RSA Public Key Representations

Representation Bytes for Reference Key
SubjectPublicKeyInfo (ASN.1) 294
RSAPublicKey (ASN.1) 270
COSE Key 268
ssh-rsa (SSH Public Key) 279
ssh-rsa without prefix string 268
ssh-rsa without prefix string or modulus length 264
4-byte exponent length 263
1-byte exponent length 260

Reference Key

Example 1 from multicodec#233 (comment).

Modulus is an integer represented by 2048 bits. Exponent is integer 65537.

JWK

Reference: RFC 7518: JSON Web Algorithms (JWA): 6.3.1. Parameters for RSA Public Keys

{
  "kty": "RSA",
  "n": "sbX82NTV6IylxCh7MfV4hlyvaniCajuP97GyOqSvTmoEdBOflFvZ06kR_9D6ctt45Fk6hskfnag2GG69NALVH2o4RCR6tQiLRpKcMRtDYE_thEmfBvDzm_VVkOIYfxu-Ipuo9J_S5XDNDjczx2v-3oDh5-CIHkU46hvFeCvpUS-L8TJSbgX0kjVk_m4eIb9wh63rtmD6Uz_KBtCo5mmR4TEtcLZKYdqMp3wCjN-TlgHiz_4oVXWbHUefCEe8rFnX1iQnpDHU49_SaXQoud1jCaexFn25n-Aa8f8bc5Vm-5SeRwidHa6ErvEhTvf1dz6GoNPp2iRvm-wJ1gxwWJEYPQ",
  "e": "AQAB"
}
Modulus ("n"), converted from base64url to hex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Representations

SubjectPublicKeyInfo (ASN.1)

Reference: RFC 5280: PKIX Certificate and CRL Profile: 4.1. Basic Certificate Fields [Page 17].

SubjectPublicKeyInfo  ::=  SEQUENCE  {
     algorithm            AlgorithmIdentifier,
     subjectPublicKey     BIT STRING  }

This format is not RSA-specific but is often used to wrap a RSAPublicKey ASN.1 data structure (in the subjectPublicKey field).

A RSAPublicKey/DER key can be converted to SubjectPublicKeyInfo/DER using the OpenSSL CLI: 

openssl rsa -RSAPublicKey_in -inform DER -in rsapk.der -outform DER -out spki.der

Reference Key

294 bytes

Base64-encoded
MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAsbX82NTV6IylxCh7MfV4hlyvaniCajuP
97GyOqSvTmoEdBOflFvZ06kR/9D6ctt45Fk6hskfnag2GG69NALVH2o4RCR6tQiLRpKcMRtDYE/t
hEmfBvDzm/VVkOIYfxu+Ipuo9J/S5XDNDjczx2v+3oDh5+CIHkU46hvFeCvpUS+L8TJSbgX0kjVk
/m4eIb9wh63rtmD6Uz/KBtCo5mmR4TEtcLZKYdqMp3wCjN+TlgHiz/4oVXWbHUefCEe8rFnX1iQn
pDHU49/SaXQoud1jCaexFn25n+Aa8f8bc5Vm+5SeRwidHa6ErvEhTvf1dz6GoNPp2iRvm+wJ1gxw
WJEYPQIDAQAB

Reference Key, ASN.1 object dump

Using dumpasn1:

  0 290: SEQUENCE {
  4  13:   SEQUENCE {
  6   9:     OBJECT IDENTIFIER rsaEncryption (1 2 840 113549 1 1 1)
 17   0:     NULL
       :     }
 19 271:   BIT STRING, encapsulates {
 24 266:     SEQUENCE {
 28 257:       INTEGER
       :         00 B1 B5 FC D8 D4 D5 E8 8C A5 C4 28 7B 31 F5 78
       :         86 5C AF 6A 78 82 6A 3B 8F F7 B1 B2 3A A4 AF 4E
       :         6A 04 74 13 9F 94 5B D9 D3 A9 11 FF D0 FA 72 DB
       :         78 E4 59 3A 86 C9 1F 9D A8 36 18 6E BD 34 02 D5
       :         1F 6A 38 44 24 7A B5 08 8B 46 92 9C 31 1B 43 60
       :         4F ED 84 49 9F 06 F0 F3 9B F5 55 90 E2 18 7F 1B
       :         BE 22 9B A8 F4 9F D2 E5 70 CD 0E 37 33 C7 6B FE
       :         DE 80 E1 E7 E0 88 1E 45 38 EA 1B C5 78 2B E9 51
       :                 [ Another 129 bytes skipped ]
289   3:       INTEGER 65537
       :       }
       :     }
       :   }

↑

RSAPublicKey (ASN.1)

Reference: RFC 8017: PKCS #1: RSA Cryptography Specifications Version 2.2: A.1.1. RSA Public Key Syntax.

