EP2168245A2

EP2168245A2 - Method for encoding information using non linear evenly distributed functions and computer software for implementing said method

Info

Publication number: EP2168245A2
Application number: EP08806003A
Authority: EP
Inventors: Marc Mouffron
Original assignee: EADS Secure Networks SAS
Current assignee: Airbus DS SAS
Priority date: 2007-06-14
Filing date: 2008-06-13
Publication date: 2010-03-31
Also published as: WO2009004235A2; WO2009004235A9; US8615537B2; US20100138462A1; FR2917549A1

Abstract

The invention relates to a method for encoding information using non-linear evenly distributed functions, that comprises the following steps for building such a function: selecting (10, 12, 14) a first natural integer n, a first set En of cardinal n, a group G operating on said first set En, a second natural integer q and a second set Eq of cardinal q; defining (15, 16, 18) a set OG of the orbits of the group G operating on vectors of n elements in the set Eq, a function ? that, for each vector of the n elements of the set Eq, associates the corresponding orbit, and a second function F allocating to each orbit of the group G a value of the set Eq; evenly distributing (20) into a number q of sections the sums of the cardinals of the orbits of the group; defining (22) a third function F comprising the first and second functions Fo?.

Description

METHOD OF ENCODING INFORMATION BY USING

EQUIDISTRIBUTED NON-LINEAR FUNCTIONS AND PROGRAMS

COMPUTER FOR IMPLEMENTING THE METHOD

The present invention relates to an information coding method using non-linear equidistributed functions. It also relates to a computer program for implementing the method.

In many fields, including cryptography and data classification, it is necessary to obtain from a large number of data a result dependent on all these data simultaneously. For this, one must have functions with a very large number of input variables and know how to calculate their result effectively in a short time or with low means. Usually, we use simple functions that we know how to build with many variables and elementary operations. Thus linear or polynomial functions of low degree are typically used. They are indeed easy to calculate but often have unacceptable properties or behaviors, such as, for example, poor distribution of results. The efficient realization of non-simplistic functions with a large number of input variables is therefore a primary need.

Equidistributed non-linear functions are very useful as: hash function for optimal classification of hash data or search tree.

The use of hash functions is, for example, described in the patent US5229768 which cites the functions of hashing "middle of square function" (middle function of the square) and "modulo function"

(modulo function). The first is non-linear but is limited to a limited number of variables because the complexity of the square increases very quickly. This function is not equidistributed and is therefore not optimal as a hash function for ranking. The second is linear and is not always equidistributed. Therefore, it can lead to biased and non-optimal hash distributions on linearly dependent conventional data. The clustering phenomenon, caused by the non-linearity or simplicity of the ranking function, must be avoided because it penalizes research on classified data. Equidistribution is a necessary condition for achieving the optimal search time, which is a logarithmic function of the number of classified data. The non-linear ranking functions achieving uniform distribution are therefore the best, but are in practice difficult to find.

With the explosion of processed data quantities the use of fast hashing functions to compute over a very large number of variables and with properties such as non-linearity and equidistribution is therefore crucial.

For cryptography, the choice of non-linear equidistributed functions is essential to ensure the quality of a cryptographic algorithm. In addition, as shown by the detailed properties described below, these functions are even better than their number of input variables n is large.

In any cryptographic process, the use of non-linear functions, that is to say of algebraic degree d strictly greater than 1, is a need fundamental. It is further necessary that these non-linear elements integrated in the cryptographic method have a number of additional properties. Thus, the use of non-linear functions with high order correlations protects a cryptographic algorithm from certain attacks. It is therefore an essential requirement analyzed by SIEGENTHALER who first indicated the compromise between the order of correlations t, the algebraic degree d and the number of input variables n of Boolean functions in the form:

d + t <n.

The Walsh transform further proves that the correlation of a Boolean function with a given number of input variables n has a minimal value of decreasing correlation with n. The extension of these results established on GF (2) to Galois body GF (q) was given by CAMION et al in the form:

d + t <(q-1) n.

