WO2011110887A1

WO2011110887A1 - Cryptographic system and method using new one-way function families

Info

Publication number: WO2011110887A1
Application number: PCT/IB2010/000936
Authority: WO
Inventors: Luis Rocha
Original assignee: Luis Rocha
Priority date: 2010-03-11
Filing date: 2010-03-11
Publication date: 2011-09-15

Abstract

Cryptographic System and method using new one-way-function families. One-way functions are functions that are easy to compute but hard to invert, where the hardness condition refers to the average complexity of the inverting task. The existence of one-way functions is the comerstone of modern cryptography. Almost all cryptographic primitives imply the existence of one-way functions, and many of them can be constructed based either on the existence of one-way functions or on related versions of this assumption.

Description

Cryptographic system and method using new one-way function families Background Art

[0001] One-way functions are functions that are easy to compute but difficult to invert, where the level of difficulty refers to the average complexity of the inverting task. The existence of one-way functions is the cornerstone of modern cryptography. Almost all cryptographic primitives imply the existence of one-way functions, and many of them can be constructed based either on the existence of one-way functions or on related versions of this assumption.

[0002] In recent years several applications of one-way functions have been used in different areas: RSA (asymmetric encryption schemes), digital signatures, public key functions, probabilistic encryption and

cryptographically secure message sending. Secure pseudo-random generation is used in order to guarantee secure encrypted transmissions. General zero-knowledge is used for interactive proofs. Private key symmetric encryption schemes are used in order to secure information against adaptive chosen-cipher text attack. Message authentication codes, where a cryptographic message authentication code (MAC) is a short piece of information, are used to authenticate a message. Digital signature schemes are used in order to secure information against adaptive chosen message attacks.

[0003] The problem with these applications is that the used functions were never demonstrated to be true one-way functions. Naor in his paper NAOR/M., Yung/M.Universal One-Way Hash Functions and their Cryptographic Applications. Proceedings of the twenty-first annual ACM symposium on Theory of computing. 989. shows that only if a real 1-to-1 one-way functions exists ( ^{x≠ z} ^ (^z)), is the one-way function-based signature scheme then secure. For this reason, in the existing systems it is very probable that collisions are encountered when for ^x * ^z =* /(*)= /(*) wherein x and z being plain text. This problem leads to chaos in the decoding process of information because one encrypted text is decoded in more than one form. Therefore the prior-art methods and systems do not guarantee a completely secure cryptographic scheme.

[0004] The document PCT/DE 102005030657 B (SIEMENS AG) 16.11.2006 , discloses a coding procedure used to recognize the post manipulation of the number on a counter. The documents PCT/DE 10200351 A

(HOETKER ANDREAS) and PCT/WO 03049363 A (ERICSSON

TELEFON AB L M [SE]) 12.06.2003 disclose other examples of symmetric cryptography which have a one-way function as a precondition. The document PCT/US 6587563 B (APPLE COMPUTER [US]) 01.07.2003 discloses a cryptographic system using chaotic dynamics for sending a message so that the message is "secure". The document PCT/US

7174017 B (BANTZ DAVID F, ; CHEFALAS THOMAS E, ; KARVE

ALEXEI A, ; MASTRIANNI STEVEN J, ; MOHINDRA AJAY, ; LENO) 04.09.2003 discloses a system which can decrypt and encrypt audio transmission for a select person or group of persons using a key FOB. The document PCT/EP 1922730 B (KONINKL PHILIPS ELECTRONICS NV [NL]) 21 .05.2008 discloses an information carrier authentication with a physical one-way function. The international application PCT/WO

2004066296 A (KONINKL PHILIPS ELECTRONICS NV [NL]; LINNARTZ JOHAN P M G [NL]) 05.08.2004 discusses a delta-contracting function for generating cryptographic data for protecting the access of cryptographic keys stored on a storage medium. The problem with these documents is that one-way functions are mentioned, but never is a proof offered that these functions are in fact true one-way functions. Therefore there is a need for a truly secure one-way function for providing highly secure cryptographic systems.

