CN104468100A - Improved sliding window modular exponentiation computing method - Google Patents
Improved sliding window modular exponentiation computing method Download PDFInfo
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- CN104468100A CN104468100A CN201410726861.8A CN201410726861A CN104468100A CN 104468100 A CN104468100 A CN 104468100A CN 201410726861 A CN201410726861 A CN 201410726861A CN 104468100 A CN104468100 A CN 104468100A
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Abstract
The invention relates to the field of cryptography and information safety and provides a precomputation method which can improve efficiency and effectively lower the computing complexity of sliding window computing. According to the technical scheme, an improved sliding window modular exponentiation computing method comprises the steps that first, a maximum window length d needs to be determined, a window is defined as a group for number grouping according to bits, the bit of each group is not larger than d, and then according to a specific strategy, an exponent e is divided into a zero window and a non-zero window; and during computing, if a scanning exponent meets the zero window, modular square operation is carried out, if the scanning exponent meets the non-zero window, modular square operation is carried out first, and then a corresponding modular exponentiation value is obtained from a precomputation storage value for modular multiplication operation. The method is mainly used for data processing and classified mail correspondence.
Description
Technical field
The present invention relates to cryptography, information security field; Be mainly used in improving the computational efficiency that in public key cryptography system, large digital-to-analogue power calculates.Specifically, the sliding window mould power computational methods of improvement are related to.
Technical background
Public-key cryptosystem is a kind of cryptographic system applied at present widely.In information system security, particularly in digital signature, Authentication and key distribution and management, public key cryptography all plays requisite role.The generation of public-key cryptosystem is derived from the two problems existed in symmetric cryptography.First is the assignment problem of key, and symmetric cryptography needs communicating pair to share same key, and needs a key distribution center, and second is exactly the problem of digital signature.Diffie and Hellman proposed all cryptographic systems before a kind of thought fundamentally differs from 1976.
Public-key cryptosystem belongs to asymmetric cryptography, depends on the decruption key that an encryption key is different with.A complete public-key cryptosystem encryption process roughly comprises six key elements: PKI, private key, cryptographic algorithm, decipherment algorithm, expressly, ciphertext.
In this cryptographic system, all users can obtain the PKI of other users, and each user produces separately and retain private key, do not need same specific user to share.User keeps the fail safe of private key, and so the exchange of information is all safe.User can change private key at any time, only the PKI distribution supervened after replacing need be replaced old PKI.
This new password thought that Diffie and Hellman proposes cryptologists are competitively proposed character that many algorithms attempt to meet public-key cryptosystem, but a lot of algorithms are wherein all unsound.1977, three scientist RonRivest, Adi Shamir and Len Adleman of MIT proposed with algorithm---the RSA Algorithm of their three people's surname initial names, and deliver in 1978 years.RSA Algorithm is one of Public Key Crypto Scheme be most widely used at present, be also first can simultaneously for the algorithm of information encryption and decryption and digital signature.The fail safe of RSA Algorithm is based on such fact---and the product calculating two large prime numbers is very easy, and it is very difficult the product of two large prime numbers to be carried out Factorization.That is, if one large several comprise two large prime factors, it will be very difficult for so carrying out Factorization to it.But the mould power calculation operations of carrying out when encryption and decryption due to public key cryptography is complicated, speed is slow, and the speed of public key cryptography enciphering and deciphering algorithm depends primarily on the efficiency of mould power and modular multiplication algorithm, this bottleneck with regard to making the computational efficiency of modulus-power algorithm become public key cryptography system, therefore how realizing modulus-power algorithm is fast password educational circles question of common concern.
The research object of mould power is: suppose that N, M, e are the binary system big integer of k position, time k very large (as k>256), calculates fast:
C＝M
emod N (1)
Formula (1) is generally all converted into a series of mould and takes advantage of and mould square calculating.As calculated M
15, by sequence:
M→M
2→M
3→M
4→M
5→…M
15
Calculate.Obviously, the efficiency of this computational methods is very low, therefore, realizes studying to have important theory significance and practical value to the acceleration of big integer mould power and modular multiplication algorithm.
In actual applications, RSA Algorithm is widely used in the encrypting and decrypting of Email.E-mail system distributes pair of secret keys to (e for each user automatically
1, e
2), wherein, e
1for the PKI of this user, e
2for the private key of this user, public key publication is gone out by system afterwards, and is properly preserved by private key.When user B sends mail to user A, by the PKI e of user A
1mail plaintext M is encrypted, obtains ciphertext C, and ciphertext C is sent.Due to the character of RSA cryptographic algorithms, the user A that this ciphertext C only has corresponding private key can decipher, thus ensure that safety and the privacy of Email.
