Cryptographic communication process and apparatus

• US 6,075,865 A
• Filed: 07/01/1998
• Issued: 06/13/2000
• Est. Priority Date: 07/01/1998
• Status: Expired due to Term

• Current Assignee: TecSec, Incorporated
• Original Assignee: TecSec, Incorporated
• Inventors: Wack, C. Jay, Scheidt, Edward M.
• Primary Examiner(s): Swann, Tod R.
• Assistant Examiner(s): JACK, TODD M
• Application Number:
  US09/108,312
• Time in Patent Office: 713 Days
• Field of Search: 380/28, 380/9, 380/29, 380/255
• US Class Current: 380/255
• CPC Class Codes: H04L 9/0631 Substitution permutation network [SPN], i.e. cipher composed of a number of stages or rounds each involving linear and nonlinear transformations, e.g. AES algorithms
• CPC Class Codes: H04L 9/0637 Modes of operation, e.g. cipher block chaining [CBC], electronic codebook [ECB] or Galois/counter mode [GCM]
• CPC Class Codes: H04L 9/08 Key distribution or management, e.g. generation, sharing or updating, of cryptographic keys or passwords network architectures or network communication protocols for supporting key management in a packet data network H04L63/06

Abstract

A communication system, which includes an origination space, a communications channel, and a destination space associated with the origination space via the communications channel. The origination space includes an encryption engine for generating an output symbol O_t based on an input symbol I_t and means for receiving an encrypt key, an encrypt text/key relation, and the input symbol. The destination space includes a decryption engine for generating a decrypted symbol I'"'"'_t based on the output symbol received from the origination space via the communications channel and means for receiving a decrypt key and a decrypt text/key relation. The encrypt text/key relation controls the encryption engine such that O_t =α_N (t)+π_N [α_N-1 (t)+π_N-1 [α_N-2 (t)+ . . . +π₂ [α₁ (t)+π₁ [I_t +α₀ (t)]] . . . ]], mod W, where α_N, α_N-1, . . . , α₁, α₀ are N+1 additive transformations defined by the encrypt key, where π_N, π_N-1, . . . , π₂, π₀ are N permutations defined by the encrypt key, and where W represents the number of possibilities for each permutation defined by the encrypt key. The decrypt text/key relation controls the decryption engine such that I'"'"'_t =π₁^-1 [π₂^-1 [π₃^-1 . . . [π_N-1^-1 [π_N^-1 [O_t -α'"'"'_N (t)]-α'"'"'_N-1 (t)]- . . . -α'"'"'₃ (t)]-α'"'"'₂ (t)]-α'"'"'₁ (t)]-α'"'"'₀ (t), mod W, where π_i^-1 is defined by the decrypt key as the inverse of the permutation π_i, where α'"'"'_N, α'"'"'_N-1, . . . , α'"'"'₁, α'"'"'₀ are N+1 additive transformations defined by the decrypt key, and where W represents the number of possibilities for each inverse permutation defined by the decrypt key.

Claims

1. A communication system, comprising:

a) an origination space;

b) a communications channel; and

c) a destination space associated with the origination space via the communications channel;

d) wherein the origination space includes;

1) an encryption engine for generating an output symbol O_t based on an input symbol I_t ; and

2) means for receiving an encrypt key, an encrypt text/key relation, and the input symbol;

e) wherein the destination space includes;

1) a decryption engine for generating a decrypted symbol I'"'"'_t based on the output symbol received from the origination space via the communications channel; and

2) means for receiving a decrypt key and a decrypt text/key relation;

f) wherein the encrypt text/key relation controls the encryption engine such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]], mod W, where α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by the encrypt key where π

_N, π

_N-1, . . . , π

₂, π

₀ are N permutations defined by the encrypt key, and where W represents the number of possibilities for each permutation defined by the encrypt key;

g) wherein the decrypt text/key relation controls the decryption engine such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

'"'"'_N (t)]-α

'"'"'_N-1 (t)]- . . . -α

'"'"'₃ (t)]-α

'"'"'₂ (t)]-α

'"'"'₁ (t)]-α

'"'"'₀ (t), mod W, where π

_i^-1 is defined by the decrypt key as the inverse of the permutation π

_i, where α

'"'"'_N, α

'"'"'_N-1, . . . , α

'"'"'₁, α

'"'"'₀ are N+1 additive transformations defined by the decrypt key, and where W represents the number of possibilities for each inverse permutation defined by the decrypt key; and

h) wherein I'"'"'_t corresponds to I_t.

2. The communication system of claim 1, wherein the encryption engine further includes W look-up tables for storing each of the possible W sets of permutations.

3. The communication system of claim 1, wherein the encryption engine further includes M<

W look-up tables for storing M available sets of the possible W sets of permutations.

4. The communication system of claim 1, wherein the encryption engine further includes N<

M<

W look-up tables for storing N sets of permutations preselected from M available sets of the possible W sets of permutations.