RSAPublicKey ::= SEQUENCE {
    modulus           INTEGER,  -- n
    publicExponent    INTEGER   -- e
}

ASN.1 is specified in ITU-T X.680.

See also SubjectPublicKeyInfo which can be used to wrap this type.

Reference Key

270 bytes

Base64-encoded
MIIBCgKCAQEAsbX82NTV6IylxCh7MfV4hlyvaniCajuP97GyOqSvTmoEdBOflFvZ06kR/9D6ctt4
5Fk6hskfnag2GG69NALVH2o4RCR6tQiLRpKcMRtDYE/thEmfBvDzm/VVkOIYfxu+Ipuo9J/S5XDN
Djczx2v+3oDh5+CIHkU46hvFeCvpUS+L8TJSbgX0kjVk/m4eIb9wh63rtmD6Uz/KBtCo5mmR4TEt
cLZKYdqMp3wCjN+TlgHiz/4oVXWbHUefCEe8rFnX1iQnpDHU49/SaXQoud1jCaexFn25n+Aa8f8b
c5Vm+5SeRwidHa6ErvEhTvf1dz6GoNPp2iRvm+wJ1gxwWJEYPQIDAQAB

Reference Key, ASN.1 object dump

Using dumpasn1:

  0 266: SEQUENCE {
  4 257:   INTEGER
       :     00 B1 B5 FC D8 D4 D5 E8 8C A5 C4 28 7B 31 F5 78
       :     86 5C AF 6A 78 82 6A 3B 8F F7 B1 B2 3A A4 AF 4E
       :     6A 04 74 13 9F 94 5B D9 D3 A9 11 FF D0 FA 72 DB
       :     78 E4 59 3A 86 C9 1F 9D A8 36 18 6E BD 34 02 D5
       :     1F 6A 38 44 24 7A B5 08 8B 46 92 9C 31 1B 43 60
       :     4F ED 84 49 9F 06 F0 F3 9B F5 55 90 E2 18 7F 1B
       :     BE 22 9B A8 F4 9F D2 E5 70 CD 0E 37 33 C7 6B FE
       :     DE 80 E1 E7 E0 88 1E 45 38 EA 1B C5 78 2B E9 51
       :     2F 8B F1 32 52 6E 05 F4 92 35 64 FE 6E 1E 21 BF
       :     70 87 AD EB B6 60 FA 53 3F CA 06 D0 A8 E6 69 91
       :     E1 31 2D 70 B6 4A 61 DA 8C A7 7C 02 8C DF 93 96
       :     01 E2 CF FE 28 55 75 9B 1D 47 9F 08 47 BC AC 59
       :     D7 D6 24 27 A4 31 D4 E3 DF D2 69 74 28 B9 DD 63
       :     09 A7 B1 16 7D B9 9F E0 1A F1 FF 1B 73 95 66 FB
       :     94 9E 47 08 9D 1D AE 84 AE F1 21 4E F7 F5 77 3E
       :     86 A0 D3 E9 DA 24 6F 9B EC 09 D6 0C 70 58 91 18
       :     3D
265   3:   INTEGER 65537
       :   }

↑

COSE Key

Reference: RFC 8230: Using RSA Algorithms with CBOR Object Signing and Encryption (COSE) Messages: Table 4: RSA Key Parameters

Reference Key

268 bytes

Hex-encoded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Dictionary output

Using python-cwt:

{1: 3, -1: b'\xb1\xb5\xfc\xd8\xd4\xd5\xe8\x8c\xa5\xc4({1\xf5x\x86\\\xafjx\x82j;\x8f\xf7\xb1\xb2:\xa4\xafNj\x04t\x13\x9f\x94[\xd9\xd3\xa9\x11\xff\xd0\xfar\xdbx\xe4Y:\x86\xc9\x1f\x9d\xa86\x18n\xbd4\x02\xd5\x1fj8D$z\xb5\x08\x8bF\x92\x9c1\x1bC`O\xed\x84I\x9f\x06\xf0\xf3\x9b\xf5U\x90\xe2\x18\x7f\x1b\xbe"\x9b\xa8\xf4\x9f\xd2\xe5p\xcd\x0e73\xc7k\xfe\xde\x80\xe1\xe7\xe0\x88\x1eE8\xea\x1b\xc5x+\xe9Q/\x8b\xf12Rn\x05\xf4\x925d\xfen\x1e!\xbfp\x87\xad\xeb\xb6`\xfaS?\xca\x06\xd0\xa8\xe6i\x91\xe11-p\xb6Ja\xda\x8c\xa7|\x02\x8c\xdf\x93\x96\x01\xe2\xcf\xfe(Uu\x9b\x1dG\x9f\x08G\xbc\xacY\xd7\xd6$\'\xa41\xd4\xe3\xdf\xd2it(\xb9\xddc\t\xa7\xb1\x16}\xb9\x9f\xe0\x1a\xf1\xff\x1bs\x95f\xfb\x94\x9eG\x08\x9d\x1d\xae\x84\xae\xf1!N\xf7\xf5w>\x86\xa0\xd3\xe9\xda$o\x9b\xec\t\xd6\x0cpX\x91\x18=', -2: b'\x01\x00\x01'}

