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L06_Repeated_Squaring_Examples

# L06_Repeated_Squaring_Examples - 2 i mod 19 I = 3 1 mod 19...

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More Repeated Squaring Examples Version of March 16, 2010

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Evaluate 2 50 mod 19 . This can be solved by repeated squaring. Set I i = 2 2 i mod 19 . I 0 = 2 1 mod 19 = 2 I 1 = 2 2 mod 19 = I 0 · I 0 mod 19 = 4 I 2 = 2 4 mod 19 = I 1 · I 1 mod 19 = 16 I 3 = 2 8 mod 19 = I 2 · I 2 mod 19 = 9 I 4 = 2 16 mod 19 = I 3 · I 3 mod 19 = 5 I 5 = 2 32 mod 19 = I 4 · I 4 mod 19 = 6
2 50 = 2 32 · 2 16 · 2 2 Evaluate 2 50 mod 19 . This can be solved by repeated squaring. Set I i = 2 2 i mod 19 . I 0 = 2 1 mod 19 = 2 I 1 = 2 2 mod 19 = I 0 · I 0 mod 19 = 4 I 2 = 2 4 mod 19 = I 1 · I 1 mod 19 = 16 I 3 = 2 8 mod 19 = I 2 · I 2 mod 19 = 9 I 4 = 2 16 mod 19 = I 3 · I 3 mod 19 = 5 I 5 = 2 32 mod 19 = I 4 · I 4 mod 19 = 6 2 50 mod 19 = I 5 · I 4 · I 1 mod 19 = 6 · 5 · 4 mod 19 = 6

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Evaluate 2 41 mod 19 . This can be solved by repeated squaring. Set I i = 2 2 i mod 19 . I 0 = 2 1 mod 19 = 2 I 1 = 2 2 mod 19 = I 0 · I 0 mod 19 = 4 I 2 = 2 4 mod 19 = I 1 · I 1 mod 19 = 16 I 3 = 2 8 mod 19 = I 2 · I 2 mod 19 = 9 I 4 = 2 16 mod 19 = I 3 · I 3 mod 19 = 5 I 5 = 2 32 mod 19 = I 4 · I 4 mod 19 = 6
2 41 = 2 32 · 2 8 · 2 1 Evaluate 2 41 mod 19 . This can be solved by repeated squaring. Set I i = 2 2 i mod 19 . I 0 = 2 1 mod 19 = 2 I 1 = 2 2 mod 19 = I 0 · I 0 mod 19 = 4 I 2 = 2 4 mod 19 = I 1 · I 1 mod 19 = 16 I 3 = 2 8 mod 19 = I 2 · I 2 mod 19 = 9 I 4 = 2 16 mod 19 = I 3 · I 3 mod 19 = 5 I 5 = 2 32 mod 19 = I 4 · I 4 mod 19 = 6 2 41 mod 19 = I 5 · I 3 · I 0 mod 19 = 6 · 9 · 2 mod 19 = 13

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Evaluate 3 50 mod 19 . This can be solved by repeated squaring. Set I i = 3
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Unformatted text preview: 2 i mod 19 . I = 3 1 mod 19 = 3 I 1 = 3 2 mod 19 = I Â· I mod 19 = 9 I 2 = 3 4 mod 19 = I 1 Â· I 1 mod 19 = 5 I 3 = 3 8 mod 19 = I 2 Â· I 2 mod 19 = 6 I 4 = 3 16 mod 19 = I 3 Â· I 3 mod 19 = 17 I 5 = 3 32 mod 19 = I 4 Â· I 4 mod 19 = 4 3 50 = 3 32 Â· 3 16 Â· 3 2 Evaluate 3 50 mod 19 . This can be solved by repeated squaring. Set I i = 3 2 i mod 19 . I = 3 1 mod 19 = 3 I 1 = 3 2 mod 19 = I Â· I mod 19 = 9 I 2 = 3 4 mod 19 = I 1 Â· I 1 mod 19 = 5 I 3 = 3 8 mod 19 = I 2 Â· I 2 mod 19 = 6 I 4 = 3 16 mod 19 = I 3 Â· I 3 mod 19 = 17 I 5 = 3 32 mod 19 = I 4 Â· I 4 mod 19 = 4 3 50 mod 19 = I 5 Â· I 4 Â· I 1 mod 19 = 4 Â· 17 Â· 9 mod 19 = 4...
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L06_Repeated_Squaring_Examples - 2 i mod 19 I = 3 1 mod 19...

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