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# Instructions | ||
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Convert a sequence of digits in one base, representing a number, into a sequence of digits in another base, representing the same number. | ||
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~~~~exercism/note | ||
Try to implement the conversion yourself. | ||
Do not use something else to perform the conversion for you. | ||
~~~~ | ||
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## About [Positional Notation][positional-notation] | ||
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In positional notation, a number in base **b** can be understood as a linear combination of powers of **b**. | ||
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The number 42, _in base 10_, means: | ||
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`(4 × 10¹) + (2 × 10⁰)` | ||
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The number 101010, _in base 2_, means: | ||
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`(1 × 2⁵) + (0 × 2⁴) + (1 × 2³) + (0 × 2²) + (1 × 2¹) + (0 × 2⁰)` | ||
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The number 1120, _in base 3_, means: | ||
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`(1 × 3³) + (1 × 3²) + (2 × 3¹) + (0 × 3⁰)` | ||
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_Yes. Those three numbers above are exactly the same. Congratulations!_ | ||
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[positional-notation]: https://en.wikipedia.org/wiki/Positional_notation |
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# Introduction | ||
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You've just been hired as professor of mathematics. | ||
Your first week went well, but something is off in your second week. | ||
The problem is that every answer given by your students is wrong! | ||
Luckily, your math skills have allowed you to identify the problem: the student answers _are_ correct, but they're all in base 2 (binary)! | ||
Amazingly, it turns out that each week, the students use a different base. | ||
To help you quickly verify the student answers, you'll be building a tool to translate between bases. |
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{ | ||
"authors": [ | ||
"erikschierboom" | ||
], | ||
"files": { | ||
"solution": [ | ||
"all-your-base.ua" | ||
], | ||
"test": [ | ||
"tests.ua" | ||
], | ||
"example": [ | ||
".meta/example.ua" | ||
] | ||
}, | ||
"blurb": "Convert a number, represented as a sequence of digits in one base, to any other base." | ||
} |
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ValidBases ← ( | ||
⍤"input base must be >= 2"≥2 | ||
⍤"output base must be >= 2"≥2 | ||
) | ||
ValidDigits ← ⍤"all digits must satisfy 0 <= d < input base" /×↧⊃(≥0|<:) | ||
Valid ← ⊃(ValidDigits|⋅ValidBases) | ||
FromInput ← /+×ⁿ:⊙(⇌°⊏): | ||
ToOutput ← ⨬([0]|◿:⌊÷⊸⊃(ⁿ⊙⋅∘|⋅∘)◡(⇌⇡+1⌊ₙ:))±. | ||
Rebase ← ToOutput FromInput ◡Valid |
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# This is an auto-generated file. | ||
# | ||
# Regenerating this file via `configlet sync` will: | ||
# - Recreate every `description` key/value pair | ||
# - Recreate every `reimplements` key/value pair, where they exist in problem-specifications | ||
# - Remove any `include = true` key/value pair (an omitted `include` key implies inclusion) | ||
# - Preserve any other key/value pair | ||
# | ||
# As user-added comments (using the # character) will be removed when this file | ||
# is regenerated, comments can be added via a `comment` key. | ||
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[5ce422f9-7a4b-4f44-ad29-49c67cb32d2c] | ||
description = "single bit one to decimal" | ||
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[0cc3fea8-bb79-46ac-a2ab-5a2c93051033] | ||
description = "binary to single decimal" | ||
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[f12db0f9-0d3d-42c2-b3ba-e38cb375a2b8] | ||
description = "single decimal to binary" | ||
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[2c45cf54-6da3-4748-9733-5a3c765d925b] | ||
description = "binary to multiple decimal" | ||
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[65ddb8b4-8899-4fcc-8618-181b2cf0002d] | ||
description = "decimal to binary" | ||
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[8d418419-02a7-4824-8b7a-352d33c6987e] | ||
