SEC1-V2 ECC标准详解：高效加密数学基础与操作指南

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"SEC1-V2 ECC规范是一份由Certicom Research发布的关于椭圆曲线密码学(Elliptic Curve Cryptography, ECC)的标准文档。该标准详细地概述了ECC算法的核心概念、数学基础以及其实现过程中的关键步骤。ECC是一种基于椭圆曲线上离散对数问题的公钥加密技术，其安全性建立在复杂的数学理论之上，特别适合在资源受限的设备上进行高效加密和身份验证。在SEC1中，作者Daniel R. L. Brown着重介绍了以下内容： 1. 简介：简要概述了ECC的概念，强调它在现代信息安全领域的应用，特别是在轻量级计算和物联网(IoT)设备中的优势。 2. 目标与合规性：明确了标准的目标是提供一套高效且安全的ECC算法实现方法，确保遵循国际和行业的标准要求，如FIPS(联邦信息处理标准)。 3. 文档发展：SEC1 Version 2.0可能包含了先前版本的更新和改进，反映了ECC算法技术的最新进展和最佳实践。 4. 知识产权：强调了复制文档的许可条件，要求所有引用或提及文档时必须注明其来源。 5. 数学基础：深入解析了ECC所需的基础数学概念，包括有限域(Finite Fields)的性质，如有限域Fp和F2m的定义，以及椭圆曲线的构造和特性。 6. 数据类型与转换：详细说明了如何在不同数据格式之间转换，例如将比特串转化为字节串，以及椭圆曲线点和有限域元素与字节串的相互转换，这对于实现算法的实际操作至关重要。 7. 加密与解密：尽管这部分内容未直接给出，但可以推测这部分会涵盖ECC算法的具体加密和解密过程，包括椭圆曲线数字签名(ECDSA)和椭圆曲线 Diffie-Hellman 密钥交换(ECDH)等关键协议的实现方法。 SEC1-V2 ECC规范对于开发人员、密码学研究者以及从事网络安全工作的专业人士来说，是一份不可或缺的技术参考文档，帮助他们理解和构建基于ECC的加密系统。这份文档不仅提供了理论指导，还提供了实用的编码和转换工具，以便于在实际应用场景中实现高效的ECC算法。"

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2.3 Data Types and Conversions SEC 1 Ver. 2.0

2.3.2 Octet-String-to-Bit-String Conversion

Octet strings should be converted to bit strings as described in this section. Informally the idea is

simply to view the octet string as a bit string instead. Formally the conversion routine is speciﬁed

as follows:

Input: An octet string M of length mlen octets.

Output: A bit string B of length blen = 8(mlen) bits.

Actions: Convert the octet string M = M

. . . M

mlen−1

to a bit string B = B

. . . B

blen−1

follows :

1. For 0 ≤ i ≤ mlen − 1, set:

8i+1

. . . B

8i+7

= M

2. Output B.

2.3.3 Elliptic-Curve-Point-to-Octet-String Conversion

Elliptic curve points should be converted to octet strings as described in this section. Informally,

if point compression is being used, the idea is that the compressed y-coordinate is placed in the

leftmost octet of the octet string along with an indication that point compression is on, and the

x-coordinate is placed in the remainder of the octet string; otherwise if point compression is oﬀ,

the leftmost octet indicates that point compression is oﬀ, and the remainder of the octet s tring

contains the x-coordinate followed by the y-coordinate. Formally the conversion routine is speciﬁed

as follows:

Setup: Decide whether or not to represent points using point compression.

Input: A point P on an elliptic curve over F

deﬁned by the ﬁeld elements a, b.

Output: An octet string M of length mlen octets where mlen = 1 if P = O, mlen = d(log

q)/8e+1

if P 6= O and point compression is used, and mlen = 2d(log

q)/8e + 1 if P 6= O and point

compression is not us ed.

Actions: Convert P to an octet string M = M

. . . M

mlen−1

as follows:

1. If P = O, output M = 00

2. If P = (x

, y

) 6= O and point compression is being used, proceed as follows:

2.1. Convert the ﬁeld element x

to an octet string X of length d(log

q)/8e o ctets using the

conversion routine speciﬁed in Section 2.3.5.

2.2. Derive from y

a single bit ˜y

as follows (this allows the y-coordinate to be represented

compactly using a single bit):

2.2.1. If q = p is an odd prime, set ˜y

= y

(mod 2).

2.2.2. If q = 2

, set ˜y

= 0 if x

= 0, otherwise compute z = z

m−1

+ ··· + z

x + z

such that z = y

−1

and set ˜y

= z

Page 10 of 138 §2 Mathematical Foundations

SEC 1 Ver. 2.0 2.3 Data Types and Conversions

2.3. Assign the value 02

to the single octet Y if ˜y

= 0, or the value 03

if ˜y

= 1.

2.4. Output M = Y kX.

3. If P = (x

, y

) 6= O and point compression is not being used, proceed as follows:

3.1. Convert the ﬁeld element x

to an octet string X of length d(log

q)/8e o ctets using the

conversion routine speciﬁed in Section 2.3.5.