5. A communication system, comprising:

a) an origination space;

b) a communications channel; and

c) a destination space associated with the origination space via the communications channel;

d) wherein the origination space includes;

1) an encryption engine for generating an output symbol O_t based on an input symbol I_t ; and

2) means for receiving an encrypt key an encrypt text/key relation and the input symbol;

e) wherein the destination space includes;

1) a decryption engine for generating a decrypted symbol I'"'"'_t based on the output symbol received from the origination space via the communications channel; and

2) means for receiving a decrypt key and a decrypt text/key relation;

f) wherein the encrypt text/key relation controls the encryption engine such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]], mod W, where α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by the encrypt key, where π

_N, π

_N-1 , . . . , π

₂, π

₀ are N permutations defined by the encrypt key, and where W represents the number of possibilities for each permutation defined by the encrypt key;

g) wherein the decrypt text/key relation controls the decryption engine such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

'"'"'_N (t)]-α

'"'"'_N-1 (t)]- . . . -α

'"'"'₃ (t)]-α

'"'"'₂ (t)]-α

'"'"'₁ (t)]-α

'"'"'₀ (t), mod W, where π

_i^-1 is defined by the decrypt key as the inverse of the permutation π

_i, where α

'"'"'_N, α

'"'"'_N-1, . . . , α

'"'"'₁, α

'"'"'₀ are N+1 additive transformations defined by the decrypt key, and where W represents the number of possibilities for each inverse permutation defined by the decrypt key; and

h) wherein α

(t) is a step function, and α

_X (t), X={0, 1, 2, . . . , N-1, N}, increments the sequence of π

_X for each value that t equals an integer multiple of R, where R is a prime number.

6. A communication system, comprising:

a) an origination space;

b) a communications channel; and

c) a destination space associated with the origination space via the communications channel;

d) wherein the origination space includes;

1) an encryption engine for generating an output symbol O_t based on an input symbol I_t ; and

2) means for receiving an encrypt key, an encrypt text/key relation, and the input symbol;

e) wherein the destination space includes;

1) a decryption engine for generating a decrypted symbol I'"'"'_t based on the output symbol received from the origination space via the communications channel; and

2) means for receiving a decrypt key and a decrypt text/key relation;

f) wherein the encrypt text/key relation controls the encryption engine such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]], mod W, where α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by the encrypt key, where π

_N, π

_N-1, . . . , π

₂, π

₀ are N permutations defined by the encrypt key, and where W represents the number of possibilities for each permutation defined by the encrypt key;

g) wherein the decrypt text/key relation controls the decryption engine such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1[O_t -α

'"'"'_N (t)]-α

'"'"'_N-1 (t)]- . . . -α

'"'"'₃ (t)]-α

'"'"'₂ (t)]-α

'"'"'₁ (t)]-α

'"'"'₀ (t), mod W, where π

_i^-1 is defined by the decrypt key as the inverse of the permutation π

_i, where α

'"'"'_N, α

'"'"'_N-1, . . . , α

'"'"'₁, α

'"'"'₀ are N+1 additive transformations defined by the decrypt key, and where W represents the number of possibilities for each inverse permutation defined by the decrypt key; and

h) wherein α

(t) is a step function, and α

_X (t), X={0, 1, 2, . . . , N-1, N}, decrements the sequence of π

_X for each value that t equals an integer multiple of R, where R is a prime number.

7. A communication system, comprising:

a) an origination space;

b) a communications channel; and

c) a destination space associated with the origination space via the communications channel;

d) wherein the origination space includes;

1) an encryption engine for generating an output symbol O_t based on an input symbol I_t ; and

2) means for receiving an encrypt key, an encrypt text/key relation, and the input symbol;

e) wherein the destination space includes;

1)a decryption engine for generating a decrypted symbol I'"'"'_t based on the output symbol received from the origination space via the communications channel; and

2) means for receiving a decrypt key and a decrypt text/key relation;

f) wherein the encrypt text/key relation controls the encryption engine such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . .]], mod W, where α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by the encrypt key where π

_N, π

_N-1, . . . , π

₂, π

₀ are N permutations defined by the encrypt key, and where W represents the number of possibilities for each permutation defined by the encrypt key;

g) wherein the decrypt text/key relation controls the decryption engine such that I'"'"'₌π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

'"'"'_N (t)]-α

'"'"'_N-1 (t)]- . . . -α

'"'"'₃ (t)]-α

'"'"'₂ (t)]-α

'"'"'₁ (t)]-α

'"'"'₀ (t), mod W, where π

_i^-1 is defined by the decrypt key as the inverse of the permutation π

_i, where α

'"'"'_N, α

'"'"'_N-1, . . . α

'"'"'₁, α

'"'"'₀ are N+1 additive transformations defined by the decrypt key, and where W represents the number of possibilities for each inverse permutation defined by the decrypt key; and

h) wherein α

(t) is a step function and α

_X (t), X={0, 1, 2, . . , N-1, N}, increments the sequence of π

_X for each value of t except when t equals an integer multiple of R, where R is a prime number.