Detailed process is as follows:
(1) ciphering process: set M as original e-mail content, e
1for the PKI of user A, N is the modulus that systematic unity specifies, then user B is by calculating M
e1the ciphertext C that mod N obtains;
(2) decrypting process: after user A receives ciphertext C, uses the private key e of oneself
2, and the N that system specifies carries out the calculating of mould power: C
e2mod N, obtains plaintext M.
Summary of the invention
In order to overcome the deficiencies in the prior art, a kind of pre-computation methods of raising the efficiency is proposed, effectively can reduce the computation complexity of sliding window algorithm, for this reason, the technical scheme that the present invention takes is, the sliding window mould power computational methods of improvement, comprise the steps: first to need to determine a maximized window length d, window definition is the grouping divided digital step-by-step, and the figure place of each grouping is not more than d, then according to specific strategy, exponent e is divided into zero window and non-zero window; During calculating, scan index runs into zero window and then carries out mould square operation, runs into non-zero window advanced row mould square operation, then from precomputation storing value, obtains respective mode values of powers carry out modular multiplication; Wherein, the window partition strategy providing slip window sampling is specially: represent zero window with ZW, and NW represents non-zero window, and length of window is set as d:
Initial condition S is ZW:
As S=ZW, scan the 1st, if 0, be included into ZW, S=ZW; If 1, be included into new NW, S=NW;
As S=NW, scanning d-1 position, this NW is terminated an in the end nonzero digit and be set to i position, ZW is included into, S=ZW in i+1 to the d-1 position of present scan.
The sliding window mould power computational methods of described improvement are further refined as:
Input: M, e, n, d;
Export: C=M
emod n;
Wherein, M represents the message that will send, and represent with binary digit, e represents index, and n represents modulus, and d represents the maximum length of sliding window, and C is result of calculation, and mod represents modular arithmetic, namely asks Me divided by the remainder of n gained;
(1) maximum length of sliding window is set to d;
(2) according to following strategy division index e from left to right:
(2.1) initial condition S is that zero window state ZW represents, non-zero Window state NW represents;
(2.2) as S=ZW, scan 1, if 0, be included into ZW, S=ZW; If 1, be included into new NW, S=NW;
(2.3) as S=NW, d-1 position is scanned backward, and forward trace to the first nonzero digit, be set to i, the 1st of present scan the to i-th (is comprised and i) be included into current NW, be included into ZW, S=ZW by i-th+1 to d-1 position;
(2.4) circulate (2.2) and (2.3), until scanned all positions of exponent e, finally obtain non-zero window F
1, F
2..., F
s, and zero window Z
1, Z
2..., Z
k, the expression that wherein s obtains scans the quantity of non-zero window, and k represents the quantity scanning the zero window obtained;
(3) according to following strategy, precomputation is carried out to non-zero window, draws addition chain:
(3.1) to F
1, F
2..., F
sby ascending order arrangement and each value only remember once, obtain sequence W
0=w
01, w
02..., w
0i;
(3.2) a is preserved
0=w
0i, calculate t
0=w
0i-w
0i-1;
If t
0at W
0middle appearance or t
0=1, then at W
0in leave out w
0i, obtain W
1=w
01, w
02..., w
0i-1;
Otherwise at W
0in use t
0replace w
0i, obtain W
1=w
01, w
02..., w
0i-1, t
0;
(3.3) with the W obtained in previous step
i-1sequence is input, repeats (3.1)-(3.2) step, until W
iin only surplus next element w
i1;
(3.4) calculate feasible method by any one and obtain w
i1an addition chain, obtain a
i, a
i+1..., a
s;
(3.5) to A=a
0, a
1..., a
sarrange by ascending order;
(4) addition chain A=a is used
0, a
1..., a
s, calculate M
fivalue, wherein F
ifor non-zero window, i=1,2 ..., s;
(5) initial C=M
f1, i circulates from 2 to s:
(5.1)C＝(C*M
Fi)mod n；
(6) to zero window Z
1, Z
2..., Z
k, i circulates from 1 to k:
(6.1) C=C
eimod n, here E
i=2
li, L
ifor window F
ilength;
(7) ciphertext C is exported.
