WO2001008350A1

WO2001008350A1 - T-sequence apparatus and method for general deterministic polynomial-time primality testing and composite factoring

Info

Publication number: WO2001008350A1
Application number: PCT/US2000/020199
Authority: WO
Inventors: Shaul O. Backal; Tie Teck Sing
Original assignee: Backal Shaul O; Tie Teck Sing
Priority date: 1999-07-26
Filing date: 2000-07-25
Publication date: 2001-02-01
Also published as: US9542158B2; US20040057580A1; US20020099746A1; AU6372400A

Abstract

Using a mathematical technique called the T-sequence, a primality testing method is disclosed. The approach is disclosed as performing factoring for numerous special cases, a method that is disclosed as extending to the general case. The same T-sequence is disclosed as constructing a prime number formula and a random number generator.

Description

T-SEQUENCE APPARATUS AND METHOD FOR GENERAL DETERMINISTIC POLYNOMIAL-TIME PRIMALITY TESTING AND COMPOSITE

FACTORING

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to prime and composite number computing and applications of the same, e.g., in the area of data security.

2. State of the Art

Prime numbers (2, 3, 5, 7, 11, 13, ..., those positive integers divisible only by themselves or 1) are the most fundamental building blocks of math, and with the invention of the public key ciphers (RSA, El Gamal and the like), they now form the backbone of computer security. Basically there are two problems related to the use of prime numbers in these fields, namely primality testing and factoring. The primal- ity testing problem is about testing and determining whether a given arbitrary positive integer is a prime number or a composite (non-prime) number. For a composite number, the factoring problem requires determining the composite number's prime factors. Practicality demands that these two problems have to be solved in polynomial time (computations being proportional to the number of digits and therefore fast), not exponential time (computations being proportional to the size of the numbers themselves and therefore too slow).

Traditionally, to decide whether a small integer is prime or composite, one can try to factor it with the smaller primes, but this trial division is too tedious for numbers greater than, say, 40 digits. Previously, experts have only been able to test for general primality up to about 2000 digits with certainty in a week of standard PC computational time. Several other faster methods have been devised to test larger integers, but they too fall short of expectations.

Presently over the Internet, record-size prime numbers over 10,000 or 100,000 digits are frequently found and published by researchers, but they are confined to special forms only (e.g., the most famous being the Lucas-Lehmer test for Mersenne numbers of the form 2^M - 1). If given an arbitrary number, however, these researchers cannot test it in polynomial time. The stringent demands of several important ciphers require testing and generating large prime number of arbitrary forms and sizes.

There are four conditions in solving these problems:

1. Polynomial-time algorithm: the algorithm's speed needs to be proportional to a small power of the number of digits of that integer, e.g. d , instead of sub-exponential or exponential time, e.g., 2 .

2. 100% generality, i.e., the primality or factors of any arbitrary number can be determined.

3. Provability, i.e., it can be shown to work in all cases mathematically and no counterexamples can be found.

4. Deterministic in nature, i.e., the algorithm can determine the primality of a number with certainty and not with statistical probability.

Present techniques are unable to satisfy all four of these conditions simultaneously.

SUMMARY OF THE INVENTION Using a new mathematical technique called the T-sequence, the inventor has discovered a powerful primality testing method that meets all four conditions above. A similar approach can be applied to perform fast factoring for numerous special cases, a method that can, in all liklihood, be extended to the general case, making possible a general and fast factoring algorithm. (Researchers heretofore have been able to factor only in sub-exponential time, never in polynomial time.) The same T- sequence can be used to construct a prime number formula (long sought after but never achieved) and a good random number generator. The former can be used to generate infinitely many prime numbers of any size efficiently, and the latter can generate non-periodic and absolutely chaotic random numbers. These numbers are widely used in all areas of industrial and scientific simulations. In general, the T- sequence can be used to handle efficiently the fundamental problems concerning prime numbers (which include primality testing, factoring, prime number formula, infinite-pattern prime problem, etc.).

Whereas previously experts have only been able to test for general primality up to about 2000 digits with certainty in a week of standard PC computational time, now with this new approach it takes only eight seconds, by comparison. On a fast computer, numbers up to a million or more digits can also be tested for primality. All other techniques become impracticable beyond 2000 or so digits for general primality testing. This new approach enables mathematicians and computer scientists to test as well as generate prime numbers of any size or form to be used in mathematical research and computer cryptography.

BRIEF DESCRIPTION OF THE DRAWING The present invention may be further understood from the following description in conjunction with the appended drawing. In the drawing:

Figure 1 is a block diagram of a prime number computing system; and Figure 2 is a flowchart illustrating a primality testing algorithm.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS T-Sequences: Definition.

Let n be a positive integer and £ ≥3 be the order. Then the general T- sequences are defined as follows:

£ £ I £ £

T₀ = 2, T_j = £ and T_{n + 1} = £ ^■ T_n - T_n_ , , where the subscript denotes the nth term while the superscript denotes the order £. Therefore the zeroth term is always 2 and the first term is always £; i.e., £ = 3 is the first T-sequence, the successive terms of which are given by

T₀ ³ = 2,T?

3 • T_n ³-T³_, .

4 .

