Cryptography on a simplified elliptical curve

Abstract

A cryptographic calculation includes obtaining a point P(X,Y) from a parameter t on an elliptical curve Y2=f(X) and from polynomials satisfying: −f(X1(t))·f(X2(t))=U(t)2 in the finite body Fq, irrespective of the parameter t, q=3 mod 4. A value of the parameter t is obtained and the point P is determined by: (i) calculating X1=X1(t), X2=X2(t) and U=U(t); (ii) testing whether the term f(X−1) is a squared term in the finite body Fq and, if so, calculating the square root of the term f(X1), the point P having X1 as abscissa and Y1, the square root of the term f(X1), as ordinate; (iii) otherwise, calculating the square root of the term f(X2), the point P having X2, as abscissa and Y2, the square root of the term f(X2), as ordinate. The point P is useful in encryption, scrambling, signature, authentication or identification cryptographic applications.

Description

PRIORITY CLAIM

This application is a 371 filing from PCT/FR2010/051191 (published as WO 2010/146303) filed Jun. 15, 2010 which claims the benefit of French Application for Patent No. 09-54043 filed Jun. 16, 2009, the disclosures of which are hereby incorporated by reference.

TECHNICAL FIELD

The present invention relates to message cryptography based on the use of points on an elliptical curve, and more particularly said cryptography of a deterministic nature.

BACKGROUND

In order to apply a cryptographic calculation to a message, conventionally algorithms are employed for inserting arbitrary values into mathematical structures. For this purpose, the elliptical curves are mathematical structures that are able to facilitate the application of such cryptographic calculations and at the same time save space in memory relative to the use of other cryptographic calculations.

However, efficient algorithms for inserting arbitrary values using elliptical curves are probabilistic. Consequently, the application time of these algorithms is not constant, it depends on the message to be encoded. Thus, if an attacker determines different application times of the algorithm applied, he can obtain information about the coded message.

In order to mask the time taken by a probabilistic insertion algorithm, it is possible to provide the addition of unnecessary steps in this algorithm so that its application always extends over a period of time of identical length, regardless of the message processed.

A point P of an elliptical curve is defined by its abscissa X and its ordinate Y, X and Y satisfying the following equation:f(X)=Y²  (1)

• • where f(X) is the polynomial f(X)=X³+aX+b

A family of polynomials is known, which satisfy Skalba's equality which makes it possible to determine such a point on an elliptical curve, as defined in the document ‘Construction of Rational Points on Elliptic curves over finite fields’ by Andrew Shallue and Christiaan van de Woestijne.

Polynomials X₁(t), X₂(t), X₃(t) and U(t) satisfy Skalba's equality if they satisfy the following equation:f(X₁(t))·f(X₂(t))·f(X₃(t))=U²(t)  (2)

where f is the function that defines the elliptical curve under consideration, and

where t is a parameter.

The polynomials that satisfy Skalba's equality can take two parameters u and t. In this case, Skalba's equality is written:f(X₁(t,u))·f(X₂(t,u))·f(X₃(t,u))=U²(t,u)

Equations of this type can be used with two parameters u and t. However, in the proposed applications, we can advantageously envisage setting u, or alternatively setting t, at any value. Thus, the value of a single parameter remains to be chosen.

Given selected parameters t and u, it is noted that X₁=X₁(t,u), X₂=X₂(t,u), X₃=X₃(t,u), U=U(t,u), where X₁, X₂, X₃ and U are elements of F_q. This equation (2) signifies that at least one of the values f(X₁), f(X₂) and f(X₃) corresponds to a squared term in the finite field F_q.

Then, once the squared term in F_q, f(X_i), is identified, we can then obtain a point on the elliptical curve P(X_i,√{square root over (f(X_i))}.

Calculation of √{square root over (f(X_i))} can be performed by means of an exponentiation calculation when the characteristic q of the field F_q satisfies:q=3 mod 4

In this case, it is known that:√{square root over (f(X_i))}=f(X_i)^(q+1)/4  (3)

In order to determine a point on the elliptical curve (1), it is therefore necessary to determine which value among the three values f(X₁), f(X₂) and f(X₃) corresponds to a squared term in the finite field F_q. For this purpose we could envisage checking firstly whether the term f(X₁) is a squared term in the finite field F_q, then, if it is not the case, apply this same check to the term f(X₂), and finally if this is still not so, check the term f(X₃) similarly. However, following this procedure, determination of a point on the elliptical curve does not always take the same time, since this determination is executed more quickly if the first term tested is a squared term than if only the third term is a squared term.