Along with the increase of index length, the length of sliding window also should corresponding increase to obtain computational efficiency better; When index length is 3, the length of sliding window should be 4, and when index length is increased to 1024, sliding window length is chosen as 6 and is, when index length is 4096, sliding window length is 7.
Compared with the prior art, technical characterstic of the present invention and effect:
The present invention, on the basis of conventional slip window technique, by window selection strategy, and the improvement of precomputation, achieves high efficiency mould power and calculates, and effectively can improve large digital-to-analogue power and calculate efficiency when running.
Accompanying drawing explanation
About Fig. 1 two flow charts are respectively the process using the inventive method to carry out public key encryption and deciphering, and wherein left figure is ciphering process, and right figure is decrypting process.In figure arrow side character representation on the output of a flow process and the input of next flow process.
Embodiment
First slip window sampling needs to determine a maximized window length d, then according to specific strategy, exponent e is divided into zero window and non-zero window F
i.Index to be divided into according to different strategies the zero window and non-zero window F that length do not wait by slip window sampling
i, during calculating, scan index runs into zero window and then carries out mould square operation, runs into non-zero window advanced row mould square operation, then from precomputation storing value, obtains respective mode values of powers carry out modular multiplication.
First, provide the window partition strategy of slip window sampling, represent zero window with ZW, NW represents non-zero window, and length of window is set as d:
Strategy slip window sampling:
Initial condition S is ZW:
As S=ZW, scan 1, if 0, be included into ZW, S=ZW; If 1, be included into new NW, S=NW.
As S=NW, scan d-1 position backward, and forward trace to the first nonzero digit, be set to i, the 1st of present scan the to i-th (is comprised and i) be included into current NW, be included into new ZW, S=ZW by i-th+1 to d-1 position;
Circulation performs above-mentioned steps, until scanned all positions of exponent e, finally obtains window F
1, F
2..., F
s;
Example 1: utilize this strategy, scanning from left to right divides e=(111010100100101), makes length of window d=4, then divide as follows:
e＝111 0 101 00 1001 0 1。
In conjunction with this strategy, the algorithm of slip window sampling is expressed as follows:
Secondly, be the improvement of precomputation.In algorithm 1, precomputation is (1) step, calculates M
wmod n, w=3,5 ..., 2
d-1, its computation complexity is 2
d-1.But the index for any given figure place determines its optimum window size, usually need to gather extensive sample, namely traditional force search method; Or test one by one within the specific limits.Both suitable consumes resources.On the other hand, when window size d selects excessive, precomputation amount exponentially type increases, and precomputation result is not probably fully utilized, and causes the wasting of resources.The present invention provides a kind of method utilizing addition sequence thought to carry out precomputation, and the result of precomputation can be fully used.Its main thought is by non-zero window F
1, F
2..., F
sthe value of middle repetition is removed, thus problem is converted into the addition sequence asking non-zero window value, that is asks one by the addition chain of all set-points.If sequence S=is (a
0, a
1..., a
s) meet following character, then claim S to be e (e=a
s) an addition chain:
(1) S is monotone-increasing sequence;
(2)a
0＝1,a
s＝e；
(3)a
i＝a
j+a
k,0≤j≤k＜i≤s。
This algorithm is expressed as follows:
Namely the A that algorithm 2 is obtained is an addition chain comprising all non-zero window value.M is calculated successively afterwards according to A
ai, i=0,1 ..., s, can obtain the mould values of powers needed for all precomputations.
Example 2: selection window length d=6, input F
1..., F
s=19,7,21,61,35,7,35,47,35,55.
(1) to F
1, F
2..., F
isequence is by ascending order arrangement, and the value of repetition is only remembered once, obtains W
0=7,19,21,35,47,55,61;
(2)a
0＝61，t
0＝61-55＝6；
(3) replace 61 with 6, and obtain W by ascending order arrangement
1=6,7,19,21,35,47,55;
(4) (1)-(3) step is repeated:
After 1st circulation:
W
1＝6,7,19,21,35,47,55，A＝61；
After 2nd circulation:
W
2＝6,7,8,19,21,35,47，A＝61,55；
After 3rd circulation:
W
3＝6,7,8,12,19,21,35，A＝61,55,47；
…
After 13rd circulation:
W
13＝2，A＝61,55,47,35,21,19,14,12,8,7,6,5,4；
(5) a last addition chain is 1,2, adds A to;
(6) obtain by ascending order arrangement:
A＝1,2,4,5,6,7,8,12,14,19,21,35,47,55,61；
(7) A is returned.