8. A communication system, comprising:

a) an origination space;

b) a communications channel; and

c) a destination space associated with the origination space via the communications channel;

d) wherein the origination space includes;

1) an encryption engine for penerating an output symbol O_t based on an input symbol I_t ; and

2) means for receiving an encrypt key, an encrypt text/key relation and the input symbol;

e) wherein the destination space includes;

1) a decryption engine for generating a decrypted symbol I'"'"'_t based on the output symbol received from the origination space via the communications channel; and

2) means for receiving a decrypt key and a decrypt text/key relation;

f) wherein the encrypt text/key relation controls the encryption engine such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]], mod W, where α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by the encrypt key, where π

_N, π

_N-1, . . . , π

₂, π

₀ are N permutations defined by the encrypt key, and where W represents the number of possibilities for each permutation defined by the encrypt key;

g) wherein the decrypt text/key relation controls the decryption engine such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

'"'"'_N (t)]-α

'"'"'_N-1 (t)]- . . . -α

'"'"'₃ (t)]-α

'"'"'₂ (t)]-α

'"'"'₁ (t)]-α

'"'"'₀ (t), mod W, where π

_i^-1 is defined by the decrypt key as the inverse of the permutation π

_i, where α

'"'"'_N, α

'"'"'_N-1, . . . , α

'"'"'₁, α

'"'"'₀ are N+1 additive transformations defined by the decrypt key, and where W represents the number of possibilities for each inverse permutation defined by the decrypt key; and

h) wherein α

(t) is a step function, and α

_X (t), X={0, 1, 2, . . . , N-1, N}, decrements the sequence of π

_X for each value of t except when t equals an integer multiple of R, where R is a prime number.

9. A communication system, comprising:

a) an origination space;

b) a communications channel; and

c) a destination space associated with the origination space via the communications channel;

d) wherein the origination space includes;

1) means for receiving an input symbol I_t, an encrypt key, and an encrypt text/key relation; and

2) an encryption engine, controllable by the encrypt text/key relation for generating an output symbol O_t based on the input symbol, such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]], mod W, where α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by the encrypt key, where π

_N, π

_N-1, . . . , π

₂, π

₀ , are N permutations defined by the encrypt key, and where W represents the number of possibilities for each permutation defined by the encrypt key;

e) wherein the destination space includes;

1) means for receiving a decrypt key and a decrypt text/key relation; and

2) a decryption engine, controllable for generating a decrypted symbol I'"'"'_t based on the output symbol received from the origination space via the communications channel, such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

'"'"'_N (t)]-α

'"'"'_N-1 (t)]- . . . -α

'"'"'₃ (t)]-α

'"'"'₂ (t)]-α

'"'"'₁ (t)]-α

'"'"'₀ (t), mod W, where π

_i^-1 is defined by the decrypt key as the inverse of the permutation π

_i, where α

'"'"'_N, α

'"'"'_N-1, . . . , α

'"'"'₁, α

'"'"'₀ are N+1 additive transformations defined by the decrypt key, and where W represents the number of possibilities for each inverse permutation defined by the decrypt key; and

f) wherein I'"'"'_t corresponds to I_t.

10. The communication system of claim 9, wherein the encryption engine further includes W look-up tables for storing each of the possible W sets of permutations.

11. The communication system of claim 9, wherein the encryption engine further includes M<

W look-up tables for storing M available sets of the possible W sets of permutations.

12. The communication system of claim 9, wherein the encryption engine further includes N<

M<

W look-up tables for storing N sets of permutations preselected from M available sets of the possible W sets of permutations.

13. A communication system, comprising:

a) an origination space;

b) a communications channel; and

c) a destination space associated with the origination space via the communications channel;

d) wherein the origination space includes;

1) means for receiving an input symbol I_t, an encrypt key and an encrypt text/key relation; and

2) an encryption engine, controllable by the encrypt text/key relation for generating an output symbol O_t based on the input symbol, such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+. . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]], mod W, where α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by the encrypt key, where π

_N, π

_N- 1, . . . , π

₂, π

₀ are N permutations defined by the encrypt key, and where W represents the number of possibilities for each permutation defined by the encrypt key;

e) wherein the destination space includes;

1) means for receiving a decrypt key and a decrypt text/key relation; and

2) a decryption engine, controllable for generating a decrypted symbol I'"'"'_t based on the output symbol received from the origination space via the communications channel, such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

'"'"'_N (t)]-α

'"'"'_N-1 (t)]- . . . -α

'"'"'₃ (t)]-α

'"'"'₂ (t)]-α

'"'"'₁ (t)-α

'"'"'₀ (t), mod W, where π

_i^-1 is defined by the decrypt key as the inverse of the permutation π

_i, where α

'"'"'_N, α

'"'"'_N-1, . . . , α

'"'"'₁, α

'"'"'₀ are N+1 additive transformations defined by the decrypt key, and where W represents the number of possibilities for each inverse permutation defined by the decrypt key

f) wherein α

(t) is a step function and α

_X (t), X={0, 1, 2, . . . , N-1, N}, increments the sequence of π

_X for each value that t equals an integer multiple of R, where R is a prime number.

