BigNum

BigNum provides a Swift wrapper for the BoringSSL BIGNUM library.

It provides most of the standard library functions

• Basic arithmetic operators (with and without modulus)
• Bitwise operators
• Powers (with and without modulus)
• Greatest common denominator
• Prime generation
• Random number generation

Examples

Factorial

Below is a function that creates factorial 1000 and then verifies that for every number from 1 to 1000 the greatest common denominator between the variable factorial and that number is equal to that number.

        var factorial = BigNum(1)
        for i in 1..<1000 {
            factorial = factorial * BigNum(i)
        }
        for i in 1..<1000 {
            assert(BigNum.gcd(i, factorial) == i)
        }

fyi factorial 1000 is quite a big number

402387260077093773543702433923003985719374864210714632543799910429938512398629020592044208486969404800479988610197196058631666872994808558901323829669944590997424504087073759918823627727188732519779505950995276120874975462497043601418278094646496291056393887437886487337119181045825783647849977012476632889835955735432513185323958463075557409114262417474349347553428646576611667797396668820291207379143853719588249808126867838374559731746136085379534524221586593201928090878297308431392844403281231558611036976801357304216168747609675871348312025478589320767169132448426236131412508780208000261683151027341827977704784635868170164365024153691398281264810213092761244896359928705114964975419909342221566832572080821333186116811553615836546984046708975602900950537616475847728421889679646244945160765353408198901385442487984959953319101723355556602139450399736280750137837615307127761926849034352625200015888535147331611702103968175921510907788019393178114194545257223865541461062892187960223838971476088506276862967146674697562911234082439208160153780889893964518263243671616762179168909779911903754031274622289988005195444414282012187361745992642956581746628302955570299024324153181617210465832036786906117260158783520751516284225540265170483304226143974286933061690897968482590125458327168226458066526769958652682272807075781391858178889652208164348344825993266043367660176999612831860788386150279465955131156552036093988180612138558600301435694527224206344631797460594682573103790084024432438465657245014402821885252470935190620929023136493273497565513958720559654228749774011413346962715422845862377387538230483865688976461927383814900140767310446640259899490222221765904339901886018566526485061799702356193897017860040811889729918311021171229845901641921068884387121855646124960798722908519296819372388642614839657382291123125024186649353143970137428531926649875337218940694281434118520158014123344828015051399694290153483077644569099073152433278288269864602789864321139083506217095002597389863554277196742822248757586765752344220207573630569498825087968928162753848863396909959826280956121450994871701244516461260379029309120889086942028510640182154399457156805941872748998094254742173582401063677404595741785160829230135358081840096996372524230560855903700624271243416909004153690105933983835777939410970027753472000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

Secure Remote Password

Another standard operation that BigNum can be used for is generating Secure Remote Password keys. Assuming we have the following

• Safe prime N
• Generator value g (very commonly 2)
• Random number a
• A hashing function H
• username and password

A value A is calculated and sent to the server

A = g.power(a, modulus: N)

The server responds with a large value B and a salt value. Then the client generates the password authentication key

// calculate u = H(A,B)
let u = BigNum(data: H(A.data, B.data))

// calculate x = H(salt , H(userId | ":" | password))
let message = Data("\(username):\(password)".utf8)
let x = BigNum(data: H(salt, H(message)))

// calculate k = H(N,g)
let k = BigNum(data: H(N.data, g.data))

// calculate S
let S = (B - k * g.power(x, modulus: N)).power(a + u * x, modulus: N)

A hashed version of S can be sent back to the server and the server can use that to verify the correct password was provided.

Compatibility

BigNum uses a vendored cutdown version of BoringSSL (Google's version of OpenSSL) so doesn't require a separate OpenSSL library. This means it can be run on iOS and on macOS and Linux platforms without requiring a separate library to be installed.