Disclosure of Invention

[0005] Accordingly, it is the object of the present invention to provide a method and system for overcoming the above mentioned problems with a new encryption algorithm using a true one-way function based on the 2 publications: ROCHA/L.On the rate of convergence of 2-term Recursion. Computing. 1997, vol.59. , and the PhD dissertation : ROCHA / L.LIber die Konvergenzordung k-stufiger Iterationen und Nullstellenbestimmung nichtlinearer Gleichungssystem. Universitat Ulm: Dissertation, 1994. and on patent PCT/DE 10351022 B (EUROPA UNIVERSITAT VIADRINA) 09.06.2005 to Luis Rocha PhD, where the invention relates to a method for encoding an electronic data quantity to be sent from an emitter device to a receiver device. This method can be implemented at a low cost and ensures a temporally efficient encoding of electronic data, easily meeting the required safety standards. The encoding is based on codes of a variable length, fitting a prefix property.

[0006] It is a further object of one embodiment of the present invention to

implement a cryptographic system and method wherein new real one-way function families render the following cryptographic applications:

encryption schemes, probabilistic encryption, secure message sending, digital signature, pseudo-random generation, zero-knowledge interactive proofs, private key encryption, message authentication coding which are rendered extremely secure and very resistant to attacks.

[0007] It is a further object of one embodiment of the present invention to

implement a cryptographic system and method wherein said new real one-way function families are proved in the publication to come from Luis Rocha PhD "A class of one way functions" of the demonstration of the validity of said classes and families of one-way functions that they do not have calculable inverse. Therefore they can only be attacked by brute attacks trying one by one all possibilities

[0008] It is a further object of one embodiment of the present invention to

implement a cryptographic system and method using new families of one-way functions in order to allow the user to select the level of security, as each user has a vast choice of one-way functions families according to the desired security requirement.

[0009] It is a further object of one embodiment of the present invention to

implement a cryptographic system and method providing a new class of one-way functions based on convergent k-terms recursions.

[0010] It is a further object of one embodiment of the present invention to

implement a cryptographic system and method using a one-way function for providing a general fixed-point for a predetermined k-terms recursion function: <P ^' M→G _wnere k>=2, s>=2, G is a compact interval in ^ and M = G x · · · x G = G^k _{f an}d wherein the initial values ^χ°·^χ' ' ^{* ' '} ' **-' of the admissible interval of ^φ represent the seeds of the Iteration {

xn₊k = C i* · · ·> „)> n≥0 f_{urtner tne se}q_Uence converges at the fixed-point ^ ⁼ . The choice of the natural number k is free so that the user can chose his level of security, all derivations of the iteration function Φ until s-1 disappear at the fixed-point , and the s-derivative of the function at is ^≠0.

[0011] It is a further object of one embodiment of the present invention to

implement a cryptographic system and method providing classes of families of one-way functions for enabling a mapping of the vector x° = ^»¾) (_the input, with < ^{e 9i} ) into the vector

r = 0¾( -, -₂ ⁽*⁰⁾⁾ (the output) in such a way as to insure that the output dependent on the input is unique, wherein ^*-¹ is fixed.

Brief summary of the invention

[0012] Certain of the foregoing and related objects are readily attained according to the present invention by the provision of a novel system and method for generating a wide range of highly secure cryptographic applications.

[0013] The advantage of this method and system for generating secure

cryptographic applications is to employ a highly secure true one-way function generator.

[0014] Said new classes of families of real one-way functions render the following cryptographic applications; encryption schemes, probabilistic encryption, secure message sending, digital signature, pseudo-random generation, zero-knowledge interactive proofs, private key encryption and message authentication coding which are extremely secure and very resistant to all attacks.

[0015] An encryption system and method is provided that may adapt the level of security of each one-way function from each class varying from k=2 with a security of

, which has a security of 100 bits, with k=3, where there is a security of 150 bits and with k=4, which has a security of

200 bits. One can achieve a security of 50k bits with a hardware system having a precision of 15 decimals using a present up-to-date laptop. It is important to note that the precision a user can achieve depends only on the precision capabilities of the hardware employed.