A potential attacker could make use of this difference in elapsed time to determine a point on the elliptical curve for breaking the secret linked to the parameter that enabled this point to be generated. Now, in the field of cryptography, these parameters must remain secret.

These parameters can in particular correspond to passwords. Thus, it is important that determination of these points does not in itself supply information that makes it possible to break the secret of the parameter, and accordingly, attacks based on an analysis of the elapsed time for determining a point on the curve are to be avoided.

To overcome this disadvantage, it would be possible to check the three terms f(X_i) systematically for i in the range from 1 to 3. Thus, the time for determining a point on the curve would no longer be a function of the point determined.

However, checking whether a term of equation (2) is a squared term in the finite field F_q is a complex operation in particular employing an exponentiation, which is costly in execution time. In the case when we wish to determine a point on an elliptical curve on the basis of Skalba's equalities, while performing these determinations in a constant time, four operations of exponentiation are required in the case described above, one exponentiation per check of each of the terms of Skalba's equation (2) and one exponentiation for calculating the square root, as described in equation (3).

The present invention aims to improve this situation.

SUMMARY

A first aspect of the present invention proposes a method of execution of a cryptographic calculation in an electronic component comprising a step of obtaining a point P(X,Y) starting from at least one parameter t, on an elliptical curve satisfying the equation:Y²=f(X); andstarting from polynomials X₁(t), X₂(t), and U(t) satisfying the following equality:f(X₁(t))·f(X₂(t))=U(t)²

• • in the finite field F_q, regardless of the parameter t, q satisfying the equation q=3 mod 4;said method comprising the following steps:

/1/ obtain a value of the parameter t;

/2/ determine point P by executing the following substeps:

• • /i/ calculate X₁=X₁ (t), X₂=X₂(t), and U=U(t)
  • /ii/ test whether the term f(X₁) is a squared term in the finite field F_q and in this case, calculate the square root of the term f(X₁), point P having X₁ as abscissa and the square root of the term f(X₁) as ordinate;
  • /iv/ otherwise, calculate the square root of the term f(X₂), point P having X₂ as abscissa and the square root of the term f(X₂) as ordinate;

/3/ use said point P in a cryptographic application of encryption or hashing or signature or authentication or identification.

It should be noted here that the determination of a point on an elliptical curve is carried out on the basis of an advantageous equation:−f(X₁)·f(X₂)=U²  (4)

This equation follows from the Skalba equality (2). In fact, this equation can be obtained by setting:f(X₃)=−1

Now, in the finite field F_q with q=3 mod 4, −1 is not a squared term. Consequently, only two terms of the equation (4) still remain to be checked in order to decide which of the two terms corresponds to a squared term in F_q.

Thanks to these arrangements, it is possible to determine a point on an elliptical curve in a manner suitable for use in the field of cryptography, since on the one hand this determination takes the same time regardless of the input parameter t and on the other hand it is efficient as the number of demanding operations is reduced.

This determination takes a constant time that does not depend on the input parameter or parameters. In fact, even if this method offers different processing options depending on the term that corresponds to a squared term in Skalba's equality, the same number of operations of the same type is performed regardless of the point on the curve that is determined. More precisely, regardless of the point on the curve that is determined, the following list of operations is executed:

• • test for a squared term in F_q;
  • determination of a square root.

Therefore it is not possible to launch an attack of the ‘timing attack’ type.

Moreover, this determination is efficient since the number of costly operations employed is limited. In fact, thanks to equation (4) only two terms instead of three in the equation (2) are to be checked in order to determine whether they correspond to the squared terms in the finite field F_q by applying a maximum of two exponentiation-type operations.

This embodiment is general and can easily be applied to any family of polynomials satisfying equality (4).

