KR100657401B1 - Method for elliptic curve cryptography for preventing side channel attack and system thereof - Google Patents

Abstract

Disclosed are an elliptic curve encryption method and a system capable of preventing side channel attacks. The method for encrypting an elliptic curve according to the present invention comprises using a point P on the elliptic curve and d for calculating Q = dP as a secret key, respectively selecting random values of odd integers for each overlapping window, and random values selected for each window. And generating each encoded sequence selected from the plurality of encoded binary numbers having the same value as the secret key d, and performing scalar multiplication using the generated sequence and P to generate Q. As a result, it is possible to encrypt an elliptic curve safely and efficiently against the existing secondary channel attack as well as the Second-Order DPA.

Elliptic Curve Encryption, Scalar Multiplication, Side Channel Attack, Windows

Description

Method for elliptic curve cryptography for preventing side channel attack and system

1 is a diagram showing an algorithm for converting to w-NAF for an odd scalar,

2 is a view for explaining an overlapping window method, and

3 is a flowchart provided to explain an elliptic curve encryption method according to the present invention;

4 is a table comparing the stability of the elliptic curve encryption method and the conventional elliptic curve encryption method according to the present invention,

5 is a table comparing the efficiency and storage space for the elliptic curve encryption method and the conventional elliptic curve encryption method according to the present invention,

6 is a table comparing the safety, efficiency, and storage space of the countermeasures safe against subchannel attacks for the elliptic curve encryption method and the conventional elliptic curve encryption method according to the present invention; and

7 is a table comparing the calculation amounts of the corresponding methods in the Jacobian coordinate system transformed by the projection coordinate system.

The present invention relates to an elliptic curve encryption method and a system thereof, and more particularly to an elliptic curve encryption method and system that can prevent attacks using side channels (Side Channel).

Since the elliptic curve cryptographic system has a smaller key length to maintain the same cryptographic stability as compared to the RSA (Rivest-Shamir-Adleman) cryptographic system, it is suitable to be implemented in an environment with limited computing power or memory such as a smart card. However, without paying attention to the implementation of the elliptic curve cryptographic system, an attacker can find secret information using subchannel information such as algorithm execution time and power consumption. An attack using such subchannel information is called a side channel attack (SCA).

On the other hand, many methods have been proposed to increase the efficiency for implementing an elliptic curve cryptosystem in a resource-limited environment such as computing power or memory. If additional memory is available, Windows-based methods can speed up the elliptic curve cryptosystem. However, since most of the Windows-based methods are analyzed by the Second-Order DPA proposed by Okeya-Sakurai, the problem of effectively preventing Second-Order DPA attacks is currently the biggest issue. The disadvantage of the countermeasures against the Second-Order DPA proposed so far is that it reduces the efficiency of computation.

Accordingly, various window-based countermeasures have been proposed to prevent side channel attacks. Okeya-Takagi classifies the window-based countermeasure into two types. The first form uses a fixed window. This method was first proposed by Moller and Okeya-Takagi proposed a method to reduce the storage space by using the w -NAF method. The second approach is to make it safe from side-channel attacks by randomly changing window-based addition chains. This method was first proposed by Liardet-Smart, and further improved by Itoh et al.

Hereinafter, as a window-based method for preventing side channel attacks, the Moller method, the Okeya-Takagi method, and the overlapping window method will be described. Both the Moller method and the Okeya-Takagi method are secured in Simple Power Analysis (SPA) by reconstructing the secret key in a fixed form. In particular, the Okeya-Takagi method focuses on reducing storage space using the w -NAF method. Finally, Itoh and three others proposed a window-based DPA (Differential Power Analysis) countermeasure including a nested window method.

라 하면, 일반적인 타원곡선 스칼라 곱셈 연산에서인 경우, 덧셈 연산을 수행하지 않게 되며 이러한 성질로 인하여 SPA에 의해서 비밀키가 노출된다. 따라서, Moller는 k_i = 0 인 경우가 발생하지 않도록 비밀키를 다음의 식과 같은 새로운 표현으로 변환하는 방법을 제안하였다.First, let's talk about Moller's method.

In this case, in general elliptic curve scalar multiplication operation, the addition operation is not performed and the secret key is exposed by the SPA due to this property. Thus, Moller proposed a method of converting the private key in the following expressions such as the expression of new avoid if k _i = 0.

The recording method is 0 ≤ c _i ≤ 2 and 0 ≤ t _i ≤ 2 ^w The temporary variables c _i and t _i that satisfy + 1 can be expressed as follows.

