RSA加密标准指南（V2.1）：公钥与私钥原理详解

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RSA加密手册（2.1版）是RSA Laboratories发布的Public-Key Cryptography Standards (PKCS) #1系列标准中的一个重要文档，该版本更新于2003年2月，旨在提供关于非对称加密算法RSA的全面指南。这份文档由Network Working Group的J. Jonsson和B. Kaliski共同编写，取代了之前的RFC 2437。它并不定义互联网标准，但作为公开资料，可供全球范围内的网络社区参考。该手册详细介绍了RSA公钥和私钥的概念与使用。在第3节中，作者着重解释了RSA的公钥（包括模数n、公共指数e）以及私钥（包括私钥d、欧拉函数φ(n)），这些都是RSA加密和解密过程的基础。4.1节和4.2节分别介绍了数据转换工具，如I2OSP（Integer to Octet String Primitive）和OS2IP（Octet String to Integer Primitive），它们用于在不同数据类型间转换，以便进行数字运算。 5.1节概述了加密和解密操作，展示了如何利用公钥加密数据，使用私钥解密，确保数据只有拥有私钥的人能够访问。同时，5.2节讨论了签名和验证机制，即如何通过RSA算法创建数字签名来保证数据完整性和来源的真实性，以及接收者如何验证这些签名。 6.1部分提供了整体的加密方案概述，可能涵盖了不同应用环境下的RSA实施策略和技术细节。对于开发者来说，这部分内容对于理解和实现RSA算法在实际通信中的应用至关重要。 7.加密章节具体阐述了如何使用RSA进行信息的加密和传输，以及安全参数的选择和处理。这部分内容深入到了实际加密技术的底层，对保护敏感信息具有实践指导意义。 RSA加密手册（2.1版）是RSA算法的权威指南，为密码学专业人士、软件开发人员和信息安全研究人员提供了关键的技术规范和实现方法，对于理解和使用RSA进行安全通信和数据保护具有重要意义。无论是初学者还是高级开发者，这份文档都是学习和理解非对称加密系统的核心参考资料。

RFC 3447

PKCS #1: RSA Cryptography Specifications February 2003

Steps:

1. If the ciphertext representative c is not between 0 and n - 1,

output "ciphertext representative out of range" and stop.

2. The message representative m is computed as follows.

a. If the first form (n, d) of K is used, let m = c^d mod n.

b. If the second form (p, q, dP, dQ, qInv) and (r_i, d_i, t_i)

of K is used, proceed as follows:

i. Let m_1 = c^dP mod p and m_2 = c^dQ mod q.

ii. If u > 2, let m_i = c^(d_i) mod r_i, i = 3, ..., u.

iii. Let h = (m_1 - m_2) * qInv mod p.

iv. Let m = m_2 + q * h.

v. If u > 2, let R = r_1 and for i = 3 to u do

1. Let R = R * r_(i-1).

2. Let h = (m_i - m) * t_i mod r_i.

3. Let m = m + R * h.

3. Output m.

Note. Step 2.b can be rewritten as a single loop, provided that one

reverses the order of p and q. For consistency with PKCS #1 v2.0,

however, the first two primes p and q are treated separately from

the additional primes.

5.2 Signature and verification primitives

A signature primitive produces a signature representative from a

message representative under the control of a private key, and a

verification primitive recovers the message representative from the

signature representative under the control of the corresponding

public key. One pair of signature and verification primitives is

employed in the signature schemes defined in this document and is

specified here: RSASP1/RSAVP1.

The primitives defined here are the same as IFSP-RSA1/IFVP-RSA1 in

IEEE 1363-2000 [

26] (except that support for multi-prime RSA has

been added) and are compatible with PKCS #1 v1.5.

Jonsson & Kaliski Informational [Page 12]

RFC 3447

PKCS #1: RSA Cryptography Specifications February 2003

parties may select RSA public and private keys for a scheme

operation. The specific additional operations and other details are

outside the scope of this document.

As was the case for the cryptographic primitives (

Section 5), the

specifications of scheme operations assume that certain conditions

are met by the inputs, in particular that RSA public and private keys

are valid. The behavior of an implementation is thus unspecified

when a key is invalid. The impact of such unspecified behavior

depends on the application. Possible means of addressing key

validation include explicit key validation by the application; key

validation within the public-key infrastructure; and assignment of

liability for operations performed with an invalid key to the party

who generated the key.

A generally good cryptographic practice is to employ a given RSA key

pair in only one scheme. This avoids the risk that vulnerability in

one scheme may compromise the security of the other, and may be

essential to maintain provable security. While RSAES-PKCS1-v1_5

(

Section 7.2) and RSASSA-PKCS1-v1_5 (Section 8.2) have traditionally

been employed together without any known bad interactions (indeed,

this is the model introduced by PKCS #1 v1.5), such a combined use of

an RSA key pair is not recommended for new applications.

To illustrate the risks related to the employment of an RSA key pair

in more than one scheme, suppose an RSA key pair is employed in both

RSAES-OAEP (

Section 7.1) and RSAES-PKCS1-v1_5. Although RSAES-OAEP

by itself would resist attack, an opponent might be able to exploit a

weakness in the implementation of RSAES-PKCS1-v1_5 to recover

messages encrypted with either scheme. As another example, suppose

an RSA key pair is employed in both RSASSA-PSS (

Section 8.1) and

RSASSA-PKCS1-v1_5. Then the security proof for RSASSA-PSS would no

longer be sufficient since the proof does not account for the

possibility that signatures might be generated with a second scheme.

Similar considerations may apply if an RSA key pair is employed in

one of the schemes defined here and in a variant defined elsewhere.

7. Encryption schemes

For the purposes of this document, an encryption scheme consists of

an encryption operation and a decryption operation, where the

encryption operation produces a ciphertext from a message with a

recipient’s RSA public key, and the decryption operation recovers the

message from the ciphertext with the recipient’s corresponding RSA

private key.

Jonsson & Kaliski Informational [Page 15]

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