In one embodiment of the present invention, it is provided at step /2/-/ii/, to carry out the following steps

• • calculate R₁ such that:

$R_1={\left(f\left(X_1\right)·f\left(X_2\right)\right)}^{\frac{q+1}{4}}$

• • if R₁² is equal to 1,
    • decide that the term f(X₁) is a squared term in the field F_q, and
    • calculate

$Y_1={f\left(X_1\right)}^{\frac{q+1}{4}}$

• • otherwise calculate

$Y_2={f\left(X_2\right)}^{\frac{q+1}{4}}$

Here, only two exponentiations are carried out, whatever the processing option applied.

In another embodiment, it is also possible to reduce the number of exponentiations, which are the most demanding operations to carry out in this method. In fact, at step /2/-/ii/, the following steps can be carried out:

• • calculate R₁′ such that:

$R_1^{\prime }={f\left(X_1\right)}^{q-1-\frac{q+1}{4}}$

• • calculate R₂′ such that:R′₂=R′₁²
  • calculate R₃′ such that:R′₃=R′₂·f(X₁)if R₃′ is not equal to 1, at step /2/-/iii/, the square root of f(X₂) is obtained according to the following equation:√{square root over (f(X₂))}=R₀·R′₁

where R₀ satisfies the following equation:

$R_0=U\left(t\right)·{\left(-1\right)}^{q-1-\frac{q+1}{4}}$

It should be noted here that, advantageously, one exponentiation is carried out in this case during execution of a method according to one embodiment of the present invention

In fact, ingenious use is made of the fact that we can finally recover the square root of f(X₂) in the case where the term f(X₂) corresponds to a squared term, without however implementing an additional exponentiation. In fact, the square root of f(X₂) obtained by:√{square root over (f(X₂))}R=R₀·R′₁;

where the term R₀ is finally obtained by a multiplication operation which is less demanding than the application of an exponentiation. Moreover, only the term U(t) is to be calculated in this embodiment, as the term

${\left(-1\right)}^{q-1-\frac{q+1}{4}}$

is an immediate calculation term. Therefore it is in no way useful to pre-calculate this last term and store it in memory. Thus memory space can be saved.

Then, if R₃ is equal to 1, then at step /2/-/iii/ the square root of f(X₁) can be obtained according to the following equation:√{square root over (f(X₁))}=R′₃·f(X₁)

This also corresponds to a multiplication.

During execution of such calculations according to one embodiment of the present invention, the time taken for carrying out the operations other than an exponentiation is negligible relative to the time taken by the application of an exponentiation. Now, owing to the characteristics of the present invention, instead of four exponentiations, as described previously in a conventional case, two exponentiations are required at most. Such a reduction in the number of exponentiations is very advantageous.

In one embodiment of the present invention, the polynomials satisfying equation (4) according to one embodiment of the present invention in X and Y are expressed in Jacobian coordinates in X′, Y′ and Z such that:X′=X·Z²,Y′=Y·Z³ and the operations of inversion are transformed into operations of multiplication.

The transformation into Jacobian coordinates makes it possible to transform the inversions into multiplications, when the term Z is correctly selected.

In one embodiment of the present invention, the polynomials are expressed in Jacobian coordinates, according to which the point P(X,Y) is written P(X′,Y′,Z) such that:X′=X·Z²,Y′=Y·Z³ where the function f is written f_X(X′) and satisfies:f_Z(X′)=X′³+a·X′·Z⁴+b·Z⁶ with the elliptical curve satisfying the equation:Y′²=f_Z(X′)in which the polynomials expressed in Jacobian coordinates are X′₁(t), X′₂(t), X′₃(t), Z(t) and U′(t) and satisfy the following equality in Jacobian coordinates:U′(t)²=−f_Z(t)(X′₁(t))·f_Z(t)(X′₂(t))and in which Z(t) is determined in such a way that the operations of inversion are transformed into operations of multiplication.

At step /1/, the value of the parameter t can be obtained as a function of a password or an identifier. It is thus possible to envisage using the password directly or a derivative of the password as parameter.

In one embodiment of the present invention, the cryptographic application is an application of authentication or identification by a checking entity, and at step /1/, the following steps are executed:

• • /a/ generate a random value;
  • /b/ obtain an encrypted value by encrypting said random value based on an encryption function using an encryption key determined from a password or identifier corresponding to the parameter; and
  • /c/ transmit the encrypted value to the checking entity.