description = "trinary to hexadecimal" | ||
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[d3901c80-8190-41b9-bd86-38d988efa956] | ||
description = "hexadecimal to trinary" | ||
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[5d42f85e-21ad-41bd-b9be-a3e8e4258bbf] | ||
description = "15-bit integer" | ||
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[d68788f7-66dd-43f8-a543-f15b6d233f83] | ||
description = "empty list" | ||
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[5e27e8da-5862-4c5f-b2a9-26c0382b6be7] | ||
description = "single zero" | ||
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[2e1c2573-77e4-4b9c-8517-6c56c5bcfdf2] | ||
description = "multiple zeros" | ||
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[3530cd9f-8d6d-43f5-bc6e-b30b1db9629b] | ||
description = "leading zeros" | ||
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[a6b476a1-1901-4f2a-92c4-4d91917ae023] | ||
description = "input base is one" | ||
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[e21a693a-7a69-450b-b393-27415c26a016] | ||
description = "input base is zero" | ||
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[54a23be5-d99e-41cc-88e0-a650ffe5fcc2] | ||
description = "input base is negative" | ||
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[9eccf60c-dcc9-407b-95d8-c37b8be56bb6] | ||
description = "negative digit" | ||
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[232fa4a5-e761-4939-ba0c-ed046cd0676a] | ||
description = "invalid positive digit" | ||
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[14238f95-45da-41dc-95ce-18f860b30ad3] | ||
description = "output base is one" | ||
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[73dac367-da5c-4a37-95fe-c87fad0a4047] | ||
description = "output base is zero" | ||
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[13f81f42-ff53-4e24-89d9-37603a48ebd9] | ||
description = "output base is negative" | ||
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[0e6c895d-8a5d-4868-a345-309d094cfe8d] | ||
description = "both bases are negative" |
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# Convert digits from base into another base | ||
# Rebased ? Digits InputBase OutputBase | ||
Rebase ← |3 ⊙⋅⋅(⍤ "Please implement Rebase" 0) |
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~ "all-your-base.ua" ~ Rebase | ||
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# Single bit one to decimal | ||
Digits ← [1] | ||
InputBase ← 2 | ||
OutputBase ← 10 | ||
Expected ← [1] | ||
⍤⤙≍ Expected Rebase Digits InputBase OutputBase | ||
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# Binary to single decimal | ||
Digits ← [1 0 1] | ||
InputBase ← 2 | ||
OutputBase ← 10 | ||
Expected ← [5] | ||
⍤⤙≍ Expected Rebase Digits InputBase OutputBase | ||
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# Single decimal to binary | ||
Digits ← [5] | ||
InputBase ← 10 | ||
OutputBase ← 2 | ||
Expected ← [1 0 1] | ||
⍤⤙≍ Expected Rebase Digits InputBase OutputBase | ||
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# Binary to multiple decimal | ||
Digits ← [1 0 1 0 1 0] | ||
InputBase ← 2 | ||
OutputBase ← 10 | ||
Expected ← [4 2] | ||
⍤⤙≍ Expected Rebase Digits InputBase OutputBase | ||
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# Decimal to binary | ||
Digits ← [4 2] | ||
InputBase ← 10 | ||
OutputBase ← 2 | ||
Expected ← [1 0 1 0 1 0] | ||
⍤⤙≍ Expected Rebase Digits InputBase OutputBase | ||
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# Trinary to hexadecimal | ||
Digits ← [1 1 2 0] | ||
InputBase ← 3 | ||
OutputBase ← 16 | ||
Expected ← [2 10] | ||
⍤⤙≍ Expected Rebase Digits InputBase OutputBase | ||
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# Hexadecimal to trinary | ||
Digits ← [2 10] | ||
InputBase ← 16 | ||
OutputBase ← 3 | ||
Expected ← [1 1 2 0] | ||
⍤⤙≍ Expected Rebase Digits InputBase OutputBase | ||
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# 15-bit integer | ||
Digits ← [3 46 60] | ||
InputBase ← 97 | ||
OutputBase ← 73 | ||
Expected ← [6 10 45] | ||
⍤⤙≍ Expected Rebase Digits InputBase OutputBase | ||
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# Empty list | ||
Digits ← [] | ||
InputBase ← 2 | ||
OutputBase ← 10 | ||
Expected ← [0] | ||
⍤⤙≍ Expected Rebase Digits InputBase OutputBase | ||
