CN107040380A - A kind of improvement mould of the elliptic curve cryptosystem based on binary field removes method - Google Patents
A kind of improvement mould of the elliptic curve cryptosystem based on binary field removes method Download PDFInfo
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
- H04L9/30—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy
- H04L9/3006—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy underlying computational problems or public-key parameters
- H04L9/3033—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy underlying computational problems or public-key parameters details relating to pseudo-prime or prime number generation, e.g. primality test
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
- H04L9/30—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy
- H04L9/3066—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy involving algebraic varieties, e.g. elliptic or hyper-elliptic curves
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L2209/00—Additional information or applications relating to cryptographic mechanisms or cryptographic arrangements for secret or secure communication H04L9/00
- H04L2209/12—Details relating to cryptographic hardware or logic circuitry
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Abstract
A kind of improvement mould of the elliptic curve cryptosystem based on binary field removes method, is related to domain operation method.A kind of improvement mould for about subtracting the fast elliptic curve cryptosystem based on binary field of efficiency high, arithmetic speed is provided and removes algorithm.According to r (t)=y (t)/x (t) mod F (t), register A, B, U, V are first assigned correspondence initial value by algorithm, again by disposably judging the value of minimum two bit binary data in register, realize correspondence about reducing, it is mould division result r (t) until the numerical value stored in register A is reduced to the numerical value stored in 1, register U.Algorithm is realized by Verilog language and emulated, contrast improved Euclidean algorithm and fermat's little theorem algorithm, the algorithm has advantage in terms of time loss, mould is effectively accelerated except calculating, available in ECC encryption and decryption IP kernels.
Description
Technical field
The present invention relates to domain operation method, more particularly, to a kind of changing for elliptic curve cryptosystem based on binary field
Progressive die removes method.
Background technology
With the development of science and technology, our quality of life has obtained huge improvement, at the same time, and information security is asked
Topic is also increasingly severe, does not threaten our property safety and individual privacy all the time.Because different scenes have different
Demand, the encryption system used is also therefore different with AES, current encryption system mainly have symmetric cryptography with
Two kinds of asymmetric encryption.
Public key encryption system was proposed jointly in 1976 from Diffie.W and Hellman.M, and it becomes grinds for cryptography
Study carefully the important topic in field, and played a significant role all the time in terms of information security.It is different from symmetric cryptography, asymmetric encryption
Communicating pair has the public key and private key of oneself respectively.It is at present to be based on three big difficult math questions, one for public key encryption system construction
It is big number Factorization intractability, two be discrete logarithm intractability, and three be Elliptic Curve Discrete Logarithm intractability.Elliptic curve is close
The foundation for security of code system is Elliptic Curve Discrete Logarithm intractability, and the system is by Miller ([1] V.S.Miller, " Use
of elliptic curves in cryptography,”Advances in Cryptology-CRYPTO’85
Proceedings.Springer, 1986, pp.417-426) and Koblitz ([2] N.Koblitz, " Elliptic curve
Cryptosystems, " Mathematics of computation, vol.48, no.177, pp.203-209,1987) carried
Go out.Elliptic curve is typically expressed as y2+ axy+by=x3+cx2+ dx+e, this kind of curve is referred to as Weierstrass equations, curve by
All points (x, y) for meeting the equation are collectively constituted, in hardware design, generally using its special shape y2+ xy=x3+ax2+
1, wherein a value is 0 or, 1, and the curve of the form is referred to as Koblitz elliptic curves.