By following this procedure, the checking entity is able to obtain the random value as a function of the encrypted value received from the password. Then it recovers the value of the parameter t by applying a suitable function.

A second aspect of the present invention proposes an electronic device comprising suitable means for applying a method of execution of a cryptographic calculation according to the first aspect of the present invention.

Other aspects, aims and advantages of the invention will become clear on reading the description of one of its embodiments.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will also be better understood with the aid of the following figures:

FIG. 1 shows the main steps of a method of execution of a cryptographic calculation according to one embodiment of the present invention;

FIG. 2 shows a method of execution of a cryptographic calculation in detail according to one embodiment of the present invention; and

FIG. 3 shows a method of execution of a cryptographic calculation in detail according to one embodiment of the present invention.

DETAILED DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the main steps of a method of execution of a calculation according to one embodiment of the present invention.

These main steps are suitable for determining a point on an elliptical curve with the aim of using said point in a cryptographic application. A cryptographic calculation of this kind can be executed in an electronic component in a secure manner, i.e. without the determination of this point giving any information on the point determined and therefore on parameter t.

This calculation comprises, in a finite field F_q, where q is equal to 3 mod 4, a step of obtaining a point P(X,Y) on an elliptical curve satisfying the equation:Y²=f(X)

A point P(X,Y) has its abscissa X which corresponds to one of X₁(t) and X₂(t), for a value of t obtained, such that:−f(X₁(t))·f(X₂(t))=U²(t)  (4)

in the finite field F_q.

Such polynomials can be a function of two parameters u and t. In the context of the present invention, one of the parameters can advantageously be set and consequently the polynomials satisfying equation (4) are then functions of a single parameter t.

Generally, in order to determine a point on the curve, we try to determine, given input parameters u and t, those among the values X₁=X₁(t,u) and X₂=X₂(t,u) that correspond to a squared term in the finite field F_q. For this purpose, at a step 11, the parameter t is taken into account and we calculate:X_i=X_i(t) for i equal to 1 or 2,andU=U(t)

At a step 12, we decide whether the term f(X₁) is a squared term on the basis of certain calculations. If the term f(X₁) is a squared term then its square root is calculated in order to obtain, at step 13, the point P on abscissa X₁ and ordinate Y₁ obtained from the calculation of the previous square root.

In the opposite case, the point P on the abscissa X₂ and the ordinate Y₂ are obtained at step 14. To this end we envisage calculating the square root of the term f(X₂).

It should be noted that reaching steps 13 or 14 for obtaining a point on the elliptical curve according to one embodiment of the present invention requires similar operations. Thus, regardless of the input parameter or parameters t and u, it is not possible to launch an attack on the basis of the time elapsed.

The point P(X_i,Y_i), for an i equal to 1 or 2, can then be used advantageously in a cryptographic application of encryption or hashing or signature or authentication or identification, since its determination has not supplied any element that can break its secret.

In the field F_q, q corresponding to 3 mod 4, it is possible to check whether a term is a squared term in various ways.

FIG. 2 illustrates the application of the method according to an embodiment of the present invention.

At step 21, we calculate:

$R_1={f\left(X_1\right)}^{\frac{q-1}{2}}$

Then, the test for checking whether the term f(X₁) is a squared term in F_q, can be carried out, at a step 22, by comparing R₁ to 1. In fact, in Fq, if R₁ is equal to 1, then f(X₁) is a squared term. In this case, at step 24, the square root of this term is calculated as follows:

$\sqrt{f\left(X_1\right)}={f\left(X_1\right)}^{\frac{q+1}{4}}$

Otherwise, the term f(X₂) is a squared term. Then, at a step 23, its square root is calculated as follows:

$\sqrt{f\left(X_2\right)}={f\left(X_2\right)}^{\frac{q+1}{4}}$

In this embodiment, it should be noted that the number and the type of operations carried out for the determination of a point P is the same whatever the processing option taken, i.e. whatever the term which corresponds to a squared term in equation (4).

FIG. 3 illustrates another embodiment of an execution method according to one embodiment of the present invention in which only one exponentiation is applied.

