14. Number systems

Conventional numbering systems consist of a base (or radix).

The primorial number system is said to be 'primoradic'; having a primorial base. The primorial number system is a mixed radix numeral system adapted to the numbering of the primorials (Table 3).

#### Modern Cryptography – Current Challenges and Solutions


1523830 ¼ 2 � 5 � 7 � 11 � 1979 ¼ ð Þ 770 ð Þ¼ 1979 ð Þ 1234�464 ð Þ 1234þ745

1521642935492617539765579106664136748401379615914⋱ 312169315386041883234627722692028711378934397966⋱

Consider each congruency and look for a factorization that is symmetrical about

30431475913593577738588710930551227419722971658953 xþ

151816659580901664885523419281115998823527019067345405631⋱

N ¼ ð Þ ð Þ aPk þ c Pk�1# þ e ð Þ ð Þ aPk þ d Pk�1# þ f k ¼ 31, P<sup>31</sup> ¼ 127, ð Þ¼ aPk þ c 1201, ð Þ¼ aPk þ d 12

N ¼ ð Þ 9P31# þ 58P30# þ e ð Þ 9P31# þ 125P30# þ f

k#

<sup>k</sup># þ ½ � a cð Þþ þ d mPk# Pk# þ ð Þ nPk# þ cd

N ¼ ð Þ 9P31# þ 58P30# þ e ð Þ 9P31# þ 125P30# þ f

N ¼ ð Þ 9P31# þ 58P30# þ 41P29# þ g ð Þ 9P31# þ 125P30# þ 46P29# þ h

k 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 … 1 P<sup>k</sup> 127 113 109 107 103 101 97 89 83 79 73 71 67 61 59 53 2 P<sup>1</sup> 9 58 41 32 43 101 13 14 60 50 54 33 3 32 12 12 1 P<sup>2</sup> 9 125 46 106 75 95 71 79 21 3 19 58 23 32 30 13 1

<sup>30</sup> þ ð Þ 1201f þ 1268e P<sup>30</sup> þ ef

<sup>31</sup> þ ð Þ a cð Þþ þ d n P<sup>31</sup> þ cd

<sup>k</sup># <sup>¼</sup> <sup>94</sup> ) <sup>a</sup> <sup>¼</sup> <sup>9</sup>, <sup>m</sup> <sup>¼</sup> <sup>13</sup>

<sup>30</sup># <sup>þ</sup> ð Þ <sup>1201</sup><sup>f</sup> <sup>þ</sup> <sup>1268</sup><sup>e</sup> <sup>P</sup>30# <sup>þ</sup> ef

<sup>2</sup> þ ð f aP <sup>ð</sup> <sup>k</sup> <sup>þ</sup> <sup>c</sup>Þ þ e aP ð ÞÞð <sup>k</sup> <sup>þ</sup> <sup>d</sup> Pk�1#Þ þ ef

<sup>2</sup> <sup>þ</sup> ð Þ f aP ð Þþ <sup>k</sup> <sup>þ</sup> <sup>c</sup> e aP ð Þ <sup>k</sup> <sup>þ</sup> <sup>d</sup> ð Þþ Pk�1# ef

2

ð Þ Pk�1# 2

<sup>2</sup> <sup>≤</sup> <sup>N</sup> ) ð Þ aPk <sup>þ</sup> <sup>c</sup> ð Þ¼ aPk <sup>þ</sup> <sup>d</sup> <sup>N</sup> � Nmod Pð Þ <sup>k</sup>�1#

<sup>1522868</sup> <sup>¼</sup> <sup>2</sup><sup>2</sup> � <sup>317</sup> � <sup>1201</sup> <sup>¼</sup> ð Þ <sup>1201</sup> ð Þ¼ <sup>1268</sup> ð Þ <sup>1234</sup> � <sup>33</sup> ð Þ <sup>1234</sup> <sup>þ</sup> <sup>34</sup>

Not symmetrical about square root [12]

DOI: http://dx.doi.org/10.5772/intechopen.84852

N ¼ ð Þ aPk þ c ð Þ aPk þ d ð Þ Pk�1#

Symmetrical about square root.