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# Single zero | ||
Digits ← [0] | ||
InputBase ← 10 | ||
OutputBase ← 2 | ||
Expected ← [0] | ||
⍤⤙≍ Expected Rebase Digits InputBase OutputBase | ||
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# Multiple zeros | ||
Digits ← [0 0 0] | ||
InputBase ← 10 | ||
OutputBase ← 2 | ||
Expected ← [0] | ||
⍤⤙≍ Expected Rebase Digits InputBase OutputBase | ||
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# Leading zeros | ||
Digits ← [0 6 0] | ||
InputBase ← 7 | ||
OutputBase ← 10 | ||
Expected ← [4 2] | ||
⍤⤙≍ Expected Rebase Digits InputBase OutputBase | ||
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# Input base is one | ||
Digits ← [0] | ||
InputBase ← 1 | ||
OutputBase ← 10 | ||
⍤⤙≍ "input base must be >= 2" ⍣(Rebase Digits InputBase OutputBase) | ||
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# Input base is zero | ||
Digits ← [] | ||
InputBase ← 0 | ||
OutputBase ← 10 | ||
⍤⤙≍ "input base must be >= 2" ⍣(Rebase Digits InputBase OutputBase) | ||
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# Input base is negative | ||
Digits ← [1] | ||
InputBase ← ¯2 | ||
OutputBase ← 10 | ||
⍤⤙≍ "input base must be >= 2" ⍣(Rebase Digits InputBase OutputBase) | ||
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# Negative digit | ||
Digits ← [1 ¯1 1 0 1 0] | ||
InputBase ← 2 | ||
OutputBase ← 10 | ||
⍤⤙≍ "all digits must satisfy 0 <= d < input base" ⍣(Rebase Digits InputBase OutputBase) | ||
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# Invalid positive digit | ||
Digits ← [1 2 1 0 1 0] | ||
InputBase ← 2 | ||
OutputBase ← 10 | ||
⍤⤙≍ "all digits must satisfy 0 <= d < input base" ⍣(Rebase Digits InputBase OutputBase) | ||
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# Output base is one | ||
Digits ← [1 0 1 0 1 0] | ||
InputBase ← 2 | ||
OutputBase ← 1 | ||
⍤⤙≍ "output base must be >= 2" ⍣(Rebase Digits InputBase OutputBase) | ||
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# Output base is zero | ||
Digits ← [7] | ||
InputBase ← 10 | ||
OutputBase ← 0 | ||
⍤⤙≍ "output base must be >= 2" ⍣(Rebase Digits InputBase OutputBase) | ||
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# Output base is negative | ||
Digits ← [1] | ||
InputBase ← 2 | ||
OutputBase ← ¯7 | ||
⍤⤙≍ "output base must be >= 2" ⍣(Rebase Digits InputBase OutputBase) | ||
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# Both bases are negative | ||
Digits ← [1] | ||
InputBase ← ¯2 | ||
OutputBase ← ¯7 | ||
⍤⤙≍ "input base must be >= 2" ⍣(Rebase Digits InputBase OutputBase) |
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# Instructions | ||
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Diffie-Hellman key exchange. | ||
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Alice and Bob use Diffie-Hellman key exchange to share secrets. | ||
They start with prime numbers, pick private keys, generate and share public keys, and then generate a shared secret key. | ||
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## Step 0 | ||
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The test program supplies prime numbers p and g. | ||
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## Step 1 | ||
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Alice picks a private key, a, greater than 1 and less than p. | ||
Bob does the same to pick a private key b. | ||
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## Step 2 | ||
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Alice calculates a public key A. | ||
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A = gᵃ mod p | ||
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Using the same p and g, Bob similarly calculates a public key B from his private key b. | ||
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## Step 3 | ||
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Alice and Bob exchange public keys. | ||
Alice calculates secret key s. | ||
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s = Bᵃ mod p | ||
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Bob calculates | ||
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s = Aᵇ mod p | ||
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The calculations produce the same result! | ||
Alice and Bob now share secret s. |
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