Elliptic curve cryptosystem is to calculate to realize in finite field, and finite field is divided into binary field GF (2m) and prime number
Domain GF (p), wherein binary field are adapted to hardware and realized, the major calculations of elliptic curve cryptosystem have point processing and domain operation,
Point processing is constituted by point plus with the point times dot product constituted.Domain operation is added by mould, mould square, modular multiplication, mould are inverse is constituted.Its
In, the inverse time loss of mould is most, and the algorithm of research modular inversion has following a few classes to represent at present：One be extension Europe it is several in
Obtain related algorithm ([3] J.H.Guo, C.L.Wang, " Systolic array implementation of Euclid's
algorithm for inversion and division in GF(2m)”.IEEE Transactions on
Computers.1998,47(10):1161-1167), two be extension Euclid's innovatory algorithm ([4] S.C.Shantz, " From
Euclid’s GCD to Montgomery multiplication to the great divide,”Tech.Rep.TR-
2001-95, Sun Microsystems, 1995), three be based on fermat's little theorem inversion algorithms ([5] T.Itoh,
S.Tsujii,“A Fast Algorithm for Computing Multiplicative Inverses in GF(2m)
Using Normal Bases,”IECE,Japan,1986,pp.31–36Paper of Technical Group,TGIT86-
44.) fermat's little theorem algorithm (M.J.Zhi, " Design and Implementation of Elliptic Curve, are improved
Cryptography over GF(2m)”,Dissertation of Shanghai Jiao Tong University,
2007)。
The content of the invention
It is fast it is an object of the invention to provide can verify that, arithmetic speed, pass through minimum two of disposable test data
Parity, reduces time loss, and a kind of based on the ellipse of binary field of the fast domain operation of efficiency high, arithmetic speed is about subtracted to realize
The improvement mould of circular curve cipher system removes method.
The improvement mould of elliptic curve cryptosystem of the invention based on binary field comprises the following steps except one of method：
1) according to the relative theory of elliptic curve cryptosystem, it is located at binary field GF (2m) in, it is known that two exponent numbers are small
In threshold value m element x (t) and y (t), respectively as two input elements, while according to NIST (American National Standard and technology
Research institute) the Koblitz elliptic curve parameters recommended, one known exponent number of selection is equal to threshold value m irreducible polynomial F
(t)；Formula r (t)=y (t)/x (t) mod F (t) are removed according to mould, mould division result r (t) are obtained, or be expressed as y (t) ≡ r (t) x
(t)mod F(t)；By using intermediate data required in four register A, B, U, V storage algorithms, reach and formula r is removed to mould
(t)=y (t)/x (t) mod F (t), or y (t) ≡ r (t) x (t) mod F (t) are iterated the purpose for about subtracting calculating, first, according to
It is secondary that initialization assignment is carried out to described four registers A, B, U, V；
2) after four registers A, B, U, V are completed with initial assignments, algorithm starts to being stored in register A, B
Numerical value, which is iterated, about to be subtracted, during about subtracting, and four registers A, B, U, V need to maintain A × y (t) ≡ U × x (t) all the time
The identity of mod F (t) and B × y (t) ≡ V × two formula of x (t) mod F (t), from A × y (t) ≡ U × x (t) mod F (t)
And B × y (t) ≡ V × formula of x (t) mod F (t) two are observed, when changing for the numerical value stored in register A, B
Afterwards, the numerical value stored in register U, V can also change therewith；
3) algorithm uses the shifting in hardware operation by the low level parity for the intermediate value for judging to be stored in register
Position and XOR complete iteration and about subtract calculating；
4) iteration Jing Guo certain round will be reduced to 1 with about subtracting the numerical value stored in calculating, register A, entirely remove
The process of method computing is terminated, if U now is UA=1, then identity now will be changed into y (t) ≡ UA=1X (t) mod F (t), i.e.,
UA=1Value it is identical with the r (t) in formula r (t)=y (t)/x (t) mod F (t), now, register U storage numerical value removed for mould
As a result r (t).
