WO2003071733A1

WO2003071733A1 - Method for generating secure keys for a cryptographic algorithm

Info

Publication number: WO2003071733A1
Application number: PCT/FR2003/000369
Authority: WO
Inventors: Eric Brier; Christophe Clavier
Original assignee: Gemplus
Priority date: 2002-02-15
Filing date: 2003-02-06
Publication date: 2003-08-28
Also published as: CN1438784A; FR2836312B1; FR2836312A1; AU2003222888A1

Abstract

The invention relates to a method and a corresponding device for generating secret secure keys for a cryptographic algorithm. The inventive method comprises the following steps: E1) the whole set (IK) of possible keys is analyzed; E2) a subset (SK) of keys is extracted; E3) the secret keys are generated on the basis of the subset of keys.

Description

SECURE KEY GENERATION METHOD FOR CRYPTOGRAPHIC ALGORITHM

The invention relates to a method for generating secure keys for a cryptographic algorithm. The invention, very general, can be used to secure any cryptographic algorithm totally or partially broken.

Cryptographic algorithms are most often used in applications where access to data or services is strictly controlled. These algorithms are used in particular in smart cards for certain applications of these. These are, for example, applications for accessing certain databases, banking applications, electronic toll applications, for example for television, petrol distribution or even the passage of motorway tolls. These algorithms are also used in so-called SIM cards, for mobile telephone applications.

Cryptographic algorithms are generally implemented in electronic components having an architecture formed around a microprocessor and memories, including a non-volatile memory which contains the secret key.

Generally and succinctly, these algorithms have the function of calculating an encrypted message from the secret key contained in the card and a clear message applied as input (of the component) by a host system (remote server, distributor bank, etc.), and provide the encrypted message obtained back to the host system. This allows the host system to authenticate the component before exchanging data. The encrypted message is accessible from the outside. However, the clear message cannot be retrieved without knowledge of the secret key used to obtain the encrypted message. The best known cryptographic algorithms are the DES, AES and RSA algorithms. As part of the mobile telephony the most used algorithm is the Compl28. This list, of course, is not exhaustive. The characteristics of cryptographic algorithms are assumed to be known: operations performed, parameters used. Only the secret key which is specific to each component remains unknown and which cannot be deduced from the mere knowledge of the clear message and / or of the encrypted message.

The invention can be applied to a symmetric algorithm (such as DES or AES) as well as to an asymmetric algorithm (such as RSA). The term "secret key" denotes both the unique key of a symmetric algorithm and the private key of an asymmetric algorithm.

In the framework of symmetric algorithms, the secret key is a binary number whose size NO depends on the algorithm used. For example, the DES algorithm uses keys of N0 = 56 bits, the Compl28 algorithm uses keys of N0 = 128 bits. For such an algorithm, there therefore exists a set comprising N = 2 ^N0 possible keys. During a component personalization phase, a secret key is generated, which is then stored in a non-volatile memory of the component. The key generation is done in a known manner from a random number generator capable of producing numbers of the desired size NO.

A component and the algorithm it uses can be vulnerable to analyzes aimed at "breaking" the algorithm, ie finding the secret key it uses. Such an action, if successful, can have serious consequences, up to the cloning of the component.

These _^ - analyzes, at least those which are currently known, are essentially of two types: cryptanalysis and hidden channel attacks (in English: side-channel attacks). A cryptanalysis consists in carrying out a mathematical or statistical process using only the knowledge of the algorithm and of one or more clear / encrypted pairs to find the secret key used by this algorithm. A hidden channel attack consists of a simple or differential (statistical) analysis of a specific physical parameter linked to the component when it executes the algorithm. This attack is based on the fact that the trace (the variation of the specific physical parameter, for example the current consumption, the electromagnetic radiation, etc.) of the component executing instructions varies according to the data which it manipulates, and therefore according to of the secret key used. In particular, when the component executes the algorithm, the trace of the component depends on the clear message, the secret key and / or the encrypted message. From measurements of this trace and statistical studies of these measurements, it is possible to find the secret key.