T_n is the second T-sequence with the following terms:

T₀ ⁴ = 2,T⁴ = 4,T⁴= 14,T⁴ = 52,...,T⁴ ₊₁ =4-T_n ⁴-T⁴_₁ , etc.

There is a general and fundamental identify relating all T terms, as expressed by:

£ £ £

T_n n, + , _ n₂ = T„ n, • T_n n, - „ n₂ where n, \ > n- I_> and n = n, \ + n- I_>.

From this can be derived the following convenient identities:

(l)T_2n+ι =T_n -Tn+i -Ti = T_n -T_n+ι -£ (odd nth terms)

£ £ £ £ £ 2

(2) T 2n = T_n -T n-To = (T_n ) -2 (even nth terms)

The T terms can grow exponentially large, but with the above identities as well as modulo arithmetic and a type of binary decomposition method described below, testing a given integer for primality is straightforward.

A numerical example serves to illustrate this approach. E.g., for n = 31 and £ = 3, binary decomposition is first performed (other forms of decomposition are feasible but are less practical):

^•π, = _τ3 Tl6 Tis -3 _τ3 Tl6 = (T| )² - 2

3 3

T³5 = Ts ^■ T7 - 3

T³s = (T₄ ³ )²-2

T³7 = T4 ^• T₃ - 3

TΪ = (T )²-2 τl = T2 ^• Ti - 3

For decomposition of odd terms D the quantity is computed. If the result is an odd number as in 31-1 = 15 , 1 is added to 15 to give 16 so that 31 16 + 15. If the result is an even number such as 37-1 = 18 , again 1 is added to 18

2 ^& to give 19 so that 37 = 18 + 19. The successive terms can then be computed by using

3 £ the above-mentioned identities. For odd nth terms such as T₃1 , the quantity Ti , or £, is always subtracted, which is 3 in this example. For even nth terms such as Ti6 ,

£ the quantity To , or 2, is always subtracted. The fundamental identify allows all these decompositions to be performed. Modulo arithmetic with respect to n and

£ recursive substitutions are then carried out using the fact that, always, To = 2 and

£ 3 3

Ti = £ ; that is, To = 2 , Ti = 3 in this example.

Computations are then started from the smallest term, that is

, 33 22 ~> 3

Ti - ( (TTii )) --22 == 33^"" --22 == 77,Tτl3 - T³2 T -3 = 7-3-3 = 18, and so on, with the residues always modulo 31 :

T31 = Tie -T³5 -3 = 3-2-3 = 3

Therefore t can be determned that n t is example the 31st term of T" (mod 31) gives a residue of 3. Of course the residue of any term of T (mod n) can be readily computed whenever needed.

There are numerous intriguing properties of T-sequences, one of which is expressed as

Tn = equivalently

Tn = (-l )^k n(n - k - l)!( )ⁿ-^2k _{k = 0} k!(n - 2k)!

From this expression one can prove that all primes p will have to satisfy the

£ £ 2 relations T = £ (mod p) and T_p- ι = 2 or £ - 2 (mod p) , as in the numerical example above for the prime p = 31 :

T31 = 3 (mod 31) and T30 = 2 (mod 30) .

By using these T-sequences in connection with the primes p, another important and useful property in primality testing and factoring can be derived, the so- called periods k(p) consisting of two types, p + 1 and p -1. The former is called the +£ type and the latter the -t type. What is meant by this terminology can be illustrated by the following numerical examples:

Take £ = 3 and p = 7. Compute every term of T³ successively modulo 7; that is, every T³ term is divided by 7 to give the respective residues, until the residues repeat themselves. Thus using R as the residue and £ always equal to 3, one obtains for the modulo of prime p = 7: R₀ = 2, R, = 3, R₂ = 0, R₃ = 4, R₄ = 5, R₅ = 4, R₆ = 0, R₇ = 3, Rg = 2, .... The next residue with 2 appears at the eighth term Rg = 2, thus the period k(7) = 8. Note that this period divides exactly into p + 1 , that is, k(p) |p + 1 => k(7) = 8 . Thus the prime 7 is said to be of the +£ type in T sequence.

Again take £ = 4 and p = 11. The residues of each T⁴ term, modulo 11 , are: R₀ = 2, R, = 4, R₂ = 3, R₃ = 8, R₄ = 7, R₅ = 9, R₆ = 7, R₇ = 8, Rg = 3, R₉ = 4, R₁₀ = 2, .... The next residue of 2 appears at the 10th term R₁₀ = 2, hence the period k(l 1) = 10. This period of 10 divides exactly into p - 1, that is, k(p) |(p — 1 ) ==> 101 11 — 1 . Thus the prime 1 1 is said to be of -£ type in T⁴ sequence.

There are no other possible patterns for prime modulo. (The £ type for composites will be shown in the following section describing the primality testing algorithm.) In essence, this unique characteristic of the T-sequences enables the primality of any positive integer to be determined, since only those numbers that are genuine primes can satisfy for appropriate £ values both

+ £ 2 + £ - £ - £

Tp- i ≡ ^ - 2 , T_p ≡ £ and T_p- ι ≡ 2 , T_p ≡ £ (all mod p) . Furthermore, this characteristic can also be used to do general polynomial time factoring of composites.

