Public key cryptosystem presented by means of a key encapsulation mechanism based on the subset sum problem

Abstract

The present invention relates to an asymmetric cryptosystem and cryptographic method with a key encapsulation mechanism based on the subset sum problem and in which the private key is described based on the initial parameters n, t and z, and made up of the positive integers pl,..., pn, g and u. The invention is characterised in that said positive integers meet the following conditions: pl,..., pn are relatively prime to each other and they are not divisible by any prime number less than z; 2q< √g, where q is the product of ql,... qt and qt, the t largest integers of the set {pl,..., pn}. 2p <√(g); where p = (z -1) ∑(i=l, t) q / qi; u is a positive integer less than g and the public key is made up of integer numbers ti,..., tn, characterised in that said positive integers meet the conditions pi(ti + u) = 1 mod g for all i = 1,..., n.

Description

[0001]PUBLIC-KEY CRYPTOSYSTEM PRESENTED BY MEANS OF A KEY ENCAPSULATION MECHANISM BASED ON THE SUBSET SUM PROBLEM TECHNIQUE SECTOR The present invention falls within the field of provisions for secret or protected communications using an encryption algorithm, in particular public-key algorithms, where the encryption algorithm is impossible to reverse by computer and the encryption keys of users are not required to be secret. Specifically, the invention relates to the subset sum problem and its use in asymmetric encryption systems. BACKGROUND OF THE INVENTION In 1976, W. Diffie and M.E.Hellman propose a new type of cryptosystem whose main characteristic is the presence of two different keys, one for the encryption process (public) and another for the decryption (private), such that it is computationally difficult to know the decryption key from the encryption key [Whitfield Diffie and Martin Hellman. ((New directions in cryptography)). In: IEEE Transactions on Information Theory 22.6 (1976), pp. 644-654. DOI: 10.1109 / TIT .1976.1055638]. This new cryptographic paradigm is called asymmetric cryptography or public-key cryptography. Given the difference in performance between public key cryptosystems and symmetric cryptosystems (AES type [National Institute of Standards and Technology. Advanced Encryption Standard (AES). Technical Report FIPSPUB 197. National Institute of Standards and Technology, Nov. 2001. DOI: 10.6028 / NIST.FIPS.197] or stream ciphers), the former are often used to transmit a session key to be used in a pre-agreed symmetric cryptosystem. One way to transmit such a key is through a key encapsulation mechanism (KEM) that works as follows: the sender randomly generates a secret e in a sufficiently large search space. This secret is sent encrypted with the receiver's public key. The receiver decrypts the secret using their private key, so both know the value of e. By using a cryptographically secure hash function K that produces K-bit strings, where K is a key length for the agreed symmetric algorithm, both receiver and receiver can derive the common key K(e). Examples of cryptographically secure hash functions are SHA-2, SHA-3 [Quynh Dang. Secure Hash Standard. en. Tech. Inf.National Institute of Standards and Technology, Aug. 2015. DOI: 10.6028 / NIST.FIPS.180-4] and SHAKE, with outputs between 224 and 522 bits. The first public-key cryptosystem [R.C. Merkle and M.E. Hellman. ((Hiding Informationand Signatures in Trapdoor Knapsacks)). In: Transactions on Information Theory 24.5 (1978), pp. 525–530] was published in 1978 by Merkle and Hellman using the knapsack problem, a well-known NP-hard problem. This was successfully attacked by Shamir in 1984 [A. Shamir. ((A polynomial time algorithm for breaking the basic Merkle-Hellman cryptosystem)). In: IEEE Transactions on Information Theory 30 (1984), pp. 699–704]. The most used today are the RSA cryptosystem [Ronald L. Rivest, Adi Shamir and Leonard Adleman. ((A Method for Obtaining Digital Signatures and Public-Key Cryptosystems)). In: Commun. ACM 21.2 (Feb. 1978), pp. 120-126. ISSN: 0001-0782. DOI: 10.1145 / 359340.359342], based on the difficulty of factoring integers with several large prime factors, and those based on the discrete logarithm problem, either on the group of units of a prime field [Taher ElGamal. ((A public key cryptosystem and a signature scheme based on discrete logarithms)). In: IEEE Transactions on Information Theory 31.4 (1985), pp. 469–472. DOI: 10.1109 / TIT.1985.1057074] or on the group of points on an elliptic curve [Victor