Let c ₀ = 0. And i = 0,... , for e + 1, t _i = k _i + c _i ,

Where c _{i + 1} · 2 ^w + d _i = t _i is always satisfied and the length of the output integer is increased by one digit at most.

Next, the method of Okeya-Takagi will be described.

In memory-constrained environments such as smart cards, cryptographic algorithms must be efficient with less storage space. The safe countermeasures proposed by Moller for SPA are based on the m-ary method and require 2 ^w of storage. The window method using less storage is the w -NAF method. The number of storage spaces required by the w -NAF method is 2 ^w-2 . Okeya-Takagi proposed a safe method for side channel attack using w -NAF method. The Okeya-Takagi method is to transform the w -NAF method into an SPA safe addition chain. This add chain creates scalars in the following fixed format:

여기서, x < 2^w 는 홀수인 정수이다.

Where x <2 ^w is an odd integer.

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The number of storage spaces required by Okeya-Takagi's method is 2 ^w-1 . Therefore, it requires less storage space than Moller method. The Okeya-Takagi method reduces the amount of storage by converting all even places to odd places. For example, when w = 4, the odd digits are 0001, 0011, 0101, 0111, 1001, 1011, 1101, and 1111. Even number | 0100 | The method of converting to odd digits is as follows.

Then | 0100 | Except even places can be converted to process as above and | 0100 | How to convert is as follows.

Using the above transformation, all even places can be converted to all odd places as follows:

1 shows an algorithm for converting w -NAF to an odd scalar using the above properties. If scalar k is even, then d = k + 1, and if odd, d = k + 2. D is always odd. Therefore, after performing scalar multiplication by applying d to the algorithm shown in FIG. 1, kP is calculated by calculating dP-P or dP-2P.

However, since the Moller method and the Okeya-Takagi method are SPA safe countermeasures, additional operations are required to defend against other attacks (DPA, Second-Order DPA, Address-DPA).

Next, the overlapping window method secured by the DPA proposed by Itoh et al. Will be described. First, the notation is described before explaining the method.

는 비밀키 값이다. r 은 윈도우의 크기이고, w_i 는 사전 계산된 테이블에 대한 인덱스 값이다. 즉 w_i 는 윈도우이다. q는 w_i의(i = 0,1, q-1) 개수이다. h_i 는 w_i와 w_i+1 사이의 중첩된 비트의 길이를 나타낸다. 즉, 0< h_i <r 이다.

Is the secret key value. r is the size of the window and w _i is the index value for the precomputed table. That is, w _i is a window. q is the number of w _i (i = 0, 1, q-1). h _i represents the length of the overlapped bits between w _i and w _{i + 1} . That is, 0 <h _i <r.

bit (a, x, ..., y) represents a bit string from the x th bit to the y th bit in the binary representation of a (x = 0, ..., 1; y = 0, 1, ...; x ≥ y) . Bits greater than the most significant bit are considered to be zero. For example, if a = 6 = (110) ₂ then bit (a, 0) = 0, bit (a, 1) = 1 and bit (a, 4,3,2,1) = (0011) ₂ = 3 .

ECADD and ECDBL represent elliptic curve addition and double operations, respectively.

The feature of the nested window method is that it allows the overlap of windows w _i and w _{i + 1} . W _i and w _{i + 1,} generated from the fixed secret key value d, change randomly with each scalar multiplication. Therefore, it is safe for DPA because it is very difficult for an attacker to predict intermediate results.

The nested window method always satisfies the following equation.

The secret key represented by the nested window method is:

However, the nested window method has the same amount of computation to create a table as compared to the k-ary method, but more operations refer to the table. That is, more ECADD operations must be performed.

Accordingly, an object of the present invention is to provide a safe and efficient elliptic curve encryption and a system for the existing sub-channel attack as well as Second-Order DPA.

In the elliptic curve encryption method according to the present invention for achieving the above object, using a point P on the elliptic curve and the d for calculating Q = dP as a secret key, selecting a random value of an odd integer for each overlapping window Generating an encoded sequence selected from a plurality of encoded binary numbers each having the same value as the secret key d according to the random value selected for each window, and generating the scalar multiplication using the generated sequence and P. Performing, generating the Q.

The random value can be calculated by the following equation.

Where u [i] is the random value, w is the window size, h is the overlapped length of the window, t is the length of the secret key, and dt = bit (d, t-1, ... t-w).

In addition, the length of the secret key can be calculated by the following equation.

Where t is the length of the secret key, s is the number of windows, h is the overlapping length of the window, and w is the size of the window.

