Method and apparatus for a robust high-speed cryptosystem

• US 6,154,541 A
• Filed: 09/30/1997
• Issued: 11/28/2000
• Est. Priority Date: 01/14/1997
• Status: Expired due to Fees

• Current Assignee: Jinglong F. Zhang
• Original Assignee: Jinglong F. Zhang
• Inventors: Zhang, Jinglong F
• Primary Examiner(s): Laufer, Pinchus M.
• Application Number:
  US08/941,350
• Time in Patent Office: 1,155 Days
• Field of Search: 380/21, 380/28, 380/30, 380/44, 380/46, 380/49
• US Class Current: 380/28
• CPC Class Codes: H04L 9/3073 involving pairings, e.g. identity based encryption [IBE], bilinear mappings or bilinear pairings, e.g. Weil or Tate pairing

Abstract

A cryptographic information and communication system of the knapsack type characterized by secret logical segregation of the key sets into sections by different construction methods, where different transformations are applied to different sections, and characterized by non-constant number of subset sum solutions to ciphertext, where resolution protocols are employed when necessary to resolve non-unique subset sum solutions at the decryptor.

Claims

1. A cryptographic method comprising the steps of:

providing a plurality of key section construction methods;

deriving a decryption key including a decryption key set of identifiables logically segregated into sections, each said section generated by one of a plurality of construction methods and said segregation is kept secret;

deriving from said decryption key an encryption key, each section of said decryption key set is transformed to a corresponding section of said encryption key set and at least two said sections are transformed under different transformation methods, and previously generated and transformed sections are transformed together with the currently generated section;

accepting a message in the form of an input sequence of integers;

converting said encryption key set to a key vector and converting said input sequence of integers to at least one data vector having the same dimensions as said key vector;

generating ciphertext by computation that includes calculating the inner product of each of said data vectors with said key vector;

accepting a ciphertext; and

decrypting said ciphertext to recover said message.

2. A cryptographic method as in claim 1 further comprising the steps of:

deriving the decryption key including a decryption fuzzy residue set;

deriving a fuzzy residue mapping whereby elements of said decryption key set and elements of said decryption fuzzy residue set are associated for decryption;

deriving from said decryption key the encryption key including an encryption fuzzy residue set from said decryption fuzzy residue set, inheriting said fuzzy residue mapping whereby elements of said encryption key set and elements of said encryption fuzzy residue set are associated for encryption;

generating the ciphertext by computation that includes applying fuzzy residues determined by the values of the integers of said data vectors and in accordance with said fuzzy residue mapping; and

decrypting said ciphertext including the removal of fuzzy residues from said ciphertext in accordance with said fuzzy residue mapping to recover said message.

3. A cryptographic method as in claim 1 which securely communicates data from a first party S to a second party R comprising the steps of:

deriving cryptographic keys at party R comprising;

deriving the decryption key to be kept secret at party R, wherein decoding by said decryption key produces one or more solutions for valid ciphertext and detects invalid ciphertext or noise; and

deriving from said decryption key the encryption key, wherein said encryption key produces a same ciphertext from different data bits;

distributing said encryption key from party R to party S;

receiving said encryption key at party S;

accepting a message at party S;

generating the ciphertext from said message using said encrption key;

transmitting said ciphertext from party S to party R over a communication channel;

receiving said ciphertext at party R; and

decoding said ciphertext at party R to obtain all possible decodes and to recover said message through the identification of valid decodes or through the elimination of any and all invalid decodes by means of a resolution method.

