What About A Padless One-Time Pad?

What About A Padless One-Time Pad?

by

Huen Y.K.

CAHRC, P.O.Box 1003, Singapore 911101
http://web.singnet.com.sg/~huens/
email: huens@singnet.com.sg
email: huens@mbox3.singnet.com.sg

(A short communication - 1st released: 22/8/97.)

Abstract

With the development of sequence algebra, I have often wondered whether it could find use in crytography. This interest is natural since number theory plays a major part in crytography. Another attraction is that human languages are formed by sequences of words and surely these can be manipulated in sequence algebra. Conventional arithmetic operations such as modular arithmetic, hash functions, stream functions, and shift-operators have already been flogged to death by cryptographers [1,2]. The author suggests that sequence algebra could be used to develop new crytographic paradigms. Using sequence algebra, there are literally hundreds of encoding variations which could be tested to find one with enhanced security. A niche application to be described is called the padless one-time pad. It is rumoured that the "hotline" between the White House and the Kremlin was encrypted with a one-time pad. It is difficult to ensure the security of one-time pad in transportation. What if a one-time pad does not come with any pad? This can be done if we use sequence exponentiation for encoding and sequence factorisation for decoding.

1. Introduction

One-time pads get their name from the pads of paper used by spies. Each page of the pad has a different codebook. Both the secret message sender and receiver must possess identical one- time pad in order to decipher it. Theoretically, it should be very secure provided the problem of security in the transportation of the pad is assured [1]. This explains why it is only used in special applications where security can be implemented even though at great cost. This defeats the simplicity of the one-time pad.

This paper describes a compromise -- a one-time pad without any pad. In other words, the deciphering algorithm is publicly known, but you can only decipher the secret message within specified time if you have a powerful computer. Messages which are time-limit sensitive would find this method useful since there will be no one-time pad to be stolen in transportation or worst still sold for profit by your trusted employees. The only drawback is that one should be in possession of a computer many times more powerful than those possessed by eavesdroppers. This would rule out its use by PC owners in the Internet.

2. The Algorithm Explained

Here is a simple but naive example which demonstrates the encoding and decoding method. Equation (1) shows the plaintext to be encoded. Equation (2) shows the result by squaring the plaintext message. You can almost guess what the plaintext is from the ciphertext. In real cases, the words themselves are encrypted by conventional means which will hide the context in addition to scrambling the plaintext by higher exponential powers. Equation (3) shows the recovered plaintext by algebraic factorisation.

Example 1:

Plaintext(z):= Mary/z^1+ has/z^2+ a/z^3+ little/z^4+ lamb/z^5;

................................................Mary...has.....a......little....lamb
..........................Plaintext(z) := ---- + --- + ---- + ------ + ---- .........................(1).
...................................................z...........2.......3........4.........5
..............................................................z........z.......z.........z

Ciphertext(z):=sort(expand(Plaintext(z)^2));

................................2............................2.......................................................................2
........................Mary......has Mary....has.........a Mary.....has a.......Mary little........has little........a
Ciphertext(z) := ----- + 2 -------- + ---- + 2 ------ + 2 ----- + 2 ----------- + 2 ---------- + ----
.............................2..............3.............4............4.............5................5.....................6..............6
............................z..............z.............z.............z.............z................z.....................z...............z

.................................................................................2.............................................2
..........Mary lamb.....has lamb........a little........a lamb.......little..........little lamb.....lamb
.....+ 2 --------- + 2 -------- + 2 -------- + 2 ------ + ------- + 2 ----------- + ----- .....(2).
..................6...................7.................7................8............8.................9.............10
................z...................z..................z................z............z.................z..............z

Decryptext(z):=sort(factor(Ciphertext(z)));

....................................................4...................3......2..........................2
.................................................(z Mary + has z + a z + z little + lamb )
.........Decryptedplaintext(z) := ----------------------------------------------- ...................(3).
..................................................................................10
................................................................................z

One can't get far using factorising algorithms provided by algebraic packages before one encounters memory problems. This practically excludes PC owners from doing decryption of realistic problems on padless one-time pad. Without supercomputers, it would be rather difficult to crack such codes. The problem given above might be misleading as it looks too straightforward. This is because the author was encountering difficulties in presenting a more realistic problem using his PC even though it is a Pentium with 64 Mbytes of RAMs.