On the other hand, in the elliptic curve encryption system according to the present invention, means for selecting a random value of an odd integer for each overlapping window using the d for calculating a point P on the elliptic curve and Q = dP as a secret key, each window Means for generating a coded sequence selected from a plurality of coded binary numbers each having the same value as the secret key d, and performing a scalar multiplication using the generated sequence and the P according to the random value selected by Means for generating Q.

Hereinafter, with reference to the drawings will be described the present invention in more detail.

In the elliptic curve encryption method according to the present invention, a method of reducing the storage space of the overlapping window method using w -NAF is used.

3 is a flowchart provided to explain an elliptic curve encryption method according to the present invention.

Referring to FIG. 3, first, an n-bit odd number d, a window size w, and a overlapped length h of a window are received (S100). In the present invention, when the size of the window is w, the overlapped length of the window uses a fixed value (h) to be safe for SPA.

When d, w and h are input, the number of windows s and the length t of the secret key are calculated (S105). The number of windows s and the length of the private key t are calculated by the following equation.

이다.For example, if w = 4 and h = 2 for 7-bit secret key d = (1001101) ₂ , s = 3 and t = 3 · (4-2) + 2 = 8. Thus, the secret key can be represented by an 8-bit integer. In addition, an example of the sequence generated from d is

to be.

When the number of windows s and the length t of the secret key are calculated, dt used for random value selection is calculated (S110). At this time, dt is calculated by the following formula.

Next, i = s-1 is set (S115), and a random value u [i] which is an odd integer in the current window (i = s-1) is selected (S120). The random value u [i] is selected within the range satisfying the following condition.

If u [i] is selected, the most significant bit is determined as u [i] in the permutation for generating a secret key, and the remaining bits are set to '0' (S125). This is expressed as follows.

Next, dt to be used in the next window is calculated (S130). At this time, dt is calculated by the following formula.

When dt is calculated, t = t-(w-h) and i = i-1 (S135) to move to the next window. Then, it is determined whether i> 0 (S140), and steps S120 to S135 are repeatedly performed until i = 1.

When steps S120 to S135 are repeatedly performed until i = 1, d _w [0] = dt (S145). Then, scalar multiplication is performed (S150). The operation of adding one point P to d times is called a scalar multiplication of one point, denoted by dP.

The final output value is expressed as the following equation.

Since the present invention uses a fixed h to prevent SPA, scalar multiplication is performed in the following format.

Then the number of precomputed points is 2 ^w-1 and the density of nonzero bits is 1 / ( wh ). In addition, the secret key is expressed by the following equation.

In the elliptic curve encryption method as described above, the number of cases where u [i] may be 2 in step S120 is generally 2 ^h . Therefore, it is safe for DPA because a different u [i] is chosen for each scalar multiplication. The difference between the present invention and the conventional overlapping window method is that u [i] selected in step S120 is an odd integer. That is, while the conventional method selects a positive integer, in the present invention, an odd integer is selected. Therefore, the storage space is cut in half because only the odd points need to be stored in the table.

If even integers are entered in the above algorithm, the problem is the least significant digit. In other words, it becomes even in step S145. Here are two ways you can turn an even scalar into an odd number.

The first method is proposed by Okeya-Takagi. If d is even, then d '= d + 1, and if odd, d' = d + 2. D 'is always odd and dP is restored by calculating d'P or d'P-2P after scalar multiplication.

The second method starts with the following assumptions.

Assumption. Suppose the points entered in the elliptic curve scalar multiplication are large primes. In other words, it is assumed that small numbers are removed before scalar multiplication is performed.

Several standard documents for protocols such as ECIES, single-pass ECDH, and single-pass ECMQV use small cofactors before small scalar multiplications to prevent small subgroup attacks. To figure out. Thus, the above assumption is justified. The point P, where the rank is a large prime, is pP = 0. If d is even, then d '= d + p, and if odd, d' = d. D 'is always odd, and d'P = dP.

,

등과 같이 다양한 방법으로 표현된다. 이와 같이 dP를 계산할 때마다 d의 표현을 임의로 변환하여 계산함으로써 부채널 공격에 안전하게 암호화를 구현할 수 있다.The binary representation of secret key d is uniquely determined. At this time, if d is expressed using -1, 0, 1, the same number can be expressed in many ways. That is, if d is 5

,

It is expressed in various ways as shown. As described above, whenever dP is calculated, the expression of d is arbitrarily converted and calculated to safely implement encryption in side channel attacks.