Survey of RSA Vulnerabilities

ð Þ aPk þ c ð Þ aPk þ d ð Þ Pk�1#

800=p30#2

In this case 1234 + 34 =1268, 1234 – 33 = 1201.

401183567090345342039152734187917869,

a ¼ 9, c ¼ 58, d ¼ 125, P<sup>31</sup> ¼ 127

P2

<sup>N</sup> <sup>¼</sup> ð Þ <sup>1201</sup> ð Þ <sup>1268</sup> <sup>P</sup><sup>2</sup>

<sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>m</sup> <sup>¼</sup> <sup>N</sup> � NmodP<sup>2</sup>

Pk# <sup>¼</sup> Pkð Þ) Pk�<sup>1</sup># <sup>N</sup> <sup>¼</sup> ð Þ <sup>1201</sup> ð Þ <sup>1268</sup> <sup>P</sup><sup>2</sup>

Repeat these steps for P29# and so on… (Table 4)

a2 P2

<sup>N</sup> <sup>¼</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>m</sup> <sup>P</sup><sup>2</sup>

N ¼ ð Þ aPk þ c ð Þ aPk þ d ð Þ Pk�1#

the square root.

Table 4.

31

P1 and P2 as base Primorial numbers.

Table 3.

Primorial radix number system.

General properties of mixed radix number systems apply to the base primorial system. The primorial number system OEIS A000040 is denoted by a subscript " Q".

Consider the following example:

Primorial to decimal, Base<sup>Q</sup> to Base10 34101<sup>Q</sup> stands for 3443120110, whose value is

$$\mathbf{x} = 3 \times \mathbf{p\_4} \mathbf{s} + 4 \times \mathbf{p\_3} \mathbf{s} + 1 \times \mathbf{p\_2} \mathbf{s} + 0 \times \mathbf{p\_1} \mathbf{s} + 1 \times \mathbf{p\_0} \mathbf{s} = 3 \times 2\mathbf{10} + 4 \times 3\mathbf{0} + 1 \times \mathbf{6} + 0 \times \mathbf{2} + 1 \times \mathbf{1} \times \mathbf{p} \mathbf{s} + 0 \times \mathbf{p} \mathbf{s} + 0 \times \mathbf{p} \mathbf{s} = ((((-3 \times 7 + 4) \times \mathbf{5} + 1) \times \mathbf{0}) + (\mathbf{0} \times \mathbf{6} + 0 \times \mathbf{7}) + \mathbf{0} \times \mathbf{p} \mathbf{s}) + (\mathbf{0} \times \mathbf{7}) \times \mathbf{p} \mathbf{s} + 0 \times \mathbf{p} \mathbf{s} + 0 \times \mathbf{p} \mathbf{s} = (((-(-3 \times 7 + 4) \times \mathbf{6}) \times \mathbf{7}) + (\mathbf{0} \times \mathbf{7}) \times \mathbf{p} \mathbf{s}) + (\mathbf{0} \times \mathbf{7}) \times \mathbf{p} \mathbf{s} + 0 \times \mathbf{p} \mathbf{s} + 0 \times \mathbf{p} \mathbf{s} + 0 \times \mathbf{p} \mathbf{s} + 0 \times \mathbf{p} \mathbf{s} + 0 \times \mathbf{p} \mathbf{s}$$

$$= ((((3 \times 7 + 4) \times \mathbf{5} + 1) \times \mathbf{3} + 0) \times \mathbf{2} + 1) \times \mathbf{1} + \mathbf{1} \times \mathbf{1} + \mathbf{1} \times \mathbf{1} + \mathbf{2} \times \mathbf{1} \times \mathbf{p} \mathbf{s} + 0 \times \mathbf{p} \mathbf{s}$$