[0016] Said highly secure true one-way-function generator of said cryptographic system can be used with different classes of families of one-way functions which when adequately chosen increase the already high level of security for e.g. k=3 in ordinary laptops or PC with 15 decimals precision, just by choosing k=5 to 250 bits, that is a level of security not presently reached in the field.

[0017] The construction of said new classes of families of true one-way functions render impossible the calculation of the inverse function because they don't have a reverse function. Only brute attacks are possible and the number of possibilities is so high due to the fact that the complexity of the k-terms recursions used increase exponentially with each iteration so that the probability to crack said encryption key constructed by the encryption generator of the invention is extremely low. A brute attack in the case of the invention means that one has to try all the initial values possible. The number of initial values is 2^50k, for k=3 it is 2¹⁵⁰. For this reason a brute attack cannot be implemented.

[0018] For the sake of comparison, two approximate calculations give the number of atoms in the observable universe to be a minimum of 10⁸⁰. Let's consider ¹⁰²⁴ = 2¹⁰ * 1000 = 10³ _ _jn this case for k=3, 2¹⁵⁰ is more than 10

3x5x3_{= 1 0}45_{i for k=4} j_t jg _{1 0}3x5x4_{= 1 0}60_{j fo|}- _R=5 j_t j_{s 1 0}3x5x5_{= 1 Q}75 _{and for k>5} the number of initial values is greater than the number of atoms in the universe.

[0019] Other objects and features of the present invention will become apparent from the following detailed description considered in connection with the accompanying drawings, which disclose an embodiment of the invention. It is to be understood, however, that the drawings are designed for the purpose of illustration only and that the particular descriptions of the invention in the context of the cryptographic applications are given by way of example only to help highlight the advantages of the current invention.

Brief Description of Drawings

[0020] FIG. 1a illustrates an example of creation and storage of a public key based on an inputted private key using the one-way function generator of the invention.

[0021] FIG. 1b illustrates an example of cryptographic authentication according to one embodiment of the invention using the one-way function generator of the invention.

[0022] FIG. 2 illustrates a preferred embodiment of a novel one-way function generator used in the cryptographic authentication process of the invention for creating public keys.

Description

[0023] Reference will now be made in detail to a specific embodiment of the

invention including a best mode contemplated by the inventor for carrying out the invention. An example of this specific embodiment is an electronic authentication which is illustrated by the accompanying drawings. While the invention is described in conjunction with this specific embodiment, it will be understood that it is not intended to limit the invention to the described embodiment. On the contrary, it is intended to cover

alternatives, modifications, and equivalents as defined by the appended claims.

[0024] The following description makes full reference to the detailed features which may form parts of different embodiments as outlined in the objects of the invention. In the following embodiment reference is made to an authentication method and system while it should be understood that the invention covers other embodiments which use other types of

cryptographic primitives such as encryption schemes, probabilistic encryption, secure message sending, digital signature, pseudo-random generation, zero-knowledge interactive proofs, private key encryption and message authentication coding.

[0025] Each embodiment may adapt the level of security of each one-way

function from our classes varying from k=2 with a security of

(l0^l3 xl0^l3)« 2¹⁰⁰ _{j wnjcn js 100 bjtSj wjtn k}_3 _tnere j_{s a secur}jty ₀f 150 bits, with k=4 there is a security of 200 bits. One can achieve a security of 50k bits with a hardware system having a precision of 15 decimals using a present up-to-date laptop. It is important to note that the precision a user can achieve depends only on the precision capabilities of the hardware employed.

[0026] It is to be understood that the example illustrates the phases of an

authentication scheme comprising two steps. First, registering a user for a specific service or range of services, wherein said user providing the

X° = (x ^■■· X ~>

authentication entity with a unique private key °' ' ^ for constructing by means of said novel one-way function generator the public key ^{r ~} ^^^{' "} ^ Yk-i ) corresponding to said unique private key and storing it in a data base. And second, the authentication itself of the same user by calculating again with the same method a private key ^y ~ ^•· > ·> *-ι ) for authenticating by comparing the new resulted public key with said created at registration time by said user and prohibiting a counterfeit private key owner to access the service said user was registered for. In order to get

0 _ 0 the same public key the user must enter the same private key ( ^{x ~ y} ), as the one-way function has no collision. This is proven in the

demonstration technical paper to be published.

