An arbitary precision unsigned integer type, also known as a “big integer”.
-
-
Operations on big integers never overflow, but they may take a long time to execute.
-The amount of memory (and address space) available is the only constraint to the magnitude of these numbers.
-
-
This particular big integer type uses base-2^64 digits to represent integers; you can think of it as a wrapper
-around Array<UInt64>. (In fact, BigUInt only uses an array if there are more than two digits.)
An arbitary precision signed integer type, also known as a “big integer”.
-
-
Operations on big integers never overflow, but they might take a long time to execute.
-The amount of memory (and address space) available is the only constraint to the magnitude of these numbers.
-
-
This particular big integer type uses base-2^64 digits to represent integers.
-
-
BigInt is essentially a tiny wrapper that extends BigUInt with a sign bit and provides signed integer
-operations. Both the underlying absolute value and the negative/positive flag are available as read-write
-properties.
-
-
Not all algorithms of BigUInt are available for BigInt values; for example, there is no square root or
-primality test for signed integers. When you need to call one of these, just extract the absolute value:
An arbitary precision signed integer type, also known as a “big integer”.
-
-
Operations on big integers never overflow, but they might take a long time to execute.
-The amount of memory (and address space) available is the only constraint to the magnitude of these numbers.
-
-
This particular big integer type uses base-2^64 digits to represent integers.
-
-
BigInt is essentially a tiny wrapper that extends BigUInt with a sign bit and provides signed integer
-operations. Both the underlying absolute value and the negative/positive flag are available as read-write
-properties.
-
-
Not all algorithms of BigUInt are available for BigInt values; for example, there is no square root or
-primality test for signed integers. When you need to call one of these, just extract the absolute value:
Initializes an integer from the bits stored inside a piece of Data.
-The data is assumed to be in network (big-endian) byte order with a first
-byte to represent the sign (0 for positive, 1 for negative)
Return a Data value that contains the base-256 representation of this integer, in network (big-endian) byte order and a prepended byte to indicate the sign (0 for positive, 1 for negative)
- This function can be unreasonably expensive for large exponents, which is why exponent is
- a simple integer value. If you want to calculate big exponents, you’ll probably need to use
- the modulo arithmetic variant.
-
-
-
See also
- BigUInt.power(_:, modulus:)
-
-
-
Complexity
- O((exponent * self.count)^log2(3)) or somesuch. The result may require a large amount of memory, too.
-
-
-
-
-
-
Declaration
-
-
Swift
-
publicfuncpower(_exponent:Int)->BigInt
-
-
-
-
-
Return Value
-
1 if exponent == 0, otherwise self raised to exponent. (This implies that 0.power(0) == 1.)
If gcd(self, modulus) == 1, the value returned is an integer a < modulus such that (a * self) % modulus == 1. If self and modulus aren’t coprime, the return value is nil.
Returns true if this integer is probably prime. Returns false if this integer is definitely not prime.
-
-
This function performs a probabilistic Miller-Rabin Primality Test, consisting of rounds iterations,
-each calculating the strong probable prime test for a random base. The number of rounds is 10 by default,
-but you may specify your own choice.
-
-
To speed things up, the function checks if self is divisible by the first few prime numbers before
-diving into (slower) Miller-Rabin testing.
-
-
Also, when self is less than 82 bits wide, isPrime does a deterministic test that is guaranteed to
-return a correct result.
An arbitary precision unsigned integer type, also known as a “big integer”.
-
-
Operations on big integers never overflow, but they may take a long time to execute.
-The amount of memory (and address space) available is the only constraint to the magnitude of these numbers.
-
-
This particular big integer type uses base-2^64 digits to represent integers; you can think of it as a wrapper
-around Array<UInt64>. (In fact, BigUInt only uses an array if there are more than two digits.)
- This function can be unreasonably expensive for large exponents, which is why exponent is
- a simple integer value. If you want to calculate big exponents, you’ll probably need to use
- the modulo arithmetic variant.
-
-
-
See also
- BigUInt.power(_:, modulus:)
-
-
-
Complexity
- O((exponent * self.count)^log2(3)) or somesuch. The result may require a large amount of memory, too.