More recent results on other attacks such as algebraic attacks show the need to use functions with high algebraic immunity.

The two constraints mentioned here as well as other similar properties therefore encourage the use of functions having many input variables. But technology limits the possibilities of carrying out such functions. The largest memories currently available contain 128 Gbits as flash memory. This means that they can only process 37 input Boolean variables, which is not very much in light of current needs: it is desirable to use a function with a hundred input variables binary or q-areas to ensure the desired properties.

Another important property of the functions used in cryptography is their equidistribution. This is also a necessary condition for the minimization of correlations previously stated.

It is therefore desirable to know how to efficiently perform a non-linear binary or equidistributed q-area function with many input variables despite the limitations of current technology.

In the prior art, various solutions have already been proposed for efficiently performing Boolean functions or on an extended Galois field GF (2 ⁿ ). For example, patents US 6397370, US 6961427 or US 6868536 propose such functions.

But the skilled person finds that these examples of the prior art:

are restricted to the case of Boolean functions,

- either do not offer constructive solutions to produce the right functions and their heuristic search processes give no guarantee of results,

either offer unsecured solutions on their equidistribution properties,

either offer unsecured solutions on algebraic properties such as the algebraic degree of the functions, or do not exactly quantify the complexity of implementing the functions,

either only offer solutions for a few dozen variables at most.

It is therefore desirable to provide methods for constructing partially symmetrical binary or q-area functions, in particular transitive, t-transitive or symmetrical functions that do not have the drawbacks mentioned above.

Thus, according to one aspect of the invention, a method of encoding information by using non-linear equidistributed functions comprises the following steps of constructing such a function:

selecting a first natural integer, and a first set comprising an element number equal to this first natural integer,

selection of a group operating on this first set,

selecting a second natural integer and a second set comprising a number of elements equal to this second natural integer,

definition of a set of orbits of the group operating on the vectors of n elements of the second set,

defining a first function which, at each of the vectors of n elements of the second set, associates the corresponding orbit, defining a second function assigning to each of the orbits of the group a value of the second set,

equidistributing in a number of sections equal to the second natural integer of the sums of the cardinals of the orbits of the group by using the mathematical properties of said cardinals,

defining a third function composed of the first and second function.

The method advantageously makes it possible systematically to obtain new functions having the desired properties. It also allows to obtain functions on any q-element sets that do not need to be GF (2) Boolean bodies, or even an extension to a GF (2 ⁿ ) Galois field, or even a extension to a body GF (p ⁿ ) while also applying to the particular cases of the prior art for which it provides new functions.

The method for producing symmetrical and partially symmetrical q-area functions proposed advantageously overcomes the limitations of the prior art, illustrated above, and allows the realization of functions of hundreds or thousands of boolean or q-ary variables with a fast program or in a negligible silicon surface compared to a flash memory of

128 Gbits. The method of the present invention may be conveniently performed by a computer or other appropriately programmed processor. Alternatively the process can be realized by a material specially designed for this.

Partially symmetric binary or q-area functions thus allow a variety of construction non-linear equidistributed functions essential for performing cryptographic methods or data classification methods.

According to other characteristics or embodiment:

the method comprises a subsequent step of optimizing the non-linearity or the algebraic degree or an algebraic property of the third function by exchange of orbits;

the method comprises a subsequent step of input composition, as well as output, with other non-linear functions equidistributed so that the global compound obtained by this step is also non-linear and equidistributed;

the first and second natural numbers are not prime between them, the second set being a commutative ring and the group being the set of permutations so that the orbits are the whole partitions whose parts are strictly smaller than the second natural integer the step of defining a first function comprises a substep of constructing a symmetric affine function:

where a is any element of the second set and b is a unit of the second set and (x _, ) " _{= i} is an n-element vector of the second set;

the first and the second natural numbers being strictly greater than 2 and first between them, the set being a commutative ring and the group being the set of permutations whose orbits have as cardinals the multinomial coefficients, the step of defining a first function comprises the selection of trivial equidistributed functions;

the selection of trivial equidistributed functions consists in applying a circular permutation with a shift of a third natural integer to a q-tuple such that, for each value of this third natural integer:

