be paired. By performing the calculation, it is possible to increase the detection rate of failure use attacks. For example, in the case where the normal Bayeux pairing is performed with the number of elements q = 2 ¹⁶⁰ of the finite field GF (q), when the third attack existence determination process is performed with N = {2, 3}, the pair The amount of calculation for obtaining the function necessary for the ring is 5/2 ¹⁶⁰ , and the failure utilization attack detection rate is 5/6. The value of each natural number included in N is preferably a prime number.

[Variation 3 of attack presence / absence determination processing]
FIG. 10 shows a fourth attack presence / absence determination process which is still another modified example of the attack presence / absence determination process. In the present modification, a fourth attack presence / absence determination process is performed instead of the third attack presence / absence determination process described with reference to FIG. Then, in the fourth attack presence / absence determination process, it is determined whether or not a failure use attack has been received by determining whether the point K is a point on the elliptic curve E before performing the attack presence / absence determination process by pairing. In the fourth attack presence / absence determination process, the public information storage unit 103 stores the number of elements q (= p ^m ) of the finite field GF (q), the equation of the elliptic curve E, and a set N of arbitrary natural numbers set in advance ( = {N ₁ , n ₂ ,..., N _t }) is assumed to be already stored. In addition, it is assumed that all values or variables substituted in the fourth attack presence / absence determination process are stored in the storage device 101.

First, the failure use attack detection unit 107 of the IC card 1 acquires the equation of the elliptic curve E stored in the public information storage unit 103 and the point K (= [d] C ₁ ) stored in the storage device 101. Then, K is substituted into the equation of E (step S161), and it is determined whether or not K is a point on E depending on whether or not the equation is established. For example, the equation of the elliptic curve E is E: y ² + a ₁ xy + a ₃ y = x ³ + a ₂ x ² + a ₄ x + a ₆
Is defined as f (x, y) = x ³ + a ₂ x ² + a ₄ x + a ₆ − (y ² + a ₁ xy + a ₃ y)
It is determined. Then, when K = (x _K , y _K ), a value f (x _K , y _K ) obtained by substituting K into the function f (x, y) is calculated. As a result, if f (x _K , y _K ) = 0, K is a point on E. If K is not a point on E (step S163: No route), the failure use attack detection unit 107 sets the attack determination flag of the storage device 101 to ON (step S165), and returns to the original process. On the other hand, when K is a point on E (step S163: Yes route), the failure use attack detection unit 107 executes the third attack presence / absence determination process of FIG. 9 (step S167) and returns to the original process.

  In this way, the control unit 111 of the IC card 1 acquires the attack determination flag set in the processing, determines that it has been subjected to a failure use attack if it is ON, and receives a failure use attack if it is OFF. It can be determined that it is not. Further, when a failure use attack is detected by determining whether the point K is on the elliptic curve E, it is not necessary to perform an attack presence / absence determination process by pairing, and the processing efficiency is increased. At the same time, the detection rate can be increased by combining the detection method of whether the point K is on the elliptic curve E and the detection method by pairing. In the present modification, the third attack presence / absence determination process is performed in step S167 (FIG. 10), but the second attack presence / absence determination process may be performed instead.

[Variation 4 of attack presence / absence determination processing]
FIG. 11 shows a fifth attack presence / absence determination process which is still another modified example of the attack presence / absence determination process. In the present modification, a fifth attack presence / absence determination process is performed instead of the second attack presence / absence determination process described with reference to FIG. Then, in the fifth attack presence / absence determination process, a point necessary for the modified pairing process is calculated using a Point Harving algorithm. Thereby, the amount of calculation can be further reduced. However, the number of elements of the finite field used in this modification must be a power of 2 and the natural number to be substituted for the variable n needs to be 2. In this modification, the public information storage unit 103 will be described assuming that the number of elements q (= 2 ^m ) of the finite field GF (q) has already been stored. In addition, it is assumed that all the values calculated in the present modification example or the values substituted for the variables are stored in the storage device 101.