Meanwhile, the safety and efficiency of the elliptic curve encryption method according to the present invention are analyzed, and the encryption method according to the present invention is compared with the conventional method with security, calculation amount and storage space as follows. First, we analyze the safety of the elliptic curve encryption method according to the present invention. Here, SPA, DPA, Second-Order DPA and Address-DPA are considered and safety is analyzed using Attenuation Ratio (AR). AR is evaluated by the ratio of peaks occurring with and without a corresponding method. ARs with values between 0 and 1 are smaller and safer. If AR = 0, the attacker cannot see the peak and the countermeasure is safe. If AR = 1, the attacker can always see the peak and the response is not safe. AR for SPA, DPA, Second-Order DPA and Address-DPA are denoted by AR _S , AR _D , AR _SD and AR _AD respectively.

The elliptic curve encryption method according to the present invention calculates a scalar multiplication in a fixed form for x ≠ 0. An attacker can distinguish between ECADD and ECDBL by measuring power consumption. However, all AD sequences appearing for all scalars occur in a fixed form:

여기서, A와 D는 각각 ECADD와 ECDBL을 의미한다.

Here, A and D mean ECADD and ECDBL, respectively.

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Regardless of the secret key, wh ECDBL and 1 ECADD operations are always performed, so there is no information that an attacker can obtain by SPA. Therefore AR _S = 0. In other words, the proposed algorithm is safe for SPA.

Since the elliptic curve encryption method according to the present invention uses a different sequence in each scalar multiplication, it is very difficult for the attacker to predict the median value that appears during the scalar multiplication.

Data representations for the same point in the projected coordinate system are different. For example for

or

When computed by, these two representations are the same in the affine coordinate system but different in the projective coordinate system. That is, X ₁ ≠ X ₂ , Y ₁ ≠ Y ₂ , and Z ₁ ≠ Z _{2 with} high probability.

In the elliptic curve encryption method according to the present invention, the number of cases where u [i] can be h and the number of non-zero windows is s.

For example, when w = 4 and h = 1 for a 160 bit scalar, AR _D = 2 ^-53 . Itoh et al.3 noted that sufficient safety can be provided if the magnitude of the peak occurring in the device to which the countermeasure is applied is reduced by 100 times the size of the peak occurring in the device implemented without the countermeasure. Therefore, this figure is sufficiently safe for DPA.

Next, look at Second-Order DPA (SO-DPA). Okeya-Sakurai proposed a second-order DPA attack on Moller's Windows method. Second-Order DPA attacks use the property of referencing the same table value when the secret key has the same place. Therefore, in order to prevent Second-Order DPA, a method of randomizing the table whenever a table is referred to should be used.

However, since the elliptic curve encryption method according to the present invention uses a different sequence in each scalar multiplication, the value referring to the table is changed every time. Therefore, it is difficult to find the secret key by the Second-Order DPA. Moreover, the elliptic curve encryption method according to the present invention has an advantage of increasing the efficiency of the operation because no additional operation is required to prevent the second-order DPA.

In the case of Address-DPA (A-DPA), toh-Izu-Takenaka stated that the countermeasure in the form of randomizing a secret key is safe for A-DPA. The random key randomization method randomly changes the representation of the secret key each time a scalar multiplication occurs, so the address of the register also changes randomly. The method according to the invention is therefore safe for A-DPA.

4 is a table comparing the stability of the elliptic curve encryption method and the conventional elliptic curve encryption method according to the present invention, showing the safety for SPA, DPA, Second-Order DPA and Address-DPA. In FIG. 4, an item labeled "Proposed" indicates an elliptic curve encryption method according to the present invention.

5 is a table comparing the efficiency and storage space for the elliptic curve encryption method according to the present invention and the conventional elliptic curve encryption method. The amount of scalar multiplication is evaluated by the number of ECDBL and ECADD, respectively. The number of storage spaces is evaluated by the number of points stored in the table. In FIG. 5, the number of ECDBL and ECADD does not include the amount of calculation required when creating the table, and indicates the amount of calculation required when performing scalar multiplication. Since the method according to the invention shows in Fig. 5 that it requires more ECADD operations than the Moller method and the Okeya-Takagi method, the calculation speed of the method according to the invention is higher than that of the Moller method and the Okeya-Takagi method. Seems slow However, if a countermeasure is added to actually prevent Second-Order DPA, the method according to the present invention will be shown to be more efficient.

Moller's countermeasures and Okeya-Takagi's countermeasures are not safe for DPA and Second-Order DPA because they are both proposed algorithms to prevent SPA. Therefore, these countermeasures should use DPA countermeasures such as random projection coordinate system or random elliptic curve isomorphism to prevent DPA, and randomize the table every time ECADD operation is performed to prevent Second-Order DPA. . For example, to randomize a table in the Jacobian coordinate system, an operation must be performed for each ECADD process. That is, four finite field multiplications and one finite field square operation are additionally required for each ECADD.