-
-
-
-
-
-
Declaration
-
-
Swift
-
publicfuncpower(_exponent:Int)->BigUInt
-
-
-
-
-
Return Value
-
1 if exponent == 0, otherwise self raised to exponent. (This implies that 0.power(0) == 1.)
If gcd(self, modulus) == 1, the value returned is an integer a < modulus such that (a * self) % modulus == 1. If self and modulus aren’t coprime, the return value is nil.
Multiply x by y, and add the result to this integer, optionally shifted shift words to the left.
-
-
Note
- This is the fused multiply/shift/add operation; it is more efficient than doing the components
-individually. (The fused operation doesn’t need to allocate space for temporary big integers.)
-
-
Returns true if this integer is probably prime. Returns false if this integer is definitely not prime.
-
-
This function performs a probabilistic Miller-Rabin Primality Test, consisting of rounds iterations,
-each calculating the strong probable prime test for a random base. The number of rounds is 10 by default,
-but you may specify your own choice.
-
-
To speed things up, the function checks if self is divisible by the first few prime numbers before
-diving into (slower) Miller-Rabin testing.
-
-
Also, when self is less than 82 bits wide, isPrime does a deterministic test that is guaranteed to
-return a correct result.
Subtract other from this integer in place, and return a flag indicating if the operation caused an
-arithmetic overflow. other is shifted shift digits to the left before being subtracted.
-
-
Note
- If the result indicates an overflow, then self becomes the twos’ complement of the absolute difference.
-
-
Subtract other from this integer, returning the difference and a flag indicating arithmetic overflow.
-other is shifted shift digits to the left before being subtracted.
-
-
Note
- If overflow is true, then the result value is the twos’ complement of the absolute value of the difference.
-
-
The number of leading zero bits in the binary representation of this integer in base 2^(Word.bitWidth).
-This is useful when you need to normalize a BigUInt such that the top bit of its most significant word is 1.
-
-
Note
- 0 is considered to have zero leading zero bits.
-
-
-
-
-
-
diff --git a/docs/badge.svg b/docs/badge.svg
deleted file mode 100644
index 09c2300..0000000
--- a/docs/badge.svg
+++ /dev/null
@@ -1,28 +0,0 @@
-
diff --git a/docs/css/295.875b8dab.css b/docs/css/295.875b8dab.css
new file mode 100644
index 0000000..de501ba
--- /dev/null
+++ b/docs/css/295.875b8dab.css
@@ -0,0 +1,9 @@
+/*!
+ * This source file is part of the Swift.org open source project
+ *
+ * Copyright (c) 2021 Apple Inc. and the Swift project authors
+ * Licensed under Apache License v2.0 with Runtime Library Exception
+ *
+ * See https://swift.org/LICENSE.txt for license information
+ * See https://swift.org/CONTRIBUTORS.txt for Swift project authors
+ */aside[data-v-3ccce809]{-moz-column-break-inside:avoid;break-inside:avoid;border-radius:var(--aside-border-radius,var(--border-radius,4px));border-style:var(--aside-border-style,solid);border-width:var(--aside-border-width,0 0 0 6px);padding:.9411764706rem;text-align:start}aside .label[data-v-3ccce809]{font-size:1rem;line-height:1.5294117647;font-weight:600;font-family:var(--typography-html-font,"Helvetica Neue","Helvetica","Arial",sans-serif)}aside .label+[data-v-3ccce809]{margin-top:var(--spacing-stacked-margin-small)}aside.deprecated[data-v-3ccce809]{background-color:var(--color-aside-deprecated-background);border-color:var(--color-aside-deprecated-border);box-shadow:0 0 0 0 var(--color-aside-deprecated-border) inset,0 0 0 0 var(--color-aside-deprecated-border)}aside.deprecated 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diff --git a/docs/css/427.ee43802d.css b/docs/css/427.ee43802d.css
new file mode 100644
index 0000000..cb076bf
--- /dev/null
+++ b/docs/css/427.ee43802d.css
@@ -0,0 +1,9 @@
+/*!