4. A resolution method as in claim 3, used in the decryption of ciphertext having unremovable fuzziness that is encrypted from a message by a first party S and sent from party S to a second party R, for eliminating sets of invalid decodes due to said unremovable fuzziness observing a non-negative integer threshold value, wherein said elimination terminates when the number of sets of invalid decodes falls below said threshold, and wherein said parties adopt a one-way characteristics function H for each cycle of the resolution method by either pre-agreement or by interactive agreement, comprising at least one cycle of the steps of:

generating a random bit sequence s of length zero or more at party R;

transmitting from party R to party S over a communication channel said random bit sequence s and p, the specification of a portion of said message;

receiving s and p at party S;

blending s with the portion of said message specified by p at party S to obtain a blended result b;

calculating at party s characteristics h=H(b);

transmitting h from party S to party R over said communication channel;

receiving h at party R; and

repeating at party R for each set of decodes C the steps of;

blending s with the portion of C corresponding to the portion of said message specified by p to obtain a blended result b'"'"';

calculating at party R characteristics h'"'"'=H(b'"'"'); and

discarding C if h'"'"' does not equal h.

5. A cryptographic method as in claim 1 which securely communicates data from a first party S to a second party R comprising the steps of:

extending said data into a set of public keys at party S;

communicating from party S to party R said extended form of said data over a communication channel; and

executing a secure protocol between party S and party R whereby party R extracts said data from its said extended form.

6. A cryptographic method as in claim 5 wherein said data are extended into encryption keys e₁, e₂, . . . , e_n (n≧

1), each of which can have one or more different corresponding decryption keys d_i1, d_i2, . . . , d_im (m≧

1), whereby d_i1, d_i2, . . . , d_im corresponding to e_i give different decipherment to ciphertext produced by e_i comprising the steps of;

extending said data at party S into a set of public encryption keys e₁, e₂, . . . , e_n ;

communicating e₁, e₂, . . . , e_n from party S to party R over a communication channel; and

executing a secure protocol between party S and party R whereby party R encrypts using each of e₁, e₂, . . . , e_n and party S uses d_ik, for some k wherein 1≦

k≦

m, for decryption in said protocol, and whereby party R extracts said data from its said extended form.

7. A cryptographic method as in claim 5 which securely communicates data from a first party S to a second party R comprising the steps of:

accepting the message in the form of a bit sequence at party S;

converting said bit sequence to n numbers a₁, a₂, . . . , a_n formed by inserting a one bit as the most significant bit to n consecutive variant size bit subsequences of said bit sequence at party S, wherein said sizes are randomly selected;

deriving the decryption key to be kept secret at party S including at least n unique or near-unique sets D₁, D₂, . . . , D_n as extended decryption key sets corresponding to said n numbers, wherein at least one of the elements in each said extended decryption set is a tangible;

deriving from said decryption key the encryption key at party S by transforming said extended decryption key sets D_i to extended encryption key sets E_i for 1≦

i≦

n, wherein one of the tangibles in each extended decryption set D_i is transformed to the corresponding number a_i ;

distributing said encryption key from party S to party R;

performing at least one cycle of the following steps until each and all a_i in E_i, for 1≦

i≦

n, are identified by party R via the identification of all elements in E_i that are not a_i ;

generating the ciphertext at party R comprising;

generating random vectors of integers V_i corresponding to E_i for 1≦

i≦

n; and

generating ciphertext s_i by computation that includes calculating the inner product of V_i and E_i for 1≦

i≦

n;

transmitting said ciphertext from party R to party S over a communication channel;

receiving at party S said ciphertext and recognizing in said s_i the presence or absence of a_i into which said tangible in corresponding D_i is transformed for 1≦

i≦

n;

reporting, via said communication channel, said recognition of the presence or absence by party S to party R; and

identifying at party R, in response to said reporting, elements in E_i that are not a_i for 1≦

i≦

n; and

reassembling and reconstructing said message at party R.