S. Miller. ((Use of Elliptic Curves in Cryptography)). In: Advances in Cryptology — CRYPTO '85 Proceedings. Ed. by Hugh C. Williams. Berlin, Heidelberg: Springer Berlin Heidelberg, 1986, pp. 417–426. ISBN:978-3-540-39799-1; Neal Koblitz. ((Elliptic curve cryptosystems)). In: Mathematics of Computation 48 (1987), pp. 203-209. DOI: 10.1090 / S0025-5718-1987-0866109-5]. These can be considered as classical asymmetric cryptosystems.However, the presumed construction of a general-purpose quantum computer in the relatively near future poses a serious threat to these cryptosystems. This is due to the existence of Shor's algorithm [Peter W. Shor. ((Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer)). In: SIAM Review 41.2 (1999), pp. 1-27. 303-332.DOI: 10.1137 / S0036144598347011], which factors integers in polynomial time on a quantum computer from a one-element period calculation in modular arithmetic.Our invention consists of a public-key cryptosystem based on the subset sum problem, for which no effective attacks on quantum computers are known.BRIEF DESCRIPTION OF THE DRAWINGSFigure 1 contains the diagram describing the steps to execute the Extended Truncated Euclidean Algorithm (ETEA).EXPLANATION OF THE INVENTIONThe following notations will be used in this document:Given a positive integer x, we will denote by |x| the absolute value of x. We will denote by x the positive square root of x. Given three integers a, b and c, where c is non-zero, we will denote by a ≡ b mod c the fact that a − b is a multiple of c, that is, there exists an integer d such that a − b = cd. Given two integers a and b, a%b denotes the positive remainder of dividing a by b, and ya / b the only integer such that a = (a / b)b + a%b. A shared secret is a random vector from which a session key will be derived to be used in a symmetric encryption algorithm. Given two integers a and b with a ≥ b > 0, we will denote by A an algorithm that given the input (a, b) of √. vuelva co√mo output, if they exist, two integers v, r verifying that vb ≡ rmod a, 0 ≤ r < a, |v| ≤ a, and v and r are relatively prime. A preferred embodiment of A is the Extended Truncated Euclidean Algorithm AEET described in Figure 1. The invention described herein is an asymmetric cryptosystem based on the subset sum problem, more specifically, the security of the cryptosystem lies in the NP-completeness of the following decision problem: Given h1, ... , h , and s positive integers, do values ​​e , ... , e ∈ {0, 1, ... , z−1} exist? ales que ∑ n 1 n} t ni=1 eihi = s?In order to describe the cryptosystem that is the object of this invention, it is necessary to detail the key generation, encryption, and decryption procedures. Description and Key Generation In a first aspect, the present invention relates to the description of the keys for use in a cryptosystem. Specifically, the public keys of the invention used in the encapsulation process and the private keys of the invention used in the decapsulation process are described below. It is assumed that the sender and receiver share a cryptographically secure hash function K : Fn → {0, 1}K for a symmetric key length K to be used in a symmetric cryptosystem. Both the hash function and the symmetric cryptosystem may be known.We consider as initial parameters a positive integer n, the length of the cryptosystem, a positive integer z, which determines the alphabet of the shared secret, and an integer t ≤ n, which determines the number of non-zero components of the shared secret. The private keys of the invention comprise n + 2 positive integers p1, ... , pn, g, u verifying the following properties: p1, ... , pn are relatively prime with each other, and are not divisible by any prime number less than z. p1, ... , pn are relatively prime with g. 2q ≤ √ g, where q is the product of q1, ..., qt−1 and qt, the t largest integers in the set {p1, ... , pn}. 