14. A communication system, comprising:

a) an origination space;

b) a communications channel; and

c) a destination space associated with the origination space via the communications channel;

d) wherein the origination space includes;

2) means for receiving an input symbol I_t, an encrypt key and an encrypt text/key relation; and

2) an encryption engine controllable by the encrypt text/key relation for generating an output symbol O_t based on the input symbol, such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]], mod W, where α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by the encrypt key, where π

_N, π

_N-1, . . . , π

₂, π

₀ are N permutations defined by the encrypt key and where W represents the number of possibilities for each permutation defined by the encrypt key;

e) wherein the destination space includes;

1) means for receiving a decrypt key and a decrypt text/key relation; and

2) a decryption engine controllable for generating a decrypted symbol I'"'"'_t based on the output symbol received from the origination space via the communications channel, such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

'"'"'_N (t)]-α

'"'"'_N-1 (t)]- . . . -α

₃ (t)]-α

'"'"'₂ (t)]-α

'"'"'₁ (t)]-α

'"'"'₀ (t), mod W, where π

_i^-1 is defined by the decrypt key as the inverse of the permutation π

_i, where α

'"'"'_N, α

'"'"'_N-1, . . . , α

'"'"'₁, α

'"'"'₀ are N+1 additive transformations defined by the decrypt key, and where W represents the number of possibilities for each inverse permutation defined by the decrypt key

f) wherein α

(t) is a step function and α

_X (t), X={0, 1, 2, . . . , N-1, N}, decrements the sequence of π

_X for each value that t equals an integer multiple of R, where R is a prime number.

15. A communication system, comprising:

a) an origination space;

b) a communications channel; and

c) a destination space associated with the origination space via the communications channel;

d) wherein the origination space includes;

1) means for receiving an input symbol I_t, an encrypt key, and an encrypt text/key relation; and

2) an encryption engine, controllable by the encrypt text/key relation for generating an output symbol O_t based on the input symbol, such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]], mod W, where α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by the encrypt key, where π

_N, π

_N-1, . . . .π

₂, π

₀ no are N permutations defined by the encrypt key, and where W represents the number of possibilities for each permutation defined by the encrypt key;

e) wherein the destination space includes;

1) means for receiving a decrypt key and a decrypt text/key relation; and

2) a decryption engine, controllable for generating a decrypted symbol I'"'"'_t based on the output symbol received from the origination space via the communications channel, such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

'"'"'_N (t)]-α

'"'"'_N-1 (t)]- . . . -α

'"'"'₃ (t)]-α

'"'"'₂ (t)]-α

'"'"'₁ (t)]-α

'"'"'₀ (t), mod W, where π

_i^-1 is defined by the decrypt key as the inverse of the permutation π

_i, where α

'"'"'_N, α

'"'"'_N-1, . . . , α

'"'"'₁, α

'"'"'₀ are N+1 additive transformations defined by the decrypt key, and where W represents the number of possibilities for each inverse permutation defined by the decrypt key

f) wherein α

(t) is a step function and α

_X (t), X={0, 1, 2, . . . , N-1, N}, increments the sequence of π

_X for each value of t except when t equals an integer multiple of R, where R is a prime number.

16. A communication system, comprising:

a) an origination space;

b) a communications channel; and

c) a destination space associated with the origination space via the communications channel;

d) wherein the origination space includes;

1) means for receiving an input symbol I_t, an encrypt key, and an encrypt text/key relation; and

2) an encryption engine, controllable by the encrypt text/key relation for generating an output symbol O_t based on the input symbol, such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]], mod W, where α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by the encrypt key, where π

_N, π

_N-1, . . . , π

₂, π

₀ are N permutations defined by the encrypt key, and where W represents the number of possibilities for each permutation defined by the encrypt key;

e) wherein the destination space includes;

1) means for receiving a decrypt key and a decrypt text/key relation; and

2) a decryption engine, controllable for generating a decrypted symbol I'"'"'_t based on the output symbol received from the origination space via the communications channel, such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

'"'"'_N (t)]-α

'"'"'_N-1 (t)]- . . . -α

'"'"'₃ (t)-α

'"'"'₂ (t)]-α

'"'"'₁ (t)]-α

'"'"'₀ (t), mod W, where π

_i^-1 is defined by the decrypt key as the inverse of the permutation π

_i, where α

'"'"'_N, α

'"'"'_N-1, . . . , α

'"'"'₁, α

'"'"'₀ are N+1 additive transformations defined by the decrypt key, and where W represents the number of possibilities for each inverse permutation defined by the decrypt key

f) wherein α

(t) is a step function and α

_X (t), X={0, 1, 2, . . . , N-1, N}, decrements the sequence of π

_X for each value of t except when t equals an integer multiple of R, where R is a prime number.