Computing Using T-Sequences

Referring now to Figure 1, a block diagram is shown of a computing system, e.g., a prime number computing system, in which T-sequences are used. The computing system includes one or more processors, random-access memory, read-only (non-volative) memory, and an I/O subsystem. The computing system is intended to be representative of all classes of computing systems, large and small, local or distributed. Within memory is stored a routine for generating T-sequence terms. The results of this routine are used by one or more other routines, e.g., a routine for primality testing, a routine for factoring, a prime number generator, a random number generator, etc. These routines find wide application, especially in data security, e.g., securely encrypting data or, by the opposite token, breaking a given encryption. The operation of various ones of these routines will now be described.

Primality Testing

Given any positive integer n, the T sequence may be used to perform primality testing (any other T^f- sequence will do but T is convenient for use here). Using binary decomposition and the above-mentioned methods, the residues are computed

3 3 3 3

Rn- i = Tn - i (mod n) and R_n = T_n (mod n) . For n to be an eligible candidate for prime, the residues have to be Rn - i = 2 or ² - 2 and Rn = 3 . Any n which does not give such residues can immediately be declared composite. As will be explained below, it can be seen readily that any n with the last digit 1 or 9 will be of the -£ type in T³' whereas any n with the last digit 3 or 7 will be ofthe +^ type in T³.

A fast and general method to determine the £ type of n in T^ (to be used in proving and determining the genuine primality of n) is as follows. Given the values of any n and £, divide n by the determinant 2£² - 8 and obtain the small residue r, that

2 is n ≡ r (mod 2£ - 8) . It can be shown that the £ type of n is the same as that of r. Since r is so small, direct computation of its residues in T^ will readily give the £

£ 2 £ type, knowing that by definition the £ type is + when R_r - 1 = £ - 1 and R_r = £

£ £

(both mod r), and is - when R_r- l = 2 and R_r = £ (both mod r). Note a few facts about the relationships between r and £:

1. The £ type is always - whenever r = £.

2. The small residue r must be coprime to the determinant, that is

(r, 2£ - 8) = 1. This means that whenever r is not coprime to 2£ - 8, that particular £ value is not used.

3. Besides r being coprime to the determinant, r needs to be greater than the value of £. Otherwise that particular £ value is not used.

4. The period k(r) must be greater than 2. When the period is 1 or 2, that £ value is not to be used.

5. Applying the above identities and binary decomposition methods to r will

£ I ( 2 I give Rr- i and R_r . Whenever R_r - 1 ≠ 2 or £ - 2 and/or R_r ≠ £ , that particular £ value will not be used. When

Rr- i = 2 or £~ - 2 and R_r = £, that particular £ value will be used.

6. The +£ type and the -£ type occur in equal proportion among all n and T . It can be shown that one £ with +£ type and another £ with -£ type can readily be found for any n.

For example, when n = 31, £ = 3 observe that r =1 since 31 =1 (mod • 3

2 • 3^" - 8 = 10 ). It is then known from the facts above that 31 is of -£ type in T . On the other hand, when n = 37, /=3 observe that r = 7 since 37 = 7 (mod 2 3 3

2 - 3 - 8 = 10 ). The quantities R7 - 1 = 7 and R7 = 3 (both mod 7) are then computed, from which it appears that 7 is of +£ type in T . Hence 37 is also of +£ type in T .

To take another example, when n = 31 , £ = 4 observe that r = 7 since 31 = 7

(mod 2 - 4 - 8 = 24 ). Direct computations like those mentioned above give

4 4 4

R₇_ ι = 7 and R7 = 4 (both mod 7). This shows that 7 is of +£ type in T and thus 31 must also be of +£ type in T⁴. On the other hand, when n = 37, £ = 4 observe that r = 13 since 37 = 13 (mod). Similar direct computations give

4 4 4

R 13 - 1 = 2 and R 13 = 4 (both mod 13). This shows that 13 is of -£ type in T and thus 37 must also be of -£ type in T . It is seen then that 31 and 37 are of opposite £ type in T and T .

Note that these small r residue computations can be skipped and the n residues computed directly for primality testing and ^-type decisions whenever r is indeterminate. The whole algorithm will still be in polynomial time owing to binary decomposition, which ensures that it is in polynomial time. The complexity is of the order of (log₂n) .

Referring now to Figure 2, a fast primality testing routine consists of the following three steps:

STEP A: For any given positive integer n, first use £ = 3. From the above, determine the £ type of n in T³, -£ type for last digit 1 or 9, +£ type for last digit 3 or 7. Then compute the two residues

3 3 3 3

Rn- i = Tn - i (mod n) and R_n = T_n (mod n) .

3 2 3

If either R_n - 1 ≠ 2 or 7 (= - 2) and/or R _n ≠ 3 , then n can be declared to be composite and the routine stops here.

Note that all composites which are not genuine primes or pseudoprimes or proper cofactors of T will be detected and sieved away in this STEP A.