Finally, in all cases, an attack by exhaustive search is possible. It consists of searching for the secret key systematically. For this, from a clear message and a known associated encrypted message, the algorithm is executed systematically with all the keys, one by one, until the secret key used is obtained. The exhaustive research requires a very significant time (time increasing exponentially with the length in bits of the keys) and / or a particularly efficient material to be carried out.

Unlike exhaustive search, the duration of a cryptanalysis or a hidden channel attack can depend on the value of the secret key.

The security or resistance of an algorithm is its capacity to resist an attack whatever it is and whatever the key it uses. An algorithm is said to be safe if the time required to break it is prohibitive (on the order of a few weeks to a few years). The security of an algorithm greatly increases with the size of the keys used. On the other hand, the security of an algorithm decreases over time because the performance of the materials likely to be used to break it increases, as does the knowledge of possible attackers.

Solutions are known to reinforce the security of an algorithm against hidden channel attacks: they consist in intervening at the level of the implementation of the algorithm and in modifying it so that its trace becomes unpredictable: for example, it is possible to invert process steps randomly, to mix data manipulated by the algorithm with one or more random parameters, etc. These solutions are effective. However, they are more or less difficult to implement because they require part modification of the implementation of the algorithm. These solutions are also costly in terms of algorithm execution time, because more often than not the total number of algorithm steps is increased.

If it proves impossible or undesirable to secure the implementation of an algorithm by these methods, it can be envisaged to replace this algorithm with a more secure algorithm. However, this replacement in an existing component also requires modifying the infrastructure in which the component is part, which may require prohibitive technical investments.

In view of the problems set out above, an object of the invention is to implement a method for securing a particularly simple and inexpensive cryptographic algorithm.

Thus, the invention relates to a method for generating secure secret keys for a cryptographic algorithm, the method being characterized in that it comprises the following steps: El: analysis of the set (IK) of possible keys; E2: extraction of a subset (SK) of keys from the set (IK);

E3: generation of secret keys from the key subset (SK).

The invention is applicable to any cryptographic algorithm sensitive to at least one identified attack, when said algorithm behaves differently with respect to the identified attack depending on the value of the key that it uses. This assumes that the concepts of strong key and weak key are relevant to the identified attacks. A key is said to be strong if the time necessary, for said key, to complete the identified attack is prohibitive. A key is said to be weak otherwise.

According to the invention, secret keys are thus generated from a subset of keys preselected for their resistance to identified attack. This greatly reduces the chances of success of this attack against a component and / or an algorithm using such secret keys.

During the analysis step E1, the strength of the keys of all of the possible keys is evaluated with respect to the attack identified. Then we classify the keys of the key set according to a decreasing order of force. In the case where several attacks are identified, the strength of the keys vis-à-vis each identified attack is determined during step E1. Then, the resulting force of a key is determined as being the minimum of the forces of this key against all of the identified attacks. Finally, we classify the keys of the set of keys according to an order of decreasing resulting force.

During the step E2 of extracting a subset of keys, in the case where a single attack is identified, a sufficient number of keys is extracted from among the strongest keys of the set of possible keys. According to a first variant, the number of keys extracted is fixed. According to one another variant, the number of keys to be extracted is a function of the average strength of the keys extracted, as will be seen more clearly below in an example.

During the step E2 of extracting a subset of keys, in the case where several attacks are identified, a sufficient number of keys is extracted among the keys whose resulting forces are the greatest among the set of possible keys. According to a first variant, the number of keys extracted is fixed. According to another variant, the number of keys to be extracted is a function of the average value of the resulting force of the keys extracted.

During the step E3 of generating the secret key, the secret key is chosen randomly from the subset of keys. The key thus obtained is finally stored in a non-volatile memory of the component to be personalized.

The secret key obtained according to the method of the invention is therefore necessarily a strong key with respect to the attack or attacks identified. The identified attack therefore does not give a result if it is applied to a component using said secret key. Furthermore, said secret key having been chosen from the subset of keys comprising a sufficient number of keys, an exhaustive search does not give any result either. The method of the invention thus makes it possible to generate keys such that the identified attack or the exhaustive search applied to a component using the key cannot succeed.