M n, Uk [T ₁ , r ₂ , ..., r _q )) = M n (n, r ,, ..., r _q )

where T ₁ , ..., Tq are multinomial coefficients; then,

optionally, a subsequent step of variation of said function is performed by exchange of orbits having identical cardinals and / or grouping of orbits having cardinals that are multiples of one another;

a third natural number is chosen and the second natural number such that the first natural number is equal to the second natural number to the power of the third natural number, said group being a transitive group, the step of defining a function comprises the sub-steps of:

Defining a transitive Boolean function having a number of variables equal to the first natural number and displaying a Boolean value to each of the orbits of the group, with the exception of the orbits and, • computation of a value, sum for all the values of the first natural integer of the product of the binomial coefficient with the first function r = ΣC ^ λ (θ "- 'l') mod (2) boolean modulo 2, that is ^{/ ≈ 1} ,

• calculation of the value of the first Boolean function for the orbits and such as:

h (θ ") = (T + 1) mod 2 and h (l") = T if p is even, and h (θ ") = T and h (l") = (T + 1) mod 2 if p is odd, -

we choose a fourth natural number, a third natural number and the second natural number such that the first natural number is equal to the sum of the second natural number to the power of the third natural number and the fourth natural number minus 1, the group being a t-transitive group, the step of defining a function comprises the substeps of:

Definition of a t-transitive group of order equal to the second natural number to the power of the third natural number,

Definition of a transitive Boolean function having a number of variables equal to the second natural number to the power of the third natural number and as a symmetry group the t-transitive group,

• definition of a boolean function t-transitive having a number of variables equal to the first natural number by extension of the transitive Boolean function of a number of operands equal to the fourth natural number minus 1.

In another aspect, a computer program product including program code instructions for performing the steps of the preceding method when said program is run on a computer.

The invention will be better understood on reading the description which follows, given solely by way of example, and with reference to the appended figures in which:

Figure 1 is a flow chart of an embodiment of the method according to the invention.

With reference to FIG. 1, an information coding method using equidistributed non-linear functions comprises the following steps of constructing such a function:

selecting a first natural integer n, and a set En having a number of elements equal to this first natural integer n, step 10;

selecting a group G operating on the set In whose degree is equal to this first natural integer n, step 12; and

selecting a second natural integer q and a set Eq having a number of elements equal to this second natural integer q, step 14;

definition of the set OG of the group G orbits operating on the vectors of n elements of Eq, step 15; and - Definition of the function II which each of these vectors associates the corresponding orbit in OG, step 16, ie π: Eq ¹¹ -> OG {view E ^, TI (u) = O _t , O _t G OG};

- definition of a function Φ of OG on Eq affecting each of the orbits of the group G a value of the set

Eq, step 18, ie Φ: OG -> Eq such that

equidistribution of the sums of the cardinals of the orbits of the group G over the set of values of Eq by using the mathematical properties of said cardinals, step 20. Thus q equidistributed sections of sums of these cardinals. The mathematical properties of the cardinals used are, for example, the binomial coefficients, the multinomial coefficients, the partitions, the symmetry of the cardinals of the orbits, the divisibility of the cardinals, the properties of the elements of E _g , and so on.

defining a function F = ΦoTl, F being the compound of the two preceding functions, step 22.

Thus, the function F = ΦoIÏ is such that for each value Wj, j = 1 to q, of Eq, the sum of the cardinals of the vectors u of Eq ¹¹ such that F (u) = Wj is constant. It is therefore equidistributed.

If necessary, step 24, the values of certain orbits are exchanged to obtain the non-linearity or a maximum algebraic degree of the function or any other algebraic property of the function F.