3 2 3

IfRn - i = 2 or 7 (= £ - 2) (mod n) and R_n = 3 (mod n) then proceed to STEP B below. STEP B: This step performs a "greatest common factor sieving" to sieve away certain pseudoprimes. For example, take a composite n = 1729 = 7 x 13 x 19 and £ = 4. The number 1729 is a pseudoprime of T⁴ since

Ti729 = Ts65 - T864 - T⁴ = 914 - 821 - 4 = 4 (mod 1729) Ts65 = T433 - T432 - Ti = 641 • 1458 - 4 = 914 (mod 1729)

4 4

Take the odd term right below T 1729 , that is T 865 • Since the residue is

₄ 914 (mod 1729), subtract from this residue To giving 912. Using the Euclidean algorithm for the greatest common factor (gcd), compute gcd (912, 1729) = 19. This shows that 1729 is composite since 19 is one of its factors. (Computing gcd by the

Euclidean algorithm is useful in factoring.) In other words, for n to be a candidate

3 3 for prime, the odd term residue Rd right under R_n when 2 is subtracted must at

1 least be copπme to n: gcd(Rd - 2, n) = 1 .

STEP B still misses some pseudoprimes or cofactor composites but when followed by STEP C, all possible exceptions in the form of proper cofactors or pseudoprimes will be sieved away, leaving only the genuine primes.

STEP C: Find an £ which is of opposite £ type to that in STEP A in T³. If in STEP A the £ type of n in T³ is -, then in this STEP C, find an £ for which the £ type of n is + in T and vice versa. This can be determined readily through the above- mentioned computations of small residue r or direct computations of

£ 2 £

Tn - i ≡ 2 or £ - 2 (mod n) and T_n ≡ ^ (mod n).

If in STEP A T_n - 1 ≡ 2 (mod n) and T_n ≡ 3 (mod n), that is, -£ type, then if for another £ in which the £ type of n in T is opposite to that in T it

£ 2 £ holds that T_n - 1 ≡ £ - 2 (mod n) and T_n ≡ £ (mod n), that is, +£ type, it follows that n must be a genuine prime. If the residues are not as just stated, that n is declared to be composite. It is assured that, when n satisfies these conditions, n must be a genuine prime, because for any composite number, n = p , ^■ p₂ say, it is impossible to satisfy all + and - divisibility conditions: P_j-ll n-1, p₂ — 11 n- 1, P_j + lj n+ 1, ρ₂+l| n+1 .

Only a genuine prime p can always satisfy these conditions when n = p. This completes STEP C.

A variation of the foregoing algorithm uses the Jacobi to avoid blind trials seeking for opposite £ types. In particular, taking JACOBIN - 4, n) gives the £ type. One strategy is to calculate the £ types beginning with £ = 3 until the lowest values of £ having opposite types have been found.

Primality Testing— Summary. Following the above method of computation ensures that this primality testing algorithm is 100% general, deterministic, provable and polynomial-time. It runs as follows:

The integer n is a genuine prime whenever n satisfies the conditions in these three steps:

STEPA:T_π-ι ≡ 2 or 7 (modn) and T_n ≡3 (modn)

STEP B: gcd(Rd - 2, n) = 1

STEPC:Tn-ι =2 or ^²-2 (modn) and Tn ≡ £ (mod n) where the £ type of n in T^ is opposite to that in T³ as in STEP A. Failing to satisfy any one or more of these conditions will render n to be composite.

As may be seen from Table 1, the time and memory requirements required to test the primality of integers is very small compared to existing methods, and remains comparatively quite small even when testing primality of integers of unprecedented size.

Table 1

Polynomial-Time Factoring Routine

A promising and viable factoring method is also based on the T-sequences. This method is unlike any previous method.

The T-sequences allow all forms of composites to be factored, without exception, in polynomial time, simply because binary decomposition modulo C is fundamentally polynomial time. So far, mathematicians have only found exponential or sub-exponential time factoring algorithms for composites less than 200 digits, in general, and no polynomial-time factoring exists for even special forms of composites like the Mersenne numbers 2 - 1, etc. A simple extension of the T sequences, however, immediately provides just such a polynomial-time factoring algorithm (PTFA) for numerous special form composites with infinite membership.

The gist of this PTFA lies in the natural mathematical interrelationships between the composite C = PιP₂, periods of its prime factors k(p|) and k(p₂), residue r and order £.

The periods of the prime factors with respect to £ can only take on the patterns pi - 1 p] + 1 and p₂ - 1_; p₂ + 1- Note that one can always flip the £ type to change p - 1 to p + 1 and vice versa by trying several pairs of £ values.

The first important thing to take advantage of in PTFA is that whenever the period residue r_p lies close to p, it can readily be factored. One numerical example will illustrate this:

Take C = 91 (= 7 x 13). The possible periods k(p_{1 2}) of 7 and 13 are, for 7, 7 - 1 = 6, 7 + 1 = 8, and for 13, 13 - 1 = 12, 13 + 1 = 14. When C = 91 is divided respectively by each of these four k(p_{1 2}) the following period residue r_p are obtained: 1, 3, 7, 7, obtained from 91 = 1 (mod 6), 91 = 3 (mod 8), 91 = 7 (mod 12), 91 = 7 (mod 14). Note how small the period residue r_p for the prime factor 7 with the -£ type is, namely only 1. This implies that the factor 7 can be sieved out by taking the greatest common factor this way: gcd(R9i - I - 2, 91 ) = 7 . When £ = 6, 1 takes on a -£ type. So T91 mod 91 is first computed, which gives 76 as residue. Now - ⁶ = , and 1 is the r_p for 7 - 1 in T91 . Thus one is able to factor by taking gcd(76

- 6, 91) = 7. Likewise T90 ≡ 72 (mod 91), therefore one can again factor by taking gcd(72 - 2, 91) = 7 as shown above.