[0027] Referring now in detail to the drawings and in particular FIG. 1a, therein illustrated is an organigram of the registration of a user to a specific service.

[0028] In step 101 the user chooses his private key as the vector ^{x =}

where k is the dimension of said vector. This information is sent to the one-way-function generator on step 102, which calculates the public key 7⁼ (Yo= - - - > r_k-2 ) |_{n s}t_ep 103 said public key ^{r = >} k-i ) j_s received from the one-way-function generator. Said received public key is stored in the database of the authorized entity (for example a bank) on step 104.

[0029] In Fig. 1 b is represented the authentication process per se. In step 110 the user chooses and inputs his private key Λ-ι) _{as a} vector, where k is the dimension of said vector. This information is sent to the one-way function generator in step 111 , which calculates the

corresponding public key ^{γ ~ >} Yk-2 ) _ |_{n S}†_ep 112 said public key

Y - (Ya,^" - ^> Yk-i ) is received from the one-way function generator. Said received newly calculated public key is compared with said stored key in the data base of said authorized entity (for example a bank) in step 113. When the stored key and the newly calculated key are the same, the user is allowed to access to the service he has registered. In the other case the access is denied.

[0030] In Fig. 2 is represented the process of the one-way function generator.

[0031] In step 200 creating first a k-term recursion function defined as follows φ : Μ→G _wnere k>=2, s>=2, G is a compact interval in ^ and

M - Gx - --x G - G^k _{j anc}j he initial values '^{" "}'**-> represent the seeds of the Iteration { *■+* ⁼ Ρ(^χ»+*-ΐ'^*■■»¾)^» »≥0j _ancj _sajd k-term recursion function converges at the fixed-point ^ ⁼ ^^{»" "}^) .

[0032] The user can choose k according to the level of security he desires such as a security of 50k bits with a hardware system having a precision of 15 decimals using a present up-to-date laptop as disclosed above, all derivations of the iteration function ^φ from 1 to s-1 is zero at the

fixed-point (€>^{' "}> and the s-derivative of the function ^ at is Q, where ^a∑k

[0033] In order to form the sequence { ^x"^{+k =} (^χ»+*-^ΐ'^{" "}'^χ»)' ^n≥ °} it is necessary, that the initial values of said sequence satisfy the following condition: Let e" ^{~ x}" ^~ . The coordinates of the vector ^x° are chosen so that the

numbers are each absolutely smaller or equal to ε^> where ^ε depending on ^φ is chosen sufficiently small.

[0034] In step 201 the user enters the precision of his system hardware defined by the number Θ e.g. 0 = 10 ¹⁶⁰ this precision is different on each hardware.

[0035] In step 202 the one-way function generator defines the normalized error

/_η = Ω \ χ_η - ξ \, η≥0

[0036] In step 203 the one-way function generator calculates the sequence xn+k = P(^J -i>^■· ·^»*„)^» "^{≥ 0} _and in step 204 f" is compared with the precision

[0037] If f* is bigger than Θ the next sequence is calculated with n + l _t else the public key is calculated where all ^1ο8(/_π) j_n step 205. . Finally the calculated public key is sent back in step 206.

[0038] As described above, the present invention of the cryptographic system and method using the new one-way function disclosed can be used with different classes of families of functions which allow only brute attacks to crack said encryption key constructed by the system and method of the invention. It should be understood however that the present disclosure is for the purpose of illustration only and does not include all modifications or improvements obvious to someone skilled in the art, which may fall within the scope of the appended claims.