It should be noted that the functions thus constructed are advantageously composable at the input and at the output with other non-linear functions equidistributed from so that the compound is also non-linear and equidistributed.

The advantage of the invention will be better understood on reading a particular case whose description is given below by way of example only. Thus, while for q = 2 and for certain n-pairs (for example 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60), there is no symmetric equidistributed non-linear function, the method makes it possible to perform for q> 2 and for any number of inputs n, a symmetric equidistributed non-linear function. For example, there is no non-linear equidistributed function of 60 binary variables among the symmetric functions. However, the method described above provides at least eight symmetric 4-ary functions of equidistributed non-linear variables, which correspond to Boolean functions of 60 equidistributed and non-linear variables.

The general method described above is implemented according to several embodiments. Four particular embodiments will now be described by way of example, one skilled in the art being able to define others from the previous general description.

In a first embodiment, values of n strictly greater than 5 and of strictly strictly greater than 2 are chosen so that they are not prime between them, that is to say that their gcd (n, q) ≠ l.

The set Eq is chosen as a commutative ring with q elements. For example, the set Eq is the ring ZIqZ or the Galois field GF (q), Any element of E _q is denoted a. Let us denote by b a unit of E _q , that is to say an element of E _g which admits an inverse for the second law of the ring commonly called multiplication. A symmetric affine function with n variables:

where the variables of the function are noted

As the number of units, or cardinal of the group of units, in E _q is u (E _q ), that equi (E _q ) equidistributed symmetric affine functions are thus constructed. For example, in Z / qZ, u (E _q ) = Φ (q) and in GF (q) it is

Φ (q) = q-1.

The group G of degree n is then defined as the set of permutations S _n and the function L defines an action of the group G, or S _n , on the set Eq ¹¹ .

A partition of an integer n is a decomposition of this integer in the form of a sum of natural numbers called parts of this partition. We can also restrict ourselves to a subset of all the partitions by imposing a limit on the number of parts in the partition, or by imposing a limit on the integer parts involved in the sum.

The symmetric functions are then constructed as indicated previously from the orbits of the elements of Eq ¹¹ . Orbits are determined by all the integer partitions of the integer n. (q-1) in n parts strictly smaller than q. Indeed, every vector u of Eqf has n elements (hence n parts) each less than

(since derived from Eq) and whose total sum is equal to or less than n. (q-1). Since n is strictly greater than 5, there are natural integers r and s, with s = 2 (mod q), such that n = 3s + r. Two numbers r and s satisfying this relation are chosen. Partitions u, z, v, w from n to at most q elements are defined from a partition of r into q- 2 elements (r ₂ , ... r _q _i):

u = s + 1, r ₂ , ... r _q _i, 2s + 1

z = 2s + 1, r ₂ , ... r _q _i, s + l

From the functions L _a , b above and particular partitions u, z, v, w, functions G _a , _b are defined by:

G _a , b (x) = L _a , b (x) if x is different from u, v or w

G _{a, b} (u) = L ₃ , _b (v)

G _a , b (V) = L _a , _b (U)

G ₃ , b (W) = L _a , _b (U).

And functions H _a , b are defined by:

H _a , b (x) = L _a , b (x) if x is different from z, v or w

H _{a, b} (Z) = L _{a, b} (V)

H _a , _fc (v) = L _a , _b (z)

H _{a, b} (w) = L _a, _b (z).

All the functions G ₃ , b, H _a , _b thus produced are non-linear equidistributed functions. It is thus possible to construct for infinitely large values of n such that pgcd (n, q) ≠ l, up to 2.qu (E _q ) functions equidistributed non-linear. In any coiranutative ring of q elements, q strictly greater than 2, the number of nonlinear equidistributed function realizations is therefore 2.qu (E _q )> 2 q> 4.

As an example applied, q is chosen equal to 16 or 2 ⁴ and Eg is the Galois body GF (2 ⁴ ). We consider

C ₁₂₁ symmetrical functions of n = 106 variables on this body, thus n = 106 = 3 * 2 + 100. Considering the formula n = 3s + r above, we obtain s = 2 and r = 100 which also satisfies the property s = 2 (mod 16) above.