Whenever the periods p + 1 or p - 1 [match?] the composite C in either the above straightforward way or a simple function like the cubic polynomial below,

±£ factoring can always be performed by taking gcd(Rf(c) ± 2 , C ) = one factor; here, -2 is used when the periods p + 1 or p - 1 divides exactly into f(C) and +2 is used whenever f(C) divided by p + 1 or p - 1 gives a residue of

, as is quite

£ £ often the case. The expression R_m stands for T_m (mod C), where m can be any arbitrary term or a function of the composite f(C) to be computed.

Again, to illustrate the this point, when +2 is added to 76 (the residue of T91 mod 91), factorization can be performed by taking gcd(R₉ι +2 , 91 ) = gcd(76 + 2, 91 ) = 13 since 13 + 1 191 - — , i.e. l4|91 - 7

There are numerous sets of composites that can be factored readily because their factors' periods bear such a simple relationship to C. For example, composites of the form C = p[l + (p + l)(p + 2)] can always be factored readily in this way: gcd(Rc + ι - 2, C) = p . For example, take p = 11, C = 11(1 + 12 x 13) = 11 x 157 = 1727. By trials, quickly select a particular £ for which 11 is of the +£ type. When £ = 5, 11 is indeed of +£ type. Thus one can compute (mod 1727) = R 1728 = 167 and can factor in this way: gcd(167 - 2, 1727) = 11.

For C of the form p[l + (p - l)(p - 2)] there results gcd(Rc- 1 - 2, C) = p , e.g., C= 23(1 + 22 x 21) = 23 x 43 = 10649, enabling the following factorization: gcd(Rιo649 - 1 - 2, 10649) = 23 . Obviously, one can easily construct or find infinitely many such composites to factor. In general for C = p[l + m(p + 1)] where m is any prime or composite, C can always be factored as fol-

+£ lows: gcd(R c+ 1 - 2, C) = p .

For C= p[l + m(p - 1)], simply take gcd(Rc- 1 - 2, C) = p .

Furthermore, often the period of one prime factor of a composite happens to divide into the other prime factor or factors with a small enough residue, e.g.,

C = 147149 = 37 x 41 x 94. In this example, factoring begins by finding by trial that when C has subtracted from it a small number 2, i.e., 147149 - 2, it gives 147147 =

41 + 1 97 + 1

— — - (mod 41 + 1) and also 147147 = — — - (mod 97 + 1). Factorization then proceeds by taking gcd(R_C-2 + 2 , C) = gcd(R 7i47 + 2 , 147149) = 41 x 97. Again it is obvious that there are infinitely many such composites. Quite often too, when C is multipled by a small integer, the period of a certain factor can divide exactly into this product of C by a small integer, allowing for ready factorization, e.g., C = 41 x 67 = 2747. Multiplying C by the small integer 3 gives 2747 x 3 = 82441. Originally, 2747 = 27 (mod 41 - 1), and 27 is too large a residue period to readily factor. However, 8241 = 1 (mod 41 - 1), and 1 is definitely small enough,

- c - £ leading to gcd( R3C- l -2 , C) = gcd(Rs240 - 2 , 2747) = 41. This constitutes another infinite set of composites that can be factored in polynomial time through PTFA by a few trials.

A powerful and very general PTFA method results from taking the cubic polynomial function of C to factor. It works as follows: f(C) = aC³ ± bC² ± cC¹ ± d where 0 < a,b,c,d < 4 . The method tests all the possible combinations; actually, there are basically just 497 combinations that need to

+ £ be computed for their respective T f(C) (mod c), because the foregoing expression

2 can be rewritten as f(C) = C(aC ± bC ± c) ± d , and since c and d will just be integers taken consecutively, the computation lends itself to use of the identity in T-

£ £ £ sequences: T_n + ι = • T _n - T_n - 1 . Since 0 <a, b < 4, and since when a = 0, also b

= 0 and c = 0 in some cases, the results is only

5 x 2 x 2 x 2 x 5 - 3 = 497 combinations basically. Factorization is performed by tak-

±£ ing gcd( Rfc ± 2, C) = one factor, where f(C) stays positive. Two arbitrary examples will illustrate. Take C = 641 x 3469 = 2223629. Note

mod 640 mod 642 mod 3468 mod 3470

C¹ 269 383 641 2829 mod 640 mod 642 mod 3468 mod 3470

C² 41 313 1657 1421

C³ 149 467 929 1749

Taking a = 1 , b = +3, c = -1 and d = -3, 641 can be factored out by selecting one £ for which 641 is of -£ type such as £ = 3: since 22233629 = 269 (mod) there results 149 + 3 x 41 - 269 - 3 = (269)³ + 3(269)² -269 - 3 = (2223629)³ + 3(2223629)² - 2223629 - 3 = f(C). Thus 641 can be factored out as follows: gcd (R³(C)- 2, C) = 641.

