Classifications

• • H—ELECTRICITY
  • H03—ELECTRONIC CIRCUITRY
  • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
  • H03M7/00—Conversion of a code where information is represented by a given sequence or number of digits to a code where the same, similar or subset of information is represented by a different sequence or number of digits
  • H03M7/30—Compression; Expansion; Suppression of unnecessary data, e.g. redundancy reduction

Definitions

• 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.
• Equidistributed non-linear functions are very useful as: hash function for optimal classification of hash data or search tree.
• hash functions are, 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"
• 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.
• non-linear equidistributed functions are essential to ensure the quality of a cryptographic algorithm.
• these functions are even better than their number of input variables n is large.
• a method of encoding information by using non-linear equidistributed functions comprises the following steps of constructing such a 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 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
• 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.
• 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 step of defining a first function comprises a substep of constructing a symmetric affine function:
• the step of defining a first function comprises the selection of trivial equidistributed functions
• T 1 , ..., Tq are multinomial coefficients
• 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:
• the step of defining a function comprises the substeps of:
• a computer program product including program code instructions for performing the steps of the preceding method when said program is run on a computer.
• Figure 1 is a flow chart of an embodiment of the method according to the invention.
• an information coding method using equidistributed non-linear functions comprises the following steps of constructing such a function:
• 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.
• 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.
• 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.
• 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.
• the set Eq is the ring ZIqZ or the Galois field GF (q)
• Any element of E q is denoted a.
• E q 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.
• u (E q ) ⁇ (q) and in GF (q) it is
• 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.
• G a , b (x) L a , b (x) if x is different from u, v or w
• G 3 , b (W) L a , b (U).
• H a , b (x) L a , b (x) if x is different from z, v or w
• q is chosen equal to 16 or 2 4 and Eg is the Galois body GF (2 4 ).
• Gg, 3 (x) L 9 , 3 (x) if x is different from u, v or w
• 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.
• dk be a circular permutation with an offset k on a q-tuple.
• M (n, dk (r x , r 2 , ..., r q )) M (n, r x , r 2 , ..., r q ) with
• g (x) f (x) if x is different from u, di (v) or dj (w);
• 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.
• This embodiment then makes it possible to construct qx (ql) ""'''non-linear equidistributed functions for infinitely large values of n.
• transitive functions such as rotationally symmetrical Boolean functions.
• 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.
• n p k , p at power k.
• a transitive group G of order n is chosen.
• the function h is a transitive equidistributed boolean function of degree n-1 so highly nonlinear.
• 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:
• the method described is feasible in software or hardware form.
• a computer program product makes it possible to execute the various steps of the method when it is executed on a computer.
• the technological execution is easy both in software and hardware.
• 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.

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• 2007
  • 2007-06-14 FR FR0704243A patent/FR2917549A1/fr not_active Withdrawn
• 2008
  • 2008-06-13 WO PCT/FR2008/051068 patent/WO2009004235A2/fr not_active Ceased
  • 2008-06-13 EP EP08806003A patent/EP2168245A2/de not_active Withdrawn
• 2009
  • 2009-12-14 US US12/637,127 patent/US8615537B2/en not_active Expired - Fee Related

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