References

[0039]

• PCT/DE 102005030657 B (SIEMENS AG) 16.11.2006

• PCT/DE 10200351 A (HOETKER ANDREAS)

• PCT/WO 03049363 A (ERICSSON TELEFON AB L M [SE]) 12.06.2003

• PCT/US 6587563 B (APPLE COMPUTER [US]) 01.07.2003

• PCT/US 7174017 B (BANTZ DAVID F, ; CHEFALAS THOMAS E, ; KARVE ALEXEI A, ; MASTRIANNI STEVEN J, ; MOHINDRA AJAY, ; LENO) 04.09.2003

• PCT/EP 1922730 B (KONINKL PHILIPS ELECTRONICS NV [NL])

21.05.2008

• PCT/WO 2004066296 A (KONINKL PHILIPS ELECTRONICS NV [NL]; LINNARTZ JOHAN P M G [NL]) 05.08.2004

• PCT/DE 10351022 B (EUROPA UNIVERSITAT VIADRINA) 09.06.2005

• NAOR/M., Yung/M.. Universal One-Way Hash Functions and their

Cryptographic Applications. Proceedings of the twenty-first annual ACM symposium on Theory of computing. 1989.

• ROCHA/L.. On the rate of convergence of 2-term Recursion.

Computing. 1997, vol.59.

• ROCHA/ L.. Clber die Konvergenzordung k-stufiger Iterationen und

Nullstellenbestimmung nichtlinearer Gleichungssystem. Universitat Ulm: Dissertation, 1994.

Claims

1. A method for registrating a user to access one or more of a plurality of services by an authorized entity wherein said user entering personal data and further generating a private key wherein said method being characterised by the steps of; choosing an integer number k according to the level of security desired by said user, wherein said private key dimension being an integer k being in relation to the level of security desired by said user wherein said security being limited by the hardware capabilities of the system employed, wherein said security being a security of 10 -d - k bits with a hardware system having a precision of 3 d decimals, wherein ≥ 1 _anc| k≥ 2 _{1 anc}j generating said

x° = (x · · · X

unique private key vector °' ' ^{k~l J} by said user (101).

2. A method for registrating a user to access one or more of a plurality of services as disclosed in claim 1 wherein; said security being a security of 50k bits with a hardware system having a precision of 15 decimals (d=5) using a present up-to-date laptop or desktop.

3. A method for registrating a user to access one or more of a plurality of services as disclosed in claim 1 further comprising the steps of; sending said private x° = (x ^■· · X )

key vector ⁰⁵ ' ^{k~l J} to a true one-way-function generator (102) for creating a corresponding unique public key ^{γ ~} ^ ^ ^' ^ Vk-i)

4. A method for registrating a user to access one or more of a plurality of services as disclosed in claim 3 further comprising the steps of; receiving from said one way function generator said corresponding unique public key ^{γ ~} ^^^{' "} ^k-i ) (103), and storing said received corresponding unique public key

7 - (Xo, - - -, y_k-2 ) (104) in the database of said authorized entity.

5. A method for registrating a user to access one or more of a plurality of services by an authorised entity as disclosed in claim 4 wherein said service being encryption schemes or probabilistic encryption or secure message sending or digital signature or pseudo-random generation or zero-knowledge interactive proofs or private key encryption or message authentication coding.

6. A method for authenticating a user to access one or more of a plurality of

services by an authorised entity wherein said user entering personal data and further generating a private key said method being characterised by the steps of; choosing an integer number k according to the level of security desired by said user, wherein said private key dimension being an integer k being in relation to the level of security desired by said user wherein said security being limited by the hardware capabilities of the system employed, wherein said security being a security of 10 d k bits with a hardware system having a precision of 3 ·^ decimals, wherein d≥ 1 and ≥ ² , and generating said unique private key vector ^{y ~} ο^{> '} "^> )Ί^<-ι ) by _s3\d user (201).

7. A method for authenticating a user to access one or more of a plurality of services by an authorised entity as disclosed in claim 6 wherein said security being a security of 50k bits with a hardware system having a precision of 15 decimals (d=5) using a present up-to-date laptop or desktop.