2p < √g, where p = (z − 1)∑ t qi=1 q i. ues a positive integer less than g. In this context, the private key generation method of the invention comprises the following steps: Random selection of n positive integers p1, ... pn, relatively prime to each other, which are not divisible by any prime number less than z. Random selection of a positive integer g, relatively prime to each of the integers p1, ... , pn, verifying: • 2q ≤ √ g, where q is the product of q1, ..., qt−1 and qt, the t largest integers in the set {p1, ... , pn}. • 2p < √g, where p = (z − 1)∑ t qi=1 q i. Random selection of a positive integer u less than g. Return of the values ​​p1, ... , pn, g and u, which form the private key. In another aspect, the present invention also relates to the public key of the invention formed by n integers t1, ... , tn such that pi(ti + u) ≡ 1 mod g for all i = 1, ... , n. Thus, a public key generation method comprising the following steps is also the object of the present invention: Calculation of positive integers h1, ... , hn, less than g, such that hipi ≡ 1 mod g for each i = 1, ... , n. Calculation of non-negative integers t1, ... , tn with = (hi − u)%g for each i = 1, ... , n. Return of the public key formed by the integers t1, ... , tn. Preferred embodiments of the description and generation of keys: In a particular embodiment of the description and generation of keys, z = 2. In another preferred embodiment of the description and generation of keys, the integers p1, ... , pn are prime. Encryption method of the invention In another aspect, the invention relates to an encryption method from a public key of the invention t1, ... , tn, comprising: 1. Randomly selecting a shared secret, that is, a random vector e = (e1, ... , en), of length n, with n − t null components, and t non-zero components containing positive integer values ​​less than z. 2. Calculating the sum c∑ n 1 = i=1 and i t i . 3. Calculate the sum c∑ n 2 = i=1 and i.The ciphertext or cryptogram will comprise the pair formed by the values ​​c1 and c2. Particular embodiments of the encryption procedure: In a particular embodiment of the encryption procedure of the invention, the initial parameter z is set to z = 2 and the encryption comprises the following steps: 1. Select a binary shared secret, that is, a random binary vector e = (e1, ... , en), of length n, with t components taking the value 1 and the rest taking the value valor 0. 2. Calculate the sum The ciphertext or cryptogram will be formed by the value c1. Decryption method of the invention In another aspect, the invention also relates to a decryption method using the private key of the invention p1, ... , pn, g, u which comprises applying Algorithm 1. Algorithm 1 Decryption algorithm Input Two positive integers (c1, c2) Output A vector e = (e1, ... , en) with integer components. 1: s ← (c1 + uc2)% g. 2: λ, ω ← A(g, s) 3: For 1 ≤ i ≤ n do 4: If pi divides λ then 5: calculate ei, the unique integer verifying ≡ ω mod pi and 0 < < pi. 6: Si no 7: ei = 0.8: Return e Particular embodiments of the decryption method: In a particular embodiment of the decryption method of the invention, the initial parameter z is set to z = 2, and the decryption method comprises applying Algorithm 2. Algorithm 2 Binary decryption algorithm Input A positive integer c1. Output A binary vector e = (e1, ... , en) 1: s ← (c1 + ut)%g. 2: λ, ω ← A(g, s) 3: For 1 ≤ i ≤ n do 4: If pi divides λ then 5: ei = 1 6: Si no 7: ei = 0 8: Return and Preferred embodiments of the decryption method:In two preferred embodiments of the decryption method of the invention, algorithm A in Algorithms 1 and 2 is the Extended Truncated Euclidean Algorithm (ETRU) described in Figure 1.Systems and apparatus of the inventionIn another aspect, the present invention also contemplates apparatus and systems that allow carrying out the previously described procedures. A clear example of an apparatus is a computer.Thus, the object of the invention is a system or apparatus suitable for processing data that comprises the necessary means to carry out any of the procedures of the invention: Public or private key generation procedures, encryption procedure or decryption procedure.More specifically, the following systems and apparatus are considered to be the object of the invention: A system or apparatus suitable for data processing comprising the means necessary to carry out the following steps: • Enter the initial parameters n, z and t into the system / apparatus. • Select integers p1, ... , pn that are relatively prime to each other and not divisible by primes less than z. • Select an integer g, relatively prime with each of the integers p1, ... , pn, verifying: ◦. 2q ≤ √ g, where q is the product of q1, ..., qt−1 and qt, the t largest integers in the set {p1, ... , p}.◦ 2p < √ ng, where p = (z − 1)∑ t qi=1 q i. •Select an integer u less than g.• Provide as output the following information giving rise to a private key:The positive integers p1, ... pn, the integer g, the integer u.A system or apparatus suitable for data processing comprising means for performing the following steps:• Inputting the following data into the system / apparatus: positive integers z, n, p1, ... pn, g and u.