The invention can be applied for the generation of secret keys for any algorithm for which at least one possible attack is identified, and for which the keys are more or less sensitive with respect to the identified attack. In an exemplary embodiment, the invention is applied to the Compl28 algorithm. The invention and the advantages which result therefrom will appear more clearly on reading the description which follows. An example of implementation of a method for generating secure keys will be given. The description should be read with reference to the accompanying drawings in which: FIG. 1 is a block diagram of an architecture of a device in which the method of the invention is installed, and FIG. 2 is a diagram of a method according to the invention.

FIG. 1 represents in the form of a block diagram an electronic device 1 capable of implementing a method of generating keys according to the invention. In the example, the device 1 is a reader intended for the personalization of smart cards of the SIM card type. The device 1 comprises a communication interface 10 and programmed calculation means composed of a central unit 2 functionally connected to a set of memories of which: - a memory 4 accessible in read only, in the example of the mask ROM type, also known by the English name "mask Read-Only Memory (mask

ROM) ", a memory 6 re-programmable electrically, in the example of the EEPROM type (from the English" electrically erasable programmable ROM "), and a working memory 8 accessible in reading and writing, in the example of RAM type (from the English "Random Access Memory"). This memory includes in particular the registers used by the device 1.

The executable code corresponding to the method of the invention for the generation of secret key intended for the card to be personalized is contained in program memory. This code can in practice be contained in memory 4, accessible in read only and / or in rewritable memory 6. The central unit 2 is connected to the communication interface 10 which ensures the exchange of signals with the card to be personalized and the supply of its chip. The communication interface 10 is in contact with the chip of the card to be personalized (not shown in FIG. 1) via a physical link (contact card) or a radio-frequency link (card without contact) .

It is assumed in the following example that the algorithm used is the Compl28. It uses keys of size N0 = 128 bits composed of 8 16-bit subkeys. The total number of possible keys for this algorithm is therefore equal to

N = (2 ¹⁶ ) ⁸ = 2 ¹²⁸ possible keys.

It is known that this algorithm is sensitive to a cryptanalysis called "collision attack". This attack consists in finding distinct clear messages providing the same encrypted message. This phenomenon is called a collision and makes it possible to find the value of a subkey. This collision search can be repeated until all 16 sub-keys are obtained, ie the value of the secret key used is obtained.

In the following description, the method of the invention produces secret keys secured against this identified attack (collision attack).

During the implementation of a key generation method according to the invention, the following steps are carried out (FIG. 2):

El: Analysis of all possible keys; - E2: Extraction of a subset of keys from said set of possible keys;

E3: Generation of a secret key from the subset of keys. The generated secret key can then be stored in a non-volatile memory of the card to be personalized. The steps E1 and E2 can be carried out once and for all. Step E3 is to be repeated for each card to be personalized.

During step E1, the set (IK) of possible keys is examined. In particular, the strength of these keys is evaluated, that is to say the force T (Ki) necessary for the outcome of the attack considered in the case of a key Ki. This effort is determined based on the technical knowledge and performance of the available equipment that the attacker is supposed to have. We then classify all the keys in decreasing order of force T (Ki), so that we have:

T (Ki)> T (Kj) for all i <j.

The first step E1 thus makes it possible to classify the keys according to their strength and thus to distinguish the strong keys from the weak keys. The evaluation of the strength of the keys does not need to be done with precision. Similarly, if the distinction between strong and weak keys is essential, the classification of keys may not be done in an absolutely rigorous manner.

In the second step E2, a subset SK of keys is extracted from the set IK of the possible keys so that: the keys of the subset SK are as strong as possible to resist the identified attack, and the keys of the sub-assembly SK are in sufficient number to withstand an exhaustive search.

In the following, a method is presented for optimizing the process of extracting the subset of keys SK described in step E2. However, such optimization is not mandatory to benefit from the invention. We can indeed extract an arbitrary and non-optimal subset of SK keys containing enough strong keys to counter both the identified attack (thanks to the strength of the keys) and the exhaustive search (thanks to the number of keys). Concretely, the average effort to provide for the identified attack to succeed is equal to the sum of the efforts to provide for all the keys of the SK subset divided by the number of keys of the SK subset, that is:

Tl = (1 / NS) * ΣT (Ki), for i varying between 1 and NS, NS being the number of elements in the subset of keys SK.