  First, the failure use attack detection unit 107 of the IC card 1 substitutes 2 which is a fixed value for a variable n (step S171), and performs a second modified pairing process (step S173).

Here, the second modified pairing process will be described with reference to FIGS. First, the failure use attack detection unit 107 assigns 1 as an initial value to j which is a counter for repeated processing (step S181). Then, the failure use attack detection unit 107 determines whether q ^j −1 is divisible by n (= 2) using q stored in the public information storage unit 103, and when it is not divisible (step S183: No route) , J is substituted for j + 1 (step S185), and the process returns to step S183. On the other hand, when q ^j −1 is divisible by n (= 2) (step S183: Yes route), q ^j is substituted into variable l (step S187).

Then, fault attack detection unit 107 uses a _C 1 stored in the storage device 101, K (= [d] C 1) and n (= 2), the point _{C 1} '= _[1 / n ] C ₁ = [1/2] C ₁ and point K ′ = [1 / n] K = [1/2] K are calculated (step S189). A point harvesting algorithm is used for this calculation.

The point hard Bing for point P number of elements is on the elliptic curve defined on a finite field is a 2 ^m, the P = [2] Q (i.e. Q = [1/2] P) This is a technique for efficiently obtaining the point Q. For example, when the elliptic curve E on the finite field GF (2 ^m ) is defined as follows,
E: y ² + xy = x ³ + ax + b (where b ≠ 0)
An algorithm for obtaining the point Q = [1/2] P = (x _Q , y _Q ) from the point P = (x _P , y _P ) on E is as follows.
(1) Solve the equation x ² + x = x _P + a and set the solution to k (2) Calculate t = y _P + x _P k (3) When Tr (t) = 0, x _Q = (t + x _P ) ^(1/2) , when Tr (t) ≠ 0, x _Q = t ^(1/2) is calculated. Tr (t) = t + t ² + t ^{(2 ^ 2)} +... + T ^{{2 ^ (m-1)}} to ₍₄₎ Note that to calculate the ^{_{y Q = x Q 2 + λx}} Q, when Tr (t) = 0, and lambda = k, when Tr (t) ≠ 0, λ = By performing the processing as described above for k + 1, Q (x _Q , y _Q ) can be obtained with a smaller amount of calculation.

Next, assuming that C ₁ ′ = (x _{C1 ′} , y _{C1 ′} ) and K ′ = (x _{K ′} , y _{K ′} ), the failure use attack detection unit 107 determines the x coordinate of C ₁ ′ and K ′ and Points σ (C ₁ ) = (x _{C1 ′} ^l , y _{C1 ′} ^l ) and σ (K ′) = (x _{K ′} ^l , y _{K ′} ^l ) obtained by raising the y coordinate to the power of ₁ are calculated (step S191). . The fault attack detection section _{107, C 1 ~ = σ (C} 1 ') -C 1' and K ~ = σ (K ') - to calculate the K' (step S193). In this modification, thus calculated C _{1 ~} and K ~ for by the input value, it is possible to calculate the pairing with preset natural number 2.

Next, the failure use attack detection unit 107 executes the function f _S derivation processing using the points C ₁ to K and the points K to stored in the storage device 101, and obtains the functions f _{C1 to} and the functions f _{K to} , respectively. Ask. First, the failure use attack detection unit 107 substitutes the points C ₁ to C stored in the storage device 101 for the variable point S (step S195), and performs a function f _S derivation process (step S197). Since the function f _S derivation processing has already been described with reference to FIG. Next, the processing shifts to the processing in FIG. 13 through the terminal C, and the failure use attack detection unit 107 substitutes the derived function f _S for the function f _C1˜ (step S201).

Next, the failure use attack detection unit 107 substitutes the point K˜ for the point S (step S203), and executes a function f _S derivation process (step S205). Since the function f _S derivation processing has already been described with reference to FIG. The fault attack detection section 107, the derived function _{f S,} the function f _K Substituting _~ (step S207).