As another arbitrary example, take C = 4567 x 10837 = 49492579. Note the fixed relationship between the period residues of each prime factors, particularly when they are the RSA form of two prime composites, e.g., when C = p_|(mod p₂ + 1 ), where p₂ is the larger prime and p_j is the smaller prime factor. This is only one of the relationships that have been found. Others abound, such as the 641 residue under mod 3468 above and also the 4567 residue under mod 10836 here.

mod 4566 mod 4568 mod 10836 mod 10838

49492579¹ 1705 2867 4567 6271

49492579² 3049 1857 9025 5177

49492579³ 2437 2299 7867 5157

Again trials show that when a = 3, b = -4, c = -3, d = -4, 456813 C -4C2 - 3C - 4 = f(C). Thus a sieve action is achieved by selecting one £ for which 4567 carries +£ type, e.g., £ = 3. Factorization is performed by taking gcd(R_{3 C}3 __4C-_ _3C_₄ - 2, C) = 4567.

This formula can be linked to the fundamental Diophantine Equation (p+ l)x - R_p y = ±d where p and R_p are related by C = R_p(mod p± 1). This kind of Diophantine Equation is always solvable, e.g., in the previous example 640 x 3075 - 269 x 73167 = +3, giving much theoretical strength to this formula. Also, this method bears a strikingly close relationship to the elliptic curve method. It is general and always polynomial time. No counterexamples have so far been found.

Also very effective are the above-mentioned small residue factoring sieve as well as a quadratic polynomial factoring sieve not described here. Composites of an arbitrary number of prime factors can be handled and factored too. A 100% complete and efficient PTFA should be based upon such a formula or similar one.

In addition to the above methods, other factoring methods have been programmed and tested such as:

£ £ £ 2 £

(A). R_n = T_n (mod n) is factored by taking a(R_n ) ± bR_n ± c (mod n). Taking the gcd of this relation minus 2 allows for factoring. Here l < a < 4, -4 < b, c < + 4 and a ≠ O . E.g., take n = 3¹⁷ - 1 - = 64570081 ( = 1871 x 34511 ) . Take 851 = R, which comes from

64570081 = 851 (mod 1871 -1) and 1870|2(851)² + 851 - 3 ( = 1870 x 775 ).Thus factorization can be performed by taking gcd (R₂. _n ^: _{+ n} _₃ - 2, n) = 1871.

£ £

(B). Compute the four possible residues: (a) R_n (mod n), (b) R_n (mod 1-n), l (

(c) R^ . _n (mod n) (d) R^ . _n (mod 1-n). Use the residues as "inputs" for one or more layers of T-sequence modulo computation. Then factor by taking gcd

±£

(R^ n ± R, ₄ + R, ₄ ± f ± 2, n) = one prime factor. E.g., take the composite n = 2077 = 31 x 67. Let = 3, n = 3 x 2077 = 6231. There result the four Rs:

(a) R2077 = 719 (mod 2077)

(b) R2077 = 2796 (mod 6231)

(c) R623i = 1190 (mod 2077)

(d) R623i ≡ 3267 (mod 6231) When e = 1, f = -1, £ = + for 67, e.g., £ = 3 there results

1 x 2077 + (3267 - 719) - 1 = 4624 = 68 x 68. Factor by taking gcd(R₄624 - 2, 2077) = (2682 - 2, 2077) = 67.

(C) Another method makes use of the recursive period pattern of certain primes, e.g., n = 2701 = 37 x 73. First compute T2701 = 1239 (mod 2701), T2702 =■ 1 49 (mod 2701). Square the larger, even term such as 2702 then subtract 2 (= T₀), always (mod n), to compute the next even term. For the odd term such as

2701, to compute the next odd term, multiply the odd term by the larger even term and then subtract £, e.g., (1) 1239 (odd), 1749 (even) - (2) 806 (odd), 1467 (even), where 1239 x 1749 - 3 = 806 (mod 2701) and (1749)² -2 = 1467 (mod 2701). Carry on these computations recursively until there emerges a repetitive pattern. Then often factoring can be performed by taking the gcd, e.g., gcd(806 - 3, 270) = 73, also gcd(1467 - 7, 2701) = 73.

(D) Whenever the ratio between the two factors of the composite n is less than 2, one can always factor by adding or subtracting from n by the nearest integer

[V4n - 4] , whereupon the residue (mod p± 1) is zero or just 1 away from p + 1 or p

43

- 1, e.g., 37 x 43 = 1591 and — = 1.1621 < 2 . Compute [J4 ^■ 1591 - 4] = 80 and 1591 - 80 = 1511. By finding £ such as £ = 4 where 37 is -£ type but not 43, factorization is made possible by taking 1511 + 1 = 0 (mod 37 -1), i.e., gcd (R,_59I _ /_{4 I SO]} _₄i ₊ ^_ 2, 1591) = 37. There are again infinite number of composites with this convenient property, or the factors can be made to lie close in size to one another by simply multiplying the composite by a suitable small integer.

Factoring— Summary. T-sequences are closely tied in to factoring. There have been described several very promising polynomial-time factoring methods. The cubic polynomial PTFA seems to work the best, but other lines of attack are feasible too. Prime Number Formula. Traditionally, a prime number formula (which has never been found) has always had these requirements:

1) It always gives a prime number for each integer input n = 1, 2, 3, 4, ....

2) It is constructive, i.e., the formula can always be computed to give prime numbers. For example, Mills' formula p 3 gives no clue how to compute a precise value for A and is therefore not constructive. 3) It is forthright, i.e., it takes little time to readily compute the prime num- ber. For example, for the polynomial equation ax + bx + c = 0, the formula x = forthright in that it gives the roots readily.