17. A communication system, comprising:

a) a first computer;

b) a communications channel; and

c) a second computer coupled with the first computer via the communications channel;

d) wherein the first computer includes;

1) a symbol input port for receiving an input symbol I_t ;

2) an encrypt key input port for receiving an encrypt key;

3) a first memory, for storing an encrypt text/key relation; and

4) a first microprocessor for generating an output symbol O_t based on the input symbol, controlled by the encrypt text/key relation such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]], mod W, where α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by the encrypt key, where π

_N, π

_N-1, . . . , π

₂, π

₀ are N permutations defined by the encrypt key, and where W represents the number of possibilities for each permutation defined by the encrypt key;

e) wherein the second computer includes;

1) a decrypt key input port for receiving a decrypt key;

2) a second memory, for storing a decrypt text/key relation; and

3) a second microprocessor for generating a decrypted symbol I'"'"'_t based on the output symbol received from the origination space via the communications channel, controlled by the decrypt text/key relation such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1 [π

_N^-1 [O_t -α

'"'"'_N (t)]-α

'"'"'_N-1 (t)]- . . . -α

'"'"'₃ (t)]-α

'"'"'₂ (t)]-α

'"'"'₁ (t)]-α

'"'"'₀ (t), mod W, where π

_i ^-1 is defined by the decrypt key as the inverse of the permutation π

_i, where α

'"'"'_N, α

'"'"'_N-1, . . . , α

'"'"'₁, α

'"'"'₀ are N+1 additive transformations defined by the decrypt key, and where W represents the number of possibilities for each inverse permutation defined by the decrypt key; and

f) wherein I'"'"'_t corresponds to I_t.

18. The communication system of claim 17, wherein the first computer further includes W look-up tables for storing each of the possible W sets of permutations.

19. The communication system of claim 17, wherein the first computer further includes M<

W look-up tables for storing M available sets of the possible W sets of permutations.

20. The communication system of claim 17, wherein the first computer further includes N<

M<

W look-up tables for storing N sets of permutations preselected from M available sets of the possible W sets of permutations.

21. A communication system, comprising:

a) a first computer;

b) a communications channel; and

c) a second computer coupled with the first computer via the communications channel;

d) wherein the first computer includes;

1) a symbol input port for receiving an input symbol I_t ;

2) an encrypt key input port for receiving an encrypt key;

3) a first memory, for storing an encrypt text/key relation; and

4) a first microprocessor for generating an output symbol O_t based on the input symbol, controlled by the encrypt text/key relation such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

_i [I_t +α

₀ (t)]] . . . ]], mod W, where α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by the encrypt key, where π

_N, π

_N-1, . . . , π

₂, π

₀ are N permutations defined by the encrypt key, and where W represents the number of possibilities for each permutation defined by the encrypt key;

e) wherein the second computer includes;

1) a decrypt key input port for receiving a decrypt key;

2) a second memory, for storing, a decrypt text/key relation; and

3) a second microprocessor for generating, a decrypted symbol I'"'"'_t based on the output symbol received from the origination space via the communications channel, controlled by the decrypt text/key relation such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

'"'"'_N (t)]-α

'"'"'_N-1 (t)]- . . . -α

'"'"'₃ (t)]-α

'"'"'₂ (t)]α

'"'"'₁ (t)]-α

'"'"'₀ (t), mod W, where π

_i^-1 is defined by the decrypt key as the inverse of the permutation π

_i, where α

'"'"'_N, α

'"'"'_N-1, . . . α

'"'"'₁, α

'"'"'₀ are N+1 additive transformations defined by the decrypt key, and where W represents the number of possibilities for each inverse permutation defined by the decrypt key; and

f) wherein α

(t) is a step function, and α

_X (t), X={0, 1, 2, . . . , N-1, N}, increments the sequence of π

_X for each value that t equals an integer multiple of R, where R is a prime number.

22. A communication system, comprising:

a) a first computer;

b) a communications channel; and

c) a second computer coupled with the first computer via the communications channel;

d) wherein the first computer includes;

1) a symbol input port for receiving an input symbol I_t ;

2) an encrypt key input port for receiving an encrypt key;

3) a first memory, for storing an encrypt text/key relation; and

4) a first microprocessor for generating an output symbol O_t based on the input symbol, controlled by the encrypt text/key relation such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]], mod W, where α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by the encrypt key where π

_N, π

_N-1, . . . , π

₂, π

₀ are N permutations defined by the encrypt key, and where W represents the number of possibilities for each permutation defined by the encrypt key;

e) wherein the second computer includes;

1) a decrypt key input port for receiving a decrypt key;

2) a second memory, for storing a decrypt text/key relation; and

3) a second microprocessor for generating a decrypted symbol I'"'"'_t based on the output symbol received from the origination space via the communications channel, controlled by the decrypt text/key relation such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

'"'"'_N (t)]-α

'"'"'_N-1 (t)]- . . . -α

'"'"'₃ (t)]-α

'"'"'₂ (t)]-α

'"'"'₁ (t)]-α

'"'"'₀ (t), mod W, where π

_i^-1 is defined by the decrypt key as the inverse of the permutation π

_i, where α

'"'"'_N, α

'"'"'_N-1, . . . , α

'"'"'₁, α

'"'"'₀ are N+1 additive transformations defined by the decrypt key, and where W represents the number of possibilities for each inverse permutation defined by the decrypt key; and

f) wherein α

(t) is a step function, and α

_X (t), X={0, 1, 2, . . . , N-1, N}, decrements the sequence of π

_X for each value that t equals an integer multiple of R, where R is a prime number.