8. A cryptographic method as in claim 1 for information and communication security with the inseparable operation of secure exchange of encryption keys and proper mutual identification and authentication between a first party X and a second party Y comprising the steps of:

generating at party X public identification encryption key K_xip, private identification decryption key K_xis, public encryption key K_xp, and private decryption key K_xs of party X;

generating at party Y public identification encryption key K_yip and private identification decryption key K_yis public encryption key K_yp, and private decryption key K_ys of party Y;

establishing a communication channel between said parties initiated by party X;

transmitting from party X to party Y over said communication channel K_yip (K_xp);

receiving K_yip (K_xp) at party Y with a timeout t;

obtaining K¹_xp at party Y through decryption;

space="preserve" listing-type="equation">K.sup.1.sub.xp =K.sub.yis (K.sub.yip (K.sub.xp))

constructing at party Y a challenge packet K¹_xp (K_xip (β

·

K_yp ·

C_y)), where C_y is a challenge generated by party Y, and β

=H(K¹_xp ·

C_y) for some one-way characteristics function H and ·

denotes some appropriate mixing or concatenating function;

sending from party Y to party X over said communication channel K¹_xp (K_xip (β

·

K_yp ·

C_y));

receiving with said timeout t and decrypting K¹_xp (K_xip (β

·

K_yp ·

C_y)) at party X;

space="preserve" listing-type="equation">β

'"'"'·

K.sup.1.sub.yp -C.sup.1.sub.y =K.sub.xis (K.sub.xs (K.sup.1.sub.xp (K.sub.xip (β

·

K.sub.yp ·

C.sub.y)));

testing at party X if β

'"'"' equals H(K_xp ·

C¹_y) and failing and aborting the identification/authentication process if β

'"'"' does not equal H(K_xp ·

C¹_y);

constructing at party X a challenge packet K¹_yp (K_yip (γ

·

R_x ·

C_x)), where C_x is a challenge by party X, R_x is the reply by party X to party Y in response to C¹_y, and γ

=H(K¹_yp ·

C_x);

sending from party X to party Y over said communication channel K¹_yp (K_yip (γ

·

R_x ·

C_x));

receiving with said timeout t and decrypting K¹_yp (K_yip (γ

·

R_x ·

C_x) at party Y;

space="preserve" listing-type="equation">γ

'"'"'·

R.sup.1.sub.x ·

C.sup.1.sub.x =K.sub.yis (K.sub.ys (K.sup.1.sub.yp (K.sub.yip (γ

·

R.sub.x ·

C.sub.x))));

testing at party Y if γ

'"'"' equals H((K_yp ·

C¹_x) and if R¹_x equals R_x, and failing and aborting the identification/authentication process if either γ

'"'"' does not equal H((K_yp ·

C¹_x) or R¹_x does not equal R_x, where R_x is the expected reply based on C_y ;

sending a reply packet K¹_xp (K_xip (R_y)) from party Y to party X over said communication channel, where R_y is the reply by party Y to party X in response to C¹_x ;

receiving and decrypting K¹_xp (K_xip (R_y)) at party X;

space="preserve" listing-type="equation">R.sup.1.sub.y =K.sub.xis (K.sub.xs (K.sup.1.sub.xp (K.sub.xip (R.sub.y))))

testing at party X if R¹_y equals R_y and failing and aborting the identification/authentication process if R¹_y does not equal R_y, where R_y is the expected reply based on C_x ;

failing and aborting the identification/authentication process whenever

1) said timeout t expires, or

2) any decryption fails; and

providing further protection and security to communication and data transfer with encryption using K¹_xp =K_xp and K¹_yp =K_yp.

9. An identification/authentication method as in claim 8 adapted for use in a cryptographic system that employs a block cipher as its cryptor, at least one time sensitive transmission block from party X is separated into m>

1 portions, p₁, p₂, . . . , P_m, according to a defined or agreed upon method of separation and sent from party X to party Y with a pause α

_j after transmitting each p_j, for 1≦

j<

m where Σ

α

_j +d≧

t and Σ

α

_i +d<

t for 1≦

i<

m-1 and d being the maximum expected delay for the transmission, and at least one time sensitive transmission block from party Y is separated into n>

1 portions, q₁, q₂, . . . , q_n, according to a defined or agreed upon method of separation and sent from party Y to party X with a pause β

