PKCS 1

The PKCS #1 standard defines the mathematical definitions and properties that RSA public and private keys must have. The traditional key pair is based on a modulus, ${\displaystyle n}$, that is the product of two distinct large prime numbers, ${\displaystyle p}$ and ${\displaystyle q}$, such that ${\displaystyle n=pq}$.

Starting with version 2.1, this definition was generalized to allow for multi-prime keys, where the number of distinct primes may be two or more. When dealing with multi-prime keys, the prime factors are all generally labeled as ${\displaystyle r_{i}}$ for some ${\displaystyle i}$, such that:

${\displaystyle n=r_{1}\cdot r_{2}\cdot ...\cdot r_{i},}$ for ${\displaystyle i\geq 2}$

As a notational convenience, ${\displaystyle p=r_{1}}$ and ${\displaystyle q=r_{2}}$.

The RSA public key is represented as the tuple ${\displaystyle (n,e)}$, where the integer ${\displaystyle e}$ is the public exponent.

The RSA private key may have two representations. The first compact form is the tuple ${\displaystyle (n,d)}$, where ${\displaystyle d}$ is the private exponent. The second form has at least five terms (p, q, dp, dq, qinv), or more for multi-prime keys. Although mathematically redundant to the compact form, the additional terms allow for certain computational optimizations when using the key. In particular, the second format allows to derive the public key.[1]

The standard defines several basic primitives. The primitive operations provide the fundamental instructions for turning the raw mathematical formulas into computable algorithms.

• I2OSP - Integer to Octet String Primitive - Converts a (potentially very large) non-negative integer into a sequence of bytes (octet string).
• OS2IP - Octet String to Integer Primitive - Interprets a sequence of bytes as a non-negative integer
• RSAEP - RSA Encryption Primitive - Encrypts a message using a public key
• RSADP - RSA Decryption Primitive - Decrypts ciphertext using a private key
• RSASP1 - RSA Signature Primitive 1 - Creates a signature over a message using a private key
• RSAVP1 - RSA Verification Primitive 1 - Verifies a signature is for a message using a public key

By themselves the primitive operations do not necessarily provide any security. The concept of a cryptographic scheme is to define higher level algorithms or uses of the primitives so they achieve certain security goals.

There are two schemes for encryption and decryption:

• RSAES-OAEP: improved Encryption/decryption Scheme; based on the Optimal Asymmetric Encryption Padding scheme proposed by Mihir Bellare and Phillip Rogaway.
• RSAES-PKCS1-v1_5: older encryption/decryption scheme as first standardized in version 1.5 of PKCS #1.

Note: A small change was made to RSAES-OAEP in PKCS #1 version 2.1, causing RSAES-OAEP in PKCS #1 version 2.0 to be totally incompatible with RSA-OAEP in PKCS #1 version 2.1 and version 2.2.

There are also two schemes for dealing with signatures:

• RSASSA-PSS: improved Probabilistic Signature Scheme with appendix; based on the probabilistic signature scheme originally invented by Bellare and Rogaway.
• RSASSA-PKCS1-v1_5: old Signature Scheme with Appendix as first standardized in version 1.5 of PKCS #1.

The two signature schemes make use of separately defined encoding methods:

• EMSA-PSS: encoding method for signature appendix, probabilistic signature scheme.
• EMSA-PKCS1-v1_5: encoding method for signature appendix as first standardized in version 1.5 of PKCS #1.

The signature schemes are actually signatures with appendix, which means that rather than signing some input data directly, a hash function is used first to produce an intermediary representation of the data, and then the result of the hash is signed. This technique is almost always used with RSA because the amount of data that can be directly signed is proportional to the size of the keys; which is almost always much smaller than the amount of data an application may wish to sign.

• Versions 1.1–1.3, February through March 1991, privately distributed.
• Version 1.4, June 1991, published for NIST/OSI Implementors' Workshop.
• Version 1.5, November 1993. First public publication. Republished as RFC 2313.
• Version 2.0, September 1998. Republished as RFC 2437.
• Version 2.1, June 2002. Republished as RFC 3447.
• Version 2.2, October 2012. Republished as RFC 8017.

Below is a list of cryptography libraries that provide support for PKCS#1:

• Botan
• Bouncy Castle
• cryptlib
• Crypto++
• Libgcrypt
• mbed TLS
• Nettle
• OpenSSL
• wolfCrypt

Multiple attacks were discovered against PKCS #1 v1.5.[2][3]

In 1998, Daniel Bleichenbacher published a seminal paper on what became known as Bleichenbacher's attack (also known as "million message attack").[3][4] PKCS #1 was subsequently updated in the release 2.0 and patches were issued to users wishing to continue using the old version of the standard.[2] With slight variations this vulnerability still exists in many modern servers.[5]

In 2006, Bleichenbacher presented a new forgery attack against the signature scheme RSASSA-PKCS1-v1_5.[6]

• Comparison of cryptography libraries

• RFC 8017 - PKCS #1: RSA Cryptography Specifications Version 2.2
• PKCS #1 v2.2: RSA Cryptography Standard at the Wayback Machine (archived April 10, 2016)
• Raising the Standard for RSA Signatures: RSA-PSS at the Wayback Machine (archived 2004-04-04)