+ * This source file is part of the Swift.org open source project
+ *
+ * Copyright (c) 2021 Apple Inc. and the Swift project authors
+ * Licensed under Apache License v2.0 with Runtime Library Exception
+ *
+ * See https://swift.org/LICENSE.txt for license information
+ * See https://swift.org/CONTRIBUTORS.txt for Swift project authors
+ */.badge[data-v-04624022]{--badge-color:var(--color-badge-default);--badge-dark-color:var(--color-badge-dark-default);font-size:.7058823529rem;line-height:1.3333333333;font-weight:400;font-family:var(--typography-html-font,"Helvetica Neue","Helvetica","Arial",sans-serif);display:inline-block;padding:2px 4px;white-space:nowrap;border-radius:var(--badge-border-radius,1px);border-style:var(--badge-border-style,none);border-width:var(--badge-border-width,1px);margin:auto;margin-left:5px;color:var(--colors-badge-text,var(--color-badge-text));background-color:var(--badge-color)}@media screen{[data-color-scheme=dark] .badge[data-v-04624022]{background-color:var(--badge-dark-color)}}@media screen and (prefers-color-scheme:dark){[data-color-scheme=auto] 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diff --git a/docs/css/documentation-topic.47a08486.css b/docs/css/documentation-topic.47a08486.css
new file mode 100644
index 0000000..53a6fc1
--- /dev/null
+++ b/docs/css/documentation-topic.47a08486.css
@@ -0,0 +1,9 @@
+/*!
+ * This source file is part of the Swift.org open source project
+ *
+ * Copyright (c) 2021 Apple Inc. and the Swift project authors
+ * Licensed under Apache License v2.0 with Runtime Library Exception
+ *
+ * See https://swift.org/LICENSE.txt for license information
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diff --git a/docs/css/tutorials-overview.ddd5b228.css b/docs/css/tutorials-overview.ddd5b228.css
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diff --git a/docs/data/documentation/bigint/bigint/isprime(rounds:).json b/docs/data/documentation/bigint/bigint/isprime(rounds:).json
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diff --git a/docs/data/documentation/bigint/bigint/isstrongprobableprime(_:).json b/docs/data/documentation/bigint/bigint/isstrongprobableprime(_:).json
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diff --git a/docs/data/documentation/bigint/bigint/iszero.json b/docs/data/documentation/bigint/bigint/iszero.json
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new file mode 100644
index 0000000..345351c
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diff --git a/docs/data/documentation/bigint/biguint/iszero.json b/docs/data/documentation/bigint/biguint/iszero.json
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diff --git a/docs/data/documentation/bigint/biguint/leadingzerobitcount.json b/docs/data/documentation/bigint/biguint/leadingzerobitcount.json
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diff --git a/docs/data/documentation/bigint/biguint/magnitude.json b/docs/data/documentation/bigint/biguint/magnitude.json
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diff --git a/docs/data/documentation/bigint/biguint/multiplyandadd(_:_:shiftedby:).json b/docs/data/documentation/bigint/biguint/multiplyandadd(_:_:shiftedby:).json
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index 0000000..ffbdb87
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diff --git a/docs/data/documentation/bigint/biguint/subscript(bitat:).json b/docs/data/documentation/bigint/biguint/subscript(bitat:).json
new file mode 100644
index 0000000..0313c6e
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diff --git a/docs/data/documentation/bigint/biguint/subtract(_:shiftedby:).json b/docs/data/documentation/bigint/biguint/subtract(_:shiftedby:).json
new file mode 100644
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diff --git a/docs/data/documentation/bigint/biguint/subtracting(_:shiftedby:).json b/docs/data/documentation/bigint/biguint/subtracting(_:shiftedby:).json
new file mode 100644
index 0000000..ee7bf9f
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diff --git a/docs/data/documentation/bigint/biguint/subtractingreportingoverflow(_:shiftedby:).json b/docs/data/documentation/bigint/biguint/subtractingreportingoverflow(_:shiftedby:).json
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diff --git a/docs/data/documentation/bigint/biguint/subtractreportingoverflow(_:shiftedby:).json b/docs/data/documentation/bigint/biguint/subtractreportingoverflow(_:shiftedby:).json
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index 0000000..e53b5dc
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diff --git a/docs/data/documentation/bigint/biguint/trailingzerobitcount.json b/docs/data/documentation/bigint/biguint/trailingzerobitcount.json