Next, the failure use attack detection unit 107 randomly selects points U and V on the elliptic curve E (step S209), and calculates V + C ₁ to and U + K to (step S211). Here, if V + C ₁ ~ = ∞ or U + K ~ = ∞ (step S213: No route), the process returns to step S209, and the points U and V are selected again. On the other hand, when V + C ₁ ~ ≠ ∞ and U + K˜ ≠ ∞ (step S213: Yes route), the process proceeds to step S215.

The fault attack detection section 107 is stored in the storage device 101, _~ function _{f C1,} the function f _{K ~,} the points U, V, using a ~ V + _{C 1} ~ and U + K, pairing _e n (C ₁ ~, K ~) is calculated (step S215), and the process returns to the original process. The formula for pairing according previously described _{(1), e n (C 1 ~,} K ~) = f C1 ~ (V + C 1 ~) / f C1 ~ (V) · f K ~ (U ) / F _{K ~} (U + K ~), and the result obtained by multiplying the calculation result by {(q-1) / n} is the output value. In this way, the second modified pairing process is performed.

  Returning to the description of the processing in FIG. 11, the failure use attack detection unit 107 determines whether the output value of pairing is “1”. When the output value of pairing is 1 (step S175: Yes route), the failure use attack detection unit 107 sets the attack determination flag of the storage device 101 to OFF (step S177), and returns to the original process. On the other hand, when the output value of pairing is not 1 (step S175: No route), the failure use attack detection unit 107 sets the attack determination flag of the storage device 101 to ON (step S179), and returns to the original process.

  In this way, the control unit 111 of the IC card 1 acquires the attack determination flag set in the processing, determines that it has been subjected to a failure use attack if it is ON, and receives a failure use attack if it is OFF. It can be determined that it is not. In addition, by using the point herving algorithm, it is possible to reduce the amount of calculation for obtaining points necessary for the deformation pairing process.

[Other variations]
As mentioned above, although embodiment of this technique was described, this technique is not limited to this. For example, the data management method in the IC card and the authentication device is not limited to that shown in FIG. Further, the functional configuration of the IC card and the authentication device is not limited to that shown in FIG. 1, and all functions may be implemented by an IC chip, or may be implemented by a processor and a program module.

  Further, as long as the processing result is the same for the processing flow, the processing flow may be performed in a different order or may be modified to be executed in parallel.

  In addition, the present technology can be applied to failure use attacks in general against scalar multiplication on an elliptic curve. In the embodiment of the present technology, the elliptical Elgamal encryption has been described as an example. However, in order to perform scalar multiplication in other elliptical curve cryptosystems, the present technology can also use an elliptical curve cryptosystem other than the elliptical Elgarmal cryptosystem. good. Similarly, any processing that performs scalar multiplication on an elliptic curve is not limited to authentication processing and encryption processing, and may be applied to electronic signatures and the like.

  Further, in the embodiment of the present technology, description has been made by taking Baye pairing as an example, but as described above, 1 is output when two points on the elliptic curve having a scalar multiplication relationship are calculated as input values. Since this is a general property of pairing, the present technology may use pairing other than Baye pairing.

Furthermore, in this embodiment, after randomly selecting the points U and V on the elliptic curve E, it is determined whether or not to select U and V again (step S75 (FIG. 5), step S133 (FIG. 8), step S213 condition (FIG. 13)) is selected by using the U and V pairing function _e n _(C 1, K) or _e n _{(C 1} ~, to calculate the K ~), the result is zero or a value It is good even if you cannot get it.

[Summary of embodiment]
This embodiment is summarized as follows.

This failure-use attack detection method calculates a scalar multiple [d] C ₁ on the elliptic curve E based on the elliptic curve E on the finite field, the point C ₁ on the elliptic curve E, and the scalar d. a scalar multiplication step of storing in a storage device, the pairing on the elliptic curve E is calculated using the point C ₁ and the scalar multiplied point [d] C _1, the failure in the case the calculation result of the pairing is not 1 A first failure use attack detection step for determining that a use attack has been received.