_j after transmitting each q_j, for 1≦

j<

n where Σ

β

_j +d≧

t and Σ

β

_i +d<

t for 1≦

β

<

n-1, party X expects q₁ within t, and party Y expects p₁ within t, and each of said parties reassembles said portions back to said time sensitive transmission block.

calculating at party S characteristics h=H(b);

transmitting h from party S to party R over said communication channel;

receiving h at party R; and

repeating at party R for each set of decodes C the steps of;

blending s with the portion of C corresponding to the portion of said message specified by p to obtain a blended result b'"'"';

calculating at party R characteristics h'"'"' =H(b'"'"'); and

discarding C if h'"'"' does not equal h.

10. A cryptographic method which securely communicates data from a first party S to a second party R comprising the steps of:

deriving cryptographic keys at party R comprising;

providing a plurality of key section construction methods;

deriving a decryption key to be kept secret at party R including a decryption key set of identifiables logically segregated into sections, each section generated by one of a plurality of construction methods and said segregation is kept secret;

deriving from said decryption key an encryption key, each section of said decryption key set is transformed to a corresponding section of said encryption key set, and at least two sections are transformed under different transformation methods, and previously generated and transformed sections are

transformed together with the currently generated section;

distributing said encryption key from party R to party S;

receiving said encryption key at party S;

generating ciphertext at party S comprising;

accepting a message in the form of an input sequence of integers;

converting said encryption key set to a key vector and converting said input sequence of integers to at least one data vector having the same dimensions as said key vector; and

generating ciphertext by computation that includes calculating the inner product of each of said data vectors with said key vector;

transmitting said ciphertext from party S to party R over a communication channel;

receiving said ciphertext at party R; and

decrypting said ciphertext at party R to recover said message.

11. A cryptographic method as in claim 10 which securely communicates data from a first party S to a second party R further comprising the steps of:

deriving the decryption key including a decryption fuzzy residue set to be kept secret at party R;

deriving a fuzzy residue mapping to be kept secret at party R whereby elements of said decryption key set and elements of said decryption fuzzy residue set are associated for decryption;

deriving at part R from said decryption key the encryption key including an encryption fuzzy residue set from said decryption fuzzy residue set, inheriting said fuzzy residue mapping whereby elements of said encryption key set and elements of said encryption fuzzy residue set are associated for encryption;

generating at party S the ciphertext by computation that includes applying fuzzy residues determined by the values of the integers of said data vectors and in accordance with said fuzzy residue mapping; and

decrypting said ciphertext at party R including the removal of fuzzy residues from said ciphertext in accordance with said fuzzy residue mapping to recover said message.

12. A cryptographic method comprising the steps of:

deriving a decryption key by generating a decryption key set of identifiables logically segregated into k>

1 sections;

##EQU5## and a decryption fuzzy residue set with corresponding sections to K_d;

space="preserve" listing-type="equation">G=G.sub.1 ∪

G.sub.2 ∪

. . . ∪

G.sub.k

and a wrinkling set with corresponding sections to K_d ;

space="preserve" listing-type="equation">B=B.sub.1 ∪

B.sub.2 ∪

. . . ∪

B.sub.k

where K_d, G and B are to be kept secret and each section i, for 1≦

i≦

k is K_d is generated by at least one of the following constructions;

1) standard

2) submerged

3) wrinkled

4) individually mapped

5) multiply mapped

and for a submerged section i

d_i,j =2^h(j-1) *m_i, for 1≦

j≦

z_i, where subscript i in m_l is defined to be;

i.tbd.i, 0.tbd.i-1, z_i-1

and for a wrinkled section i

σ

=sum(η

.sub.θ

* |θ

|)+r

d_i,j =σ

+Ψ

_i-1 (b_i,j), for 1≦

j≦

z_i

and for a section of any of the other types;