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new file mode 100644
index 0000000..168663f
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diff --git a/docs/data/documentation/bigint/swift/string/init(_:).json b/docs/data/documentation/bigint/swift/string/init(_:).json
new file mode 100644
index 0000000..d03a698
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diff --git a/docs/data/documentation/bigint/swift/string/init(_:radix:uppercase:)-247k6.json b/docs/data/documentation/bigint/swift/string/init(_:radix:uppercase:)-247k6.json
new file mode 100644
index 0000000..d268654
--- /dev/null
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diff --git a/docs/data/documentation/bigint/swift/string/init(_:radix:uppercase:)-2sg1t.json b/docs/data/documentation/bigint/swift/string/init(_:radix:uppercase:)-2sg1t.json
new file mode 100644
index 0000000..cd51a29
--- /dev/null
+++ b/docs/data/documentation/bigint/swift/string/init(_:radix:uppercase:)-2sg1t.json
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diff --git a/docs/docsets/BigInt.docset/Contents/Info.plist b/docs/docsets/BigInt.docset/Contents/Info.plist
deleted file mode 100644
index 220c951..0000000
--- a/docs/docsets/BigInt.docset/Contents/Info.plist
+++ /dev/null
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-
-
-
-
- CFBundleIdentifier
- com.jazzy.bigint
- CFBundleName
- BigInt
- DocSetPlatformFamily
- bigint
- isDashDocset
-
- dashIndexFilePath
- index.html
- isJavaScriptEnabled
-
- DashDocSetFamily
- dashtoc
- DashDocSetFallbackURL
- https://attaswift.github.io/BigInt/reference/
-
-
diff --git a/docs/docsets/BigInt.docset/Contents/Resources/Documents/Extensions.html b/docs/docsets/BigInt.docset/Contents/Resources/Documents/Extensions.html
deleted file mode 100644
index d64d667..0000000
--- a/docs/docsets/BigInt.docset/Contents/Resources/Documents/Extensions.html
+++ /dev/null
@@ -1,173 +0,0 @@
-
-
-
- Extensions Reference
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
An arbitary precision unsigned integer type, also known as a “big integer”.
-
-
Operations on big integers never overflow, but they may take a long time to execute.
-The amount of memory (and address space) available is the only constraint to the magnitude of these numbers.
-
-
This particular big integer type uses base-2^64 digits to represent integers; you can think of it as a wrapper
-around Array<UInt64>. (In fact, BigUInt only uses an array if there are more than two digits.)
An arbitary precision signed integer type, also known as a “big integer”.
-
-
Operations on big integers never overflow, but they might take a long time to execute.
-The amount of memory (and address space) available is the only constraint to the magnitude of these numbers.
-
-
This particular big integer type uses base-2^64 digits to represent integers.
-
-
BigInt is essentially a tiny wrapper that extends BigUInt with a sign bit and provides signed integer
-operations. Both the underlying absolute value and the negative/positive flag are available as read-write
-properties.
-
-
Not all algorithms of BigUInt are available for BigInt values; for example, there is no square root or
-primality test for signed integers. When you need to call one of these, just extract the absolute value:
An arbitary precision signed integer type, also known as a “big integer”.
-
-
Operations on big integers never overflow, but they might take a long time to execute.
-The amount of memory (and address space) available is the only constraint to the magnitude of these numbers.
-
-
This particular big integer type uses base-2^64 digits to represent integers.
-
-
BigInt is essentially a tiny wrapper that extends BigUInt with a sign bit and provides signed integer
-operations. Both the underlying absolute value and the negative/positive flag are available as read-write
-properties.
-
-
Not all algorithms of BigUInt are available for BigInt values; for example, there is no square root or
-primality test for signed integers. When you need to call one of these, just extract the absolute value:
Initializes an integer from the bits stored inside a piece of Data.
-The data is assumed to be in network (big-endian) byte order with a first
-byte to represent the sign (0 for positive, 1 for negative)
Return a Data value that contains the base-256 representation of this integer, in network (big-endian) byte order and a prepended byte to indicate the sign (0 for positive, 1 for negative)
- This function can be unreasonably expensive for large exponents, which is why exponent is
- a simple integer value. If you want to calculate big exponents, you’ll probably need to use
- the modulo arithmetic variant.