Note: RFC 7049: Concise Binary Object Representation (CBOR): 3.9. Canonical CBOR needed?

↑

ssh-rsa (SSH Public Key)

Reference: RFC 4253: SSH Transport Layer Protocol: 6.6. Public Key Algorithms [Page 14]

The "ssh-rsa" key format has the following specific encoding:

string    "ssh-rsa"
mpint     e
mpint     n

string and mpint are defined in RFC 4251: SSH Protocol Architecture: 5. Data Type Representations Used in the SSH Protocols as containing a length prefix as a 4-byte unsigned integer in network byte order. Signed integer data follows for mpint.

SSH public key format is used in the SSH protocols, including SSH Agent protocol, and in file formats (public key files, authorized keys files).

Reference Key

279 bytes

Base64-encoded
AAAAB3NzaC1yc2EAAAADAQABAAABAQCxtfzY1NXojKXEKHsx9XiGXK9qeIJqO4/3sbI6pK9OagR0
E5+UW9nTqRH/0Ppy23jkWTqGyR+dqDYYbr00AtUfajhEJHq1CItGkpwxG0NgT+2ESZ8G8POb9VWQ
4hh/G74im6j0n9LlcM0ONzPHa/7egOHn4IgeRTjqG8V4K+lRL4vxMlJuBfSSNWT+bh4hv3CHreu2
YPpTP8oG0KjmaZHhMS1wtkph2oynfAKM35OWAeLP/ihVdZsdR58IR7ysWdfWJCekMdTj39JpdCi5
3WMJp7EWfbmf4Brx/xtzlWb7lJ5HCJ0droSu8SFO9/V3Poag0+naJG+b7AnWDHBYkRg9
Annotated hex
00000007        (4 bytes) length of "ssh-rsa"
7373682d727361  (7 bytes) "ssh-rsa" (ASCII)
00000003        (4 bytes) length of e
010001          (3 bytes) e
00000101        (4 bytes) length of n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     (257 bytes) n

↑

ssh-rsa without prefix string

Like the "ssh-rsa" key format, but without the static "ssh-key" string prefix.

mpint     e
mpint     n

Reference Key

268 bytes

Base64-encoded
AAAAAwEAAQAAAQEAsbX82NTV6IylxCh7MfV4hlyvaniCajuP97GyOqSvTmoEdBOflFvZ06kR/9D6
ctt45Fk6hskfnag2GG69NALVH2o4RCR6tQiLRpKcMRtDYE/thEmfBvDzm/VVkOIYfxu+Ipuo9J/S
5XDNDjczx2v+3oDh5+CIHkU46hvFeCvpUS+L8TJSbgX0kjVk/m4eIb9wh63rtmD6Uz/KBtCo5mmR
4TEtcLZKYdqMp3wCjN+TlgHiz/4oVXWbHUefCEe8rFnX1iQnpDHU49/SaXQoud1jCaexFn25n+Aa
8f8bc5Vm+5SeRwidHa6ErvEhTvf1dz6GoNPp2iRvm+wJ1gxwWJEYPQ==

↑

ssh-rsa without prefix string or modulus length

Like ssh-rsa without prefix string, but also remove the length prefix for "n". The length of "n" is implicit as the amount of bytes remaining in the buffer. So the format consists of:

  1. Length of e (4 bytes)
  2. e (3 bytes for reference key)
  3. n (257 bytes for reference key)

Reference Key

264 bytes

Base64-encoded
AAAAAwEAAQCxtfzY1NXojKXEKHsx9XiGXK9qeIJqO4/3sbI6pK9OagR0E5+UW9nTqRH/0Ppy23jk
WTqGyR+dqDYYbr00AtUfajhEJHq1CItGkpwxG0NgT+2ESZ8G8POb9VWQ4hh/G74im6j0n9LlcM0O
NzPHa/7egOHn4IgeRTjqG8V4K+lRL4vxMlJuBfSSNWT+bh4hv3CHreu2YPpTP8oG0KjmaZHhMS1w
tkph2oynfAKM35OWAeLP/ihVdZsdR58IR7ysWdfWJCekMdTj39JpdCi53WMJp7EWfbmf4Brx/xtz
lWb7lJ5HCJ0droSu8SFO9/V3Poag0+naJG+b7AnWDHBYkRg9

↑

4-byte exponent length

Like ssh-rsa without prefix string or modulus length, but integers are unsigned and so do not have any leading zeros, saving up to 2 bytes.