In addition, the effort required to carry out an exhaustive search on the basis of the SK subset of keys is equal to:

T2 = NS * T0, where T0 is the execution time of the algorithm. Insofar as an attacker can choose to conduct an exhaustive search or the identified attack, the average effort required to obtain a key of the SK subset is given by the formula:

T3 = Min [Tl; T2] = Min [(l / NS) * ∑ (Ki); NS * T0] To harden the algorithm, we seek to maximize the average effort T3. To do this, keys are inserted into the subset SK in decreasing order of force T (Ki), until the number of keys in SK reaches an optimal number NSO.

The function Tl = (l / NS) * ∑T (Ki) is a decreasing function of NS, insofar as the keys inserted in the subset SK have a decreasing force. Conversely, the function T2 = NS * T0 is an increasing, linear function of NS.

A quick mathematical study shows that in this case, T3 is maximum when T1 = T2. This allows in the general case to calculate the optimum number NSO of keys to be inserted in the subset of keys SK according to the relation:

T0 * (NS0) ² = ΣT (Ki), for i varying between 1 and NSO. During step E3, a secret key is then randomly chosen from the subset of keys obtained during step E2. The secret key chosen is finally stored in a non-volatile memory of the component to be personalized.

In the practical case of a component using the Compl28 algorithm, a key is composed of 8 16-bit subkeys and a key is strong if and only if the 8 subkeys are themselves strong.

During step E1, the 2 ¹⁶ sub-keys of

16 bits and 769 of them are identified as strong subkeys. These 769 sub-keys are those presenting the property of not giving rise to collisions considered by the identified attack.

Step E2 then consists in defining the sub-set SK as the set of keys of which all the sub-keys are part of the set of 769 strong sub-keys identified in step El.

During step E3, 8 sub-keys are randomly chosen from the subset of the 769 strong sub-keys, to finally form a strong secret key. The 8 sub-keys being strong, the secret key thus obtained is resistant to the identified attack (collision attack). Furthermore, the secret key obtained is also resistant to an exhaustive search because the size of the subspace SK (from which it comes) is equal to 769 ⁸ # 2 ⁷⁷ .

Claims

1. Method for generating secure secret keys for a cryptographic algorithm, facing an attack which makes it considered broken, comprising the following steps: El: analysis of the set (IK) of possible keys,

E2: extraction of a subset (SK) of keys from the set (IK),

E3: generation of secret keys from the subset (SK) of keys, method characterized in that, during the analysis step (E1), the strength of the keys of the set (IK) is evaluated possible keys vis-à-vis an identified attack, which makes the algorithm unusable.

2. Method according to claim 1, characterized in that, during the step (E2) of extraction of a subset (SK), a sufficient number of keys are extracted from the set of keys (IK) strong keys to the identified attack.

3. Method according to claim 2 characterized in that the sufficient number (NSO) is fixed.

4. Method according to claim 3 characterized in that the sufficient number is determined according to the average force of the keys of the key subset (SK).

5. Method according to claim 1 characterized in that, faced with several attacks identified during the analysis step (El), the strength of the keys of the set of keys (IK) is evaluated vis-à-vis of each of these so-called attacks.

6. Method according to claim 5 characterized in that the resultant force of a key is defined as being the minimum of the forces of said key against all of said identified attacks.

7. Method according to any one of claims 1, 5 or 6 characterized in that, during the step (E2) of extraction of a subset (SK), one extracts from all the keys (IK) a sufficient number of strong keys against said identified attacks.

8. Method according to claim 7 characterized in that the sufficient number (NSO) is fixed.

9. Method according to claim 7 characterized in that the sufficient number (NSO) is determined as a function of the average value of the forces resulting from said keys with respect to said identified attacks.

10. Method according to one of claims 1 to 4 characterized in that, during the analysis step (El), the keys of the set of keys (IK) are then classified according to a decreasing order of force .

11. Method according to any one of claims 1, 5 to 9 characterized in that, during the analysis step (El), the keys of the set of keys are then classified

(IK) in decreasing order of resulting force.

12. Device for personalizing an electronic component by a secret key, chosen randomly from a subset (SK) of keys, characterized in that it comprises programmed means (1) for the implementation of a method according to any one of claims 1 to 11, the programmed means comprising in particular a central unit (2) and a program memory.