Using Repeating Decimals As An Alternative To Prime Numbers In Encryption
aa r X i v : . [ c s . CR ] J u l Abstract
This article is meant to provide an additional point of view, applying known knowledge, tosupply keys that have a series of non-repeating digits, in a manner that is not usually thought of.Traditionally, prime numbers are used in encryption as keys that have non-repeating sequences.Non-repetition of digits in a key is very sought after in encryption. Uniqueness in a digit sequencedefeats decryption by method. In searching for methods of non-decryptable encryption as well asways to provide unique sequences, other than using prime numbers, the idea of using repeatingdecimals came to me. Applied correctly, a repeating decimal series of sufficient length will standin as well for a prime number. This is so, because only numbers prime to each other will producerepeating decimals and; within the repeating sequence there is uniqueness of digit sequence. sing Repeating Decimals As An Alternative To Prime Numbers InEncryption Givon ZirkindJune 22, 2010
Prime numbers are often used and sought after, in encryption. One of the several reasons forthis is, that prime numbers can be used as keys that have a sequence of non-repeating digits. [2](This is a fundamental concept of encryption and a pre-requisite for understanding the applicationof the number theory disccused in this paper. This fundamental concept of encryption will notbe explained in this paper. The author refers the reader to [2] for an in depth explanation of thisconcept.) One of the many other ways of producing a sequence of non-repeating digits, besidesusing prime numbers, is to use the sequences of repeating decimals. Like prime numbers, repeatingdecimals, within their sequence, do not have repeating (sub)sequences.Ex. = 0.142857142857 within the repeating sequence 142857, there is uniqueness. Similar toa prime number. [3]Furthermore, for any given message, of any size, we have a formula, Fermats Little Theorem,to select a denominator that will produce a repeating sequence with a specific number of digits.[10 ( p − ≡ =0.444 While neither 4 nor 9 is prime, they are prime to each other. Hence, they cannever fully divide each other. [1] [5]So, although we wish we could generate prime numbers and can not; we can generate repeatingdecimals and choose their size. All we have to do, is choose an appropriate size and select the ap-propriate factors. This makes the use of repeating sequences an attractive option for key generation.2 The Infinity of Repeating Decimals
The number of prime numbers is infinite. [6] Every prime number will generate several repeat-ing decimals. So, the number of repeating decimals generated by prime numbers is infinite.
In fact, as there are many repeating decimals generated by each prime number, there is a greaterthan a one-to-one correspondence between primes and repeating decimals generated by prime num-bers. Hence, the number of repeating decimals is greater than the number of prime numbers.
There are more co-primes than primes. Because there are more odd numbers than prime num-bers. For while every prime number is odd, not every odd number is prime. [5] And, every pairof odd numbers that do not share another odd number as a factor are co-prime. This includes allprime numbers (which are co-prime with each other) plus all odd multiples of prime numbers thatdo not share the same multipliers and any pairs of combination of primes and multipliers that donot share the same primes and multipliers. Therefore, the number of co-primes is greater than thenumber of primes.Also, even and odd numbers are co-prime; while prime numbers can be only odd numbers.Therefore, there are more co-prime numbers than prime numbers. And, since prime numbers areinfinite and; there are more co-prime numbers than prime numbers; therefore co-prime numbersare also infiinite.In addition, since the number of co-primes are greater than the number of primes and; sinceevery pair of co-primes produces a repeating decimal, therefore, there are more repeating decimalsthan prime numbers (and the number of repeating decimals is infinite as deduced above).
It is clear how many pairs of odd numbers share common factors. One way of demonstratingthis is to analyze the series of odd numbers and their corresponding ordinal numbers.3 .2 Analysis of the Series of Odd Numbers
Series of Odd NumbersPosition 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19Odd Number 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39Every odd number will share a common factor with every other odd number in the series ofodd numbers, whose ordinal number is a multiple of itself plus one.Ex. The number 3 is the first odd number. It will share a factor with the 4th odd number as4 = (1 ∗
3) + 1. The 4th odd number is 9. The 7th odd number, 7 = (2 ∗
3) + 1, is 15. This too hasa common factor with 3.Ex. The number 39 is the 19 th ordinal odd number. 19 = (6 th *3) + 1 and; (3 rd *6) +1. Thenumber 39 has the 1st and 6th ordinal odd numbers as factors or: 39 = 13 * 3. For the 1 st odd number, 3, within every period of 3 numbers, will have 2 co-prime numbers. Threewill be co-prime with of the series of odd numbers. The 2 nd odd number, 5, within every periodof 5 numbers, will have 4 co-prime numbers. Five will be co-prime with of the series of oddnumbers. And, so forth. Odd numbers that are multiples of a given odd number, will reduce to asmaller fraction. Thus reducing the number of co-primes for a given odd number. The end resultwill be that the number of co-primes for any odd number ’n’ is equal to [ ( n − n * O]. Where ’O’equals the number of odd numbers. As the number ’n’ of odd numbers increases, the number of pairs of numbers co-prime (n cp )with an odd number approaches the number of odd numbers.Eq. limn → ⊲⊳ ( n − n → nn → limn → ⊲⊳ [ ( n − n ] = 1= O.The sum of pairs of odd numbers that are co-primes approaches the number of odd numbers,an odd number of times. Or; the number of odd numbers (O) squared (O ).Eq. limn cp → ⊲⊳ = O .While the number of primes (n p ), is less than the number odd numbers. [n p < O].This also proves that there are more co-primes than primes.Eq. [n p < O]Eq. [n cp = O ]Eq. [O < O ]Eq. [n p < n cp ] 4his calculation does not include the number of co-primes that the odd numbers will have witheven numbers.Ex. The number 33 besides being co-prime with all odd numbers not multiples of 3 and 11; willbe co-prime with every even number. In arithmetic terms:Every odd number is equal to (2n+1).f = The 1 st position a particular odd number ’o’ occurs within the series of odd numbers.o = an odd numberm = a multiplier of an odd numberThen, for every multiple of an odd number ’o’ that will be odd, n = f + (o*m).Not every multiple of an odd number is odd.Ex. 15, 30, 45, 60, 75 are all multiples of the odd number 15.But, only 15, 45 and 75 are odd.Because any odd number times an even number is even.Only an odd number times an odd number is odd.Using the formula to calculate the odd multiples of 15:f = 7o = 15For m= {
1, 2, 3, 4 } Odd First Position Multiplier (m) n=f+(o*m) 2n+1 ON = ReducedF raction NO
Number (Occurence)15 7 1 22=7+(15*1) 45 =
315 7 2 37=7+(15*2) 75 =
515 7 3 52=7+(15*3) 105 =
715 7 4 67=7+(15*4) 135 = As demonstrated in the table above, odd numbers that are multiples of other odd numbers, re-duce to the series of inverse odd numbers. This is true for every odd number. Thus many co-primeswill produce the same repeating decimal. This cyclic pattern of common factors to odd numbersdoes reduce the number of co-primes and possible repeating decimals. Still, an infinity of co-primeswill produce an infinity of repeating decimals.
It is obvious that for any number of integers where there are at least 2 primes greater than halfthe number of integers in the series, there are more co-primes than primes and; more repeatingdecimals than primes.Ex. From 1 to 100, there is 51 and 67. Each will produce a repeating decimal for numerator.That is a total of 118 different sequences. And, there are more primes within the series from whichto generate repeating sequences. Yet, obviously, the number of primes between 1 and 100 must beless than 100.
In addition, when searching for co-primes, one must also reckon the repeating sequences fromnumbers that are prime to each other, but not necessarily prime. Again, the total number of deci-mal repeating sequences will be larger than the number of primes within the given series of numbers.Ex. – Primes, Co-Primes & Repeating Decimals Less Than 101. The number of primes less than 10 is just 4 {
1, 3, 5, 7 } .2. Ignore any number that is a power of 2 or 5 in the denominator {
2, 4, 5, 8 } .3. The number of decimal repeating sequences using the digits from 1 – 10, is 14, which, isgreater than 4, the number of primes from 1 – 10.4. The number of decimal repeating sequences from the prime numbers { } that are greaterthan half of the number of elements in the series is 6 { , , , , , } .5. Note, the fractions { , , } have even numbers as numerators and yet, produce repeatingdecimals. Because these even numbers are co-prime with the denomiators.6. As demonstrated by the series of fractions produced by the prime number 7 { , , , , , } ; If the prime number is greater than half the number of numbers in the range:6a) Then, the number of numerators is also greater than half the number of numbers in therange.(b) Then, the number of numerators are greater than half the number of prime numbers inthe range.7. The number of repeating decimals from numbers prime to each other with a non-prime de-nomiator is 6 { , , , , , } . This is also greater than the number of primes in theseries.8. The total number of repeating sequences is 14 { , , , , , , , , , , , , , } Thismore than 3 times the number of primes in the series.9. The repetitive fractions and are redundant to and which have already been counted.These fractions are not counted twice. In addition, the entire discussion above, is only of a pair of single factor numbers that areco-prime. One can also use multiplicands of odd numbers that are co-prime. Thus, another infinityof co-primes can be generated.Ex.[3/(17 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Not all the repeating decimals will be of use for encryption. In fact, many will be useless forencryption. However, it becomes apparent that there are enough repeating decimal sequences toprovide quality encryption. It is also apparent that there are more repeating decimals that willprovide non-repetitive digit sequences for keys, than there are prime numbers that will providenon-repetitive digit sequences for keys.Most messages are 500 characters or less. (This is a statistic from standard radio-telegraphyand studies of typing.) To generate a unique key, for encryption, with enough digits, that do notrepeat, for a message of 500 characters, only requires a prime number with as many digits as themessage. A repeating decimal sequence of 500 characters or less, can be easily generated. It toowill not have repeating sequences within it. It will contain a unique series of non-repeating digits.It will be just like a prime number in that regard.7lso, using the unique sequences of repeating decimals adds another possibility to check for,when deciphering. (Presumably, decryption methods worth their salt, automatically check forprime numbers.) For individual and low grade traffic, using a repeating sequence as a key, wouldbe an acceptable, secure method of encryption.
To form an equation for the total number of co-primes within a given range of numbers. (This isa current work in progress.)
The number of odd numbers and prime numbers have the transfinite number of ℵ . Does thenumber of co-primes also have the transfinite number of ℵ ? More importantly, is the numberof repeating decimals from co-primes equal to ℵ ? Or, is the transfinite number of the repeatingdecimals from co-primes greater than ℵ since there is a one-to-many relationship of repeatingdecimals to a pair or group of co-primes?6 Bibliography