The improvement mould of elliptic curve cryptosystem of the invention based on binary field removes the two of method, comprises the following steps：
1) when minimum two of register A are 00, register A will be carried out continuously and move to left twice；Then judge register U's
Numerical value, if minimum two of register U are 00, register U will be carried out continuously and move to left twice；If minimum the two of register U
Position is 10, and register U value will be changed into register U and continuously move to left the data sum moved to left twice with F (t) once；If deposit
Minimum two of device U are 01, register U value will be changed into register U continuously move to left moved to left twice with F (t) data twice it
With；If minimum two of register U are 11, register U value will be changed into register U and continuously move to left to move to left two with F (t) twice
Secondary data move to left data sum once with F (t)；
2) when minimum two of register A are 10, register A will be moved to left once；Then register U number is judged
Value, if register U is even number, then register U will be moved to left once；If register U is odd number, register U value
1/2nd of register U and F (t) sums will be changed into；
3) when minimum two of register B are 00, register B will be carried out continuously and move to left twice；Then judge register V's
Numerical value, if minimum two of register V are 00, register V will be carried out continuously and move to left twice；If minimum the two of register V
Position is 10, and register V value will be changed into register V and continuously move to left the data sum moved to left twice with F (t) once；If deposit
Minimum two of device V are 01, register V value will be changed into register V continuously move to left moved to left twice with F (t) data twice it
With；If minimum two of register V are 11, register V value will be changed into register V and continuously move to left to move to left two with F (t) twice
Secondary data move to left data sum once with F (t)；
4) when minimum two of register B are 10, register B will be moved to left once；Then register V number is judged
Value, if register V is even number, then register V will be moved to left once；If register V is odd number, register V value
1/2nd of register V and F (t) sums will be changed into；
5) when register A is more than register B, A=(A+B)/2 and U=U+V operations are completed first；Then deposit is judged
Device U value, if register U is even number, then register U will be moved to left once, if register U is odd number, then post
Storage U value will be changed into 1/2nd of register U and F (t) sums；
6) during remaining situation, B=(A+B)/2 and V=U+V operations are completed first；Then register V value is judged, if
Register V is even number, then register V will be moved to left once, if register V is odd number, register V value will be changed into posting
Storage V and 1/2nd of F (t) sums；
7) register U value is finally returned to, its value stored is mould division result r (t).
A kind of improvement mould of elliptic curve cryptosystem based on binary field designed by the present invention removes algorithm, right
Shantz moulds are improved except algorithm, and specific improved procedure is algorithm to be iterated during about subtracting, and will be sentenced every time
The parity of the minimum double figures value of the numerical value stored in disconnected register, on the premise of increase is not many hardware resources,
Accelerate calculating process.
The present invention is also based on the design of binary field progress.
In order to meet the demand that every field communicates for actual time safety, the security of AES should be strengthened, again
Improve the arithmetic speed of AES.
Brief description of the drawings
Fig. 1 is register A, U operational block diagram of inventive algorithm.
Fig. 2 is register B, the V operation block diagram of inventive algorithm.
Fig. 3 is the emulation comparative result that inventive algorithm consumes clock number under 50MHz clocks with other mould algorithm for inversions.
Fig. 4 is emulation comparative result of the inventive algorithm under 50MHz clocks with other mould algorithm for inversion throughputs.
Embodiment
Embodiments of the present invention are described further below with reference to Figure of description.
The present invention is that a kind of improvement mould of the elliptic curve cryptosystem based on binary field removes algorithm, is entered using the present invention
The algorithm structure block diagram of row Modulo division refers to Fig. 1 and Fig. 2, and algorithm includes procedure below：
1. initiation parameter：Inventive algorithm is designed is based on binary field GF (2 with checking implementationm), user according to
The Koblitz elliptic curve parameters that NIST recommends, two exponent numbers of setting are less than threshold value m element x (t) and y (t), respectively as
The molecule denominator of input, then, one exponent number of setting are equal to threshold value m irreducible polynomial F (t).
2. initialization register：Four registers A, B, U, V will be used in the present invention, following initialization is carried out respectively and is assigned
Value：A←x(t),B←F(t),U←y(t),V←0.
3. iteration about subtracts：
Complete after initial assignment, algorithm starts to be iterated input and about subtracted, and about subtracts process by judging institute in register
The low level parity of the numerical value of storage, to complete corresponding displacement and xor operation, is embodied as：
1) as A [1:0]==00, A=A/4.U value is judged again, if U [1:0]==00, U=U/4；If U [1:0]=
=10, U=U/4+F (t)/2；If U [1:0]==01, U=U/4+F (t)/4；If U [1:0]==11, U=U/4+F (t)/4+
F(t)/2。
2) as A [1:0]==10, A=A/2.U value is judged again, if U is even number, U=U/2；If U is odd number, U=(U+
F(t))/2。
3) as B [1:0]==00, B=B/4.V value is judged again, if V [1:0]==00, V=V/4；If V [1:0]=
=10, V=V/4+F (t)/2；If V [1:0]==01, V=V/4+F (t)/4；If V [1:0]==11, V=V/4+F (t)/4+
F(t)/2。
4) as B [1:0]==10, B=B/2.V value is judged again, if V is even number, V=V/2；If V is odd number, V=(V+
F(t))/2。
5) as A ＞ B, A=(A+B)/2 and U=U+V.U value is judged again, if U is even number, U=U/2, if U is odd number,
U=(U+F (t))/2.