-
-
-
See also
- BigUInt.power(_:, modulus:)
-
-
-
Complexity
- O((exponent * self.count)^log2(3)) or somesuch. The result may require a large amount of memory, too.
-
-
-
-
-
-
Declaration
-
-
Swift
-
publicfuncpower(_exponent:Int)->BigInt
-
-
-
-
-
Return Value
-
1 if exponent == 0, otherwise self raised to exponent. (This implies that 0.power(0) == 1.)
If gcd(self, modulus) == 1, the value returned is an integer a < modulus such that (a * self) % modulus == 1. If self and modulus aren’t coprime, the return value is nil.
Returns true if this integer is probably prime. Returns false if this integer is definitely not prime.
-
-
This function performs a probabilistic Miller-Rabin Primality Test, consisting of rounds iterations,
-each calculating the strong probable prime test for a random base. The number of rounds is 10 by default,
-but you may specify your own choice.
-
-
To speed things up, the function checks if self is divisible by the first few prime numbers before
-diving into (slower) Miller-Rabin testing.
-
-
Also, when self is less than 82 bits wide, isPrime does a deterministic test that is guaranteed to
-return a correct result.
An arbitary precision unsigned integer type, also known as a “big integer”.
-
-
Operations on big integers never overflow, but they may take a long time to execute.
-The amount of memory (and address space) available is the only constraint to the magnitude of these numbers.
-
-
This particular big integer type uses base-2^64 digits to represent integers; you can think of it as a wrapper
-around Array<UInt64>. (In fact, BigUInt only uses an array if there are more than two digits.)
- This function can be unreasonably expensive for large exponents, which is why exponent is
- a simple integer value. If you want to calculate big exponents, you’ll probably need to use
- the modulo arithmetic variant.
-
-
-
See also
- BigUInt.power(_:, modulus:)
-
-
-
Complexity
- O((exponent * self.count)^log2(3)) or somesuch. The result may require a large amount of memory, too.
-
-
-
-
-
-
Declaration
-
-
Swift
-
publicfuncpower(_exponent:Int)->BigUInt
-
-
-
-
-
Return Value
-
1 if exponent == 0, otherwise self raised to exponent. (This implies that 0.power(0) == 1.)
If gcd(self, modulus) == 1, the value returned is an integer a < modulus such that (a * self) % modulus == 1. If self and modulus aren’t coprime, the return value is nil.
Multiply x by y, and add the result to this integer, optionally shifted shift words to the left.
-
-
Note
- This is the fused multiply/shift/add operation; it is more efficient than doing the components
-individually. (The fused operation doesn’t need to allocate space for temporary big integers.)
-
-
Returns true if this integer is probably prime. Returns false if this integer is definitely not prime.
-
-
This function performs a probabilistic Miller-Rabin Primality Test, consisting of rounds iterations,
-each calculating the strong probable prime test for a random base. The number of rounds is 10 by default,
-but you may specify your own choice.
-
-
To speed things up, the function checks if self is divisible by the first few prime numbers before
-diving into (slower) Miller-Rabin testing.
-
-
Also, when self is less than 82 bits wide, isPrime does a deterministic test that is guaranteed to
-return a correct result.
Subtract other from this integer in place, and return a flag indicating if the operation caused an
-arithmetic overflow. other is shifted shift digits to the left before being subtracted.
-
-
Note
- If the result indicates an overflow, then self becomes the twos’ complement of the absolute difference.
-
-
Subtract other from this integer, returning the difference and a flag indicating arithmetic overflow.
-other is shifted shift digits to the left before being subtracted.
-
-
Note
- If overflow is true, then the result value is the twos’ complement of the absolute value of the difference.
-
-
The number of leading zero bits in the binary representation of this integer in base 2^(Word.bitWidth).
-This is useful when you need to normalize a BigUInt such that the top bit of its most significant word is 1.
-
-
Note
- 0 is considered to have zero leading zero bits.