8. A method for authenticating a user to access one or more of a plurality of services by an authorised entity as disclosed in claim 6 wherein said method further comprising the steps of; sending, said private key ^y ~ (^yo^{> ' " >} yk-i ) to a true one-way-function generator (11 1) wherein said true one-way-function generator calculating the corresponding unique public key ^ ^~ βΌ^{>' " >}/*-²) _ and receiving said calculated unique public key ^γ ~ ^^^{•" >■} Yk-i f_{rom δ3}^ true one-way-function generator (1 12).

9. A method for authenticating a user to access one or more of a plurality of services by an authorised entity as disclosed in claim 8 wherein said method further comprising the steps of; comparing said received calculated unique public key ^{Y = <}^^^{" '}^^^ the stored unique public key ^{x =} (*o^>-^{" >}¾-ⁱ) j_n the data base of said authorised entity, corresponding to said authorised user (1 13), and if said stored key and the new calculated key are the same then allowing said user to access said one or more services he has registered for, otherwise denying said user to access to said one or more services.

10. A method for authenticating user to access one or more of a plurality of

services by an authorised entity as disclosed in claim 9 wherein said service being encryption schemes or probabilistic encryption or secure message sending or digital signature or pseudo-random generation or zero-knowledge interactive proofs or private key encryption or message authentication coding.

1 1. A method for generating a unique public key by a true one-way-function generator for providing access to one or more of a plurality of services being characterized by the steps of; receiving an integer number k according to the level of security desired by a user wherein said security being limited by the hardware capabilities of the system employed by said true one-way-function generator, wherein said security being a security of l -d-k bits with a hardware system having a precision of ³ d decimals, wherein d≥\ and k≥2₁ and receiving a unique private key vector in the form of a vector

x = Οο, · · · , ¾-ι ) generated by said user (101).

12. A method for generating a unique public key by a true one-way-function

generator for providing access to one or more of a plurality of services as disclosed in claim 11 wherein said security being a security of 50k bits with a hardware system having a precision of 15 decimals (d=5) using a present up-to-date laptop or desktop.

13. A method for generating a unique public key by a true one-way-function

generator for providing access to one or more of a plurality of services as disclosed in claim 11 further comprising the steps of; creating a k-term recursion function (200) <P ^' M^→G wherein k>=2 and s>=2, G being a compact interval in 9¾ , M = Gx---xG = G^k and the initial values *o *ⁱ ' ^{" '}' **-ⁱ ₀f a private key forming the seed of an recursion function ^φ , and feeding said initial values ^X°-^Xl' ^{", Xk~l} to said recursion function ^ wherein {

xn₊k =^„_+Η,···>¾)> «≥0^ _{and sajd k}._term recursion function < - ^M→G converging at the fixed-point ^{= <} ( >^{' "}> ) _t and the initial values ^⁰ *¹'^{' "}'**-' of said sequence { ^{Xn+k ~} ^( ^»+*-^{i > ' ' '}=^■* ^>)' "^≥ °} wherein said sequence further satisfying the condition that the coordinates of said initial vector ^x° , and wherein the numbers ^e° ^e*^{~2 e}*-' being each absolutely smaller or

1

ε =

equal to ⁴^ , wherein the constant K depending on ^φ and wherein e" ^{~ x}" ^~^ , and all derivatives of said recursion function ^ from 1 to s-1 being zero at said fixed-point ^ , and the s-derivative of said recursion function ^ at being ^&*

14. A method for generating a unique public key by a true one-way-function generator for providing access to one or more of a plurality of services as disclosed in claim 13 further comprising the steps of; entering the precision of the user system hardware (201), defined by the number Θ wherein said precision being different on each hardware, and receiving said private key in

X° = ( X ^■■· X )

the form of a vector of the form °' ' ' , serving as the seed of said recursion, and defining the normalized error f^{» ~ s} ' ^{x» ~ n≥} , by said true one-way-function generator (202), where is the fixed point of said recursion.

15. A method for generating a unique public key by a true one-way-function

generator for providing access to one or more of a plurality of services as disclosed in claim 14 further comprising the steps of; calculating the sequence x_n+k =

···^»≠»)^» *^≥ 0 by _sajd true one-way-function generator (203), testing ^^n+k with said precision # wherein if f^»+k is bigger than Θ then calculating, by said means for calculating, the next sequence with «+ ¹ (204) otherwise said unique public key being ^{γ ~} ^^^{" '} erein the coefficients of said unique public key being

calculated with the last value of said recursion.