23. A communication system, comprising:

a) a first computer;

b) a communications channel; and

c) a second computer coupled with the first computer via the communications channel;

d) wherein the first computer includes;

1) a symbol input port for receiving an input symbol I_t ;

2) an encrypt key input port for receiving an encrypt key;

3) a first memory, for storing an encrypt text/key relation; and

4) a first microprocessor for generating an output symbol O_t based on the input symbol, controlled by the encrypt text/key relation such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]], mod W, where α

_N, α

_N-1, . . . ,α

₁, α

₀ are N+1 additive transformations defined by the encrypt key, where π

_N, π

_N-1, . . . π

₂, π

₀ are N permutations defined by the encrypt key, and where W represents the number of possibilities for each permutation defined by the encrypt key;

e) wherein the second computer includes;

1) a decrypt key input port for receiving a decrypt key;

2) a second memory, for storing a decrypt text/key relation; and

3) a second microprocessor for generating a decrypted symbol I'"'"'_t based on the output symbol received from the origination space via the communications channel, controlled by the decrypt text/key relation such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

'"'"'_N (t)]-α

'"'"'_N-1 (t)]- . . . -α

'"'"'₃ (t)]-α

'"'"'₂ (t)]-α

'"'"'₁ (t)]-α

'"'"'₀ (t), mod W, where π

_i^-1 is defined by the decrypt key as the inverse of the permutation π

_i, where α

'"'"'_N, α

_N-1, . . . , α

'"'"'₁, α

'"'"'₀ are N+1 additive transformations defined by the decrypt key, and where W represents the number of possibilities for each inverse permutation defined by the decrypt key; and

f) wherein α

(t) is a step function, and α

_X (t), X={0, 1, 2, . . . , N-1, N}, increments the sequence of π

_X for each value of t except when t equals an integer multiple of R, where R is a prime number.

24. A communication system, comprising:

a) a first computer;

b) a communications channel; and

c) a second computer coupled with the first computer via the communications channel;

d) wherein the first computer includes;

1) a symbol input port for receiving an input symbol I_t ;

2) an encrypt key input port for receiving an encrypt key;

3) a first memory, for storing an encrypt text/key relation; and

4) a first microprocessor for generating an output symbol O_t based on the input symbol, controlled by the encrypt text/key relation such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]], mod W, where α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by the encrypt key, where π

_N, π

_N-1, . . . π

₂, π

₀ are N permutations defined by the encrypt key, and where W represents the number of possibilities for each permutation defined by the encrypt key;

e) wherein the second computer includes;

1) a decrypt key input port for receiving a decrypt key;

2) a second memory, for storing a decrypt text/key relation; and

3) a second microprocessor for generating a decrypted symbol I'"'"'_t , based on the output symbol received from the origination space via the communications channel, controlled by the decrypt text/key relation such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

'"'"'_N (t)]-α

'"'"'_N-1 (t)]- . . . -α

'"'"'₃ (t)]-α

'"'"'₂ (t)]α

'"'"'₁ (t)]-α

'"'"'₀ (t), mod W, where π

_i^-1 is defined by the decrypt key as the inverse of the permutation π

_i, where α

'"'"'_N, α

'"'"'_N-1, . . . , α

'"'"'₁, α

'"'"'₀ are N+1 additive transformations defined by the decrypt key, and where W represents the number of possibilities for each inverse permutation defined by the decrypt key; and

f) wherein α

(t) is a step function, and α

_X (t), X={0, 1, 2, . . . , N-1, N}, decrements the sequence of π

_X for each value of t except when t equals an integer multiple of R, where R is a prime number.

25. A process of communicating between an origination space and a destination space, comprising:

receiving an input symbol I_t at the origination space;

generating an output symbol O_t based on the input symbol, such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]], mod W, where α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 predetermined additive transformations, where π

_N, π

_N-1, . . . , π

₂, π

₀ are N predetermined permutations, and where W represents the number of possibilities for each permutation;

receiving the output symbol at the destination space; and

generating a dccrypted symbol I'"'"'_t based on the received output symbol, such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

'"'"'_N (t)]-α

'"'"'_N-1 (t)]- . . . -α

'"'"'₃ (t)]-α

'"'"'₂ (t)]-α

'"'"'₁ (t)]-α

'"'"'₀ (t), mod W, where π

_i^-1 is an inverse of the predetermined permutation π

_i, where α

'"'"'_N, α

'"'"'_N-1, . . . , α

'"'"'₁, α

'"'"'₀ are N+1 predetermined additive transformations, and where W represents the number of possibilities for each inverse permutation;

wherein I'"'"'_t corresponds to I_t.

26. The process of claim 25, further including retrieving the possible W sets of permutations from W look-up tables, prior to generating the output symbol.

27. The process of claim 25, further including retrieving M available sets of the possible W sets of permutations from M<

W look-up tables, prior to generating the output symbol.

28. The process of claim 25, further including retrieving N sets of permutations preselected from M available sets of the possible W sets of permutations from N<

M<

W look-up tables, prior to generating the output symbol.

