openssl - 使用 RSA 公钥在 Openssl 中生成私钥？

Question

我知道可以使用 this link在 OpenSSL 中为自签名证书生成公钥和私钥。但是对于给定的公钥，我是否有可能找出相应的私钥？我一直在使用 1024 位 RSA 公钥。
因为我在作业中有这个问题说:
为句子“我的名字是 .我的声音就是我的护照。”使用具有以下 1024 位 RSA 公钥的 OpenSSL 进行正确验证。 (提示:模数可能不像正常的 RSA 模数那样生成。):

-----BEGIN PUBLIC KEY-----
MIGdMA0GCSqGSIb3DQEBAQUAA4GLADCBhwKBgQCgF35rHhOWi9+r4n9xM/ejvMEs
Q8h6lams962k4U0WSdfySUevhyI1bd3FRIb5fFqSBt6qPTiiiIw0KXte5dANB6lP
e6HdUPTA/U4xHWi2FB/BfAyPsOlUBfFp6dtkEEcEKt+Z8KTJYJEerRie24y+nsfZ
MnLBst6tsEBfx/U75wIBAw== 
-----END PUBLIC KEY-----

Answer 1

解决方案的关键(双关语)在提示中:

The modulus might not have been generated like a normal RSA modulus.

这是我采取的方法。
第 1 步:导出素数

我先把你的文件保存为public.pem .

为了获得模数 ( n ) 和公共(public)指数 ( e ) 的值，我运行:openssl rsa -pubin -in public.pem -text -noout

我将十六进制模数和指数转换为十进制数，得到n=112420265940019545385580931264662691888876377549063413938338239508058300548918731393322848876821656910452908064089039911552450302375557565600923056341141750687524704844725632296552824986371719004485250857447936962589230504662333990648942759862805127715014382377701044586628936249950092121536791020138692688871e=3

我将该号码插入 online factorialization calculator发现用于计算模数的两个素数之一相对较小:p=55685342628135644993q=2018848419246646476894946094575564515176862561629979956283227393349426117194195173357244644821277073710795134539986018769393928719340504755806449531413017314396784334912136112253736003497362080917517151753555605597776865614151048604681116557282512513238254935296910445878892354969335089447

第 2 步:计算其他所需值

我计算了n的总和如ϕ(n) = (p - 1) * (q - 1)使用python REPL:ϕ(n)=112420265940019545383562082845416045411981431454487849423161376946428320592635503999973422759627461737095663419267762837841655167835571546831529127621801245931718255313312614982156040651459582892231514853950574881671713352908778385051165894248654079110333265820418532073390681314653181675602213322541221954432

然后我使用了 this answer 中的 python 脚本。计算私有(private)指数 ( d ) 和系数 ( c )，得到:d=74946843960013030255708055230277363607987620969658566282107584630952213728423669333315615173084974491397108946178508558561103445223714364554352751747867497287812170208875076654770693767639721928154343235967049921114475568605852256700777262832436052740222177213612354715593787543102121117068142215027481302955c=1040291110785843997

第三步:创建私钥的ASN.1结构
然后，我使用计算值创建了 ASN.1名为 asn 的文件中的结构如 this answer 中所述:

asn1=SEQUENCE:rsa_key

[rsa_key]
version=INTEGER:0
modulus=INTEGER:112420265940019545385580931264662691888876377549063413938338239508058300548918731393322848876821656910452908064089039911552450302375557565600923056341141750687524704844725632296552824986371719004485250857447936962589230504662333990648942759862805127715014382377701044586628936249950092121536791020138692688871
pubExp=INTEGER:3
privExp=INTEGER:74946843960013030255708055230277363607987620969658566282107584630952213728423669333315615173084974491397108946178508558561103445223714364554352751747867497287812170208875076654770693767639721928154343235967049921114475568605852256700777262832436052740222177213612354715593787543102121117068142215027481302955
p=INTEGER:55685342628135644993
q=INTEGER:2018848419246646476894946094575564515176862561629979956283227393349426117194195173357244644821277073710795134539986018769393928719340504755806449531413017314396784334912136112253736003497362080917517151753555605597776865614151048604681116557282512513238254935296910445878892354969335089447
e1=INTEGER:37123561752090429995
e2=INTEGER:903312890059631
coeff=INTEGER:1040291110785843997

第四步:创建私钥
基于ASN.1结构，我生成的私钥如下:

以 DER 格式创建私钥:openssl asn1parse -genconf asn -out private.der

将私钥转换为 PEM 格式:openssl rsa -in private.der -inform der -out private.pem -outform pem

这导致 private.pem正在使用以下内容创建文件:

-----BEGIN RSA PRIVATE KEY-----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-----END RSA PRIVATE KEY-----

第 5 步:验证结果
为了检查创建的私钥(在 private.pem 中)是否与提供的公钥匹配，我刚刚从私钥生成了一个新的公钥:

openssl rsa -in private.pem -pubout

这会产生以下输出:

writing RSA key
-----BEGIN PUBLIC KEY-----
MIGdMA0GCSqGSIb3DQEBAQUAA4GLADCBhwKBgQCgF35rHhOWi9+r4n9xM/ejvMEs
Q8h6lams962k4U0WSdfySUevhyI1bd3FRIb5fFqSBt6qPTiiiIw0KXte5dANB6lP
e6HdUPTA/U4xHWi2FB/BfAyPsOlUBfFp6dtkEEcEKt+Z8KTJYJEerRie24y+nsfZ
MnLBst6tsEBfx/U75wIBAw==
-----END PUBLIC KEY-----

此输出与您提供的公钥完全匹配。