-
-
This repository provides integer types of arbitrary width implemented
-in 100% pure Swift. The underlying representation is in base 2^64, using Array<UInt64>.
Two big integer types are included: BigUInt and BigInt,
-the latter being the signed variant.
-Both of these are Swift structs with copy-on-write value semantics, and they can be used much
-like any other integer type.
-
-
The library provides implementations for some of the most frequently useful functions on
-big integers, including
The implementations are intended to be reasonably efficient, but they are unlikely to be
-competitive with GMP at all, even when I happened to implement an algorithm with same asymptotic
-behavior as GMP. (I haven’t performed a comparison benchmark, though.)
-
-
The library has 100% unit test coverage. Sadly this does not imply that there are no bugs
-in it.
BigInt 4.0.0 requires Swift 4.2 (The last version with support for Swift 3.x was BigInt 2.1.0.
-The last version with support for Swift 2 was BigInt 1.3.0.)
-
-
-
-
Swift Version
-
last BigInt Version
-
-
-
-
3.x
-
2.1.0
-
-
-
4.0
-
3.1.0
-
-
-
4.2
-
4.0.0
-
-
-
5.x
-
5.3.0
-
-
-
-
BigInt deploys to macOS 10.10, iOS 9, watchOS 2 and tvOS 9.
-It has been tested on the latest OS releases only—however, as the module uses very few platform-provided APIs,
-there should be very few issues with earlier versions.
-
-
BigInt uses no APIs specific to Apple platforms, so
-it should be easy to port it to other operating systems.
-
-
Setup instructions:
-
-
-
Swift Package Manager:
-Although the Package Manager is still in its infancy, BigInt provides experimental support for it.
-Add this to the dependency section of your Package.swift manifest:
BigUInt is a MutableCollectionType of its 64-bit digits, with the least significant
-digit at index 0. As a convenience, BigUInt allows you to subscript it with indexes at
-or above its count. The subscript operator returns 0 for out-of-bound gets and
-automatically extends the array on out-of-bound sets. This makes memory management simpler.
-
-
BigInt is just a tiny wrapper around a BigUInt [absolute value][magnitude] and a
-sign bit, both of which are accessible as public read-write properties.
The types provided by BigInt are not parametric—this is very much intentional, as
-Swift generics cost us dearly at runtime in this use case. In every approach I tried,
-making arbitrary-precision arithmetic operations work with a generic Digit type parameter
-resulted in code that was literally ten times slower. If you can make the algorithms generic
-without such a huge performance hit, please enlighten me!
-
-
This is an area that I plan to investigate more, as it would be useful to have generic
-implementations for arbitrary-width arithmetic operations. (Polynomial division and decimal bases
-are two examples.) The library already implements double-digit multiplication and division as
-extension methods on a protocol with an associated type requirement; this has not measurably affected
-performance. Unfortunately, the same is not true for BigUInt‘s methods.
-
-
Of course, as a last resort, we could just duplicate the code to create a separate
-generic variant that was slower but more flexible.
Let’s start with a simple function that generates a random n-bit prime. The module
-includes a function to generate random integers of a specific size, and also an
-isPrime method that performs the Miller–Rabin primality test. These are all we need:
-
func generatePrime(_ width: Int) -> BigUInt {
- while true {
- var random = BigUInt.randomInteger(withExactWidth: width)
- random |= BigUInt(1)
- if random.isPrime() {
- return random
- }
- }
-}
-
-let p = generatePrime(1024)
-==> 13308187650642192396256419911012544845370493728424936791561478318443071617242872
- 81980956747087187419914435169914161116601678883358611076800743580556055714173922
- 08406194264346635072293912609713085260354070700055888678514690878149253177960273
- 775659537560220378850112471985434373425534121373466492101182463962031
-
-let q = generatePrime(1024)
-==> 17072954422657145489547308812333368925007949054501204983863958355897172093173783
- 10108226596943999553784252564650624766276133157586733504784616138305701168410157
- 80784336308507083874651158029602582993233111593356512531869546706885170044355115
- 669728424124141763799008880327106952436883614887277350838425336156327
-
-
-
Cool! Now that we have two large primes, we can produce an RSA public/private keypair
-out of them.