Here, advantageously, the number of exponentiations can be further reduced, by not using the same test for a squared term 12 of FIG. 1

In one embodiment of the present invention, when trying to determine whether a term A is a squared term in F_q, the following steps can be executed:

$\begin{array}{cc}{W}_1=\frac{1}{A^{\frac{q+1}{4}}}=A^{q-1-\frac{q+1}{4}}& \left(i\right)\\ W_2=W_1^2& \left(\mathrm{ii}\right)\\ W_3=W_2·A& \left(\mathrm{iii}\right)\end{array}$

Finally, if term A is a squared term then:

• • W₁ corresponds to the reciprocal of the square root of A, i.e. 1/√{square root over (A)}, since an exponentiation at (q−1) corresponds to an inversion and an exponentiation at (q+1)/4 corresponds to a square root in the finite field F_q;
  • W₂ corresponds to the inverse of A; and
  • W₃ corresponds to the value 1.

Thus, when W₃ is equal to the value 1, it is concluded from this that the term A is a squared term in the finite field F_q. If A is not a squared term then W₃ is not equal to 1.

The following sections describe an embodiment based on this type of test. In one embodiment of the present invention, at a step 311, the following multiplication is performed:

$R_1^{\prime }={f\left(X_1\right)}^{q-1-\frac{q+1}{4}}$

Then it is checked whether this term R₀ is a squared term as stated previously. Thus in a step 312, we calculateR′₂=R′₁²

Then in a step 313, we calculateR′₃=R′₂·f(X₁)

Then, we decide whether the term R′₃ is equal to 1 at step 314. If this is the case, then the following term corresponds to the square root of the term f(X₁):R′₄=R′₃·f(X₁)

If the test 314 is not satisfied, then the term f(X2) is a square root in F_q. The square root of this term is thus obtained at step 316 according to the following equation:R″₄=R₀·R′₁

where R₀ satisfies the following equation

$R_0=U\left(t\right)·{\left(-1\right)}^{q-1-\frac{q+1}{4}}$

It should be noted that the above equation makes it possible to obtain advantageously the square root of f(X₂) but without carrying out an operation of exponentiation such as that carried out at step 23 or also at step 311. In fact, here it is, ingeniously, a matter of performing a multiplication instead of an exponentiation.

We then obtain R₄″, which corresponds to the term f(X₂). Thus, a point P on the elliptical curve has been determined which has X₂ as abscissa and R₄″ as ordinate.

In the embodiment described previously with reference to FIG. 3, like that described with reference to FIG. 2, regardless of the determination of point P, i.e. whether this determination is based on the value X₁ or X₂, similar calculations are employed, thus ensuring determination of a point on the elliptical curve in a constant time.

In one embodiment of the present invention, it is possible to select polynomials that satisfy equation (4) according to one embodiment of the present invention, by basing it on Ulas polynomials as defined in the document “Rational points on certain hyperelliptic curves over finite fields” by Macie Ulas, dated 11 Jun. 2007.

In this document, the polynomials satisfying Skalba's equation (2) are described:

$X_1\left(t,u\right)=-\frac{b}{a}\left(1+\frac{1}{t^{4}f\left(u\right)+t^{2}f\left(u\right)}\right)$ $X_2\left(t,u\right)=t^{2}f\left(u\right)X_1\left(t,u\right)$ $X_3\left(t,u\right)=u$ $U\left(t,u\right)=t^3{f\left(u\right)}^{4}f\left(X_1\left(t,u\right)\right)$

• • where f(u)=u³+au+b

where a and b are elements of F_q such that their product is not zero.

Thus, the equations can be rewritten by settingf(u)=−1

without it being necessary to calculate a value of parameter u for which this last equation is satisfied. We then obtain

$X_1\left(t\right)=-\frac{b}{a}\left(1+\frac{1}{t^4-t^2}\right)$ $X_2\left(t\right)=-t^2{X}_1\left(t\right),\mathrm{and}$ $U\left(t\right)=t^{3}f\left(X_1\left(t\right)\right)$

Advantageously, these polynomials satisfy the following equation:−f(X₁(t))·f(X₂(t)=U(t)²

In one embodiment of the present invention, the use of Jacobian coordinates is advantageously envisaged. This transformation into Jacobian coordinates makes it possible to transform the operations of inversion into operations of multiplication which are quicker and easier to apply.