On the face of it, these requirements seem natural enough. Seekers of prime number formula have always exerted their best efforts to find a prime number formula that satisfies these three requirements. The continuing failure to find such a prime number formula has caused many researchers to conclude no such formula exists.

While it appears doubtful that a prime number formula of this type can be constructed, upon reflection, it may be seen that the third requirement is inconsistent with the very definition of prime numbers, namely that they cannot be divided exactly by any other numbers other than themselves and 1. The implication is that the primality of a positive integer n needs to be determined by a legitimate polynomial-time primality testing algorithm. Whether n is prime or composite cannot be ascertained right away. Rather, n must be tested for primality. A prime number formula which is supposed to generate primes and not composites also needs to obey such a fundamental requirement.

Now redefine a prime number formula as one that satisfies the three requirements:

1) It always gives a prime number for each integer input n = l, 2, 3, ....

2) It is always constructive.

3) It possesses polynomial-time complexity.

Since a prime number formula is in essence one version of a primality testing algorithm; whereas the traditional formulation of a prime number formula is an NP problem, the foregoing formulation recast the problem such that NP → P.

A new prime number formula of the type described may be aπived at by making use of a revised version of the Fortune Conjecture, i.e., P₁₊₁ - PιP₂P₃...Pi is always a prime. This can be shown to be equivalent to the conjecture that the small- est gap between two consecutive primes P₁₊₁ and P, is (InP, lnlnP,)². If this gap is simplified to In P,, then following Euclid's celebrated proof for the infinity of prime numbers, one can easily show that Fortune Conjecture is equivalent to this smallest gap conjecture. The validity of these two conjectures are well substantiated empirically as well as theoretically. It is known that the maximum gap between two consecutive primes must be rounded by the order of InP,. Any such logarithmic gap will do just fine for the following prime number formula gap or range: g = ( lnP^P₂... P, ) . According to Fortune/Smallest Gap Conjecture there is at least one prime between Q and Q + g. The method therefore needs to compute only these sums: Q + P₁₊], Q + P₁₊ , ... Q + P_J; where P_j is the largest prime smaller than g. There is at least one prime among these sums, and by applying the primality testing method described previously, the primality of each sum can be determined rapidly. Actually, it is also useful to compute the differences: Q - P,₊₁, Q - P₁₊₂, ... Q - P.. It turns out that practically all such differences give not just one but many primes within the range.

One numerical example illustrates this formula clearly: Let P, = 2, P₂ = 3, P₃ = 5, and a = 2, b = 1, x = 2. Then

Q = PΪP₂P₃ = 2² - 3 - 5² = 300, (In 300)² « 32.5 . That means there is a need to compute only these numbers 300 ± 7, 300 ± 11, 300 ± 13, 300 ± 17, 300 ± 19, 300 ± 23, 300 ± 29, 300 ± 31. (The numbers 300 + 1 are not computed here). Among these 16 numbers, the foregoing primality testing algorithm or a similar algorithm enables us to sieve out 11 prime numbers. The conjecture tells us that there are at least two primes. The method obtains 1 1 out of 16; this is a high yield of primes. In fact, even for big numbers this formula or sieve will still yield large quantities of primes consistently, with an estimated or 36.78% of the sums and differences being prime. One condition which must be observed at all times is that InP , P₂... P, must always be smaller than the next prime after P,, that is P₁₊₁.

One can also add or subtract a large product with a small product e.g. 2² • 3 • 5² = 300 and 2^{2 •} 3² = 36 , giving 300 + 36 = 336. The smallest gap in this case will be determined by (In 36)² = 12.84. Among the differences 336 - 5 = 331 is found to be prime, in line with the conjecture. In short the possible number of candidates for primes can always be minimized so that the greatest number of composites is filtered out beforehand.

Note that Q + P, can always be arranged in such a way that will best minimize the number of computations needed to sieve out all the primes in any given range. E.g., there is a relatively large gap between 114 and 127. Computing

2 3

2 - 3 - 5 + 2 - 3 = 114 would require computation of 114 + 5, 7, 11, 13, i.e., four steps too many. Instead, choose 2 • 3 • 5 = 120 which allows computation of the immediate primes as 120 + 7 = 127 and 120 - 7 = 113. This gives all primes within that range readily while skipping all the composite candidates simultaneously. In fact, using a few trials and checks beforehand, one can always manage to optimize the yield of primes within the range efficiently. Large primes can then be chunked out much faster and consistently, all the time based on this prime number sieving algorithm.

This approach makes it possible to compute a large prime. This formula, along with adding or subtracting suitable sums or differences, will readily generate many other primes around this large prime.

Random Number Generator

Mathematically a good random number generator (RNG) should be infinitely non-periodic, such that no generated number can be deduced from any previous number. Of course, statistical tests like the chi-square test can be applied to ensure that all digits are distributed 100% randomly with no bias whatsoever. Admittedly, if only math is concerned, a fixed input will always yield a fixed output. Only physical systems like the quantum mechanical systems can give "dynamically genuine" random numbers. Coupling these two notions together, it is possible to construct a pow- erful and convenient RNG.

