public void uploadToAmazonS3 () {
    String filename = this.record.File_Name__c;
    String attachmentBody = EncodingUtil.base64Encode(fileBlob);
    String formattedDateString = DateTime.now().formatGMT('EEE, dd MMM yyyy HH:mm:ss z');
    String bucketname = 'test-salesforce-upload';
    String host = 's3.amazonaws.com';

    String key = credentials.key;
    String secret = credentials.secret;
    String stringToSign = 'PUTnnn'+formattedDateString+'nn/'+bucketname+'/'+filename;
    Blob mac = Crypto.generateMac('HMACSHA1', blob.valueOf(stringToSign), blob.valueof(secret));
    String signed = EncodingUtil.base64Encode(mac);
    String authHeader = 'AWS' + ' ' + key + ':' + signed;

    HttpRequest req = new HttpRequest();
    req.setMethod('PUT');
    req.setHeader('Authorization', authHeader);
    req.setEndpoint('https://s3.amazonaws.com/'+bucketname);
    req.setHeader('Host', bucketname + '.' + host);
    req.setHeader('Content-Length', String.valueOf(attachmentBody.length()));
    req.setHeader('Content-Type', contentType);
    req.setHeader('Connection', 'keep-alive');
    req.setHeader('Date', formattedDateString);
    req.setHeader('ACL', 'public-read-write');
    Blob blobBody = EncodingUtil.base64Decode(attachmentBody);
    req.setBodyAsBlob(blobBody);

    Http http = new Http();
    HTTPResponse resp = http.send(req);
    System.debug(resp);
}

DEBUG|System.HttpResponse[Status=Forbidden, StatusCode=403]