With s = 2, we obtain s + 1 = 3 and 2s + l = 5, 2s + 2 = 6. Consequently,

u = 3 times 1, 5 times 0 and the other 98 values are arbitrary,

z = 5 times 1, 3 times 0 and the other 98 values are arbitrary,

v = 2 times 1, 6 times 0 and the other 98 values are arbitrary,

w = 6 times 1, 2 times 0 and the other 98 values are arbitrary.

We then construct a nonlinear equidistributed function of 106 variables in E _q = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, f} which coincides with the function j + 9 except when the number of 1s or zeros in the input 106-tuple is 2, 3, 5, 6, in which case the value is:

Gg, 3 (x) = L ₉ , ₃ (x) if x is different from u, v or w

G ₉ , ₃ (u) = L ₉ , ₃ (v) G ₉ , ₃ (V) = L ₉ , ₃ (U) G ₉ , ₃ (W) = L _{9 (3} (U).

In another exemplary embodiment, n = 16 = 3 * 2 + 10 is selected. Thus, we define s = 2 and r = 10 which satisfies the property s = 2 (mod 16).

Since s = 2, s + l = 3, 2s + i = 5 and 2s + 2 = 6. So,

u = 3 times 1.5 times 0 and the other 8 values are arbitrary;

z = 5 times 1, 3 times 0 and the other 8 values are arbitrary;

v = 2 times 1, 6 times 0 and the other 8 values are arbitrary;

w = 6 times 1, 2 times 0 and the other 8 values are arbitrary.

In this example, we construct two non-linear equidistributed functions of 16 variables in

E _q = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, f} which coincide;

- with the function L ₉ - ~ (x) = 3 \ ^ _d x _l J + 9 except when the number of 1 or zeros in the input 16-tuple is 2, 3, 5, 6 in which case the value is that of:

H _{9; 3} (x) = Lg, 3 (x) if x is different from u, v or w

H ₉ , ₃ (u) = L ₉ , ₃ (v)

H _9, 3 (V) L = _9, 3 (Z) H _9, ₃ (w) = L _9, ₃ (z) or with the function 11

Gn, 5 (x) - Lii, 5 (x) if x is different from u, v, w

Gn, ₅ (u) = Lu, ₅ (v)

Gii, ₅ (v) = Lu, s (u)

Gu, ₅ (w) = Lu, ₅ (u).

In a second embodiment, values of n strictly greater than 2 and q strictly greater than 2 are chosen such that n and q are prime between them. The set Eq is chosen to be a commutative ring. The starting functions are then no longer the affine functions, as in the first embodiment, but trivial equidistributed functions f which exist thanks to the choice n and q prime between them.

The group G of degree n is, in this embodiment, the set of permutations S _n , and its orbits have as cardinal the multinomial coefficients.

These functions are constructed from relations on multinomial coefficients.

Let dk be a circular permutation with an offset k on a q-tuple.

For each value of k we have the relation:

M (n, dk (r _x , r ₂ , ..., r _q )) = M (n, r _x , r ₂ , ..., r _q ) with

When n and q are prime between them, we get different shifts of q ri because the ri can not be all equal. Because if the laughs were all equal to d then qd = n which contradicts the primacy of q and n.

We thus succeed in defining, equidistributed symmetrical functions f.

It should be noted that for q strictly greater than 2, v and w can not be obtained by an offset of one another but by a transposition. Thus in this case where n and q are prime between them, by the construction of the trivial equidistributed functions f, these realize the equality such that for each i exists j so that f (di (v)) = dj (f (w)). Thus a new function g can be established by:

g (x) = f (x) if x is different from u, di (v) or dj (w);

g (u) = f (di (v));

g (di (v)) = f (u);

g (dj (w)) = f (u).