Let’s try out our new keypair by converting a string into UTF-8, interpreting
-the resulting binary representation as a big integer, and encrypting it with the
-public key. BigUInt has an initializer that takes an NSData, so this is pretty
-easy to do:
Well, it looks encrypted all right, but can we get the original message back?
-In theory, encrypting the cyphertext with the private key returns the original message.
-Let’s see:
-
let plaintext = encrypt(cyphertext, key: privateKey)
-==> 83323446846105976078466731524728681905293067701804838925389198929123912971229457
- 68818568737
-
-let received = String(data: plaintext.serialize(), encoding: NSUTF8StringEncoding)
-==> "Arbitrary precision arithmetic is fun!"
-
-
-
Yay! This is truly terrific, but please don’t use this example code in an actual
-cryptography system. RSA has lots of subtle (and some not so subtle) complications
-that we ignored to keep this example short.
Another fun activity to try with BigInts is to generate the digits of π.
-Let’s try implementing Jeremy Gibbon’s spigot algorithm.
-This is a rather slow algorithm as π-generators go, but it makes up for it with its grooviness
-factor: it’s remarkably short, it only uses (big) integer arithmetic, and every iteration
-produces a single new digit in the base-10 representation of π. This naturally leads to an
-implementation as an infinite GeneratorType:
-
func digitsOfPi() -> AnyGenerator<Int> {
- var q: BigUInt = 1
- var r: BigUInt = 180
- var t: BigUInt = 60
- var i: UInt64 = 2 // Does not overflow until digit #826_566_842
- return AnyIterator {
- let u: UInt64 = 3 * (3 * i + 1) * (3 * i + 2)
- let y = (q.multiplied(byDigit: 27 * i - 12) + 5 * r) / (5 * t)
- (q, r, t) = (
- 10 * q.multiplied(byDigit: i * (2 * i - 1)),
- 10 * (q.multiplied(byDigit: 5 * i - 2) + r - y * t).multiplied(byDigit: u),
- t.multiplied(byDigit: u))
- i += 1
- return Int(y[0])
- }
-}
-
-
-
Well, that was surprisingly easy. But does it work? Of course it does!
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"}}
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- BigInt Reference
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This repository provides integer types of arbitrary width implemented
-in 100% pure Swift. The underlying representation is in base 2^64, using Array<UInt64>.
Two big integer types are included: BigUInt and BigInt,
-the latter being the signed variant.
-Both of these are Swift structs with copy-on-write value semantics, and they can be used much
-like any other integer type.
-
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The library provides implementations for some of the most frequently useful functions on
-big integers, including
The implementations are intended to be reasonably efficient, but they are unlikely to be
-competitive with GMP at all, even when I happened to implement an algorithm with same asymptotic
-behavior as GMP. (I haven’t performed a comparison benchmark, though.)
-
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The library has 100% unit test coverage. Sadly this does not imply that there are no bugs
-in it.
BigInt 4.0.0 requires Swift 4.2 (The last version with support for Swift 3.x was BigInt 2.1.0.
-The last version with support for Swift 2 was BigInt 1.3.0.)
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Swift Version
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last BigInt Version
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3.x
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2.1.0
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4.0
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3.1.0
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4.2
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4.0.0
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5.x
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5.3.0
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BigInt deploys to macOS 10.10, iOS 9, watchOS 2 and tvOS 9.
-It has been tested on the latest OS releases only—however, as the module uses very few platform-provided APIs,
-there should be very few issues with earlier versions.
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BigInt uses no APIs specific to Apple platforms, so
-it should be easy to port it to other operating systems.
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Setup instructions:
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Swift Package Manager:
-Although the Package Manager is still in its infancy, BigInt provides experimental support for it.
-Add this to the dependency section of your Package.swift manifest:
BigUInt is a MutableCollectionType of its 64-bit digits, with the least significant
-digit at index 0. As a convenience, BigUInt allows you to subscript it with indexes at
-or above its count. The subscript operator returns 0 for out-of-bound gets and
-automatically extends the array on out-of-bound sets. This makes memory management simpler.
-
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BigInt is just a tiny wrapper around a BigUInt [absolute value][magnitude] and a
-sign bit, both of which are accessible as public read-write properties.