##EQU6## where r is a non-zero positive ranged random number, η

.sub.θ

is the maximum applicable multiple of θ

ε

Θ

_i-1, and Θ

_i is defined to be ##EQU7## where C_u,v ({μ

₁, μ

₂, . . . , μ

_n })={C_u,v (μ

₁), C_u,v (μ

₂), . . . , C_u,v (μ

_n)} denotes a complementation function for the construction of the entire encryption key and C_u,v (μ

), for any number μ

, is either defined to be;

##EQU8## and gε

G is a ranged random number;

deriving fuzzy residue mapping FM( ), whereby elements of said key sets are associated with elements of said fuzzy residue set, and FM( ), for a section i having at least one element individually mapped, is defined to be;

##EQU9## satisfying IM(μ

-1)=(μ

-1)*d_i,j <

IM(μ

)+μ

*d_i,j <

IM(μ

+1)+(μ

+1)*d_i,j, where X=X₁ ∪

X₂ ∪

. . . ∪

X_k is a set of integers in the ranges of [0, 2^h) input to FM( ) and x_i,j corresponds to e_i,j and IM(x_i,j) equals the value of applied residues mapped to by x_i,j,

and FM( ), for a section i having elements multiply mapped, is defined to be;

space="preserve" listing-type="equation">FM(X)=FM(X.sub.1 ∪

X.sub.2 ∪

. . . ∪

X.sub.k)=FM(X.sub.1)+FM(X.sub.2)+ . . . +FM(X.sub.k)

where FM(X_i)=SM(Σ

IM(x_i,j)) for 1≦

j≦

z_i and IM(x_i,j) equals x_i,j if e_i,j is mapped and equals zero otherwise, and SM(μ

) returns α

*g_i,j for some integer α

corresponding to the value of μ

satisfying;

space="preserve" listing-type="equation">SM(μ

-1)<

SM(μ

)<

SM(μ

+1);

deriving from said decryption key an encryption key θ

_k, inheriting said fuzzy residue mapping;

generating ciphertext comprising;

accepting a message in the form of an input sequence of integers in the range of [0, 2^h);

converting said encryption key set to a key vector and converting said input sequence of integers to at least one data vector X having the same dimensions as said key vector; and

generating ciphertext by computation that includes calculating the inner product of each of said data vectors with said key vector defined by;

space="preserve" listing-type="equation">γ

=K.sub.e ×

X+FM(X)

if fuzzy residues have been generated for said encryption key, and defined by;

space="preserve" listing-type="equation">γ

=K.sub.e ×

X

otherwise;

obtaining a decryption key;

decrypting said ciphertext to recover said message via decrypting each of the ciphertext block of said ciphertext, and corresponding to a standard section i of said encryption key set, the decryption is carried out according to;

##EQU10## and after obtaining all y_i,j for 1≦

j≦

z_i, is adjusted by;

##EQU11## and corresponding to a submerged section i, the decryption is carried out according to;

##EQU12## where Ω

is the set of elements in θ

_i-1 that are tranformed from said decryption key set and Y'"'"' is the decrypted {y₁,1, y₁,2, . . . y_i-1,μ

}, where μ

=z_1-1,

and corresponding to a wrinkled section i, the decryption is carried out according to;

##EQU13## where b'"'"'_i,j is the decoded multiple of identifiable b_i,j contained in γ

'"'"', and after obtaining q_i, γ

'"'"' is adjusted by;

γ

'"'"'<

γ

'"'"'-q_i *d_i,1.