If f (di (v)) is different from f (u), there is (q-1) case where it is different.

g is a new equidistributed function. Indeed, among all the circular permutations dk (v), k varying from 0 to q-1, all the values of the parts of the partition do not appear because some are repeated and others are absent.

Thus for each function f, (q-1) functions g can be established.

This embodiment then makes it possible to construct qx (ql) ""'''non-linear equidistributed functions for infinitely large values of n. In a third embodiment, we consider transitive functions such as rotationally symmetrical Boolean functions. In particular, they make it possible to perform curved Boolean functions that are not quadratic, unlike curved symmetric Boolean functions, which are always quadratic. The realization of such functions extends the choice of functions to transitivity beyond mere symmetry or rotation symmetry. A large number of authors have studied the Boolean symmetric functions but the generalization allowed by the present embodiment on the one hand to the q-ary functions when q is strictly greater than 2 and on the other hand to the binary or q-ary functions. partially symmetrical provides new possibilities.

We choose k natural integer and p prime natural integer and we calculate n = p ^k , p at power k. A transitive group G of order n is chosen. A transitive boolean function h of n = p ^k variables is defined by assigning a Boolean value of either 0 or 1 to each of the orbits of group G except for the two orbits represented by (0 ⁿ ) and d ⁿ ). nl

Calculating the value T = C ^ _d ^ _ _ι h (θ '"' V) modf2j.

If p is even, we define h ((0 ⁿ )) = (T + 1) mod (2) and h ((l ⁿ )) = T and if p is odd h ((l ⁿ )) = (T + l) mod (2) and h ((0 ⁿ )) ≈ T.

The function h is a transitive equidistributed boolean function of degree n-1 so highly nonlinear.

In a fourth embodiment, a fourth natural number, a third natural number and the second natural number are chosen, such as the first number. natural is equal to the sum of the second natural number to the power of the third natural number and the fourth natural number minus 1, the group being a t-transitive group, the step of defining a function comprises the substeps of:

* definition of a t-transitive group of order equal to the second natural number to the power of the third natural number,

* definition of a transitive Boolean function having a number of variables equal to the second natural number to the power of the third natural number and as a symmetry group the t-transitive group,

* Definition of a boolean function t-transitive having a number of variables equal to the first natural number by extension of the transitive Boolean function of a number of operands equal to the fourth natural number minus 1.

The method described is feasible in software or hardware form. In particular, a computer program product makes it possible to execute the various steps of the method when it is executed on a computer.

For the functions obtained by the proposed method, the technological execution is easy both in software and hardware. In addition, the properties of the functions are interesting for different applications including cryptography or data classification.

All constructed functions are integrable as a constituent element or as a replacement for a component of any cryptographic process or hash data classification.

The described method is particularly useful in cryptographic and / or data classification applications.

It finds application particularly in the field of secure telecommunications as well as in other areas such as optimized computer storage, telecommunication protocols, electronic voting, surveys or electronic auctions or biometrics. It is adapted to the realization of cryptographic mechanisms in terminal equipment, fixed and / or mobile, operating on any type of network or in cryptographic modules. In these applications, it provides enhanced efficiency for performing a cryptographic solution or a hash or search tree ranking solution.

Claims

A method of encoding information by using non-linear, equidistributed functions comprising the following steps of constructing such a function:

selecting (10) a first natural integer (n), and a first set (En) having an element number equal to this first natural integer (n),

selecting (12) a group (G) operating on this first set (En),,

selection (14) of a second natural integer (q) and a second set (Eq) comprising a number of elements equal to this second natural integer (q),

defining (15) a set (OG) of the G-group orbits operating on vectors, having the first natural number (n) number of elements of the second set (Eq),

defining (16) a first function (FI) which, at each of said vectors of the second set (Eq), associates the corresponding orbit,

defining (18) a second function (Φ) assigning to each of the orbits of the group (G) a value of the second set (Eq),

equirepartition (20) in a number of sections equal to the second natural integer (q) of the sums of the cardinals of the orbits of the group by using the mathematical properties of said cardinals, defining (22) a third function (F) composed of the first and second function

(Φoii).