First, note the fact that the last digits of all primes, except 2 and 5, can only be 1, 3, 7 and 9. They are distributed absolutely randomly among the infinite set of positive integers. The very definition of prime number demands this, since prime numbers can only be divided exactly by 1 and themselves. Thus by taking the last digits only and ignoring the trivial 2 and 5, from the prime set 3, 7,11, 13, 17, 19, 23, 29, 31, etc., one obtains the random digits 3, 7, 1, 3, 7, 9, 3, 9, 1, 7, 1, 3, 7, 3, 9, 1, 7, 1, 3, 9, 3, 9, 7 for primes from 3 to 97. These digits form an infinite set, and no digit can be derived from the previous or succeeding ones. Each one of the four digits appears 25% of the time. Above all, they are absolutely non-periodic.

The prime-number formula based on the T-sequence polynomial-time primality testing algorithm provides infinitely many variations of these random prime digits, e.g., take 2 x 3 x 5 x 7 = 210. Based on the formula presented previously, add or subtract all the primes between 7 < 11 and 47 < 7 , to test each sum or difference for primality. From the seed 210 onward one obtains the sums +1, +11, +13, +17, ... + 47 which give this set of random digits: 1, 3, 7, 9, 3, 9, 1, 1, 7 from 211 to 257. The differences -1, -11, -13, -17, ... -47 give another set of random digits: 9, 7, 3, 1, 1, 9, 3, 3, 7, 3 from 199 back to 163. Of course, the foregoing primality testing algorithm can be used generate an abundance of large primes such as cannot be generated in any other way.

2 ~>

Since the seeds such as 2 - 3 - 5 or 2 ^• 3^" • 5 • 7 , etc. can be varied in infinitely many ways, the set of last prime digits can also be generated and aπanged in all sorts of arbitrary ways. The seeds can be added or subtracted in any which way too. Without a complete knowledge of the exact seeds and their mathematical operations, no one can reproduce or deduce this type of random digits of the primes. These random digits of primes behave in just as chaotic fashion as the physical subatomic particles in their distribution. Therefore this method can conveniently generate any length of random digits or numbers desired to use in mathematical research or industrial simulation. This generator of random digits can be implemented easily and efficiently in both hardware and software. Conventional RNGs such as linear or non-linear feedback shift registers always carry period patterns which are inherent. Non-periodicity is inherent in the foregoing random prime digit generator.

This RNG can also be easily modified into a simple but innovative cipher: a function Fi (such as transposition, shuffling, etc.) that operates on the last prime digit and another function F₂ that computes and determines the seeds are both kept secret. F₂ is coupled to a simple but chaotic physical system such as dice-throwing, radioactive matter, etc., for the first random input as seeds. The functions F₂ and Fi are used to generate a truly random string of digits such as 9, 7, 3, 1, 1, 9, 3, 3, 7, 3, 1, 3, 7, 9, 3, 9, 1, 1, 7. This string of random digits can be used as a one-time pad for encryption. The receiver who is informed only of the starting seeds (from the physical system input) can decrypt the ciphertext to obtain the plaintext since he also possesses F_j and F₂ as well as the relevant table of primes like the sender. As long as F_] and F₂ are kept secret, no eavesdropper can decrypt the ciphertext. The cipher can even be timed accordingly so that the functions Fj and F₂ change according to time changes or context changes. In any event, math theory about primes guarantees that the string of random digits thus generated are absolutely chaotic. No fixed inter-relationship can be derived from among themselves.

It will be appreciated by those of ordinary skill in the art that the invention can be embodied in other specific forms without departing from the spirit or essential character thereof. The presently disclosed embodiments are therefore considered in all respects to be illustrative and not restrictive. The scope of the invention is indicated by the appended claims rather than the foregoing description, and all changes which come within the meaning and range of equivalents thereof are intended to be embraced therein.

Claims

What is claimed is:

1. A computer-implemented method, comprising: determining at least one element of a non-montonic sequence, the non-montonic sequence being one of a family of related non-montonic sequences; using at least said element, determining at least one property of a number; and depending on said property, taking an action the effect of which is to enhance or degrade data security within a computer system or network.

2. The method of Claim 1 , wherein said property is primality.

3. The method of Claim 1, wherein said number is a composite number, and said property is a factor of said number.

4. The method of Claim 1 , wherein said family of related non-montonic sequences is defined as follows:

T₀' = 2, Tf = 1 and l[ ₊ , = / . T_n ^f - T_n'_ , , where the subscript denotes the nth term while the superscript denotes the order £.

5. A prime number generator, comprising: means for generating candidate numbers by forming at least one of sums and differences of a given number and a series of prime numbers; and means for deterministically evaluating primality of each of the candidate numbers in polynomial time.

6. The apparatus of Claim 5, wherein said means for deterministically evaluating primality comprises means for determining at least one element of a non- montonic sequence, the non-montonic sequence being one of a family of related non-montonic sequences.

7. The apparatus of Claim 6, wherein said family of related non-montonic sequences is defined as follows:

T = 2, Tf = £ and l[ ₊ , = £ - T_n' - T_n'_ , , where the subscript denotes the nth term while the superscript denotes the order £.

8. A random number generator, comprising: means for determining a seed number; means for forming at least one of sums and differences of the seed number and a series of prime numbers; and means for outputting last digits of the series of prime numbers to produce a set of random digits.