6) in the case of remaining, B=(A+B)/2, V=U+V operations.V value is judged again, if V is even number, V=V/2, such as
Fruit V is odd number, V=(V+F (t))/2.
4. output result：Iteration by certain round about subtracts, and register A numerical value is reduced to 1, if now U is UA=1, then
There are y (t) ≡ UA=1X (t) mod F (t), i.e. UA=1It is equal with the r (t) in r (t)=y (t)/x (t) mod F (t), therefore register U
The numerical value of storage is mould division result r (t).Wherein inventive algorithm carries out the phase that minimum two bits parity judges to register A, U
Operation (register B, V are similarly) is closed, reference can be made to table 1.
Table 1
Table 2
| Frequency | Area | Critical Path Delay | Cell |
| 250MHz | 0.253mm2 | 3.78ns | 11864 |
Table 3
| Degree(m) | 163 | 233 | 283 | 409 |
| Time(ns) | 4480 | 6240 | 7580 | 11320 |
| Clock | 224 | 312 | 379 | 566 |
5. simulation result：With reference to Fig. 3, it can be seen that inventive algorithm is under 50MHz clocks, when being consumed with other mould algorithm for inversions
Clock
Several comparing results.With reference to Fig. 4, it can be seen that inventive algorithm is handled up under 50MHz clocks with other mould algorithm for inversions
The comparing result of rate.
Synthesis result of the inventive algorithm under 0.18CMOS techniques, referring to table 2, inventive algorithm is in 50MHz clocks
Under different threshold values when consume clock number, referring to table 3.It can be seen that comprehensive knot of the inventive algorithm under 0.18CMOS techniques
Really.It can be seen that inventive algorithm consumes clock number in 50MHz clocks, different threshold values are lower.
The present invention is according to r (t)=y (t)/x (t) mod F (t), and it is initial that register A, B, U, V are first assigned correspondence by algorithm
Value, then by disposably judging the value of minimum two bit binary data in register, correspondence about reducing is realized, until register
It is mould division result r (t) that the numerical value stored in A, which is reduced to the numerical value stored in 1, register U,.Realized and calculated by Verilog language
Method is simultaneously emulated, and contrasts improved Euclidean algorithm and fermat's little theorem algorithm, and the algorithm has advantage in terms of time loss,
Mould is effectively accelerated except calculating, available in ECC encryption and decryption IP kernels.
Claims (2)
1. a kind of improvement mould of the elliptic curve cryptosystem based on binary field removes method, it is characterised in that including following step
Suddenly：
1) according to the relative theory of elliptic curve cryptosystem, it is located at binary field GF (2m) in, it is known that two exponent numbers are less than threshold value
M element x (t) and y (t), respectively as two input elements, while according to NIST (National Institute of Standards and Technology)
The Koblitz elliptic curve parameters recommended, one known exponent number of selection is equal to threshold value m irreducible polynomial F (t)；According to
Mould removes formula r (t)=y (t)/x (t) mod F (t), obtains mould division result r (t), or be expressed as y (t) ≡ r (t) x (t) mod F
(t)；By using intermediate data required in four register A, B, U, V storage algorithms, reach and formula r (t)=y is removed to mould
(t)/x (t) mod F (t), or y (t) ≡ r (t) x (t) mod F (t) are iterated the purpose for about subtracting calculating, first, successively to institute
State four registers A, B, U, V and carry out initialization assignment；
2) after four registers A, B, U, V are completed with initial assignment, algorithm starts the numerical value to being stored in register A, B
It is iterated and about subtracts, during about subtracting, four registers A, B, U, V needs to maintain A × y (t) ≡ U × x (t) mod all the time
The identity of F (t) and B × y (t) ≡ V × two formula of x (t) mod F (t), from A × y (t) ≡ U × x (t) mod F (t) and B
× y (t) ≡ V × formula of x (t) mod F (t) two are observed, it is changed when the numerical value stored in register A, B
Afterwards, the numerical value stored in register U, V can also change therewith；
3) algorithm is by the low level parity for the intermediate value for judging to be stored in register, using the displacement in hardware operation and
XOR completes iteration and about subtracts calculating；
4) iteration Jing Guo certain round will be reduced to 1, whole division fortune with about subtracting the numerical value stored in calculating, register A
The process of calculation is terminated, if U now is UA=1, then identity now will be changed into y (t) ≡ UA=1X (t) mod F (t), i.e. UA=1
Value it is identical with the r (t) in formula r (t)=y (t)/x (t) mod F (t), now, register U storage numerical value for mould except knot
Fruit r (t).