3. Summary

There are many possibilities in increasing the difficulties of decryption but the price is an increased length of the message and increased decoding time which may or may not be a blessing depending on your superiority in computing power which is hardware dependent. It is easy to photocopy a one-time pad but no one would contemplate duplicating a whole supercomputer. It is too bulky to be stolen. Unauthorised usage on site is the problem one would have to address but that is an adminstrative problem common to all secret sites. Having said that, padless one-time pads are more suitable for large establishments.

4 References

(Not all references are directly relevent. References in sequence algebra are all included as these are not generally available outside this URL site.)

1. Garfinkel S. (1995): Encryption for Everyone, PGP Pretty Good Privacy, O'Reilly & Associates, Inc. January Ed., 1995, pp 33 to 40.

2. Schneier Bruce : Applied Cryptography, Protocols, Algorithms, and Source Code in C, second edition,, John Wiley & Sons Inc., 1966, pp 15-17.

3. Huen Y.K.: A Simple Introduction To Sequence Algebra, URL site: http://web.singnet.com.sg/~huens/

4. Huen Y.K.: The Canonical Generating Function or CGF(z) - a Swiss-knife function. URL site: http://web.singnet.com.sg/~huens/ .

5. Huen Y.K.: Information Contents Of Number Theoretic Functions. URL site: http://web.singnet.com.sg/~huens/ .

6. Huen Y.K.: In Search Of Exotic Arithmetic Operators, URL site: http://web.singnet.com.sg /~huens/ .

7. Huen Y.K.: Visual Solutions Of Number Theoretic Functions in Multidimensional Sequence Space, URL site: http://web.singnet.com.sg /~huens/ .

8. Huen Y.K.: Final Value Theorems Applied To Number Sequences -- its strengths and weaknesses, URL site: http://web.singnet.com.sg /~huens/ .

9. Huen Y.K.: Unsolved Problems In Sequence Algebra, URL site: http://web.singnet.com.sg /~huens/ .

10. Huen Y.K.: Explicit Formulation For Modular Arithmetic In Sequence Algebra, URL site: http://web.singnet.com.sg /~huens/ .

11. Huen Y.K.: Cyclic Generating Functions In Sequence Algebra, URL site: http://web.singnet.com.sg /~huens/ .

12. Huen Y.K. : Methods Of Developing Sequence Algebraic Formulations For Comp(z) and Prime(z). URL site: http://web.singnet.com.sg /~huens/

. 13. Huen Y.K. : Information Contents Of Hypothetical DNA Sequences. URL site: http://web.singnet.com.sg/~huens/ .

14. Huen Y.K. : Composite Number Sequence Challenge 1/97. URL site: http://web.singnet.com.sg/~huens/ .

15. Huen Y.K. : Lemmata, Corollaries, And Theorems In Sequence Order Analysis. URL site: http://web.singnet.com.sg/~huens/ .

16. Huen Y.K. : Is it a number line or a number generator? URL site: http://web.singnet.com.sg/~huens/ .

17. Huen Y.K. : What is so special about GS1(109):0? URL site: http://web.singnet.com.sg/~huens/ .

18. Huen Y.K. : Some Interesting Contiguity Properties Of Odd(z)^2. URL site: http://web.singnet.com.sg/~huens/ .

19. Huen Y.K.: A Matrix Map for Prime and Non-prime Numbers, INT. J. Math. Educ. Sci. Technol., 1994, VOL. 25, NO.6, pp 913-920.

20. Huen Y.K.: Some Interesing Properties Of The Natural Number System, Int. J. Math. Educ. Sci. Technol., 1996, VOL.27, NO. 5, 685-691.

21. Huen Y.K.: Visual algebra and its applications, INT. J. Math. Educ. Sci. Technol.,1997, VOL.28 NO.3, 333-344.

22. Huen Y.K.: Twin primes revisited: INT. J. Math. Educ. Sci. Technol., 1997, VOL.??,NO.?, ???-???. (In the press as proof paper mes 100488).

23. Huen Y.K.: Is Pie Periodic? : INT. J. Math. Educ. Sci. Technol., 1997, VOL??,NO.?,???- ???,(In the press as proof paper mes100495).

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