• Calculating positive integers t1, ... , tn using the following procedure:(1) Calculate integers h1, ... , hn such that hipi ≡ 1 mod gy 0 < hi < g for each i = 1, ... , n.(2) Calculate integers t1, ... , tn with = (hi − u)%g for each i = 1, ... , n.• Provide the integers as output that give rise to the public key associated with the previous private key. A system or apparatus suitable for data processing that includes the necessary means to carry out an encryption procedure based on a public key by performing the following steps: • Enter the following data into the system / apparatus: n, t1, ... , tn, t and z. • Select a random vector e = (e1, ... , en) with t non-zero components and values ​​less than z. • Calculate the sum c1 = and i t i . •Calculate the sum and i. •Provide the cryptogram (c1, c2) as output. A system or apparatus suitable for data processing that includes the necessary means to carry out a decryption procedure using a private key by performing the following steps: • Enter the following data into the system / apparatus: integers c1 and c2. • Apply the decryption algorithm (Algorithm 1) to the pair (c1, c2), obtaining the vector. •Provide the vector e as output.IMPLEMENTATION OF THE PROCEDURES OF THE INVENTIONIn another aspect, the invention relates to computer programs comprising instructions for causing a computer, or a system or apparatus suitable for data processing, to carry out any of the procedures of the invention (Processes for generating a public or private key, encryption procedure, decryption procedure) when loaded into said computer, system or apparatus suitable for data processing.The invention encompasses computer programs that may be in the form of source code, object code, or code intermediate between source code and object code, such as in partially compiled form, or in any other form suitable for use in implementing the processes according to the invention. In particular, computer programs also encompass cloud applications that implement any of the methods of the invention. These programs may be arranged on or within a medium suitable for reading, hereinafter referred to as a "carrier medium" or "carrier." The carrier medium may be any entity or device capable of carrying the program. When the program is embodied in a signal that can be transported directly by a cable or other device or medium, the carrier medium may consist of said cable or other device or medium.Alternatively, the carrier medium could be an integrated circuit on which the program is embedded, the integrated circuit being adapted to execute, or to be used in the execution of, the corresponding processes. For example, the programs could be embedded in a storage medium, such as a ROM memory, a CD-ROM memory or a semiconductor ROM memory, a USB memory stick, or a magnetic recording medium, for example, a floppy disk or a hard disk. Alternatively, the programs could be supported on a transmissible carrier signal. For example, this could be an electrical or optical signal that could be transported through electrical or optical cable, by radio, or by any other means.In this sense, another object of the invention is a computer-readable storage medium comprising program instructions capable of causing a computer or a system or apparatus suitable for data processing to carry out any of the methods of the invention when loaded into said computer, system or apparatus suitable for the process. de datos.Finally, a final object of the invention relates to a transmissible signal comprising program instructions capable of causing a computer or a system or apparatus suitable for data processing to carry out any of the methods of the invention when loaded into said computer, system or apparatus suitable for data processing. MODE OF EMBODIMENT OF THE INVENTION An implementation of an example of how to select the cryptosystem keys, the encryption process and the decryption process is described below. It has been implemented in a Jupyter Notebook of the Sagemath computational algebraic system (Python) on a computer with an eight-core Apple M1 processor under the macOS Monterey operating system version 12.3.1.1. Parameter selection. For our example we will work with n = 20, z = 7 and t = 2.2. To build the private key: We select relatively prime integers between them and not divisible by smaller primes. que 7:p1 = 41 p2 = 31 p3 = 67 p4 = 131 p5 = 89 p6 = 191 p7 = 79 p8 = 53 p9 = 73 p10 = 163 p11 = 103 p12 = 109 p13 = 113 p14 = 37 p15 = 151 p16 = 83 p17 = 173 p18 = 7 p19 = 149 p20 = 181 We choose g = 544620996436291, whose square root is bounded below. te por√g ≥ 23337116. • g = 359 · 563 · 1217 · 1483 · 1493 is its prime factorization, so g is relatively prime to the integers p1,..., pn. • q1 = 181 and q2 = 191 are the largest values of the set {p1, 2, 2q = 2 × 34571 ≤ 23337116 ≤ √,..., pn}. Then q = q1q2 = 34571, which satisfies g. • p = (z − 1)(q1 + q2) = 2232, which satisfies 2p = 2 × 2232 <2333711>6 ≤ √g. We select the positive integer u = 29578846142407, which is less than g. 3. For the construction of the public key: We calculate the integer hi as the inverse of pi modulo g for i = 1,..., n.