The equation of an elliptical curve:X³+aX+b=Y²

can be written in Jacobian coordinates:X′³+aX′Z⁴+bZ⁶=Y′²

It should be noted that the coordinates of a point (X,Y) can be written in Jacobian coordinates (X′,Y′,Z′) such that:X′=X·Z² andY′=Y·Z³

We should therefore determine a polynomial Z(t,u) in such a way that the Jacobian coordinates X′, Y′ and Z can be written without inversion.

In the following sections, this transformation into Jacobian coordinates is applied to a particular case of polynomials as stated previously.

In this context, any operation of inversion is eliminated by taking:Z(t)=a(t⁴−t²)

In fact, the polynomials can then be written in the following form in Jacobian coordinates:X′₁(t)=−b·Z(t)(t⁴−t²+1)X′₂(t)=t²·X′₂(t)

It should therefore be noted that there is no longer any inversion in Jacobian coordinates. As this operation can be as costly as an exponentiation, these coordinates permit a significant improvement in calculation time.

Then, to obtain the Jacobian coordinate Y′, it is advisable to calculate U′(t,u), the equivalent of U(t,u) in Jacobian coordinates.

We can then write in Jacobian coordinates:U′(t)=t³·f_Z(X₂′(t))with:f_Z(t)(X′)=X′³+a·X′·Z(t)⁴+b·Z(t)⁶

By way of example only, the equations below make it possible to no longer have to carry out inversion operations. Under these conditions an execution method is then obtained which is more efficient and quick, while ensuring an execution still in a constant time.

The present invention can advantageously be implemented in any type of cryptographic calculation using elliptical curves. It can in particular be advantageous in protocols for authentication by password, such as PACE (Password Authenticated Connection Establishment). In this case, it allows an improvement in calculation performance, while not allowing any attack linked to the execution time of the cryptographic calculation.

The present invention can also be applied advantageously in the context of privacy protocols, such as those used for checking electronic identity documents, such as electronic passports.

Claims

1. Method of execution of a cryptographic calculation in an electronic component comprising a step of obtaining a point P(X,Y) from at least one parameter t, on an elliptical curve satisfying the equation: Y2=f(X); and

2. Method of execution of a cryptographic calculation according to claim 1, in which the following steps are executed at step /2/-/ii/: calculating R1 such that:

3. Method of execution of a cryptographic calculation according to claim 1, in which at step /2/-/ii/, the following steps are carried out: calculating R1′ such that:

4. Method of execution of a cryptographic calculation according to claim 3, in which, if R3′ is equal to 1, at step /2/-/iii/, obtaining the square root of f(X1) according to the following equation: √{square root over (f(X1))}=R3′·f(X1)

5. Method of execution of a cryptographic calculation according to claim 1, in which the polynomials are expressed in Jacobian coordinates according to which the point P(X,Y) is written P(X′,Y′,Z) such that: X′=X·Z2,Y′=Y·Z3 where the function f is written fZ(X′) and satisfies: fZ(X′)=X′3+a·X′·Z4+b·Z6 with the elliptical curve satisfying the equation: Y′2=fZ(X′)in which the polynomials expressed in Jacobian coordinates are X′1(t), X′2(t), Z(t) and U′(t) and satisfy the equality in Jacobian coordinates: U′(t)2=−fZ(t)(X′1(t))·fZ(t)(X′2(t)))and in which Z(t) is determined in such a way that the operations of inversion are transformed into operations of multiplication.

6. Method of execution of a cryptographic calculation according to claim 1, in which, at step /1/, the value of the parameter t is obtained as a function of a password or an identifier.

7. Method of execution of a cryptographic calculation according to claim 1, in which the cryptographic application is an application of authentication or identification by a checking entity, and in which, at step /1/, the following steps are executed: /a/ generating a random value;/b/ obtaining an encrypted value by encrypting said random value based on an encryption function using an encryption key determined from a password or identifier corresponding to the parameter; and/c/ transmitting the encrypted value to the checking entity.