2. a kind of improvement mould of the elliptic curve cryptosystem based on binary field removes method, it is characterised in that including following step
Suddenly：
1) when minimum two of register A are 00, register A will be carried out continuously and move to left twice；Then register U number is judged
Value, if minimum two of register U are 00, register U will be carried out continuously and move to left twice；If minimum two of register U
For 10, register U value will be changed into register U and continuously move to left the data sum moved to left twice with F (t) once；If register U
Minimum two be 01, register U value will be changed into register U and continuously move to left the data sum moved to left twice with F (t) twice；
If minimum two of register U are 11, register U value will be changed into register U and continuously move to left to move to left twice with F (t) twice
Data and F (t) move to left data sum once；
2) when minimum two of register A are 10, register A will be moved to left once；Then register U numerical value is judged, such as
Fruit register U is even number, then register U will be moved to left once；If register U is odd number, register U value will be changed into
Register U and 1/2nd of F (t) sums；
3) when minimum two of register B are 00, register B will be carried out continuously and move to left twice；Then register V number is judged
Value, if minimum two of register V are 00, register V will be carried out continuously and move to left twice；If minimum two of register V
For 10, register V value will be changed into register V and continuously move to left the data sum moved to left twice with F (t) once；If register V
Minimum two be 01, register V value will be changed into register V and continuously move to left the data sum moved to left twice with F (t) twice；
If minimum two of register V are 11, register V value will be changed into register V and continuously move to left to move to left twice with F (t) twice
Data and F (t) move to left data sum once；
4) when minimum two of register B are 10, register B will be moved to left once；Then register V numerical value is judged, such as
Fruit register V is even number, then register V will be moved to left once；If register V is odd number, register V value will be changed into
Register V and 1/2nd of F (t) sums；
5) when register A is more than register B, A=(A+B)/2 and U=U+V operations are completed first；Then judge register U's
Value, if register U is even number, then register U will be moved to left once, if register U is odd number, then register U
Value will be changed into 1/2nd of register U and F (t) sums；
6) during remaining situation, B=(A+B)/2 and V=U+V operations are completed first；Then register V value is judged, if deposit
Device V is even number, then register V will be moved to left once, if register V is odd number, register V value will be changed into register
V and 1/2nd of F (t) sums；
7) register U value is finally returned to, its value stored is mould division result r (t).
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Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| WO2019120066A1 (en) * | 2017-12-20 | 2019-06-27 | 云图有限公司 | Fast mode reduction method and medium suitable for sm2 algorithm |
| CN110999207A (en) * | 2017-08-15 | 2020-04-10 | 区块链控股有限公司 | Computer-implemented method of generating a threshold library |
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2017
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Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN110999207A (en) * | 2017-08-15 | 2020-04-10 | 区块链控股有限公司 | Computer-implemented method of generating a threshold library |
| CN110999207B (en) * | 2017-08-15 | 2024-05-31 | 区块链控股有限公司 | Computer-implemented method of generating a threshold library |
| WO2019120066A1 (en) * | 2017-12-20 | 2019-06-27 | 云图有限公司 | Fast mode reduction method and medium suitable for sm2 algorithm |
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Application publication date: 20170811 |