<h2 style=";text-align:left;direction:ltr">In other words, h1 = 518054118561350 h2 = 245957869358325 h3 = 430819594195872 h4 = 515519111130535 h5 = 538501659397681 h6 = 173936548600072 h7 = 358484706515027 h8 = 524069260721714 h9 = 402870326130955 h10 = 30071097962740 h11 = 486457589049891 h12 = 439694015471501 h13 = 457867209393342 h14 = 529901510046121 h15 = 530193950173078 h16 = 111548878788156 h17 = 78702456132412 h18 = 311211997963595 h19 = 51172442618175 h20 = 436298588305316 We calculate the values ​​as (hi − u)%g for i = 1, ... , n. In this case, t1 = 488475272418943 t2 = 216379023215918 t3 = 401240748053465 t4 = 485940264988128 t5 = 508922813255274 t6 = 144357702457665 t7 = 328905860372620 t8 = 494490414579307 t9 = 373291479988548 t10 = 492251820333 t11 = 456878742907484 t12 = 410115169329094t13 = 428288363250935 t14 = 500322663903714 t15 = 500615104030671t16 = 81970032645749 t17 = 49123609990005 t18 = 281633151821188t19 = 21593596475768 t20 = 4067197421629094.We randomly select the shared secret, a vector e of length n = 20 with t = 2 non-zero components whose value is less than z = 7. (0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0). In this case, the non-zero components correspond to positions 10 and 19.5. Encryption. We encrypt the message by performing the sums c1 = 1 · 373291479988548 + 2 · 21593596475768 = 43679444771869. c2 = 1 + 2 = 36. Decryption. The receiver applies the Decryption Algorithm (Algorithm 1), using AEET to obtain λ and ω, to the pair (c1, c2), obtaining the vector e and decrypting the message. Specifically, we calculate s = c1 + uc2 = 132415983199090. Running AEET(g, s), we obtain as output λ = 24287 and ω = 475. Since λ = 149·163, the non-zero components are found in the positions. 10 y 19. We then have that e10 = (475 · 128) mod 163 = 1 and e19 = (475 · 32) mod 149 = 2. An example of the particular realization of this cryptosystem is described below when z = 2.Parameter selection. For this example we will work with n = 10 and t = 3. To build the private key: We select relatively prime integers: p1 = 48 p2 = 31 p3 = 47 p4 = 11 p5 = 25 p6 = 49 p7 = 41 p8 = 17 p9 = 29 p10 = 13 We select g = 1689602223409, whose square root is √ g ≥ 1299847 and is co-prime with all the integers pi for i = 1, ... , 10. q1 = 47, q2 = 48 and q3 = 49 are the largest values ​​in the set {p1, ... , p}. 1 2 3 and 2q <√ is verified 10Then q = qqq = 110544, qug. p= q1q2 + q1q3 + q2q3 = 6911, which verifies 2p <√ g. We select the positive integer u = 1456271001391, which is less than g. For the construction of the public key: We calculate integer hi as the inverse of pi modulo g for i = 1... , n. In this case, h1 = 1654402177088 h2 = 163509892588 h3 = 1186316454734h4 = 1536002021281 h5 = 743424978300 h6 = 1551675511294h7 = 824196206541 h8 = 894495294746 h9 = 291310728174h10 = 1299694018007We calculate the ti values ​​as (hi − u)%g for i = 1, ... , n. In this case, t1 = 198131175697 t2 = 396841114606 t3 = 1419647676752 t4 = 79731019890 t5 = 976756200318 t6 = 95404509903 t7 = 1057527428559 t8 = 1127826516764 t9 = 524641950192 t10 = 1533025240025 We randomly select the shared secret, a vector e, of length n = 10 with t = 3 non-zero components, (1, 0, 0, 0, 0, 0, 0, 1, 1, 0). In this case, the non-null components correspond to positions 1, 8 and 9. Encryption.We encrypt the message by performing summation 1 = 198131175697 + 1127826516764 + 524641950192 = 1850599642653. Decryption. The receiver applies the Decryption Algorithm (Algorithm 2), using AEET to obtain λ and ω, to the integer c1, obtaining the vector e and decrypting the message. Specifically, we calculate s = (c1 + ut)%g = 1150605976599. Running AEET(g, s), we obtain as output λ = 23664 and ω = 2701. Since λ = 48·17·29, the non-zero components are found in the positions. 1,8 y 9. Therefore, the shared secret decrypts to (1, 0, 0, 0, 0, 0, 0, 1, 1, 0).