and corresponding to an individually mapped section i, the decryption is carried out, for j=z_i down to 1, according to;

space="preserve" listing-type="equation">γ

'"'"'<

Γ

.sub.i.sup.-1 (γ

'"'"')

space="preserve" listing-type="equation">y.sub.i,j <

q,

satisfying;

space="preserve" listing-type="equation">FM(q-1)+(q-1)*d.sub.i,j ≦

FM(q)+q*d.sub.i,j <

FM(q+1)+(q+1)*d.sub.i,j

and after obtaining each y_i,j, γ

'"'"' is adjusted by;

space="preserve" listing-type="equation">γ

'"'"'<

γ

'"'"'-FM(y.sub.i,j)-y.sub.i,j *d.sub.i,j

and corresponding to a multiply mapped section i, the decryption is carried out, for j=z_i down to 1, according to;

space="preserve" listing-type="equation">γ

'"'"'<

Γ

.sub.i.sup.-1 (γ

'"'"')

space="preserve" listing-type="equation">y.sub.i,j <

q, satisfying;

space="preserve" listing-type="equation">FM(q)+q*d.sub.i,j ≦

γ

'"'"'

and after obtaining each y_i,j, γ

'"'"' is adjusted by;

##EQU14## where for the decryption of section k, the last generated section of the decryption key set, γ

'"'"' equals the ciphertext γ

to be decrypted and

space="preserve" listing-type="equation">Γ

.sub.i.sup.-1 =Γ

.sub.i,1.sup.-1 °

Γ

.sub.i,2.sup.-1 °

. . . °

Γ

.sub.i,t.sbsb.i.sup.-1

space="preserve" listing-type="equation">Γ

.sub.u,v.sup.- (μ

)=C.sub.u,v.sup.-1 ((w.sub.u,v.sup.-1 *μ

)Ξ

m.sub.u,v)

C_u,v^-1 denotes the reverse of C_u,v, where C_u,v^-1 (μ

), for any number μ

, is defined as;

##EQU15##

13. A cryptographic system comprising:

means for providing a plurality of key section construction methods;

means for deriving a decryption key including a decryption key set of identifiables logically segregated into sections, each said section generated by one of a plurality of construction methods and said segregation is kept secret;

means for deriving from said decryption key an encryption key, each section of said decryption key set is transformed to a corresponding section of said encryption key set and at least two said sections are transformed under different transformation methods, and previously generated and transformed sections are transformed together with the currently generated section;

means for accepting a message in the form of an input sequence of integers;

means for converting said encryption key set to a key vector and converting said input sequence of integers to at least one data vector having the same dimensions as said key vector;

means for generating ciphertext by computation that includes calculating the inner product of each of said data vectors with said key vector.

means for accepting a ciphertext; and

means for decrypting said ciphertext to recover said message.

14. A cryptographic system as in claim 13 further comprising:

means for deriving the decryption key including a decryption fuzzy residue set;

means for deriving a fuzzy residue mapping whereby elements of said decryption key set and elements of said decryption fuzzy residue set are associated for decryption;

means for deriving from said decryption key the encryption key including an encryption fuzzy residue set from said decryption fuzzy residue set, inheriting said fuzzy residue mapping whereby elements of said encryption key set and elements of said encryption fuzzy residue set are associated for encryption;

means for generating the ciphertext by computation that includes applying fuzzy residues determined by the values of the integers of said data vectors and in accordance with said fuzzy residue mapping; and

means for decrypting said ciphertext including the removal of fuzzy residues from said ciphertext in accordance with said fuzzy residue mapping to recover said message.

15. A cryptographic system which securely communicates data from a first party S to a second party R comprising:

means for deriving cryptographic keys at party R comprising;

means for providing a plurality of key section construction methods;

means for deriving a decryption key to be kept secret at party R including a decryption key set of identifiables logically segregated into sections, each section generated by one of a plurality of construction methods and said segregation is kept secret;

means for deriving from said decryption key an encryption key, each section of said decryption key set is transformed to a corresponding section of said encryption key set, and at least two sections are transformed under different transformation methods, and previously generated and transformed sections are transformed together with the currently generated section;

means for distributing said encryption key from party R to party S;

means for obtaining said encryption key at party S;

means for generating ciphertext at party S comprising;

means for accepting a message in the form of an input sequence of integers;

means for converting said encryption key set to a key vector and converting said input sequence of integers to at least one data vector having the same dimensions as said key vector; and

means for generating ciphertext by computation that includes calculating the inner product of each of said data vectors with said key vector;

means for transmitting said ciphertext from party S to party R over a communication channel;

means for receiving said ciphertext at party R; and

means for decrypting said ciphertext at party R to recover said message.