29. A process of communicating between an origination space and a destination space, comprising:

receiving an input symbol I_t, at the origination space;

generating an output symbol O_t based on the input symbol, such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]], mod W, where α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 predetermined additive transformations, where π

_N, π

_N-1, . . . , π

₂, π

₀ are N predetermined permutations, and where W represents the number of possibilities for each permutation;

receiving the output symbol at the destination space;

generating a decrypted symbol I'"'"'_t based on the received output symbol, such that I'"'"'_t =π

₁^-1 [π

₂^-7 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

'"'"'_N (t)]-α

'"'"'_N-1 (t)]- . . . -α

'"'"'₃ (t)]-α

'"'"'₂ (t)]-α

'"'"'₁ (t)]-α

'"'"'₀ (t), mod W, where π

_i^-1 is an inverse of the predetermined permutation π

_i, where α

'"'"'_N, α

'"'"'_N-1, . . . , α

'"'"'₁, α

'"'"'₀ are N+1 predetermined additive transformations, and where W represents the number of possibilities for each inverse permutation; and

using α

_X (t), X={0, 1, 2, . . . , N-1, N}, to increment the sequence of π

_X for each value that t equals an integer multiple of R, where R is a prime number and α

(t) is a step function.

30. A process of communicating between an origination space and a destination space, comprising:

receiving an input symbol I_t at the origination space;

generating an output symbol O_t based on the input symbol, such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]]mod W, where α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 predetermined additive transformations, where π

_N, π

_N-1, . . . , π

₂, π

₀ are N predetermined permutations, and where W represents the number of possibilities for each permutation;

receiving the output symbol at the destination space;

generating a decrypted symbol I'"'"'_t, based on the received output symbol, such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

'"'"'_N (t)]-α

'"'"'_N-1 (t)]- . . . -α

'"'"'₃ (t)]-α

'"'"'₂ (t)]-α

'"'"'₁ (t)]-α

'"'"'₀ (t), mod W, where π

_i^-1 is an inverse of the predetermined permutation π

_i, where α

'"'"'_N, α

'"'"'_N-1, . . . , α

'"'"'₁, α

'"'"'₀ are N+1 predetermined additive transformations, and where W represents the number of possibilities for each inverse permutation; and

using α

_X (t), X={0, 1, 2, . . . , N-1, N}, to decrement the sequence of π

_X for each value that t equals an integer multiple of R, where R is a prime number and α

(t) is a step function.

31. A process of communicating between an origination space and a destination space, comprising:

receiving an input symbol I_t at the origination space;

generating an output symbol O_t based on the input symbol, such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]], mod W, where α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 predetermined additive transformations, where π

_N, π

_N-1, . . , π

₂, π

₀ are N predetermined permutations, and where W represents the number of possibilities for each permutation;

receiving the output symbol at the destination space;

generating a decrypted symbol I'"'"'_t based on the received output symbol, such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

'"'"'_N (t)]-α

'"'"'_N-1 (t)]- . . . -α

'"'"'₃ (t)]-α

'"'"'₂ (t)]-α

'"'"'₁ (t)]-α

'"'"'₀ (t), mod W, where π

_i^-1 is an inverse of the predetermined permutation π

_i, where α

'"'"'_N, α

'"'"'_N-1, . . . , α

'"'"'₁, α

'"'"'₀ are N+1 predetermined additive transformations, and where W represents the number of possibilities for each inverse permutation; and

using α

_X (t), X={0, 1, 2, . . . , N-1, N}, to increment the sequence of π

_X for each value of t except when t equals an integer multiple of R, where R is a prime number and α

(t) is a step function.

32. A process of communicating between an origination space and a destination space, comprising:

receiving an input symbol I_t at the origination space;

generating an output symbol O_t based on the input symbol, such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]], mod W, where α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 predetermined additive transformations, where π

_N, π

_N-1, . . . , π

₂, π

₀ are N predetermined permutations, and where W represents the number of possibilities for each permutation;

receiving the output symbol at the destination space;

generating a decrypted symbol I'"'"'_t based on the received output symbol, such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

'"'"'_N (t)]-α

'"'"'_N-1 (t)]- . . . -α

'"'"'₃ (t)]-α

'"'"'₂ (t)]-α

'"'"'₁ (t)]-α

₀ (t), mod W, where π

_i^-1 is an inverse of the predetermined permutation π

_i, where α

'"'"'_N, α

'"'"'_N-1, . . . , α

'"'"'₁, α

'"'"'₀ are N+1 predetermined additive transformations, and where W represents the number of possibilities for each inverse permutation; and

using α

_X (t), X={0, 1, 2, . . . , N-1, N }, to decrement the sequence of π

_X for each value of t except when t equals an integer multiple of R, where R is a prime number and α

(t) is a step function.

33. A storage medium, comprising:

interface means, and

means for controlling a microprocessor, via the interface means, to produce an output symbol O_t, such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]], mod W,

where I_t is an input symbol, α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by a key, π

_N, π

_N-1, . . . , π

₂, π

₀ are N permutations defined by the key, and W represents the number of possibilities for each permutation defined by the key, and

where α

(t) is a step function, and α

_X (t), X={0, 1, 2, . . . , N-1, N}, increments the sequence of π

_X for each value that t equals an integer multiple of R, where R is a prime number.