2. Encoding method according to claim 1, characterized in that it further comprises a subsequent step of optimizing the non-linearity or the algebraic degree or an algebraic property of said third function (F) by exchange of 'orbits.

3. Encoding method according to one of claims 1 and 2, characterized in that it further comprises a subsequent step of input composition, as output, with other non-linear functions equidistributed so that the overall compound obtained by this step is also non-linear and equidistributed.

4. Encoding method according to one of claims 1 and 2, characterized in that the first and second natural numbers are not prime between them, the second set (Eq) being a commutative ring and the group (G) being the set of permutations in such a way that the orbits are the whole partitions whose parts are strictly smaller than the second natural integer (q), the step of defining a first function, comprises a sub-step of constructing a symmetric affine function (L _a , - _D ):

where a is any element of the second set and b is a unit of the second set and 0 0 " _{= 1} is an n-element vector of the second set.

Coding method according to one of Claims 1 to 2, characterized in that the first and the second natural integers being strictly greater than 2 and prime between them, the set being a commutative ring and the group being the set of permutations whose orbits have as cardinals the multinomial coefficients, the step of defining a first function comprises the selection of trivial equidistributed functions.

6. Encoding method according to claim 4, characterized in that the selection of trivial equidistributed functions consists in applying a circular permutation (dk) with a shift of a third natural integer (k) on a q-tuple such that, for each value of this third natural integer (k):

M (n, dk (r _x , r ₂ , ..., r _q )) = M (n, η, ..., r _q )

Where ri, ..., r _q are multinomial coefficients.

7. Coding method according to claim 6, characterized in that it further comprises a subsequent step of variation of said function by exchange of orbits having identical cardinals and / or grouping of orbits having cardinals multiple of others.

8. Encoding method according to one of claims 1 to 2, characterized in that a third natural number (k) and the second natural number (p) are chosen such that the first natural number (n) is equal to the second number natural (p) to the power of the third natural number (k), said group (G) being a transitive group, the step of defining a function comprises the substeps of:

defining a transitive Boolean function (h) having a number of variables equal to the first natural number (n) and displaying a value Boolean at each of the group's orbits, with the exception of orbits (On) and (In),

computation of a value (T), sum for all the values of the first natural integer of the product of the binomial coefficient with the first function nI boolean (h) modulo 2, ie T =] TC _n ' _Λ hψ ^~ ' \ ^; ) mod (2),

calculating the value of the first Boolean function for the orbits (On) and (In) such that:

h (θ ") = (T + 1) mod 2 and h (l") = T if p is even, and h (θ ") = T and Λ (r) = (7 + 1) mod2 if p is odd .

9. Coding method according to one of claims 1 to 2, characterized in that a fourth natural number (t), a third natural number (k) and the second natural number (p) such as the first number are chosen. natural (n) is equal to the sum of the second natural number (p) to the power of the third natural number (k) and the fourth natural number (t) minus 1, the group (G ') being a t-transitive group, the step of defining a function comprises the sub-steps of:

- definition of a group (G 't) t-transitive order equal to the second natural number (p) to the power of the third natural number (k)

defining a transitive Boolean function

(ht) having a number of variables equal to the second natural number (p) to the power of the third natural number (k) and as a symmetry group the t-transitive group (G 't) according to the function definition step of method of claim 8, definition of a boolean function t-transitive (h) having a number of variables equal to the first natural number (n) by extension of the transitive Boolean function (ht) of a number of operands equal to the fourth natural number (t) less 1.

A computer program product comprising program code instructions for performing the steps of the method according to any one of claims 1 to 9 when said program is run on a computer.

11. Cryptography method using a non-linear equidistributed function characterized in that it comprises an information coding method according to any one of claims 1 to 9.

12. Method of classification of data by hashing or search tree characterized in that it comprises an information coding method according to any one of claims 1 to 9.