6 is a table comparing safety, efficiency, and storage space of countermeasures that are safe against subchannel attacks with respect to the elliptic curve encryption method and the conventional elliptic curve encryption method according to the present invention. In FIG. When both DPA and Second-Order DPA are configured securely, they show the amount of computation and storage for the 160-bit scalar and window 4 and in the proposed algorithm. In FIG. 6, Additional Mul. Is an amount of computation required when performing (ADD ³ r, R ³ Y, rZ) before the ECADD operation in the Jacobian coordinate system. By randomizing the address of the register proposed by Address-DPA, it can be easily prevented without any additional operation.

7 is a table comparing the calculation amounts of the corresponding methods in the Jacobian coordinate system transformed by the projection coordinate system. In FIG. For the countermeasures against SPA, DPA, and Second-Order DPA, the 160-bit scalar and the window with 4 and h = 1 show the amount of computation in the various coordinate systems defined in GF (p).

In GF (p), the squared amount is generally consumed by 80% of the multiply amount (S = 0.8M). The elliptic curve encryption method according to the present invention can be calculated when the ECADD is calculated, it can be seen that the operation is efficient. Moreover, in the elliptic curve encryption method according to the present invention, the selection of the coordinate system is free, but the Moller method and the Okeya-Takagi method are not. If the Moller method and the Okeya-Takagi method perform ECADD operation on Jacobian Chundonovsky coordinate system, five coordinates need to be randomized to prevent Second-Order DPA. Accordingly, the method according to the present invention can freely select coordinates that are effective for each operation for ECADD and ECDBL operations, thereby increasing the efficiency of the operation.

On the other hand, the elliptic curve encryption method according to the present invention can also be embodied as computer executable code on a computer-readable medium. Computer readable media includes all kinds of media readable by a computer system, and examples thereof include ROM, RAM, CD-ROM, magnetic tape, floppy disk, optical disk, and the like.

As described above, according to the present invention, by applying the w- NAF method to the overlapping window method, it is possible to reduce the storage space and improve the efficiency of the operation while being safe against the existing side channel attack. In addition, if a mixed coordinate system is used to increase the efficiency, the conventional methods require additional calculation according to the coordinate system, but the encryption method according to the present invention can increase the efficiency by freely selecting the coordinate system without additional calculation. Therefore, the elliptic curve encryption method according to the present invention can be efficiently operated using a small storage space while safe for the existing sub-channel attack.

In addition, although the preferred embodiment of the present invention has been shown and described above, the present invention is not limited to the specific embodiments described above, but the technical field to which the invention belongs without departing from the spirit of the invention claimed in the claims. Of course, various modifications can be made by those skilled in the art, and these modifications should not be individually understood from the technical spirit or the prospect of the present invention.

Claims (8)

In the elliptic curve encryption method using the d as a secret key for calculating a point P on the elliptic curve and Q = dP, Selecting random values of odd integers for each overlapping window; Generating an encoded sequence selected from a plurality of encoded binary numbers having the same value as the secret key d according to the random value selected for each window; And And performing the scalar multiplication using the generated sequence and the P, and generating the Q. 11. The method of claim 1, The random value is an elliptic curve encryption method characterized in that it is calculated by the following formula:

Where u [i] is the random value, w is the window size, h is the overlapped length of the window, t is the length of the secret key, and dt = bit (d, t-1, ... t-w). The method of claim 2, The length of the secret key is an elliptic curve encryption method characterized by the following equation:

Where t is the length of the secret key, s is the number of windows, h is the overlapping length of the window, and w is the size of the window. The method of claim 3, wherein The h is an elliptic curve encryption method, characterized in that using a fixed value. In the elliptic curve encryption system using the d as a secret key for calculating a point P on the elliptic curve and Q = dP, Means for selecting a random value of an odd integer for each overlapped window; Means for generating an encoded sequence selected from a plurality of encoded binary numbers each having the same value as the secret key d, according to the random value selected for each window; And And means for generating the Q by performing a scalar multiplication using the generated sequence and the P. 11. The method of claim 5, Elliptic curve encryption system, characterized in that the random value is calculated by the following formula:

Where u [i] is the random value, w is the window size, h is the overlapped length of the window, t is the length of the secret key, and dt = bit (d, t-1, ... t-w). The method of claim 6, The length of the secret key is an elliptic curve encryption system, characterized by the following equation:

Where t is the length of the secret key, s is the number of windows, h is the overlapping length of the window, and w is the size of the window. The method of claim 7, wherein The h is an elliptic curve encryption system, characterized in that using a fixed value.

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