  1. Length of e (4 bytes)
  2. e (3 bytes for reference key)
  3. n (256 bytes for reference key)

Reference Key

263 bytes

Base64-encoded
AAAAAwEAAbG1/NjU1eiMpcQoezH1eIZcr2p4gmo7j/exsjqkr05qBHQTn5Rb2dOpEf/Q+nLbeORZ
OobJH52oNhhuvTQC1R9qOEQkerUIi0aSnDEbQ2BP7YRJnwbw85v1VZDiGH8bviKbqPSf0uVwzQ43
M8dr/t6A4efgiB5FOOobxXgr6VEvi/EyUm4F9JI1ZP5uHiG/cIet67Zg+lM/ygbQqOZpkeExLXC2
SmHajKd8Aozfk5YB4s/+KFV1mx1HnwhHvKxZ19YkJ6Qx1OPf0ml0KLndYwmnsRZ9uZ/gGvH/G3OV
ZvuUnkcInR2uhK7xIU739Xc+hqDT6dokb5vsCdYMcFiRGD0=

↑

1-byte exponent length

Like 4-byte exponent length, but one byte instead of four for the length the public exponent.

This was proposed in multicodec#235.

This representation can only encode exponents whose size can be represented in a one-byte integer, i.e. whose length is a less than 256 bytes. This is not a problem if we are following NIST Digital Signature Standard (DSS) (FIPS 186-4), as that constrains the exponent as 216 < e < 2256 (B.3.1 Criteria for IFC Key Pairs - Page 52), effecting a maximum length (exclusive) of 256 bits; that can be represented with 32 bytes. So one byte is more than enough to represent the exponent length: the length will range from 3 to 32. The two high bits will be always zero so could even be repurposed if we wanted to do that.

  1. Length of e (1 byte)
  2. e (3 bytes for reference key)
  3. n (256 bytes for reference key)

Reference Key

260 bytes

Base64-encoded
AwEAAbG1/NjU1eiMpcQoezH1eIZcr2p4gmo7j/exsjqkr05qBHQTn5Rb2dOpEf/Q+nLbeORZOobJ
H52oNhhuvTQC1R9qOEQkerUIi0aSnDEbQ2BP7YRJnwbw85v1VZDiGH8bviKbqPSf0uVwzQ43M8dr
/t6A4efgiB5FOOobxXgr6VEvi/EyUm4F9JI1ZP5uHiG/cIet67Zg+lM/ygbQqOZpkeExLXC2SmHa
jKd8Aozfk5YB4s/+KFV1mx1HnwhHvKxZ19YkJ6Qx1OPf0ml0KLndYwmnsRZ9uZ/gGvH/G3OVZvuU
nkcInR2uhK7xIU739Xc+hqDT6dokb5vsCdYMcFiRGD0=
Annotated hex
03          (1 byte) length of exponent
010001      (3 bytes) exponent
b1b5fcd8d4d5e88ca5c4287b31f578865caf6a78826a3b8ff7b1
b23aa4af4e6a0474139f945bd9d3a911ffd0fa72db78e4593a86c91f9da8
36186ebd3402d51f6a3844247ab5088b46929c311b43604fed84499f06f0
f39bf55590e2187f1bbe229ba8f49fd2e570cd0e3733c76bfede80e1e7e0
881e4538ea1bc5782be9512f8bf132526e05f4923564fe6e1e21bf7087ad
ebb660fa533fca06d0a8e66991e1312d70b64a61da8ca77c028cdf939601
e2cffe2855759b1d479f0847bcac59d7d62427a431d4e3dfd2697428b9dd
6309a7b1167db99fe01af1ff1b739566fb949e47089d1dae84aef1214ef7
f5773e86a0d3e9da246f9bec09d60c705891183d    (256 bytes) n

↑

About this document

Latest version: https://celehner.com/2021/11/rsapk/
This version: https://celehner.com/2021/11/rsapk/19.html (November 19, 2021, 4:16pm EST)
Previous version: https://celehner.com/2021/11/rsapk/16-11.html (November 16, 2021, 11:51am EST)

Changes since previous version

Author: Charles E. Lehner (Spruce Systems, Inc.)