16. A method for generating a unique public key by a true one-way-function

generator for providing access to one or more of a plurality of services as disclosed in claim 15 further comprising the steps of: sending said calculated unique public key ^ ^~ ^^^{' "}' V^k-ⁱ) (206) to the provider of said requested one or more services or using said unique public key locally.

17. A method for generating a unique public key by a true one-way-function

generator for providing access to one or more of a plurality of services as disclosed in claim 16 wherein said service being encryption schemes or probabilistic encryption or secure message sending or digital signature or pseudo-random generation or zero-knowledge interactive proofs or private key encryption or message authentication coding.

18. A system of registration of a user to access one or more of a plurality of

services by an authorised entity wherein said user entering personal data and generating a private key wherein said system being characterised by: means for choosing an integer number k according to the level of security desired by said user wherein said security level being limited by the hardware capabilities of the system employed, wherein said security level being a security of 1 d k bits with a hardware system having a precision o ^ -d decimals, wherein d≥ 1 and k≥2 _{t anc}j means for allowing said user to generate said unique private

|₍gy ^X ~ (^X0> ^{' "} >^Xk-l ) _

19. A system of registration of a user to access one or more of a plurality of

services as disclosed in claim 18 wherein said security being a security of 50k bits with a hardware system having a precision of 15 decimals (d=5) using a present up-to-date laptop or desktop.

20. A system of registration of a user to access one or more of a plurality of

services as disclosed in claim 18 further comprising a true one-way-function generator for calculating a corresponding unique public key to said private key

^{X =} (^■¾»^{" " "}> *Jt-l )

21. A system of registration of a user to access one or more of a plurality of

services as disclosed in claim 20 wherein said authorised entity further

x° = (x ^■■■ X

comprising; means for sending said private key ⁰⁵ ' , to said true one-way-function generator, and means for receiving said calculated unique public key ^{γ _} ^°'^{' "}'^*-²^ from said true one-way-function generator, and a database for storing a plurality of said received unique public keys

Y - (yo,^' ~, y_k-2 ) calculated by said true one-way-function generator each corresponding to one of a plurality of users or subscribers to access one or more of said services.

22. A system of registration of a user to access one or more of a plurality of

services by an authorised entity as disclosed in claim 21 wherein said service being encryption schemes or probabilistic encryption or secure message sending or digital signature or pseudo-random generation or zero-knowledge interactive proofs or private key encryption or message authentication coding.

23. A system for authenticating a user to access one or more of a plurality of

services by an authorised entity characterised by: means for entering a private key (^o^{> ' " *}' -ⁱ ) by a user, wherein said private key dimension being an integer k being in relation to the level of security desired by said user wherein said security being limited by the hardware capabilities of the system

employed, wherein said security being a security of 1° d k bits with a hardware system having a precision of 3-d decimals, wherein d≥ 1 and k≥2 ₁ and means for sending said private key to a true one-way-function generator.

24. A system for authenticating a user to access one or more of a plurality of

services by an authorised entity as disclosed in claim 23 wherein said security being a security of 50k bits with a hardware system having a precision of 15 decimals (d=5) using a present up-to-date laptop or desktop.

25. A system for authenticating a user to access one or more of a plurality of

services by an authorised entity as disclosed in claim 23 further comprising; a true one-way-function generator further comprising means for receiving said private key ^{y "}^⁰'^{' "}'^-¹' from said authorised entity, and means for calculating the corresponding unique public key ^ ^~~ 0O> ^{, , -} > /*-2 )

26. A system for authenticating a user to access one or more of a plurality of

services by an authorised entity as disclosed in claim 25 wherein said true one-way-function generator further comprising; means for comparing said received calculated unique public key with the stored unique public key corresponding to said user in the data base of said authorized entity, and if said stored key and the new calculated key are the same then said authorised entity further comprising means for allowing said user to access the one or more services he has registered for, otherwise said authorised entity further comprising means for denying said user the access to said one or more service.