First, when a failure use attack is received in the scalar multiplication operation step, the calculation of the scalar multiple [d] C ₁ may be wrong, and C ₁ and [d] C ₁ may not have a scalar multiplication relationship. On the other hand, (a) (P1 + P2, Q) = (P1, Q). (P2, Q), (b) (P, Q) = (Q, P), (c) (P) , P) = 1, it can be derived that when the pairing calculation is performed using two points having a scalar multiplication relationship as input values, the output value becomes 1. Based on this property, pairing is calculated using C ₁ and [d] C _1, and it is determined whether or not the calculation result is 1, so that C ₁ and [d] C ₁ are multiplied by a scalar. You can check if they are in a relationship. Thereby, the presence or absence of a failure use attack can be determined more strictly.

As for the calculation of pairing, there are a method of calculating pairing with C ₁ and [d] C ₁ as inputs, and a method of calculating pairing after converting C ₁ and [d] C _1. It can be adopted. If it is the former, a detection rate can be raised.

Then, the first failure utilization attack detection step includes the point C ₁ ′ = [1 / n] C ₁ (n is an arbitrary natural number) = (x _{C1 ′} , y _{C1 ′} ) and the point [d] C ₁ ′ = Calculating [1 / n] [d] C ₁ = (x _{[d] C1 ′} , y _{[d] C1 ′} ) and the point σ (C ₁ ′) = (x _{C1 ′} ^l , y _{C1 ′} ^l ) (L = q ^m, q is the number of elements of the finite field, m is the smallest number that q ^m −1 is divisible by n) and point σ ([d] C ₁ ′) = (x _{[d] C 1 ′} ^l , y _{[d] C1 ′} ^l ), points C ₁ ~ = σ (C ₁ ′) −C ₁ ′ and points [d] C ₁ ~ = σ ([d] C ₁ ′) − [d ] A step of calculating C ₁ ′ and a step of calculating pairing on the elliptic curve E using the points C ₁ to C and the point [d] C ₁ to as input values. By thus converting the point at which the input values of the pairing, than when the input value points C ₁ and [d] C _1, it is possible to reduce the calculation amount required for the calculation of the pairing .

  Further, the first failure use attack detection step calculates pairing for each of a plurality of natural numbers n, and determines that a failure use attack has been received if a plurality of pairing calculation results include a value other than 1. A step may be included. Thus, the detection rate can be increased by performing the pairing calculation a plurality of times.

In addition, this failure use attack detection method determines whether the point [d] C ₁ is on the elliptic curve E before performing the first failure use attack detection step, and is determined not to be on the elliptic curve E. In such a case, a second failure use attack detection step for determining that a failure use attack has been received may be further included. If the second failure utilization attack detection step with a smaller amount of computation can be detected, it is not necessary to perform the first failure utilization attack detection step, and the processing efficiency can be increased.

Further, in the first failure utilization attack detection step, when q is a power of 2 and an arbitrary natural number n is 2, a point C ₁ ′ = [1 / n] C ₁ = (X _{C1 ′} , y _{C1 ′} ) and calculating the point [d] C ₁ ′ = [1 / n] [d] C ₁ = (x _{[d] C 1 ′} , y _{[d] C 1 ′} ) It may be included. Thereby, the amount of calculation required for calculation of the point C ₁ ′ and the point [d] C ₁ ′ can be reduced.

  The following additional notes are further disclosed with respect to the embodiments including the above examples and modifications.

(Appendix 1)
Scalar multiplication means for calculating a scalar multiple [d] C ₁ on the elliptic curve E based on the elliptic curve E on the finite field, the point C ₁ on the elliptic curve E, and the scalar d;
Pairing on the elliptic curve E is calculated using the point C ₁ and the scalar multiple [d] C _1, and if the calculation result of the pairing is not 1, it is determined that a failure use attack has been received. First failure-use attack detection means;
An elliptic curve calculation device.