16. A cryptographic system as in claim 15 which securely communicates data from a first party S to a second party R further comprising:

means for deriving the decryption key including a decryption fuzzy residue set to be kept secret at party R;

means for deriving a fuzzy residue mapping to be kept secret at party R whereby elements of said decryption key set and elements of said decryption fuzzy residue set are associated for decryption;

means for deriving at party R from said decryption key the encryption key including an encryption fizzy residue set from said decryption fuzzy residue set, inheriting said fuzzy residue mapping whereby elements of said encryption key set and elements of said encryption fuzzy residue set are associated for encryption;

means for generating at party S the ciphertext by computation that includes applying fuzzy residues determined by the values of the integers of said data vectors and in accordance with said fuzzy residue mapping; and

means for decrypting said ciphertext at party R including the removal of fuzzy residues from said ciphertext in accordance with said fuzzy residue mapping to recover said message.

17. A cryptographic system comprising:

means for deriving a decryption key by generating a decryption key set of identifiables logically segregated into k>

1 sections;

##EQU16## and a decryption fuzzy residue set G=G₁ ∪

G₂ ∪

. . . ∪

G_k and a wrinkling set B=B₁ ∪

B₂ ∪

. . . ∪

B_k with sections corresponding to those in K_d, wherein K_d, G and B are to be kept secret and each section i, for 1≦

i≦

k of K_d is generated by at least one of the following constructions;

1) standard

2) submerged

3) wrinkled

4) individually mapped

5) multiply mapped

and for a submerged section i

d_i,j =2^h(j-1) *m_i, for 1≦

j≦

z_i, where subscript i and m_i is defined to be;

i.tbd.i, 0.tbd.i-1, z_i-1

and for a wrinkled section i

σ

=sum(η

.sub.θ

*|θ

|)+r

d_i,j =σ

+Ψ

_i-1 (b_i,j), for 1≦

j≦

z_i

and for a section of any of the other types;

##EQU17## where r is a non-zero positive ranged random number, η

.sub.θ

is the maximum applicable multiple of θ

ε

Θ

_i-1, and Θ

_i is defined to be ##EQU18## where C_u,v {μ

₁, μ

₂, . . . , μ

_n })={C_u,v (μ

₁), C.sub.μ

,ν

(μ

₂), . . . , C_u,v(μ

_n)} denotes a complementation function for the construction of the entire encryption key and C_u,v (μ

), for any number μ

, is either defined to be;

##EQU19## and g ε

G is a ranged random number;

means for deriving fuzzy residue mapping FM( ), whereby elements of said key sets are associated with elements of said fuzzy residue set, and FM( ), for a section i having at least one element individually mapped, is defined to be;

##EQU20## satisfying IM(μ

-1)+(μ

-1)*d_i,j <

IM(μ

)+μ

*d_i,j <

IM(μ

+1)+(μ

+1)*d_i,j, where X=X₁ ∪

X₂ ∪

. . . ∪

X_k is a set of integers in the ranges of [0, 2^h) input to FM( ) and x_i,j corresponds to e_i,j and IM(x_i,j) equals the value of applied residues mapped to by x_i,j,

and FM( ), for a section i having elements multiply mapped, is defined to be;

FM(X)=FM(X₁ ∪

X₂ ∪

. . . ∪

X_k)=FM(X₁)+FM(X₂)+ . . . +FM(X_k) where FM(X_i)=SM(Σ

IM(x_i,j)) for 1≦

j≦

z_i and IM(x_i,j) equals x_i,j if e_i,j is mapped and equals zero otherwise, and SM(μ