Claims

CLAIMS1. Private key, described from the initial parameters n, t and z, and formed by positive integers p1, ... , pn, g and u, characterized in that said positive integers fulfill the following conditions: p1, ... , pn are relatively prime among themselves, and that they are not divisible by any prime number less than z. 2q ≤ √ g, where q is the product of q1, ..., qt−1 and qt, the t largest integers in the set {p1, ... , pn}. 2p < √g, where p = (z − 1)∑ t qi=1 q i. ues a positive integer less than g.

2. Private key according to claim 1 characterized in that z = 2.

3. Private key according to claim 1 characterized in that the numbers p1, ... , pn are prime.

4. Private key generation method according to any of the preceding claims, comprising the following steps: Selection of initial parameters: a positive integer n, which determines the length of the shared secret, a positive integer z, which determines the alphabet of the shared secret, and an integer t, which determines the number of non-zero components of the shared secret. Selection of n positive integers p1, ... pn, relatively prime to each other, which are not divisible by any prime number less than z. Selection of a positive integer g greater than the integers p1, ... , pn, and relatively prime to all of them, verifying that (1) 2q ≤ √g, where q is the product of q1, ..., qt−1 and qt, the t largest integers in the set {p1, ... , pn}. (2) 2p <√g, where p = (z − 1)∑ t qi=1 q i . Selection of a positive integer u less than g. Return of the values ​​p1, ... , pn, g and u, which form the private key.

5. Method according to claim 4, characterized in that z = 2.

6. Method according to claim 4, characterized in that the numbers p1, ... , pn are prime.

7. Public key, associated with a private key according to any of claims 1 to 3, formed by integers t1, ... , tn, characterized in that said positive integers fulfill the conditions pi(ti + u) ≡ 1 mod g for all i = 1, ... , n.

8. Method for generating a public key according to claim 7 from a private key according to any of claims 1 to 3, comprising the following steps: Calculation of positive integers h1, ... , hn, where hipi ≡ 1 mod gy 0 < hi < g for all i = 1, ... , n. Calculation of non-negative integers t1, ... , tn with = (hi−u)%g for all i = 1, ... , n. Returning the public key formed by the integers t1, ... , tn.

9. Encryption method from a public key according to claim 7, comprising: Randomly selecting a shared secret, that is, (e1, ... , en), verifying that n − t components are zero and t components are positive integers less than or equal to z − 1. Calculating the positive integerCalculating the positive integer 10. Decryption method comprising applying a decryption algorithm using the private key according to any one of claims 1 to 3.

11. Decryption method, using a private key according to claim 1, of a cryptogram encrypted with the method of claim 9, comprising the following steps: Calculate s = (c1 + uc2)%g. Apply algorithm A with input g and s. Denote the output of A by (λ, ω). Establish the shared secret as e = (e1, ... , en), where, for each i =1, ... , n, The integer ei verifies that ≡ ω mod pi and 0 if pi divides λ. divides λ,12. Method according to claim 11 characterized in that algorithm A is the Extended Truncated Euclidean Algorithm (EEET).

13. Computer program comprising instructions for causing a computer to carry out any of the methods according to claims 4 to 6.

14. Computer program comprising instructions for causing a computer to carry out a method according to claim 8.

15. Computer program comprising instructions for causing a computer to carry out a method according to claim 9.

16. Computer program comprising instructions for causing a computer to carry out any of the methods according to claims 10 to 12.

17. Computer-readable storage medium comprising program instructions capable of causing a computer to carry out any of the methods according to claims 4 to 6.18.Computer-readable storage medium comprising program instructions capable of causing a computer to carry out a method according to claim 8.

19. Computer-readable storage medium comprising program instructions capable of causing a computer to carry out a method according to claim 9.

20. Computer-readable storage medium comprising program instructions capable of causing a computer to carry out any of the methods according to claims 10 to 12.

21. Transmissible signal comprising program instructions capable of causing a computer to carry out any of the methods according to claims 4 to 6.

22. Transmissible signal comprising program instructions capable of causing a computer to carry out a method according to claim 8.23.Transmissible signal comprising program instructions capable of causing a computer to carry out a method according to claim 9.

24. Transmissible signal comprising program instructions capable of causing a computer to carry out any of the methods according to claims 10 to 12.

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