(Appendix 2)
The first failure use attack detection means includes:
Point C ₁ ′ = [1 / n] C ₁ (n is an arbitrary natural number) = (x _{C1 ′} , y _{C1 ′} )
And the point [d] C ₁ ′ = [1 / n] [d] C ₁ = (x _{[d] C 1 ′} , y _{[d] C 1 ′} )
Point σ (C ₁ ′) = (x _{C1 ′} ^l , y _{C1 ′} ^l ) (l = q ^m, q is the number of elements of a finite field, m is the smallest number that q ^m −1 is divisible by n)
And the point _{σ ([d] C 1 '} ) = (x [d] C1' l, y [d] C1 'l) is calculated,
Point C ₁ ~ = σ (C ₁ ′) −C ₁ ′
And the point [d] C ₁ ~ = σ ([d] C ₁ ′) − [d] C ₁ ′,
The elliptic curve calculation device according to appendix 1, wherein pairing on the elliptic curve E is calculated with the points C ₁ to and the points [d] C ₁ to be input.

(Appendix 3)
The first failure use attack detection means includes:
Calculating the pairing for each of the plurality of natural numbers n;
The elliptic curve calculation device according to appendix 2, wherein the calculation result of the pairing includes that which is not 1 is determined to have been subjected to a failure use attack.

(Appendix 4)
Before the first failure utilization attack detection means performs processing, it is determined whether the point [d] C ₁ is on the elliptic curve E, and if it is determined that the point [d] C ₁ is not on the elliptic curve E The elliptic curve calculation device according to any one of supplementary notes 1 to 3, further comprising: a second failure use attack detection unit that determines that a failure use attack has been received.

(Appendix 5)
The first failure use attack detection means includes:
When q is a power of 2 and the arbitrary natural number n is 2,
In the point harvesting algorithm, the point C ₁ ′ = [1 / n] C ₁ = (x _{C1 ′} , y _{C1 ′} ) and the point [d] C ₁ ′ = [1 / n] [d] C ₁ = (x _{[d] C1 ', y [} d] C1') elliptic curve calculation device according to note 2, wherein the calculating the.

(Appendix 6)
The computer calculates a scalar multiple [d] C ₁ on the elliptic curve E based on the elliptic curve E on the finite field, the point C ₁ on the elliptic curve E, and the scalar d, and stores it in the storage device. A scalar multiplication step to store;
The computer calculates a pairing on the elliptic curve E using the point C ₁ and the scalar multiple [d] C _1, and if the calculation result of the pairing is not 1, a failure use attack is performed. A failure-use attack detection step for determining that it has been received;
Elliptic curve calculation method.

(Appendix 7)
A scalar multiple [d] C ₁ on the elliptic curve E is calculated based on the elliptic curve E on the finite field, the point C ₁ on the elliptic curve E, and the scalar d, and stored in the storage device. A doubling step,
Pairing on the elliptic curve E is calculated using the point C ₁ and the scalar multiple [d] C _1, and if the calculation result of the pairing is not 1, it is determined that a failure use attack has been received. A failure-use attack detection step;
Elliptic curve calculation program that makes the computer execute.

It is a system outline figure in an embodiment of this art. It is a figure showing a processing flow of an embodiment of this art. It is a figure which shows the process flow which concerns on the attack presence determination process of this technique. It is a diagram depicting a processing flow relating to the function f _S deriving process of the present technology. It is a figure which shows the process flow which concerns on the attack presence determination process of this technique. It is a figure showing the processing flow concerning the 2nd attack existence judging processing of this art. It is a figure which shows the process flow which concerns on the deformation | transformation pairing process of this technique. It is a figure which shows the process flow which concerns on the deformation | transformation pairing process of this technique. It is a figure showing the processing flow concerning the 3rd attack existence judging processing of this art. It is a figure showing the processing flow concerning the 4th attack existence judging processing of this art. It is a figure showing the processing flow concerning the 5th attack existence judging processing of this art. It is a figure showing the processing flow concerning the 2nd modification pairing processing of this art. It is a figure showing the processing flow concerning the 2nd modification pairing processing of this art.