Calculate all non-zero window value according to addition chain A to need altogether to calculate for 14 times, conventional method then needs to calculate whole value, and totally 2
6-1calculate for=32 times.
Analyze the effect of the method below by experiment, the index choosing 512,1024,2048,4096 4 kinds of length contrasts, and chooses shorter 3,6,7,8 and longer 12 7 kinds of length of window.1000000 numbers are produced at random to each unit and carries out its various actual complex degree of division acquisition, utilize formula program calculation to obtain various theoretical complexity.Experimental data is as table 1, table 2.
In table 1, BC represents binary law complexity, WLEN represents most left window length, NWIN represents non-zero window number, OPRE represents conventional slip window technique precomputation complexity, OC represents the total complexity of conventional slip window technique, and APRE represents the precomputation complexity of improving one's methods, and AC represents total complexity of improving one's methods.
Table 1 is improved one's methods and to be contrasted with conventional slip window technique
Table 2 precomputation amount improves percentage
| Position long d | 3 | 6 | 7 | 8 | 12 |
| 512 | 0.00％ | 5.83％ | 30.58％ | 57.53％ | 96.09％ |
| 1024 | 0.00％ | 0.13％ | 10.53％ | 37.04％ | 93.92％ |
| 2048 | 0.00％ | 0.00％ | 0.65％ | 14.78％ | 90.54％ |
| 4096 | 0.00％ | 0.00％ | 0.00％ | 1.98％ | 84.25％ |
Experiment shows, when the length of window selected is less than optimum length of window, efficiency of improving one's methods is suitable with traditional slip window sampling efficiency; When the length of window selected is greater than optimum length of window, can find out that the growth of complexity of improving one's methods is comparatively mild, and the exponentially type growth of conventional slip window technique complexity.That is, when optimum length of window the unknown, application enhancements method need not worry that the complexity caused because window selection is excessive rises suddenly and sharply.
Therefore, the present invention, on the basis of conventional slip window technique, by window selection strategy, and the improvement of precomputation, achieves high efficiency mould power and calculates, and effectively can improve large digital-to-analogue power and calculate efficiency when running.
1. the selection of length of window
The index that 128,256,512,1024,2048,4096 6 kinds of length are chosen in experiment contrasts, and chooses 3 ~ 86 kinds of length of window.Produce 1000000 numbers at random to each unit and carry out division acquisition Practical Calculation complexity, experimental result is as following table.
The actual complex degree of table 3 slip window sampling
| D position long | 128 | 256 | 512 | 1024 | 2048 | 4096 |
| 3 | 161.19 | 321.19 | 641.18 | 1281.18 | 2561.17 | 5121.15 |
| 4 | 157.92 | 311.51 | 618.71 | 1233.11 | 2461.90 | 4919.50 |
| 5 | 160.72 | 310.05 | 608.72 | 1206.05 | 2400.72 | 4790.06 |
| 6 | 172.70 | 318.99 | 611.56 | 1196.70 | 2366.97 | 4707.52 |
| 7 | 201.44 | 345.44 | 633.44 | 1209.44 | 2361.44 | 4665.47 |
| 8 | 262.67 | 404.90 | 689.33 | 1258.24 | 2395.97 | 4671.53 |
As seen from the above table, along with the increase of index length, the length of sliding window also should corresponding increase to obtain computational efficiency better.Such as when index length is 3, the length of sliding window should be 4, and when index length is increased to 1024, sliding window length is chosen as 6 and is, when index length is 4096, sliding window length is that 7 effects are best.
2. the algorithm implemented
Claims (3)
1. the sliding window mould power computational methods improved, is characterized in that, comprise the steps: first to need to determine a maximized window length d, then according to specific strategy, exponent e is divided into zero window sequence Z
1..., Z
iwith non-zero series of windows F
1..., F
i; During calculating, scan index runs into zero window and then carries out mould square operation, runs into non-zero window advanced row mould square operation, then from precomputation storing value, obtains respective mode values of powers carry out modular multiplication; Wherein, the window partition strategy providing slip window sampling is specially: represent zero window with ZW, and NW represents non-zero window, and length of window is set as d:
Initial condition S is ZW:
As S=ZW, scan the 1st, if 0, be included into ZW, S=ZW; If 1, be included into new NW, S=NW;
As S=NW, scanning d-1 position, this NW is terminated an in the end nonzero digit and be set to i position, ZW is included into, S=ZW in i+1 to the d-1 position of present scan.