) returns α

*g_i,1 for some integer α

corresponding to the value of μ

satisfying;

space="preserve" listing-type="equation">SM(μ

-1)<

SM(μ

)<

SM(∥

+1);

means for deriving from said decryption key an encryption key Θ

_k, inheriting said fuzzy residue mapping,

means for obtaining an encryption key;

means for generating ciphertext comprising;

means for accepting a message in the form of an input sequence of integers in the range of [0, 2^h);

means for converting said encryption key set to a key vector and converting said input sequence of integers to at least one data vector having the same dimensions as said key vector; and

means for generating ciphertext by computation that includes calculating the inner product of each of said data vectors with said key vector defined by;

space="preserve" listing-type="equation">γ

=K.sub.e ×

X+FM(X)

if fuzzy residues have been generated for said encryption key, and defined by;

space="preserve" listing-type="equation">γ

=K.sub.e ×

X

otherwise.

means for accepting a ciphertext;

means for obtaining a decryption key corresponding to said encryption key;

means for decrypting said ciphertext to recover said message via decrypting each of the blocks of said ciphertext, and corresponding to a standard section i of said encryption key set, the decryption is carried out according to;

##EQU21## and after obtaining all y_i,j for 1≦

j≦

z_i, γ

'"'"' is adjusted by;

##EQU22## and corresponding to a submerged section i, the decryption is carried out according to;

##EQU23## where Ω

is the set of element in Θ

_i-1 that are transformed from said decryption key set and Y'"'"' is the decrypted {y₁,1, y₁,2, . . . , y_i-1,u }, where u=z_i-1,

and corresponding to a wrinkled section i, the decryption is carried out according to;

##EQU24## where b'"'"'_i,j is decoded multiple of identifiable b_i,j contained in γ

'"'"', and after obtaining q_i, γ

'"'"' is adjusted by;

γ

'"'"'<

γ

'"'"'-q_i *d_i,1.

and corresponding to an individually mapped section i, the decryption is carried out, for j=z₁ down to 1, according to;

space="preserve" listing-type="equation">γ

'"'"'<

Γ

.sub.i.sup.-1 (γ

'"'"')

space="preserve" listing-type="equation">y.sub.i,j <

q, satisfying;

space="preserve" listing-type="equation">FM(q-1)+(q-1)*d.sub.i,j ≦

FM(q)+q*d.sub.i,j <

FM(q+1)+(q+1)*d.sub.i,j

and after obtaining each y_i,j, γ

'"'"' is adjusted by;

space="preserve" listing-type="equation">γ

'"'"'<

γ

'"'"'-y.sub.i,j *d.sub.i,j -FM(y.sub.i,j)

and corresponding to a multiply mapped section i, the decryption is carried out, for j=z_i down to 1, according to;

space="preserve" listing-type="equation">γ

'"'"'<

Γ

.sub.i.sup.-1 (γ

'"'"')

space="preserve" listing-type="equation">y.sub.i,j <

q, for the largest q satisfying;

space="preserve" listing-type="equation">FM(q)+q*d.sub.i,j ≦

γ

'"'"'

and after obtaining each y_i,j, γ

'"'"' is adjusted by;

##EQU25## where fore the decryption of section k, the last generated section of the decryption key set, γ

'"'"' equals the ciphertext γ

to be decrypted and

space="preserve" listing-type="equation">Γ

.sub.i.sup.-1 =Γ

.sub.i,1.sup.-1 °

Γ

.sub.i,2.sup.-1 °

. . . °

Γ

.sub.i,y.sbsb.i

space="preserve" listing-type="equation">Γ

.sub.u,v.sup.- (μ

)=C.sub.u,v.sup.-1 ((w.sub.u,v.sup.-1 *μ

)Ξ

m.sub.u,v)

C_u,v^-1 denotes the reverse of C_u,v, where C_u,v^-1 (μ

), for any number μ

, is defined as;

##EQU26##