Explanation of symbols

1 IC Card 3 Authentication Device 101 Storage Device 103 Public Information Storage Unit 105 Secret Key Storage Unit 107 Failure Utilization Attack Detection Unit 109 Decryption Processing Unit 111 Control Unit 113 Communication Unit 301 Storage Device 303 Public Information Storage Unit 305 Random Number Generation Unit 307 Authentication Processing unit 309 Encryption processing unit 311 Control unit 313 Communication unit

Claims (7)

Scalar multiplication means for calculating a scalar multiple [d] C ₁ on the elliptic curve E based on the elliptic curve E on the finite field, the point C ₁ on the elliptic curve E, and the scalar d;
Pairing on the elliptic curve E is calculated using the point C ₁ and the scalar multiple [d] C _1, and if the calculation result of the pairing is not 1, it is determined that a failure use attack has been received. First failure-use attack detection means;
An elliptic curve calculation device. The first failure use attack detection means includes:
Point C ₁ ′ = [1 / n] C ₁ (n is an arbitrary natural number) = (x _{C1 ′} , y _{C1 ′} )
And the point [d] C ₁ ′ = [1 / n] [d] C ₁ = (x _{[d] C 1 ′} , y _{[d] C 1 ′} )
Point σ (C ₁ ′) = (x _{C1 ′} ^l , y _{C1 ′} ^l ) (l = q ^m, q is the number of elements of a finite field, m is the smallest number that q ^m −1 is divisible by n)
And the point σ ([d] C ₁ ′) = (x _{[d] C1 ′} ^l , y _{[d] C1 ′} ^l ),
Point C ₁ ~ = σ (C ₁ ′) −C ₁ ′
And the point [d] C ₁ ~ = σ ([d] C ₁ ′) − [d] C ₁ ′,
The elliptic curve calculation device according to claim 1, wherein pairing on the elliptic curve E is calculated by using the points C ₁ ~ and the points [d] C ₁ ~ as inputs. The first failure use attack detection means includes:
Calculating the pairing for each of the plurality of natural numbers n;
The elliptic curve calculation device according to claim 2, wherein if the calculation result of the plurality of pairings includes a value other than 1, it is determined that a failure use attack has been received. Before the first failure utilization attack detection means performs processing, it is determined whether the point [d] C ₁ is on the elliptic curve E, and if it is determined that the point [d] C ₁ is not on the elliptic curve E The elliptic curve calculation device according to any one of claims 1 to 3, further comprising second failure use attack detection means for determining that a failure use attack has been received. The first failure use attack detection means includes:
When q is a power of 2 and the arbitrary natural number n is 2,
In the Point Halving algorithm, the point C ₁ ′ = [1 / n] C ₁ = (x _{C1 ′} , y _{C1 ′} ) and the point [d] C ₁ ′ = [1 / n] [d] C The elliptic curve calculation device according to claim 2, wherein ₁ = (x _{[d] C1 ′} , y _{[d] C1 ′} ) is calculated. The computer calculates a scalar multiple [d] C ₁ on the elliptic curve E based on the elliptic curve E on the finite field, the point C ₁ on the elliptic curve E, and the scalar d, and stores it in the storage device. A scalar multiplication step to store;
The computer calculates a pairing on the elliptic curve E using the point C ₁ and the scalar multiple [d] C _1, and if the calculation result of the pairing is not 1, a failure use attack is performed. A failure-use attack detection step for determining that it has been received;
Elliptic curve calculation method. A scalar multiple [d] C ₁ on the elliptic curve E is calculated based on the elliptic curve E on the finite field, the point C ₁ on the elliptic curve E, and the scalar d, and stored in the storage device. A doubling step,
Pairing on the elliptic curve E is calculated using the point C ₁ and the scalar multiple [d] C _1, and if the calculation result of the pairing is not 1, it is determined that a failure use attack has been received. A failure-use attack detection step;
Elliptic curve calculation program that makes the computer execute.

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