34. A storage medium, comprising:

interface means, and

means for controlling a microprocessor, via the interface means, to produce an output symbol O_t, such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]], mod W,

where I_t is an input symbol, α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by a key, π

_N, π

_N-1, . . . , π

₂, π

₀ are N permutations defined by the key, and W represents the number of possibilities for each permutation defined by the key, and

where α

(t) is a step function, and α

_X (t), X={0, 1, 2, . . . , N-1, N}, increments the sequence of π

_X for each value that t equals an integer multiple of R, where R is a prime number.

35. A storage medium, comprising:

interface means, and

means for controlling a microprocessor, via the interface means, to produce an output symbol O_t such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [I_t +α

₀ (t)]] . . . ]], mod W,

where I_t is an input symbol, α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by a key, π

_N, π

_N-1, . . . π

₂, π

₀ are N permutations defined by the key, and W represents the number of possibilities for each permutation defined by the key, and

where α

(t) is a step function, and α

_X (t), X={0, 1, 2, . . , N-1, N}, increments the sequence of π

_X for each value of t except when t equals an integer multiple of R, where R is a prime number.

36. A storage medium, comprising:

interface means, and

means for controlling a microprocessor, via the interface means, to produce an output symbol O_t, such that O_t =α

_N (t)+π

_N [α

_N-1 (t)+π

_N-1 [α

_N-2 (t)+ . . . +π

₂ [α

₁ (t)+π

₁ [+α

₀ (t)]] . . . ]], mod W,

where I_t is an input symbol, α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by a key, π

_N, π

_N-1, . . . , π

₂, π

₀ are N permutations defined by the key, and W represents the number of possibilities for each permutation defined by the key, and

where α

(t) is a step function, and α

_X (t), X={0, 1, 2, . . . , N-1, N}, decrements the sequence of π

_X for each value of t except when t equals an integer multiple of R, where R is a prime number.

37. A storage medium, comprising:

interface means, and

means for controlling a microprocessor, via the interface means, to produce a generated symbol I'"'"'_t, such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

_N (t)]-α

_N-1 (t)]- . . . -α

₃ (t)]-α

₂ (t)]-α

₁ (t)]-α

₀ (t), mod W,

where O_t is a received symbol, α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by a key, π

₁^-1, π

₂^-1, π

₃^-1 . . . , π

_N-1^-1, π

_N^-1 are N inverse permutations defined by the key, and W represents the number of possibilities for each inverse permutation defined by the key; and

where α

(t) is a step function, and α

_X (t), X={0, 1, 2, . . . , N-1, N}, increments the sequence of π

_X for each value that t equals an integer multiple of R, where R is a prime number.

38. A storage medium, comprising:

interface means, and

means for controlling a microprocessor, via the interface means, to produce a generated symbol I'"'"'_t, such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

_N (t)]-α

_N-1 (t)]- . . . -α

₃ (t)]α

₂ (t)]-α

₁ (t)]-α

₀ (t), mod W,

where O_t is a received symbol, α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by a key, π

₁^-1, π

₂^-1, π

₃^-1 . . . , π

_N-1^-1, π

_N^-1 are N inverse permutations defined by the key, and W represents the number of possibilities for each inverse permutation defined by the key; and

where α

(t) is a step function, and α

_X (t), X={0, 1, 2, . . . , N-1, N}, decrements the sequence of π

_X for each value that t equals an integer multiple of R, where R is a prime number.

39. A storage medium, comprising:

interface means, and

means for controlling a microprocessor, via the interface means, to produce a generated symbol I'"'"'_t, such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

_N (t)]-α

_N-1 (t)]- . . . -α

₃ (t)]-α

₂ (t)]-α

₁ (t)]-α

₀ (t), mod W,

where O_t is a received symbol, α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by a key, π

₁^-1, π

₂^-1, π

₃^-1 . . . , π

_N-1^-1, π

_N^-1 are N inverse permutations defined by the key, and W represents the number of possibilities for each inverse permutation defined by the key; and

where α

(t) is a step function, and α

_X (t), X={0, 1, 2, . . . , N-1, N}, increments the sequence of π

_X for each value of t except when t equals an integer multiple of R, where R is a prime number.

40. A storage medium, comprising:

interface means, and

means for controlling a microprocessor, via the interface means, to produce a generated symbol I'"'"'_t such that I'"'"'_t =π

₁^-1 [π

₂^-1 [π

₃^-1 . . . [π

_N-1^-1 [π

_N^-1 [O_t -α

_N (t)]-α

_N-1 (t)]- . . . -α

₃ (t)]-α

₂ (t)]-α

₁ (t)]-α

₀ (t), mod W,

where O_t is a received symbol, α

_N, α

_N-1, . . . , α

₁, α

₀ are N+1 additive transformations defined by a key, π

₁^-1, π

₂^-1, π

₃^-1 . . . , π

_N-1^-1, π

_N^-1 are N inverse permutations defined by the key, and W represents the number of possibilities for each inverse permutation defined by the key; and

where α

(t) is a step function, and α

_X (t), X={0, 1, 2, . . . , N-1, N}, decrements the sequence of π

_X for each value of t except when t equals an integer multiple of R, where R is a prime number.