2. the sliding window mould power computational methods improved as claimed in claim 1, it is characterized in that, the sliding window mould power computational methods of described improvement are further refined as:
Input: M, e, n, d;
Export: C=M
emod n;
Wherein, M represents the message that will send, and represent with binary digit, e represents index, and n represents modulus, and d represents the maximum length of sliding window, and C is result of calculation, and mod represents modular arithmetic, namely asks M
edivided by the remainder of n gained;
(1) maximum length of sliding window is set to d;
(2) according to following strategy division index e from left to right:
(2.1) initial condition S is that zero window state ZW represents, non-zero Window state NW represents;
(2.2) as S=ZW, scan 1, if 0, be included into ZW, S=ZW; If 1, be included into new NW, S=NW;
(2.3) as S=NW, d-1 position is scanned backward, and forward trace to the first nonzero digit, be set to i, the 1st of present scan the to i-th (is comprised and i) be included into current NW, be included into ZW, S=ZW by i-th+1 to d-1 position;
(2.4) circulate (2.2) and (2.3), until scanned all positions of exponent e, finally obtain non-zero window F
1, F
2..., F
s, and zero window Z
1, Z
2..., Z
k, the expression that wherein s obtains scans the quantity of non-zero window, and k represents the quantity scanning the zero window obtained;
(3) according to following strategy, precomputation is carried out to non-zero window, draws addition chain:
(3.1) to F
1, F
2..., F
sby ascending order arrangement and each value only remember once, obtain sequence W
0=w
01, w
02..., w
0i;
(3.2) a is preserved
0=w
0i, calculate t
0=w
0i-w
0i-1;
If t
0at W
0middle appearance or t
0=1, then at W
0in leave out w
0i, obtain W
1=w
01, w
02..., w
0i-1;
Otherwise at W
0in use t
0replace w
0i, obtain W
1=w
01, w
02..., w
0i-1, t
0;
(3.3) with the W obtained in previous step
i-1sequence is input, repeats (3.1)-(3.2) step, until W
iin only surplus next element W
i1;
(3.4) calculate feasible method by any one and obtain w
i1an addition chain, obtain a
i, a
i+1..., a
s;
(3.5) to A=a
0, a
1..., a
sarrange by ascending order;
(4) addition chain A=a is used
0, a
1..., a
s, calculate M
fivalue, wherein F
ifor non-zero window, i=1,2 ..., s;
(5) initial C=M
f1, i circulates from 2 to s:
(5.1)C＝(C*M
Fi)mod n；
(6) to zero window Z
1, Z
2..., Z
k, i circulates from 1 to k:
(6.1) C=C
eimod n, here E
i=2
li, L
ifor window F
ilength;
(7) ciphertext C is exported.
3. the as claimed in claim 2 sliding window mould power computational methods improved, is characterized in that, along with the increase of index length, the length of sliding window also should corresponding increase to obtain computational efficiency better; When index length is 3, the length of sliding window should be 4, and when index length is increased to 1024, sliding window length is chosen as 6 and is, when index length is 4096, sliding window length is 7.
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| CN107040370A (en) * | 2015-11-04 | 2017-08-11 | 恩智浦有限公司 | Use the Montgomery Algorithm of random addition chain |
| CN108055126A (en) * | 2017-12-11 | 2018-05-18 | 哈尔滨理工大学 | The method of anti-power consumption attack based on random addition chain |
| CN113627202A (en) * | 2021-07-21 | 2021-11-09 | 大唐互联科技(武汉)有限公司 | System for binding production data and products through code scanning |
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Cited By (5)
| Publication number | Priority date | Publication date | Assignee | Title |
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| CN107040370A (en) * | 2015-11-04 | 2017-08-11 | 恩智浦有限公司 | Use the Montgomery Algorithm of random addition chain |
| CN107040370B (en) * | 2015-11-04 | 2021-07-27 | 恩智浦有限公司 | Apparatus for generating code implementing modular exponentiation |
| CN108055126A (en) * | 2017-12-11 | 2018-05-18 | 哈尔滨理工大学 | The method of anti-power consumption attack based on random addition chain |
| CN113627202A (en) * | 2021-07-21 | 2021-11-09 | 大唐互联科技(武汉)有限公司 | System for binding production data and products through code scanning |
| CN113627202B (en) * | 2021-07-21 | 2024-03-29 | 大唐互联科技(武汉)有限公司 | System for binding production data with product through code scanning |
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