The types provided by BigInt are not parametric—this is very much intentional, as
-Swift generics cost us dearly at runtime in this use case. In every approach I tried,
-making arbitrary-precision arithmetic operations work with a generic Digit type parameter
-resulted in code that was literally ten times slower. If you can make the algorithms generic
-without such a huge performance hit, please enlighten me!
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This is an area that I plan to investigate more, as it would be useful to have generic
-implementations for arbitrary-width arithmetic operations. (Polynomial division and decimal bases
-are two examples.) The library already implements double-digit multiplication and division as
-extension methods on a protocol with an associated type requirement; this has not measurably affected
-performance. Unfortunately, the same is not true for BigUInt‘s methods.
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Of course, as a last resort, we could just duplicate the code to create a separate
-generic variant that was slower but more flexible.
Let’s start with a simple function that generates a random n-bit prime. The module
-includes a function to generate random integers of a specific size, and also an
-isPrime method that performs the Miller–Rabin primality test. These are all we need:
-
func generatePrime(_ width: Int) -> BigUInt {
- while true {
- var random = BigUInt.randomInteger(withExactWidth: width)
- random |= BigUInt(1)
- if random.isPrime() {
- return random
- }
- }
-}
-
-let p = generatePrime(1024)
-==> 13308187650642192396256419911012544845370493728424936791561478318443071617242872
- 81980956747087187419914435169914161116601678883358611076800743580556055714173922
- 08406194264346635072293912609713085260354070700055888678514690878149253177960273
- 775659537560220378850112471985434373425534121373466492101182463962031
-
-let q = generatePrime(1024)
-==> 17072954422657145489547308812333368925007949054501204983863958355897172093173783
- 10108226596943999553784252564650624766276133157586733504784616138305701168410157
- 80784336308507083874651158029602582993233111593356512531869546706885170044355115
- 669728424124141763799008880327106952436883614887277350838425336156327
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Cool! Now that we have two large primes, we can produce an RSA public/private keypair
-out of them.
Let’s try out our new keypair by converting a string into UTF-8, interpreting
-the resulting binary representation as a big integer, and encrypting it with the
-public key. BigUInt has an initializer that takes an NSData, so this is pretty
-easy to do:
Well, it looks encrypted all right, but can we get the original message back?
-In theory, encrypting the cyphertext with the private key returns the original message.
-Let’s see:
-
let plaintext = encrypt(cyphertext, key: privateKey)
-==> 83323446846105976078466731524728681905293067701804838925389198929123912971229457
- 68818568737
-
-let received = String(data: plaintext.serialize(), encoding: NSUTF8StringEncoding)
-==> "Arbitrary precision arithmetic is fun!"
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Yay! This is truly terrific, but please don’t use this example code in an actual
-cryptography system. RSA has lots of subtle (and some not so subtle) complications
-that we ignored to keep this example short.
Another fun activity to try with BigInts is to generate the digits of π.
-Let’s try implementing Jeremy Gibbon’s spigot algorithm.
-This is a rather slow algorithm as π-generators go, but it makes up for it with its grooviness
-factor: it’s remarkably short, it only uses (big) integer arithmetic, and every iteration
-produces a single new digit in the base-10 representation of π. This naturally leads to an
-implementation as an infinite GeneratorType:
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func digitsOfPi() -> AnyGenerator<Int> {
- var q: BigUInt = 1
- var r: BigUInt = 180
- var t: BigUInt = 60
- var i: UInt64 = 2 // Does not overflow until digit #826_566_842
- return AnyIterator {
- let u: UInt64 = 3 * (3 * i + 1) * (3 * i + 2)
- let y = (q.multiplied(byDigit: 27 * i - 12) + 5 * r) / (5 * t)
- (q, r, t) = (
- 10 * q.multiplied(byDigit: i * (2 * i - 1)),
- 10 * (q.multiplied(byDigit: 5 * i - 2) + r - y * t).multiplied(byDigit: u),
- t.multiplied(byDigit: u))
- i += 1
- return Int(y[0])
- }
-}
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Well, that was surprisingly easy. But does it work? Of course it does!