3.2. Convert the ﬁeld element y

to an octet string Y of length d(log

q)/8e octets using the

conversion routine speciﬁed in Section 2.3.5.

3.3. Output M = 04

kX kY .

2.3.4 Octet-String-to-Elliptic-Curve-Point Conversion

Octet strings should be converted to elliptic curve points as described in this section. Informally

the idea is that, if the octet string represents a compressed point, the compressed y-coordinate is

recovered from the leftmost octet, the x-coordinate is recovered from the remainder of the octet

string, and then the point compression process is reverse d; otherwise the leftmost octet of the octet

string is removed, the x-coordinate is recovered from the left half of the remaining octet string,

and the y-coordinate is recovered from the right half of the remaining octet string. Formally the

conversion routine is speciﬁed as follows:

Input: An elliptic curve over F

deﬁned by the ﬁeld elements a, b, and an octet string M which is

either the single octet 00

, an octet string of length mlen = d(log

q)/8e + 1, or an octet string of

length mlen = 2d(log

q)/8e + 1.

Output: An elliptic curve point P , or “invalid”.

Actions: Convert M to an elliptic curve point P as follows:

1. If M = 00

, output P = O .

2. If M has length d(log

q)/8e + 1 octets, proceed as follows:

2.1. Parse M = Y kX as a single octet Y followed by d(log

q)/8e octets X.

2.2. Convert X to a ﬁeld element x

of F

using the conversion routine speciﬁed in Sec-

tion 2.3.6. Output “invalid” and stop if the routine outputs “invalid”.

2.3. If Y = 02

, set ˜y

= 0, and if Y = 03

, set ˜y

= 1. Otherwise output “invalid” and

stop.

2.4. Derive from x

and ˜y

an elliptic curve point P = (x

, y

), where:

2.4.1. If q = p is an odd prime, compute the ﬁeld element α ≡ x

+ ax

+ b(mod p), and

compute a square root β of α modulo p. Output “invalid” and stop if there are

no square roots of α modulo p, otherwise set y

= β if β ≡ ˜y

(mod 2), and set

= p −β if β 6≡ ˜y

(mod 2).

2.4.2. If q = 2

and x

= 0, output y

= b

m−1

in F

§2 Mathematical Foundations Page 11 of 138

2.3 Data Types and Conversions SEC 1 Ver. 2.0

2.4.3. If q = 2

and x

6= 0, compute the ﬁeld element β = x

+ a + bx

−2

in F

, and

ﬁnd an element z = z

m−1

+ ···+ z

x + z

such that z

+ z = β in F

. Output

“invalid” and stop if no such z exists, otherwise set y

= x

z in F

if z

= ˜y

and set y

= x

(z + 1) in F

if z

6= ˜y

2.5. Output P = (x

, y

3. If M has length 2d(log

q)/8e + 1 octets, proceed as follows:

3.1. Parse M = W kX kY as a single octet W followed by d(log

q)/8e octets X followed by

d(log

q)/8e octets Y .

3.2. Check that W = 04

. If W 6= 04

, output “invalid” and stop.

3.3. Convert X to a ﬁeld element x

of F

using the conversion routine speciﬁed in Sec-

tion 2.3.6. Output “invalid” and stop if the routine outputs “invalid”.

3.4. Convert Y to a ﬁeld element y

of F

using the conversion routine speciﬁed in Sec-

tion 2.3.6. Output “invalid” and stop if the routine outputs “invalid”.

3.5. Check that P = (x

, y

) satisﬁes the deﬁning equation of the elliptic curve.

3.6. Output P = (x

, y

2.3.5 Field-Element-to-Octet-String Conversion

Field elements should be converted to octet strings as described in this section. Informally the

idea is that, if the ﬁeld is F

, convert the integer to an octet string, and if the ﬁeld is F

, view the

coeﬃcients of the polynomial as a bit string with the highest degree term on the left and convert

the bit string to an octet string. Formally the conversion routine is speciﬁed as follows:

Input: An element a of the ﬁeld F

Output: An octet string M of length mlen = d(log

q)/8e octets.

Actions: Convert a to an octet string M = M

. . . M

mlen−1

as follows:

1. If q = p is an odd prime, then a is an integer in the interval [0, p −1]. Convert a to M using

the conversion routine speciﬁed in Section 2.3.7 (with a and mlen as inputs). Output M.

2. If q = 2

, then a = a

m−1

+ ··· + a

x + a

is a binary polynomial. Convert a to M as

follows :

2.1. For 0 < i ≤ mlen − 1, let:

= a

7+8(mlen−1−i)

6+8(mlen−1−i)

. . . a

8(mlen−1−i)

2.2. Let M

have its leftmost 8(mlen) −m bits set to 0, and its rightmost 8 −(8(mlen) −m)

bits set to a

m−1

m−2

. . . a

8(mlen)−8

2.3. Output M.

Page 12 of 138 §2 Mathematical Foundations

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