27. A system for authenticating a user to access one or more of a plurality of

services by an authorised entity as disclosed in claim 26 wherein said service being encryption schemes or probabilistic encryption or secure message sending or digital signature or pseudo-random generation or zero-knowledge interactive proofs or private key encryption or message authentication coding.

28. A true one-way-function generator for generating a unique public key for

providing an access to a user to one or more of a plurality of services being characterized by: means for receiving an integer number k according to the level of security desired by said user wherein said security being limited by the hardware capabilities of the system employed by said true one-way-function generator, wherein said security being a security of 1 -d -k bits with a hardware system having a precision of 3 -d decimals, wherein d≥ \ and £≥ 2 ,

Y° ₌ ( _x . . . v

and means for receiving said unique private key vector °' '

generated by said user.

29. A true one-way-function generator for generating a unique public key for

providing an access to a user to one or more of a plurality of services as disclosed in claim 28 wherein said security being a security of 50k bits with a hardware system having a precision of 15 decimals (d=5) using a present up-to-date laptop or desktop.

30. A true one-way-function generator for generating a unique public key for

providing an access to a user to one or more of a plurality of services as disclosed in claim 28 further comprising means for creating a k-term recursion function (200) <P ^' M^→G wherein k>=2 and s>=2, G being a compact interval in 9? _f M = Gx - --x G = G^k and the initial values ^ν"^{' "}'^ΪΗ of a private key forming the seed of an recursion function Φ , and means feeding said initial values ^X°-^Xi' '^*^-1 to said recursion function wherein {

xn_+k =

^M→G converging at the fixed-point ^{= (}P(€ and the initial values ^" ^"'^^"1 of said sequence { ^Xn+k ~ ^^x»+k-i > ^{" '}' ^xnX ^n≥ °j wherein said sequence further satisfying the condition that the coordinates of said initial vector ^x° , wherein the numbers

being each absolutely smaller or equal to ε - 1

— r 4^ , wherein the constant K depending on ^φ , and wherein ^e" ^{~ n ζ} , and all derivatives of said recursion function ^ from 1 to s-1 being zero at said fixed-point and the s-derivative of said recursion function ^ at being ^0∑k

31. A true one-way-function generator for generating a unique public key for

providing an access to a user to one or more of a plurality of services as disclosed in claim 30 further comprising; means for providing said private key in the form of a vector of dimension k ^{x "} ^⁰'^{' "}'¹'-¹' for serving as the seed of a recursion, and means for entering the precision of the user system hardware, defined by the number Θ wherein said precision being different on each hardware, and means for defining the normalized error

f_n - Q^s I ^xn - ζ I^{» n≥ 0} ₁ by said true one-way-function generator, where ^ being the fixed point of said recursion.

32. A true one-way-function generator for generating a unique public key for

providing an access to a user to one or more of a plurality of services as disclosed in claim 31 further comprising; means for calculating the sequence xn₊k = <P(^x _n+k-i> - - -,x„), n≥0 _{by sajd true one}._way_f_unction generator (203), means for testing f"^+k with said precision & wherein if f"^+k is bigger than Θ then calculating by said means for calculating the next sequence with ^{M +}l

(204) otherwise said unique public key being ^ ^_ ^^^{' "}^^k-ⁱ) _> wherein the coefficients of said unique public key being

calculated with the last values of said recursion.

33. A true one-way-function generator for generating a unique public key for

providing an access to a user to one or more of a plurality of services as disclosed in claim 32 further comprising means for sending said calculated unique public key ^{γ ~} ^^^{' "}^^-ι) t_ne provider of said requesting entity and/or means for using said unique public key locally.

34. A true one-way-function generator for generating a unique public key for

providing an access to a user to one or more of a plurality of services as disclosed in claim 33 wherein said service being encryption schemes or probabilistic encryption or secure message sending or digital signature or pseudo-random generation or zero-knowledge interactive proofs or private key encryption or message authentication coding.